p-Adic Analysis (Lecture Notes in Mathematics)

Lecture Notes in Mathematics Edited by A. Dold, B. Eckmann and E Takens

1454 E Baldassarri S. Bosch B. Dwork (Eds.)

p-adic Analysis Proceedings of the International Conference held in Trento, Italy, May 29-June 2, 1989

Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona

Editors

Francesco Baldassarri Dipartimento di Matematica Pura e Applicata Universit& di Padova Via Belzoni 7, 35131 Padova, Italy Siegfried Bosch Mathematisches Institut der Universit&t Einsteinstr. 62, 4400 MQnster, Federal Republic of Germany Bernard Dwork Department of Mathematics, Princeton University Princeton, N.J. 08544, USA

Mathematics Subject Classification (1980): 12Jxx, 14Fxx, 12Hxx, 46PO5 ISBN 3-540-53477-6 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-53477-6 Springer-Verlag NewYork Berlin Heidelberg

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its current version, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1990 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210 - Printed on acid-free paper

INTRODUCTION

The present volume contains the Proceedings of the Congress on (~p-adic Analysis~ held at Trento from May 28 to June 3, 1989. The idea of organizing a meeting on this subject in Italy was first promoted by Philippe Robba, whose visits to Italy were always welcomed by his Italian colleagues for both the warmth and the illumination which he brought with him, He died prematurely on October 12, 1988, leaving a profound sense of loss in the world of p-adic analysis. We believe we have expressed the feelings of that whole community by dedicating this Meeting to him. At the opening of the Conference, Elhanan Motzkin commemorated Robba's exceptional character in a touching reminiscence, that will appear in the Seminars of the Groupe d'Etude d'Analyse Ultram6trique, of which Robba was one of the founders. The conference was organized by the Centro Internazionale per la Ricerca Matematica (CIRM), of Trento, and was also sponsored by the Diparfimento di Matematica Pura e Applicata of the University of Padova. We are grateful to both these institutions. We wish to express our gratitude in particular to Mr. Augusto Micheletti for his indefatigable efforts on behalf of the conference.

F. Baldassarri, S. Bosch, B. Dwork

>__

c~ O0 C~

CO

0

CONTENTS

B, Dwork

-

Work of Philippe Robba

A. Adolphson, Y. Andr6

-

S. S p e r b e r -

............................................................................

p-Adic estimates f o r exponential sums ........................................

p - A d i c B e t t i lattices ...................................................................................

J. A r a u j o , J. M a r t i n e z - M a u r i c a

- The nonarchimedean Banach-Stone theorem ............................

P. B e r t h e l o t - Cohomologie rigide et thdori~ des D - m o d u l e s

I 11 23 64

................................................

80

D . B e r t r a n d - Extensions de D-modules et groupes de Galois diffdrentiels ...............................

125

B. Chiarellotto R.F. Coleman M.J.

Coster

-

-

Duality in rigid analysis ......................................................................

142

On the Frobenius matrices o f Fermat curves ..............................................

173

Supercongruences

...............................................................................

194

V . C r i s t a n t e - Witt realization o f the p-adic Barsotti-Tate groups; some applications .................... 2 0 5 J. D e n e f , F. L o e s e r - Polyddres de Newton et poids de sommes exponentielles . . . . . . . . . . . . . . . . . . . . . . . . . 2 1 7 E . - U . G e k e l e r - De Rham cohomoIogy and the Gauss-Manin connection f o r Drinfeld modules ...... 2 2 3 F. H e r r l i c h - The nonarchimedean extended Teichmiiller space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 5 6 Z . M e b k h o u t , L. N a r v a e z - M a c a r r o

vari~tLs

- Sur les coefficients de De Rham-Grothendieck des

alg~briques

................................................................................

267

D . M e u s e r - On a functional equation o f Igusa" s local zeta function .......................................

309

Y . M o r i t a - On vanishing o f cohomologies o f rigid analytic spaces .......................................

314

A . O g u s - A p - a d i c analogue o f the Chowla-Selberg f o r m u l a ..............................................

319

W . H . S c h i k h o f - The complementation property o f f ~ in p-adic Banach spaces ......................... 3 4 2 D.S. Thakur

-

Gross-Koblitz f o r m u l a f o r f u n c t i o n f i e l d s ...................................................

351

L. V a n H a m m e - Three generalizations of Mahler's expansion f o r continuous functions on Zp ....... 3 5 6 H. VoskuilList

of

p-Adic

Participants

symmetric

d o m a i n s ......................................................................

...............................................................................................

362 379

Work of Philippe Robha B. Dwork Department of Mathematics Princeton University Among the subjects studied by Philippe Robba were: 1. domains of analyticity 2. p-adic Mittag-Leffier 3. index of differential operators 4. factorization of differential operators corresponding to radii of convergence and to order of logarithmic growth 5. effective estimates for logarithmic growth 6. weak frobenius (dimension one, precursor of work of ChristoI) 7. L-functions and exponential sums 8. application of p-adic methods to questions of irrationality and transcendence. Robba's work was so involved with the p-adic theory of ordinary differential equations that it may be useful in an article devoted to his work to give a survey of the present status of this subject. Let K be a field of characteristic zero, complete under a rank one valuation extending the ordinary p - a d i c valuation of Q. Let E be the completion of K ( X ) under the gauss norm. Elements of E are admissible (resp: superadmissible) if they are analytic elements on the complement of a finite set of residue classes (resp: a finite set of disks of local radius strictly less than unity). A p - a d i c Liouville number is an element a E Zp (necessarily transcendental over Q) such that either liminf la - ml 1/'' < 1 or

These conditions are not equivalent and for the operator L = x ~

- a the first condition

gives difficulties at x = 0 while the second gives difficulties at x = oo. We recall the notion of a generic point t in a universal domain ~ which is algebraically closed and complete relative to a valuation extending that of K . We insist that It[ = 1 and the residue class of t be transcendental over the residue class field of K. The disks and ~nnuli appearing in our theory involve subsets of ~.

Let .40 = {~ 6 K[[x]] [ ~ converges in D(O, 1-)} ~0 = {~ • Jl0 I ~ is bounded on D(O, 1-)} A t ( r ) = {~ • K(t)[[x - t ] ] l ~

converges in D(0, r - ) }

W [ '~ = {~ = E A i ( x - a ) i • K ( a ) [ [ z - a]] l S u p A i r J / ( i + j)Z < cxD}. J The theory started in 1937 with Lutz's solution of the Cauchy problem: Let f ( x , - y ) be an element of K[[X, Y 1 , . . . , Yn]]n converging on a polydisk in n + 1 space. Then the equation d-y = f ( x ( ' y ) dx ~(0) = 0 has a unique solution in (xK[[x]]) ~ converging on a non-trivial disk about the origin. Lutz estimated the radius of convergence and applied it to the study of rational points on elliptic curves. 1 3, t t , x ) dates to the late 1950's and involved the (long unpubOur own interest in 2F 1(7, lished) calculation of Tate's constant (of. Dwork 1987). Our interest in tile general theory of linear equations goes back to our study of the variation of cohomology of hypersurfaces (Dwork 1964, 66). Clark's work on linear equations at a singular point appeared in 1966. It was here that the question of p-adic Liouville exponents was first discussed. Adolphson [1976a, 1976b] investigated symmetric powers of 2FI ($, 1 3, 1 1, x) and studied index in the early 1970's. Robba's work started in 1974. We will restrict our attention to linear equations but I cannot refrain from mentioning the splendid result of Sibuya and Sperber [1981]. THEOREM. Let Yo • K[[X]] be a forrnaJ solution of a non-linear polynomial differentiM equation, P ( x , y, y ' , . . . , y(,O) = O, where P is a polynomial in n + 2 variables with coefficients in K . Substituting y = Yo + u we obtain the tangent linear operator OP ~ OP ~ L(u) = -~y (X, y o ) u + -~Ty,(z, y o ) V u + - - - +

OP ~ O-~7~(x, y o ) e

u,

defined over K[[X]], If the exponents of L at x = 0 are p-adically non-Liouville then Yo has a nontrivied p-adic radius of convergence. The theory of ordinary p-adic differential equations addresses such questions as: 1. What axe the radii of formal local solutions? 2. How do solutions grow as the boundary of the circle of convergence is reached? 3. W h a t are the filtrations of the solution spaces relative to the growth and radii of convergence? 4. Index.

I. Order of growth The main result of Robba on this question does not directly refer to differential equations. THEOREM. (Robba 1980b) Let ul,.. • Un • .Ao and let the wronskian

W

:

U 1

.

tt~

. . .

(,-I)

U 1

.

.

...

U

n

# tt n

u(n-1)

never vanish in D(0, 1-). Then each element v = EasxS in the K space spanned by u t , . . . u , satisfies the condition Ia, I _<

Sup

lad. { s , n - - 1}

0
where {s,n

-

1} = 1/Inf tzlz2""z.-ll

(< ,,-1),

the inf being over all 1 < zl < z2 < • .- < z , - i _< s. This type of logarithmic estimate first appeared in the study of eigenvectors in our dual theory [Dwork 1964]. This type of estimate is a natural consequence of a strong Frobenius structure [Dwork, 1969]. For example if --y = E a s x ~ • K[[X]] n and if for some A • K x, we have A'-yy(x p) = X-y(x) where A • A,tn(Bo) and is bounded by unity on D(0, 1 - ) then--y must converge in D(0, 1 - ) and for s _> 1

1~1 _< ~01l-tl -~°g"/~°s'l More recently effective bounds for solutions at a regular singular point have been found by Adolphson, et al. 1982 and by Christol,Dwork 1990, the former if the local monodromy is nilpotent of maximal rank, the second without the restriction of maximality of rank. II. Filtration by growth and radius of convergence For the second type of filtration we have THEOREM. (Robba 1977a). Let L be a differential operator of order n with coefficients in

K ( X ) (or more generally with superadmissible coefficients). For r • (0,1], Ker(L,.At(r)) defines a monic factor, Lr, of L in E[D] whose coefficients are indeed superadmisible. In the case of a Frobenius structure, filtration by growth should correspond to filtration by magnitude of eigenvalue. THEOREM. (Robba 1975a) Let L C E[D], dimKer(L, At) = orderL = n. Then this kernel /ies in W : ''~-1 and for each t3 e [0, n - 1], Ker(L, Wa, '/3) determines a monic factor L~ of L in E[D]. The great contribution of Robba to these questions was to view 92 = E[D] as a subspace of the Banach space £.(141['t~, W['~). Thus he considered ~-'L, the completion in T~ of the ideal

TgL under the Banach space norm. This ideal has a generator shown by Robba to be the factor of L corresponding to Ker(L, W['~). There are a number of unanswered questions. 1. If the coefficients of L (in this last theorem) axe admissible (or even superadmissible) then the coefficients of L~ need not be superadmissible. But are they admissible? Conjecture: Yes,

2. Let L have coefficients which are analytic elements on D(0, 1-). We may construct a Newton polygon for L at t whose slopes are the exceptional values fl such that Ker(L, W~I ' z - ' ) C Ker(L, W) '~+~) for an infinite sequence of ~ --* 0, and whose vertices have abscissas given by l i m ~ 0 dimKer(L, W ) '~-~) for fl exceptional. We may construct a similar polygon at x = 0. CONJECTURE. The polygon at x = 0 lies above the polygon at x = t. It is known (Dwork 1973, Robba 1975a) dim Ker(L, W}'°)) _> dim Ker(L,

W0l'°)

dimKer(L,.4t(1)) > 1 implies dim ker(L, W] '°) _> 1. A geometric example of the filtration by growth is given by 2 F l ( i1, 71, 1 , x ) .

This was

analyzed (Dwork 1969, 1971) in two ways: (a) by directly demonstrating the admissibility of F ( X ) / F ( X ) b via congruences associated with the p-adic gamma function (b) by constructing a unit root crystal from the given superadmissible two dimensional crystal. For 2 F l ( a , b , l , x ) Robba [1976(b)] gave a treatment based on a weak form of the Hahn Banaeh theorem. He avoided all references to Frobenius structure. Dwork (1983) discussed 2Fl(a, b, c, x) on the basis of Frobenius structure. The nature of the factorization subject to geometric type hypotheses have been investigated for hypersurfaces, (Dwork 1973) for kloosterman sums (Adolphson, Sperber 1984) and hyperkloosterman sums (Sperber 1980, Sibuya, Sperber 1985). Subject to geometric type hypotheses, Sperber and the author [Dwork, Sperber 1990] have found the coefficients of the factor corresponding to the bounded solutions to have mittag-leffier decompositions in which the components are of the form E A d / ( x - a) j with o r d A j > klog(1 + j ) for some k > 0. This has played a role in investigating the unit root zeta function. III. Index This question had great interest for Robba. At least four of his articles mention index in the title while others are devoted to apphcations of index. In his early work (1975, 76) there were no indications of applications but these appeared subsequently (1982c). His 1984 Asterisque article was dominated by the application to one dimensional cohomology and by 1986 he began studying symmetric powers of the Bessel differential equation.

Both Robba and Adolphson used patching arguments to reduce the question of index to the case of L e K[X][D] and to the calculation of either A o / L A o or Bo/Ll3o. For the applications it made no difference which one was finite. We consider only this elementary form. For his application Adolphson was able to reduce to the case of order one and more explicitly

tO X ddg -- a, a 6 Q. Robba [1975a] showed If ker(L, A~(1)) = 0 then x(L, A0(1)) = x ( L , B0(1)). This result is of interest as it seems to capture the essential point of dagger type cohomology involving over convergent series. Unfortunately this has not been extended to the case of several variables. Of course if order L = dim Ker(L, A0(1)) then L has index as operator on A0 (but not on B0). In particular, by the transfer principle: If L has no singularity on D ( 0 , 1 - ) and if order L = dim(KerL, . a t ( l ) ) then L has index on

,40(1). By means of Christol's transfer theorem [Christol 1984] we may extend this last result to the case in which L has just one regular singularity in D(0, 1 - ) with non-LiouviUe exponents. The operator (Robba 1977a) L = p(1 - x ) D 2 - x D - a where l i m i n f [a - m[ 1/m = +1, l i m i n f la + ml 1/m < 1 is an example of an operator with no singularity in D ( 0 , 1 - ) b u t without index in A0. For operators of the first order L = aD + b, a, b E K[X], Robba (1985a) gave a beautiful formula

logp(n,r)

= x - ( L , r) + o r d - ( a , r )

where X- = dim Ker - dim cokernel for L as operator on H(D(O, 1 - ) )

p(L, r) = radius of convergence of the solution at tr, o r d - ( a , r) = abcissa of point of contact of the Newton polygon of a with the line of support of slope - log r / l o g p , i.e. if a ----E A , X n, v minimal such that lalo(r) = IA~,I~'~ then v = o r d - ( a , r ) . This formula showed how the Turritin form may be used to compute the index if the origin is an irregular singular point. It gives htrther motivation for extending Christol's transfer theorem to the case of irregular singular points. This index need not be equal to the algebraic index. In view of the failure of crystalline cohomology to provide a proof finiteness of cohomology~ the question of finiteness of index in the sense of this section must still be viewed as pertinent. It is our opinion that the critical case is that in which D ( 0 , 1 - ) contains more t h a n one regular singularity and order L = dim(Ker L, A t ( l ) ) . We mention a few aspects of Robba's mathematical personality.

He was a very clear expositor and did much to popularize p-adic analysis. Together with Amice and Escassut he organized the GEAU. He gave a total of 24 written expos~s~ three in the first year, five in the second year. He had many beautiful ideas. One was his abstract construction of the generic disk (1977c), a second was his explanation of Turrittin's theorem by means of the valuation polygon (1980a), a third was his method for removal of apparent singularities (cf. Christol 1981, Theorem 8.3), a fourth was his construction of a transcendental 7r, ordTr = 1 / ( p - 1) which had the property that ord 7r(x - x p) converges for ordx > - l i p . Bibliography of Ph. Robba [1966]

"D6riv6e en moyenne de fonction moyennable", Communication an Congr~s des Mathematiciens, Moscow, 2p.

[1967]

"Solutions presque-periodiques d'~quations differentieUes aux differences", SeminMre Lions-Schwartz, 25p.

[1969a]

(with E. Motzkin), "Ensembles satisfaisant au principe du prolongement analytique en analyse p-adique" C.R. Acad. Sc. Paris 269, p. 126-129.

[1969b]

, "Ensembles d'analyticit~ en analyse p-adique", C.R. Acad. Sc. Paris 269, p. 450-453.

[1970]

"Decomposition en ~lements simple d'une fonction analytique sur un corps valu6 ultrara~trique. Application au prolongement analytique", Sere. Delange-PisotPoitou, (1969/70), No. 22, 14p.

[1971]

"Prolongement analytique pour les fonction de plusieurs variables sur un corps valu~ complet', Sere. Delang~Pisot-Poitou, (1971/72).

[1972a]

"Decomposition en ~lements simple d'un element analytique sur un corps ultram6trique", C.R. Acad. Sc. Paris 274, p. 532-535.

[1972b]

"Fonctions analytiques sur un corps ultram~trique", C.R. Acad. Sc. Paris 274, p. 721-723.

[1972c]

"Prolongement analytique en analyse ultram6trique", C.R. Acad. Sci. Paris 274, p. 830-833.

[1973a]

"Fonctions analytiques sur les corps values ultram6triques complets", A s t e r isque, 10, p. 109-220.

[1973b]

"Prolongement analytique pour les fonctions de plusieurs variables sur un corps valu6 complet", Bull. Soc. M a t h . ~ a a c e 101, pp. 193-217.

[1974a]

"Croissance des solutions d'une 6quation differentille homog6ne", GEAU 73/74, exp. 1, 15p.

[1974b]

"Indice d'un op~rateur diff6rentiel", GEAU 73/74, exp. 2, 14p.

[1974c]

"Prolongement des solutions d'un 6quation diff6rentielle p-adique', GEAU 73/74 exp 11, 3p. (also C.R. Acad. Sc. 279, p. 153-154).

[1974d]

"Quelques propridtes des $16ments et fonctions analytiques au sens de Krasner", B a l l Soc. Math. France MSmoire 39-40, p. 341-349.

[1974e]

"Un classe d'ensembles analytiques $ plusieurs dimensions", Bull. Soc. Math France, M~moire 39-40, p. 351-357.

[1975a]

"On the index of p--adie differential operators I", Annals of Math. 101 p. 280316.

[1975b]

"Equations diffSrentielles p-adiques", Sere. Delange-Pisot-Pitou, 74/75, 15p.

[1975e]

"Factorisation d'un op~rateur diff6rentiel", GEAU 74/75 exp. 2, 16p.

[1975d]

"Factorisation d'un opdrateur diff~rentiel", Applications, GEAU 74/75 exp. 10, 6p.

[1975e]

"Caracterisation des d6riv~es logarithmiques", GEAU 74/75 exp. 12, 6p.

[1975f]

"Lemma de Hensel pour les op~rateurs diff6rentielles" GEAU 74/75 exp. 16, llp.

[1975g]

"Structure de Frobenius faible pour les $quations diff6rentielles de premier ordre", GEAU 74/75 exp. 20, llp.

[1976a]

"On the index of p--adic differential operators II", Duke Math. J. 43, p. 19-31.

[1976b]

"Solutions born~es des systems diffSrentiels lindaires, Application aux fonctions hyperg~om~triques", GEAU 75/76 exp. 5, 16p.

[1976c]

"Factorisation d'un op~rateur diffSrentiel II", GEAU 75/76 exp. 6, 6p.

[1976(t]

"Prolongement alg6brique et equations diffSrentielles", Proc. Conference on padic analysis, Nijmegen (1978), p. 172-184. Math. Inst. Kathotieke Univ. Toernooivetd.

[1976e]

"Th~orie de Galois p-adique', GEAU 75/76 exp. 14, 8p.

[1976f]

"Lemme de Schwarz p-adique pour plusieurs variables" GEAU 75/76 exp. Jg, 12p.

[1977a]

(with B. Dwork) "On ordinary linear p-adic differential equations", Trans. Amer. Math. Soc. 231, p. 1-46.

[1977b]

"Nouveau point de vue sur le prolongement alg6brique", GEAU 76/77 exp. 5, 14p.

[1977c]

"Disque g6n6rique et 6quations diff6rentielles", GEAU 76/77 exp. 8, 7p.

[1977d]

"Nombre de z6ros des fonctions exponentielles-polyn6mes", GEAU 76/77 exp. 9, 3p.

[1978a]

"Lemmes de Schwarz of lemmes d'approximation p-adiques en plusieurs variables", Invent. Math. 48, 245-277.

[1978b]

"Z6ros de suites r6currentes lin6aires", GEAU 77/78 exp. 13, 5p.

[1979a]

(with B. Dwork) "On natural radii ofp--adic convergence", Trans. Amer. Math. Soc. 256, p. 199-213.

[1979b]

"Points singuliers d'equations diff6rentielles, d'apres F. Baldassarri", GEAU 78/79 exp. 5, 2p.

[1979c]

(with B. Dwork) "Majorations effective", GEAU 78/79 exp. 18, 2p.

[1979(t]

"C16ture alg~brique relative de certain anneaux de s~ries enti~res", GEAU 78/79 exp. 21, 2p.

[1979e]

"Lemmes de Hensel pour les opSrateurs differ~ntielles', GEAU 78/79 exp. 23, 8p.

[19s0a]

"Lemmes de Hensel pour les ol~rateurs diff~rentielles, Application a la reduction formeUe des equations diff~rentielles", Ens. Math. 26, p. 279-311.

[1980b]

(with B. Dwork) "Effective p-adic bounds for solutions of homogeneous linear differential equations", Trans. Amer. Math. Soc. 259, p. 559-577.

[i980c]

(with S. Bosch and B. Dwork) "Un theorem de prolongement pour les fonctions analytiques", Math. Ann. 242, p. 165-173.

[1980(1]

"Factorisation d'op~rateurs diff~rentiel £ coefficients rationnels', GEAU 79/81 exp. 12, 3p.

[i98i]

"Une propri~t~ de specialisation continue, d'apr~s Artin, Bosch, Lang, Van den Dries", GEAU 79/81 exp. 26, llp.

[igs2a]

"Sur tes equations diff~rentielles lin~aires p-adiques II", Pacit~c J. Math. 98, p. 393-418.

[1982b]

"Calcul des residus en analyse p-adique, d'apres Gerritzen et Van der Put", GEAU 81/82 exp. 28, 8p.

[i982c]

"Indice d'un ol~rateur diff6rentiel lin6aire p-adique d'ordre 1 et cohomologies p-adiques", GEAU 81/82 exp. J15, 10p.

[i983]

"Cohomologie de Dwork, I", GEAU 82/83 exp. 3, 10p.

[1984a]

"Une Introduction naive aux cohomologies de Dwork", BuI1. Soc. Math. F~'ance 114 (Supp) Memoire 23, p. 61-105.

[198461

"Index of p-adic differential operators III, Application to twisted exponential sums", Ast@risque, 119-120, p. 191-266.

[1984c]

"Quelques remarques sur les Ol~xateurs diff6rentiels d'order 1", GEAU 83/84 exp. 12, 8p.

[1985a]

"Indice d'un ol~rateur diff6rentiel p-adique IV, Cas des syst@ms. Mesure d'irregularit6 dans un disque", Arm. Inst. Fourier 35, p. 13-35.

[1985b]

"Conjecture sur les equations diff~rentielles p~adiques lin6aires", GEAU exp. 2, 8p.

[1985c]

"Product symmetrique de l'equation de Bessel", GEAU exp. 7, 3p.

[1986a]

"Symmetric powers of the p-adic Bessel Function", Crelle 366, p. 194-220.

[1986b]

(with J. P. Bezivin) "Rational solutions of linear differential equations", Proe. of corLference on p-adic analysis, Hengelhoef, Belg. p. 11-18.

[t98;]

"Propriete d'approximation pour les elements alg6briques", Comp. Math. 63, p. 3-t4.

[1989a]

(with J. P. Bezivin) "A new p-adic method for proving irrationality and transcendence results", Ann. of Math. 129, p. 151-160.

[1989b]

"Rational solutions of linear differential equations", J. Australian Math. Soc. 46, p. 184-196. General Bibliography

[1976a]

Adolphson, A., "An index theorem for p- adic differential operators", Trans. Amer. Math. Soc. 216, 279-293.

[1976b]

, "A p--adic theory of Hecke polynomials", Duke Math. d. 43, 115145.

[1982]

, Dwork, B., Sperber, S., "Growth of solutions of linear differential equations at a logarithmic singularity", Trans. Amer. Math. Soc. 271, 245252.

[1984]

, Sperber, S., "Twisted kloosterman sums and p--adic Bessel functions", Amer. J. Math. 105, 549--591.

[1981]

Christol, "Syst~nes diff@rentiels lin@alres p-adiques", Structure de Frobenius Faible, Buff. Soc. Math. France 109, 83-122.

10 , "Un theorem de transfert pour les disques singuliers r~gtfliers",

[1984]

Ast6risque 119-120, 151-168.

[1990

, Dwork, B. "Effective p-adic bounds at regular singular points", Duke Math. J. (submitted).

[1966]

Clark, D. "A note on the F a d i c convergence of solutions of linear differential equations", Proc. Amer. Math. Soc. 17, 262-269.

[1964]

Dwork, B. "On the zeta function of a hypersufface II", Ann. Math. 80, 227-299.

[1966]

, "On the zeta function of a hypersufface III", Arm. Math. 83, 457-519.

[19691

, "p-adic cycles", Pub. Math. IHES 37, 27-116,

[1971]

, "Normalized period matrices I", Arm. Math. 93, 337-388.

[19731

, "Normalized period matrices II", Ann. Math. 98, 1-57.

[1982]

, Lectures on p-adic differentiM equations, Springer-Vertag.

[1987]

, "On the Tate constant", Comp. Math. 61, 43-59.

[1990]

, Sperber, S., "Logarithmic decay and over convergence of the unit root and associated zeta functions", Ann. Sc. ENS, to appear.

[1973]

Katz, N.,Travaux de Dwork, Sere. Bourbuki 71/72 exp. 409 (Lecture Notes in Math., Springer--Verlag).

[1937]

Lutz, E., "Sur l'equation y2 = z3 _ Ax - B dans les corps p-adiques", J. Rehae Angew. Math. 177, 238-247.

[19811

Sibuya, Y., Sperber, S., "Arithmetic properties of power series solutions of algebraic differential equations", Ann. Math. 113, 111-157.

[1985]

, On the p-adic continuation of the logarithmic derivative of certain hypergeometric functions, GEAU 84/85 exp. 4, 5p.

[198o]

Sperber, S., "Congruence properties of the hyperkloosterman sum", Comp. Math. 40, 3-32.

p-Adic estimates for exponential sums Alan Adolphson*

Steven Sperber t

Department

of Mathematics

School of Mathematics

Oklahoma

State University

University of Minnesota

Stillwater, Oklahoma

74078

Minneapolis,

USA

1

Minnesota

55455

USA

Introduction

Let k be the finite field of q = p~ dements, f = Eies aJ xj E k [ z l , . . . , z,,, (xl .-- z , ) -~] where 0 < r < n, ~I' : k ~ Q(~p)x be a nontrivial additive character, and X~, . . . , X , : k x ~ Q({q-1) x be multiplicative characters. Define

&(x~,...,x,,f) =

)_7

xl(,~)-..x,(,,)~(I(~)).

(1)

z~(kx)- xk,,-,-

The first problem we consider in this article is: P r o b l e m 1: Find a p-adic estimate for $1. Let km be the extension of k of degree rn. We can define for each m an exponential sum related to (1):

S,~,(X,,...,X,,I) =

~_,

xi(N,,(xl))

~(Tr,,~(f(z))),

=~(kx).xt~. . . .

(2)

where Tr,,, : k,,~ -+ k is the trace map and Arm : km --+ k is the norm map. This data can be encapsulated in an L-function:

L(X1,...,X,,f;t)

:-exp

XI,*..,Xr,

[rn

.

(3)

The following result is well-known: T h e o r e m 1 (Dwork, Grothendieck) L ( x ~ , . . . , x , , f ; t )

is a rational]unction, i. e.,

L ( X ~ , . . . , x , , /; t) = I 1 ~ ° ( 1 - '~,0 I]n_.~it~(1 - flit)" *Partially supported by NSF Grant No. DMS-8803085. tPartially supported by NSF Grant No. DMS-8601461.

(4)

12 By logarithmic differentiation of (4),

~ m ( ~ l , . . . , X r , f ) =- E J ~ ? -- E O ~ n" ./ i

(5)

Thus a more precise version of Problem 1 is: P r o b l e m 2: Find p-adic estimates for the ai and flj. For Problem 1, we use an elementary approach due to Ax[6], namely, reduction to Stickelberger's calculation of the p-ordinals of Gauss sums. Ax's theorem (an improvement of the theorem of Chevalley-Warning[13]) was generalized by Katz[9], Sperber[11], and Adolphson-Sperber[3] using Dwork's trace formula and the theory of completely continuous operators on p-adic Banach spaces. Recently, however, ~Van[12] showed that Katz's improvement can be obtained using Ax's original method. We then realized that the results of [11] and [3] can also be obtained by Ax's method. Our theorem is a generalization of the main result of [3], where only an additive character was considered. Problem 2 is much harder, and good results are known only in those cases where the L-function or its reciprocal is a polynomial. In such cases, Problem 2 can be interpreted as asking for the p-adlc Newton polygon of that polynomial. In [4] we dealt with the case where all the multiplicative characters are trivial. We gave conditions on f that insure that L(f; t) (-1)~-1 is a polynomial and found a lower bound for its p-adic Newton polygon. And we conjectured that this lower bound is attained generically for primes p in certain residue classes. We are now able to handle the case of nontrivial multiplicative characters using similar arguments. These results are described in section 3. Since the proofs are like those in [4], we only sketch them here.

2

Problem

1

We assume throughout this section that f is not a polynomial in some proper subset of the variables x l , . . . , z , . This involves no loss of generality, since it is trivial to reduce the sum (1) to one of this type. First we embed SI(X1,... ,X~, f) into a p-adic field. Let K be the unramified extension of Qp of degree a and let T be the set of (q - 1)-st roots of unity in K together with 0. We regard our characters as taking values in K(~p). Our sum may then be written as S(dl,..., d,, f) = ~ t~-I-d' ... t~-l-d" ~(f(t-)), (6) tE(T×)~×T

*-~

where dl,...,d, E { 0 , 1 , . . . , q - 2} and t- E k is the reduction of t mod p. For later convenience we set di = q - 1 for i : r + 1 , . . . , n. For each d E { 0 , 1 , . . . , q - 1}, define d' = ~ 0 if d = 0 least positive residue of pd modulo q 1 if d ~ 0. t

(7)

We denote t h e / - f o l d iteration of this operation by d(0. Note that since q = p", we have d(') = d. P u t d = ( d : , . . . , d,,) E R" and d' = ( d : , . . . , d ' ) E R". Define A(f), the Newton polyhedron of f, to be the convex hull of g, the exponents of the monomials appearing in

13

f considered as points in R", together with the origin. For i = 0 , . . . , a - 1, let w(d (1), f ) be the least nonnegative real number w such that w A ( f ) ~f ( w z [ x E A ( f ) } contains a point of (q - 1)-~d (0 + (Z" x N~-'), where N denotes the nonnegative integers. (The condition that f involve all n variables ensures that the set of such w's is nonempty.) Let ordq be the additive p-adic valuation on K(~v) normMized by ordq q = 1. T h e o r e m 2 ordq S ( d , f ) > -1 a-t ~ w(d (/), f) a i=0

Ezample. For 0 < d < q - 2, consider the Gauss sum gd = ~ t e T x tq-l-dff2(t-) • Then A ( f ) = [0,1] C_ R 1 and w(d(0, f ) = (q - 1)-1d('). The theorem implies (after an easy calculation) that o r e g~ > ~(d)/a(p 1), (8) -

where ~r(d) is the sum of the digits in the p-adic expansion of d. (It is well-known by Stickelberger that the two sides of (8) are equal for 1 < d < q - 2.) It will be seen that the estimate (8) is all that is needed to prove the theorem, so the theorem is, in fact, equivalent to (8). Proof of the theorem. Let F = E j e J A j xj C I f [ z , , . . . , z , , ( z , . . . x , ) -I} be the Teichmfiller lifting of f , i. e., A~ C T and Aj ~- a~ (rood p). Define a polynomial p q-1 = Era=0 cm um C K({p)[u] by the condition P(t) = ~(t-) e K(~p) for all t E T. One checks that co = 1, %-1 = - q / ( q - 1), and that for 1 _< m < q - 2, c,~ = g,~/(q - 1), where gm is the Gauss sum defined above. Then Stickelberger implies that for 0 < m < q - 1, ordq em = ~r(m)la(P - 1). (9) Let e = ( 1 , . . . , 1 ) E R". We have S(d, f )

=

~

t(q-1)e-d~(f(t-))

te (T ×)" x T"-"

=

II *(,'A

te(T×)'xT n-"

=

jEJ

t(q-1)e-d 1-I P ( A f l j)

~ tE(TX )" x T ~'-"

jeJ q-1

=

tE(TX)'xT ~-"

jEJ

rn=O

Now expand the product: Let ~ be the set of all functions from J to { 0 , 1 , . . . , q - 1}. Then

S(d,f)

L.~

~(q-1)e-d

=

A¢(J)~ 2 _ . ,/i~l l" e+(J)J j ( ¢C@ " j E J

z_, \ll <-,s,u r - ~

ee~l' j E J

tE(T×)'xT ~-"

"

E t e ( T × ) , × T ,~-.

)"

t(q-1)e-dT~'eJ¢(J)J) •

14

Since the Aj's are roots of unity,

qSeeat

"jEJ

"

te(T×)'xT

'~-"

The second factor on the right-hand side of (10) can be evaluated explicitly. For i•Z ti={ taT×

0 q--1

if(q-1)Xi i f ( q - - 1 ) li,

and for i • N ~t~ = taT

0 q q--1

if(q-1)/i if/= 0 ifi>0and(q-1)]i.

~tence

t(q-1)e-d+~,ej*Ci)J" = te(T×),xT ~-,

0 if ( q - 1)e - d + E i e J ¢ ( J ) J ¢ ( q - 1)(Z" x N n-r) Cl1~ q. . . . . (¢)(q -- 1) "+re(e) if (q -- 1)e d + ~de.r ¢(J)J • (q 1)(7" x N'~-'), ' - ' where re(e) is the number of nonzero entries in the last n - r coordinates of the vector (q - 1)e - d + ~ j e J ¢(J)JWe show that the minimum on the right-hand side of (10) occurs when re(e) = n - r . Say the (r + 1)-st coordinate of (q - 1)e - d + ~ i e J ¢(J)J is zero. Since f is assumed to involve all the variables, there exists j0 • J whose (r + 1)-st coordinate is nonzero, so we must have ¢(j0) = 0. Define ¢ • O by ~ ( j ) = { ¢(j) q-1

ifjCj0 if j = j0.

Then (q - 1)e - d + ~ j e J ¢(J)J is in (q - 1)(Z r x N"-r). In the last n - r coordinates, it has a nonzero entry wherever (q - 1)e - d + 2 i e J ¢(J)J does and, in addition, it is nonzero in the (r + 1)-st coordinate. Hence m(~;) _> re(e) + 1. It follows from (9) and (11) that

°rd' (j~jc¢(J))Qe(rx)~xT._t(q-1)e-d+EieJ*s(J)J) >-x./CJ

t(/(T x ), XT '~-"

hence the minimum on the right-hand side of (10) occurs when the last n - r coordinates of (q - 1)e - d + ~ j e J ¢(J)J are nonzero. This implies by (10) and (11) that ordq S(d, f ) >__~ie'~{ ~ ordq c¢,(j)} , "jeJ

where 00 = {¢ • • I ~ j e s ¢(J)J • d + (q - 1)(Z" x N'~-')}.

(12)

15

For ¢ E ¢, define ¢' C • by

¢'(J) =

0 if ¢(j) = 0 least positive residue of pC(j) modulo q - 1 if ¢(j) ¢ O.

We denote the/-fold iteration of this operation by ¢(0. In terms of p-adic expansions, if ¢(j) = m o + m l p + . . . + m , - l p "-1, where 0 < mi _< p - l , then ¢'(j) = rn,,_x+rnop+mxp2+ • .-+ma_2p "-1. Hence ~i=0 ,-1 ¢(i)(j) = q ( ¢ ( j ) ) ( l + p + . . . + p " - ~ ) = ~r(¢(j))(q-1)/(p-1). It then follows from (9) and (12) that

oral, S(d,i)> .~n~ ~-~

- ee¢o(a i ~ o ~

~(')(J) ~ q-1

J"

(13)

Let ql be the set of all functions ¢ from J to the nonnegative reM numbers such that ~ e J ¢ ( J ) J C (q - 1)-1d (0 + (Z" × N'~-'). If ¢ E q'0, then (q - 1)-t¢(0 E q~ for i=0,1,...,a-1. So if we put

then (13) implies

la--I ordq S(d, f ) > a ~ P(d(0' f)"

The theorem is an immediate consequence of the following lemma. L e m m a 1 Let J be a finite subset of R", A(J) its convex hull, and L an arbitrary

subset of R '~. Define

~(J,L) = i n f { ~ n _ > 0 t ~ ( j ) n L # 0 } , p(J,L)

=

i n f { Z ~ b ( j ) ¢ : J --+ H>__oand X ¢ ( J ) J C L}. -jeJ jeJ

Then w(J, L) = p(J, L). Proof. If x E wA(J), then by properties of the convex hull,

x =., Z: ¢(J)J, jEJ

where ¢ : J ~ R>0 and E~eJ ¢(J) = 1. Put ¢ = we. If z E L also, then E i e J ¢(J)J E L and ~ j e J ¢(J) = w. This argument is reversible, so the two sets defined in the statement of the lemma are identical.

16

3

Problem

2

In this section, we restrict attention to sums where all variables are assumed to be nonzero: Let f = ~ ajfgj e k [ z l , . . . , Zn, (~gl """ ~n)--l], (14) jEJ

sm(x,,...,x.,s) =

(II

(15)

(~ ....... )c(k~)- "/--1-

Let d = ( d l , . . . ,d,~) C R" be the vector associated to the multiplicative characters as in (6), a~d let L(d, f ; t) denote the L-function associated to the sums (15). Dwork's theory provides us with a p-adic Banach space B(d, f ) and a completely continuous operator a on B(d, f ) such that L(d, f; t) (-1)"-' = det(I - t~ [ B(d, f ) ) ~ ,

(16)

where 6 is the operator on formal power series with constant term 1 defined by g(t) 6 = g(t)/g(qt). Futhermore, d e t ( I - t a I B(d, f)) is a p-adic entire function of the variable t. Equation (16) shows that there is a close connection between the zeros and poles of L ( d , / ; t) = d the zeros of det(S - t~ 1 B(d, f)). We recall the lower bound we gave in [5] for the Newton polygon of det(I - *c~ I B(d, f)). Let C ( f ) be the cone over A ( f ) , i. e., C ( f ) is the union of all rays emanating from the origin and passing through A ( f ) . For r E C ( f ) , let w(r) be the least nonnegative real number w such that r E w A ( f ) . It is clear that there exists a positive integer D such that for all r 6 (q - 1 ) - I Z n, Dw(r) is a nonnegative integer. For i = 0 , . . . ,a - 1, put R (1) = (q - 1)-Id (1) + Z", and for k a positive integer let

W(i)(k) = card {r 6 R (i) N C ( f ) I w(r) = k / D } .

(17)

Order the elements of R (1) f) C ( f ) =- I-(i) ~(i),---} so that w(r~')) < w(r~ )) for j < j'. t-1 ,-2 We define a--1

i.=o

The main technical result of [5] is: T h e o r e m 3 The Newton polygon of d e t ( / - tcK I B ( d , f ) ) with respect to the valuation ordq lien on or above the polygon with vertieen (0,0) and W(k),~kW(k =

,

M=0,1,

....

k=0

In other words, it ia bounded below by the Newton polygon of I]k~0(1 -- qk/~Dt)w(k). Theorem 2 can be deduced from this theorem just as [3, Theorem 1.2] was deduced from [2, Proposition 3.13], although, of course, this proof is less elementary than the one in section 2 since it relies on the theory of p-adic Banach spaces. Roughly speaking, Theorem 2 comes from looking at the slope of the first side of the lower bound of Theorem 3. Further information about L(d, f ; t) can be derived from Theorem 3. See [5] for the proofs of the following two theorems.

17 T h e o r e m 4 Let V ( f ) be the volume of A ( f ) with respect to Lebesgue measure on R n. Then 0 _< degree L(d, f; t) (-1)"-I ~ n! V ( f ) ,

where the degree of a rational function is the degree of its numerator minus the degree of its denominator. Suppose that d i m A ( f ) = n. If G is an (n - 1)-dimensional face of A ( f ) containing the origin and E(cr) is the (n - 1)-dimensional subspace of R'~ generated by ~r, let S(cr) be the (n - 1)-dimensional volume of cr relative to d, i. e., S(G) = 0 if E(~r) f3 R(i) = 0 for some (and hence for all) i = 0 , 1 , . . . , a - 1; otherwise, S(o') is the volume of o" with respect to Haar measure on E(G) normalized so that a fundamental domain for the lattice E ( a ) n Z'* has volume 1. Let S ( f ) = ~ , S(~), where the sum is over all (n - 1)-dimensional faces o" of A ( f ) that contain the origin. T h e o r e m 5 Suppose that d i m A ( f ) = n and that L ( d , f ; t ) ( - a P -~ and its dual Z ( - d , - f ; t)(-')"-' are polynomials of degree n! V ( f ) . ~,e~ ..r tPili=l .°:v<:) (resp. (:,};L[ (+)) be the reciprocal roots of L ( d , f ; t ) ( - ~ ) "-' (resp. L ( - d , - f ; t ) ( - 1 ) " - ~ ) . Then

,~ v(l) ordqpi +

i=l

,~ v(l) ~ ordq/~i > n . n! V ( f ) - (n - 1 ) ! S ( f ) . i=l

In particular, if the origin is an interior point of A ( f ) or if E(~r) n R (1) = 0 for every ( n - 1)-dimensional face o" of A ( f ) that contains the origin, then ,~ v(f) ,,! v(l) E ordq j0i "~- E ordq fii _~ n . n] V(f)o i=l i=l At this point, we are unable to say anything more precise about Problem 2 unless we make assumptions about f that ensure that L(d, f ; t)( -1)"-t is a polynomial. For f as in (14) and ~ a face of A ( f ) , let f~ = ~je~nJ aj zj. Retail that f is said to be nondegenerate with respect to A ( f ) if for every face cr of A ( f ) that does not contain the origin, the Laurent polynomials Of~./Ox~,..., Of~,/Oz,, have no common zero in (~x),~, where k denotes the Mgebraic closure of k. The proof of the following theorem will be sketched later. T h e o r e m 6 Suppose f is nondegenerate and d i m A ( f ) = n. Then L ( d , f ; t) (-1P-1 is a polynomial of degree n[ V ( f ) . Suppose that dim A ( f ) = n and consider the generating series }2~=0 W(i)(k) tk/D, where the W(O(k) are as defined in (17). The proof of the following lemma will be sketched later. L e m m a 2 For i = O , . . . , a - 1 there exists polynomials P(0(t) E Z[t] of degree <_ n D with nonnegative coefficients satisfying P(i)(1) = n! V ( f ) and

W(¢)(k )t kID = P(i)(tVD) k=o

(1 -- t)"

(19)

18

Since the P(1)(t) have nonnegative integral coefficients they may be written as P(')(t) =

~! v(f) a(i) ~

t~ ,

(20)

j=l

where the a~i) are nonnegative integers ordered so that

a~0 < a~0 < ... < --

--

--

.( 0 ~n!

V(f)"

We use this data to define a new polynomial P(t) E Z[t] of degree < anD with nonnegalive coefficients satisfying P(1) = n! V(f):

e(t)=

n, v(f) 2_. t ~ ' ° ° ' -

(21)

j=l

Rewrite P(t) as anD

P(t) = ~ bit j,

(22)

j=O

where the bj are nonnegative integers. T h e o r e m 7" Suppose f is nondegenerate and d i m A ( f ) = n. Then the Newton polygon of L(d, f ; t)( -1)"-' (which is a polynomial of degree n! V ( f ) by Theorem 6) with respect to the valuation ordq lies on or above the Newton polygon of the polynomial r I j=0 a n d [~1 __ qj/aDt)~i and their endpoints coincide.

Remark. D. Wan has pointed out that under the hypothesis of Theorem 7, the inequMity of Theorem 5 is actually an equality. Since the endpoints of the Newton polygons coincide, this amounts to checking that 1 anD h D ~ j(bj + ~ ) = n . ~! u ( f ) - (n - 1)! s ( y ) , j=0

where the bj are associated to - d in the same manner the b~ are associated to d. Outline of proof. Consider the ring k[{x ~ I u E Z" N C(f)}] generated over k by all monomials x u with u E Z'* n C(f). The weight function w defines a filtration on this ring, whose associated graded ring we denote by A. These two rings are identical as vector spaces over k, but multiplication in A satisfies the rule: x~'" z~" =

( x ~'+~'' if w ( u ) - l u and w(u')-lu ' fie on the same face of A ( f ) 0 otherwise.

(23)

Let M~ be the vector space over k generated by all z ~' with u C R(ONC(f). Equation (23) defines a structure of graded A-module on Mi. The generating series ~ = 0 W(1)(k) tk/D is then the Poincar~ series of the graded module Mi. For j = 1 , . . . , n, let fj be the d e m e n t of A of degree 1 that is the image of Of/Oxj when one passes to the associated graded. The method of Kouchnirenko[10] shows that the Koszul complex on Mi determined by { f j } j ~ is acyclic in positive dimension, i. e., 0 --, (M,)(:) -a-5. . . - ~ (M,)(~) ~ M,

(24)

19

is exact, and dimk Mi/O~(M~)(':) = n!V(f). Since the boundary maps in (24) have degree 1, this establishes Lemma 2. The sequence (24) can be lifted to a sequence of p-adic Banach spaces with action of Frobenius, still acyclic in positive dimension and with H°-term of dimension n! V(f). (It takes some work to fill in the details of this step: see [4, section 2].). Theorem 6 then follows from equation (16) by passing to homology. Finally, the lower bound of Theorem 7 follows by the method of [8, section 7]. We conjecture that, except in finitely many characteristics, if p -=- 1 (mod D) then the lower bound of Theorem 7 is equal to the Newton polygon of L(d, f; t) (-i)"-~ for all f in a nonempty Zariski-open subset of the set of all Laurent polynomials having the same Newton polyhedron A(f). In fact, we believe that it would suffice to choose D so that Dw(r) is a nonnegative integer for all r E e - l l " N C ( f ) , where e is the least common multiple of the orders of the multiplicative characters X i , . . . , X-.

4

Examples

For z E k x consider the Kloosterman sum

ItET x

In [1], we determined precisely the generic Newton polygon for the associated L-function. It can lie strictly above the lower bound of Theorem 7. However, if all the digits in the p-adic expansion of d are > (p - 1)/2 (or if they are all _< (p - 1)/2), then [1, Theorem 1] and a short calculation show that the lower bound of Theorem 7 equals the actual Newton polygon for generic x. In particular, the conjecture in the last paragraph of section 3 is true for this example because the condition p = 1 (rood D) forces all digits in the p-adic expansion of d to be equal. See Carpentier[7] for the determination of the Newton polygons of some hyperkloosterman sums with multiphcative characters. The next example illustrates a simpler situation where everything can be worked out explicitly using Gauss sums. Assume that 3 [ (q - 1), so that k has multiplicative characters of order 3. In this section we illustrate the above results by calculating the Newton polygons of the sums

+

y4),

(25)

x,yEk x

where X~ = X~ - 1. We assume that p is a prime k 5, which implies p - 1,5,7,11 (rood 12). (The answer win turn out to depend on p (rood 12).) Referring to equations (4) and (5), we see that we may always extend scalars when calculating the Newton polygon since ordq at = ordq., ct~. To evaluate (25) explicitly, it will be necessary to work over a field with multiplicative characters of order 12; we therefore take q = p2,

which ensures that 12 I (q - 1). The results are summarized in Table 1, where as usual w denotes the Teichmfiller character. We carry out the calculation for Xi = X2 = w -(q-i)~3, p = 7 (mod 12), the other cases being similar. In all cases, the point is that the number of terms appearing in the additive character equals the number of variables, hence the given sum

20 Xi 1

X~

p (mod 12)

Lower bound

Actual slopes

1

1,5,7,11

0,1,1,1

0,1,1,1

(M--(q--1)/3

1,5,7,11

1,1,1,1

1,1,1,1

w-2(¢-1)/3

1,5,7,11

w-(q-i)/~

1,1,1,1

1,1,1,1

1

1

1

4

4

1

4

4

3' 3' 3' 3

5' 5' 5' 5

5

1 1 3 ~,~,1,~

I

I

7

I__I_4_4

1

5

3 5

4

3' 3 ~ 3' 3

11 w-(q--1)/3

'~--(q-1)/3

w-2(q-1)/3

1

5' 5' 5' 5

1

1

4

5

1

1

3

1

1

3

7

1

1

4

1 5 5 3'6}6'3

4

3,5,3,3 1 1 ~,~,

13~

i ~,1,1,1

1

4

4

5

1 1, 1, ~ 3 ~,

7

5' 5' 3' 3 i , I , I , ~3 ~

1

4 4

4

1

4

4

4

1

4

4

4

5, 5, 5, 5 1,1,1,1 5' ~' 5' 5

1,1,1,1 2

2 2 2 5 3' 3' 3' 3

5'5'5'5

1 1 3 ~,~,1,~

I 1 ~,~,

2 2 2 5 3' 3' 3' 3

2 2 7 7 3' 3' 6' 6

11

1

1

3

1 ~,1,1,1

1

2

2

5

3' 3' 3' 3

2 2 2 5 3' 3' 3' 3

5

I 3 ~,1,1,~ 2 2 2 5

1,1,1,1 2 2 2 5

37 3 ' 3 ' 5

3' 3 ~ 3' 3

1

3

1,1,1,1

5

2 2 2 5 3' 3' 3' 3

7

11 Oj-2(q-1)/3

4

5, 5, 5,5

w-~(¢-i)/~

w-(q-1)/3

4

4

1

11

t.O-2(q-1)/3

½,1,1,1 1 1 4 4 3'3'3'3

11 w-(~-i)/3

~1 , ~1 , l , g3

w-2(q-1)/3

2

2

2 2

~, 5, 5, 5 1 1 3 ~,~,1,~

11

1

2

2

13

3

1

2

5

7

7

2 2 2 5 3' 3' 3' 3

3' 3' 6' 6

I I 3 ~,~,1,~

½,1,1,1

Table 1: Comparison of the lower bound of Theorem 7 with the actual slopes of the Newton polygon for E ~ . ~ x Zl(z)X2(y)~(¢ + ~y4).

21

can be evaluated explicitly in terms of Gauss sums. One then uses Stickelberger to find the exact p-ordinal. Note that the cases p - 7 (rood 12), (X~,X2) = ( w - ( q - ~ ) / s , 1 ) , (w -(q-~)/a, w-(q-~)/a), (w -2(q-~)/a, 1), (w -2(q-1)/3, w-2(q-~)/a), do not contradict the conjecture at the end of section 3. One has D = 3 and p --= 1 (mod3), but the sum (25) is not generic. The generic polynomial with this Newton polyhedron has the form aox, -b a l x y + a 2 x y 2 + a a x y 3 + a4xy 4 + as.

We have ,,-(q-~)/3(~)co-(q-~)/3(y)~2( x + :cy4) = x~yEk x

~_,

O,$--(q-1)13(X.)O3--(q-1)112(y4)~(X. .~_~y4)

:z,yc: k x

=

~

w-(q-~)13(x)w-Cq-')ll2(y)(1

+

w-Cq-1)/4(y) +

x,yEk ×

+ =

+

x + +

+

(26)

where the last line is obtained by the change of variable y -+ y / x and where for typographical convenience we have written g ( d ) for the Gauss sum denoted by gd in the example following Theorem 2. We have for p -- 7 (rood 12) that ( q - 1 ) / 4 = ( 3 p - 1 ) / 4 - k p ( p - 3)/4, so by Stickelberger ordq g(~-~) = ((3p - 1)/4 + (p - 3 ) / 4 ) / 2 ( p - 1) = 1/2. The ordinals of the other Gauss sums are evaluated similarly. The terms appearing in (26) have ordq equal to 516 , 1/3, 5/6, and 4/3, respectively. References

[1] A. Adolphson and S. Sperber: Twisted Kloosterman sums and p-adic Bessel functions, II: Newton polygons and analytic continuation. Amer. J. Math. 109,723-764 (1987) [2] A. Adolphson and S. Sperber: Newton polyhedra and the degree of the L-function associated to an exponential sum. Invent. Math. 88, 555-569 (1987) [3] A. Adolphson and S. Sperber: p-Adie estimates for exponential sums and the theorem of Chevalley-Warning. Ann. Scient. E. N. S. 20, 545-556 (1987) [4] A. Adolphson and S. Sperber: Exponential sums and Newton polyhedra: Cohomology and estimates. Ann. of Math. 130,367-406 (1989) [5] A. Adolphson and S. Sperber: On twisted exponential sums (preprint) [6] J. Ax: Zeroes of polynomials over finite fields. Amer. J. Math. 86, 255-261 (1964) [7] M. Carpentier: Sommes exponentielles dont la g6om~trie est tres belle: p-adic estimates. Pac. J. Math. 141,229-277 (1990) [8] B. Dwork: On the zeta function of a hypersurface, II. Ann. of Math. 80, 227-299 (1964) [9] N. Katz: On a theorem of Ax. Amer. J. Math. 93, 485-499 (1971)

22 [10] A. G. Kouchnirenko: Poly&dres de Newton et nombres de Milnor. Invent. Math. 32, 1-31 (1976) [11] S. Sperber: On the p-adic theory of exponentia/sums. Amer. J. Math. 108,255-296 (1986) [12] D. Wan: An elementary proof of a theorem of Katz. Amer. J. Math. 111, 1-8 (1989) [13] E. Warning: Bemerkung zur vorstehenden Arbeit yon Herr Chevalley. Abh. Math. Sere. Univ. Hamburg 11, 76-83 (1936)

p-ADIC BETTI LATTICES Yves Andr6 Intitut H. Poincar6, 11 rue P. et M. Curie, F-75231 Paris 5, France

Under the label "p-adic Betti lattices", we shall discuss two kinds of objects. The first type of lattices arises via Artin's embedding of integral Betti cohomology into p-adic ~tale cohomology for complex algebraic varieties; there are comparison theorems with algebraic De Rham cohomology both over the complex numbers (Grothendieck) and p-adically (Fontaine--Messing-Faltings). The second type of lattices, which we believe be new, arises in connection with p-adic tori. Although its definition is purely p-adic, it is closely tied to the classical Betti lattice of some related complex torus, and can be viewed as a bridge between the Dwork and Fontaine theories of p-adic periods; "haiti' of this lattice is provided by the cohomology of the rigid analytic constant sheaf ~ . In fact, both themes of this paper are motivated by a question of Fontaine about the p-adic analog of the Grothendieck period conjecture, as follows.

1. Let X be a proper smooth variety over the field of rational numbers ~ . The singular rational cohomology space

n HB := Hn(x~:,Q)

carries a rational Hodge structure (for any

structure is defined by a complex one-parameter subgroup of

n ); this

GL(H~ ® ~:), whose rational n

Zariski closure in GL(H~) is the so-called Mumford-Tate group of H B . n Let HDR denote the canonical isomorphism

nth --

algebraic De Rham cohomology group of X . There is a

n ®Q~

: HDR

,~

n®Q~

,~H B

provided by the functor GAGA and the analytic Poincar~ lemma. The entries in ~: of a matrix of n

n

w.r.t, some bases of HDR , H B , are usually called periods. One variant of the Grothendieck period conjecture [G1] predicts that the transcendence degree of the extension of Q generated by the periods is the dimension of the Mumford-Tate group.

n 2. On the other hand, let Het := Het(X~,~p ) denote the n th p-adic ~tale cohomology group of

24 X ~ , where ~ stands for the complex algebraic closure of Q. ^ Let us choose an embedding 7 of ~ into the fieJd q:p = ~ p . The successive works of Fontaine, Messing and Faltings [FM] [Fa] managed to construct a canonical isomorphism of filtered Gal(~p/~p)-modules:

n

HDR

®9

BDR

~

~Her(X(: p ),Q~

®{~p BDR

where BDR denotes the quotient field of the universal pro--infinitesimal thickening of ( p . Via Artin/s comparison theorem and the theorem of proper base change for ~tale cohomology (applied to 7 ) [SGA 4] III, this supplies us with an isomorphism

n @~

~ 7 : HDR

BDR

~ ~Hn @Qp et

BDR ~

~

n@~

HB

BDR "

The entries in BDR of a matrix of ~ 7 w.r.t, some bases of HDR n , HB n , a r e called (7)-P-adic periods. Fontaine asked whether the analog of Grothendieck's conjecture for p---adic periods holds true. The answer turns out to be negative; indeed, we shall prove: Proposition 1. Let X be the elliptic modular curve X0(ll ) , and n = 1, p = 11 . There are two choices of 7 for which the transcendence degree of the extension of Q generated by the respective p-adic periods differ. Nevertheless, one can still ask in general whether the property holds true for "sufficiently general" 7 • This would be a consequence of a standard conjecture on "geometric p---adic representations": Proposition 2. Let G be the image of Gal(~/Q)

......J GL(Hnt ) 2 GL(H~3) IQp. Assume that the

rational Zariski closure of G in GL(H~3 ) contains the Mumford-Tate group. Then for "sufficiently general" 7, the transcendence degree of the extension of Q generated by the p-adic periods is not smaller than the dimension of the Mumford-Tare group; if moreover n = 1, there is equality.

3. Let us next turn to p-adic Betti lattices of the second kind, the construction of which i~ modelled on the following pattern. Let us assume that over some finite extension E of ~ in ~:p, X E acquires semi-stable reduction, i.e. admits locally a model over the valuation ring of the p-adic completion K of E , which is smooth over the scheme defined by an equation XlX2...x n = some uniformizing parameter of K . In this situation Hyodo and Kato showed the

25 n (as was conjectured by Jannsen and Fontaine): existence of a semi-stable structure on HDR namely an is®crystal (H0,
H 0 ®K0 K depending on the choice of a branch

/3 of the p-adic logarithm on K × (here K 0 denotes the maximal absolutely unramified subfield of K ) . On the other side, one can sometimes use the combinatorics of the intersection graph of the reduction to provide lattices, well-behaved under ~o, in suitable twited graded (w.r.t. the "p--adic monodromy" N ) forms of HDR, n n . For and then use ~ in order to lift them to HDR instance, this works pretty well when X E = A is an Abelian variety with multiplicative reduction at p .

4. Before we describe this situation, let us remind the classical situation

(E fi ~:) : A(~:) is a

complex toms ¢ g / L , where L is a lattice of rank 2g ; furthermore H1 R ® £ ~ Hom(L,£). Composition with a suitably normalized exponential map yields the Jacobi parametrization: A(~:) 2 £xg/M where M is a lattice of rank g ; thus L appears as an extension of M by 2it M ~v where M ~ denotes the character group of £xg The bilinear map on M say q obtained by composing any "polarization" M r ; M with the bilinear map M x M / ..+ m (the multiplicative group) describing M ........., £ x g , enjoys the following property: -log] q l is a scalar product. Similarly, at any place of multiplicative reduction above p , there is the Tate parametrization: A(~p) q::g/M where M is again a lattice of rank g ; there is an analogous bilinear map qt on M such that ~ o g [ q [ p is a scalar product. Using the semi-stable structure, we construct the "p-adic" lattice L3 of rank 2g, formed of ~---invariants and depending on /~, which sits in an exact sequence like L (in this new context, 2it has to be understood as a generator of gp(1) inside BDR ). Setting

KHT := K [2i~,(2i~) - 1 ] ,

5~fl : H1R ®E KHT ~

we

have

moreover

a

canonical

isomorphism:

H°mg(L#KHT) "

5. We call the entries in KHT of a matrix of 5~B w.r.t, some basis of H~R, Lj3, "(~-p--adic periods". We may now state a more rigid p-adic transcendence conjecture: Conjecture 1: for suitable choice of /~, the transcendence degree of the extension of E generated by the ~ p-adic periods equals the dimension of the Mumford-Tate group of H I .

26 This conjecture splits into two parts: We first prove the inequality tr.deg E E [ 5~fl] (_ dim M.T. under some extra hypothesis (*) (theorem 3); this amounts to showing the rationality of Hodge classes w.r.t. Lfl. (The hypothesis (*) concerns the Shimura variety associated to A , but we think it is unnecessary, or even always satisfied). On the other side, we use G-function methods to prove inequalities of the type "boundary tr.deg E E [ ~fl] )_ dim M.T." refering to polynomial relations of bounded degree between periods (theorems 4 and 5). Roughly speaking, this is made possible because, when A varies in a degenerating family defined over E , the fl--p-adic periods involve the ~--logarithm and p-adic evaluations of Taylor series with coefficients in E , whose complex evaluations give the usual periods (theorem 2).

6. The previous considerations suggest the possibility of a purely p-adic definition of (absolute) Hodge classes on A . 1

Conjecture 2: Let E I be any extension of E , and let ~ be a mixed tensor on H~R ® E f lying in the 0-step of the Hodge filtration. Then ~ is an absolute Hodge class (i.e. rational w.r.t. L for every E 1 ,- * C ) if and only if ~ is rationalw.r.t. every branch fl of the p-adic logarithm. I II III IV V

Lfl for every E / "

,~:p and

The p-adic comparison isomorphic Hodge classes Covanishing cycles and the monodromy filtration Frobenius and the p-adic Betti lattice p-Adic lattice and Hodge classes

Convention: In this text, a smooth separated commutative group scheme will be called semi-abetian if each fiber is an extension of an abelian variety by a torus. Acknowledgements: I first thank the organizers of the Trento conference for inviting me to such a nice congress. This article was written out during a stay at the Max-Planck-Institut at Bonn, partially supported by the Humboldt Stiftung; I thank both institutions heartily. The paper benefited very much from several conversations with W. Messing, J.M. Fontaine and especially J.P. Wintenberger.

27 I. The p-adic comparison isomorphism

1. The "Barsotti rings" BDR and Bcris Let K be a p-adic field, i.e. a finite extension of ~p with valuation v]p. Let K0 , resp. K, ~:p denote the maximal nonramified extension of Qp inside K , resp. an algebraic closure of K , and its completion. Let R, R0, l~, ~p denote the respective rings of integers, and let

~ denote the

Galois group Gal(iZ/K). Fontaine has constructed a universal p-adic pro-infinitesimal thickening of ~p, see e.g. IF1] IF2]. It is denoted by B~)R and can be constructed as follows. Let us consider the Witt ring W of the perfection

lim

l~/pl~ of the residual ring R/pl~. It

4

sits in an exact sequence 0 defined by the diagram:

x~-~x p IR )0, where the ring homomorphism is P

~F1-----* W ~

W

- - ~

tim I~/pl~

Iss [Vn]

E

Witt(1 i m 1~)

(Tei chmiiller lifting

l;s ........~ lira 1~

~

0 vn1''''j Vn=~/vn'

Z pn [Vn]

of Vn) ~Rp

*vT0 Z pn

This provides a continuous surjective homomorphism B+R, - - ~ ~:p, where B~)R denotes the Fl-adic completion of W[ 1] . The fraction field BDR of B+ R is a K[ ~ ]-module, endowed with the Fl-adic (called Hodge) filtration F, and GrF BDR 2 r ~ ~:p(r) (Tate twists).

On the other hand, there is a universal PD-thickening of ~:p, denoted by B +cns" " It is obtained n

by inverting p in the p-adic completion of the subalgebra of W [1] generated by the nP-T's. For instance,

if

e = (e0,el,...)

is

a

generator

of

~p(1) = lira $z n(i~), p

tp:=log[e]__ = ~ ( - ) n - l ( n [ e ] - l ) n e B + "cns " The Frobenius

~o of

W

then extends to

28 Boris

= B + [1 ] cris Lrpj

(7~tp= ptp) and commutes with the

® K ~ - a c t i o n . Moreover Bcris K 0

imbeds into BDR.

2. The comparison theorem for Abelian varieties Let A = A K be an Abelian variety over K . According to Fontaine-Messing [F1] [FM], there is a canonical isomorphism of filtered ~-modules: N

F.M.: HDR(A ) K ® BDR ~

Het(AI~,Qp) ®QpBDR •

In particular HDR can be recovered from Het as the space of This isomorphism can be reformulated as a pairing: ttI~R(A ) ® Tp(A~-) ~

~Qnvariants in the R.H.S.

BDR.

[Faltings and Wintenberger have generalized this pairing to the relative case [W] ; the relative . HDR and BDR are endowed with connections and the relative comparison isomorphism is horizontal.] In order to describe part of this pairing in down-to-earth terms, let us assume that A has semi-stable reduction, i.e. extends to a semi-abelian scheme A R over R. (By a fundamental result of Grothendieck this always happens after replacing K by a finite extension). Let '~R be the formal group attached to A R ; then Tp(,~R)(K----) is the "fixed part" of Tp(AI~ ) [G2]. Now the restricted pairing H 1 R(A) ® Tp(~R)(lk~

® K is easily described as follows: ~Bcris K0

a)

It factorizes through the quotient H1R(AR)K ® Tp(,~R)(K---) .

b)

Using the formal Poincar~ lemma, write any fE " • OAK

c)

For any 7 = (70,71,..) E Tp(AR)(R--) = Wp(,~R)(l~, lift every 7n e ~ to '~n E Scris-

d)

® K is then given by lim The coupling constant <w,7> E Bcris K0 n pnf(~n) . See [Co].

w E H R(AR)K

in the form

w = df,

3. The crystalline and semi-stable structures a)

Let us first assume that A has good reduction, i.e. extends to an Abelian scheme A R over

29 R , and let us denote the special fiber of A R by ~ . In this case HDR(A ) carries a natural * 0) ® K0-strueture, namely tt *0 := Hcris(~/R R0 K 0 ; moreover this K0--space is a crystal: it is canonicalty endowed with a semi~inear "Frobenius" isomorphism !o. The Fontaine--Messing isomorphism is then induced by an isomorphism of filtered ~ and ~ - modules:

H0 ~(0 Bcris

) Het(A]~'~p) ®Qp Bcris "

In particular Het can be recovered from tt 0 as the space of ~-invariants in the F0-subspace of the L.H.S. b)

If contrawise A has bad reduction, let us use Grothendieck's theorem to reduce to the case ,

of semi-stable reduction. [Jannsen had the idea that there is still a fine structure on

ttDR,

,

involving some 'monodromy operator", and such that Het could be recovered in a similar way as in the good reduction case [J]. Fontaine then formulated a precise conjecture and proved it in the ease of Abelian varieties]. The result is IF2] : Choose a branch /3 of the v-adic logarithm. Then bl) there exists a canonical K0-structure H 0 on HDR(A ) , endowed with a nilpotent endomorphism N ; N = 0 iff A has good reduction. b2) H 0 is naturally endowed with a semi-linear "Frobenius" ~ = ~/3' related to N by means of the formula: Nto = p~oN. b3) there exists ufl E BDR such that Bss := Bcris [ufl] is ~---stable, and such that N = d/duff and the extension of to to Bss given by toufl = puff commute with the J-action. b4) the p-adic comparison isomorphism is induced by an isomorphism of filtered K0( ~ )-modules compatible with to and N : *

tI0 ~0

N

Bss~

*

tiet(AK)

®QpBss "

* * ®Bss) ]~o=l,N=0 In particular Het can be recovered from H *0 as the space [F 0 (Het For a concrete description of the semi-stable structure due to Raynand, see below III 4, IV 1 and [P~].

4. Rigid 1-motives and Fontaine's LOG In the study of the comparison isomorphism, it is useful to embed Abelian varieties into the bigger category of 1-motives [D1].

30 a)

Recall i) ii) iii)

that a smooth 1-motive [M ¢ ~,G] on a scheme S consists in an 6tale sheaf M locally defined by a free abelian group of finite rank a semi-abelian scheme G over S a morphism ¢ : M ) G.

For each prime p , one attaches to

[M

¢ --*G G]

6tale cohomology (= 4tale realization of [M

a (Barsotti-Tate) p-divisible group, and its

~ ..~ G] ).

On the other hand, the universal vectorial extension M__ De

Rham

realization

H1R[M

) G ~ of M

) G] := Colic G4 ,

with

; G provides the

its

Hodge

filtration

F1H1 R = Colic G . b) There is a notion of duality for 1-motives. We shall only consider symmetrizable 1-motives, i.e. 1-motives isogeneous to their duals (the isogeny inducing a polarization of the Abelian quotient of G ). This amounts to giving i) a polarized Abelian scheme (A,~_) over S ii)

a morphism X ; M

) A , where M is an 6tale sheaf of lattices; let ,X"v = ,~ o X

iii) a symmetric trivialization of the inverse image by biextension of A × A ~ .

(X,Xv )

of the Poincar4

-1 0 c) It is convenient to view 1-motives as complexes in degree ( - 1 , 0 ) : M ....). G . When S = Spec K , K = p-adic field, it is more convenient, according to Raynand [R2], to identify 1-motives which are quasi-isomorphic in the rigid analytic category; for instance, if A is isomorphic to the rigid quotient G / M , we consider A (or [0 ) A] ) and [M ) G] as two incarnations of the same rigid 1-motive. Indeed, the associated p--~visible groups, resp. filtered De Rham realizations, are isomorphic; furthermore this isomorphism is compatible with the Fontaine---Messing comparison isomorphism, which extends to the case of 1-motives (its semi-stable refinement also extends to this case (Fontaine-Raynand)).

d)

Let us illustrate this in the simple case [g ~ . Gin] (when q is not a unit in K , this is

1 ,---~ q the 1-motive attached to the Tate curve K×/qg ). The Tate module sits in an exact sequence

0

) gp(1)

) Tp

...). qg ®g Zrp

)0

Let tp be a generator of ~p(1), and let Up 6 Tp lift q. Let moreover /~ be a generator of the character group X(Gm)

so that d#/l+/~ generates the K-space II1 '

•m

. At last let us repre-

31 sent Up by a sequence (q,ql,...) with qP+l = q n ' and let ~n lift qn in BDR. The p-adic periods of []~ ~h ) Gm] are given by: <tp,d/~/l+~> = ~" tp in BDR , ~p = l i m l o g qn

n

/q "

n

By abuse language, one denotes this limit by LOG q ; its class rood ~p(1) depends only on q. If one requires more rigidity, one may embed K into ~: somehow, and choose Up in the Z/-lattice given by the Betti realization of the corresponding complex 1-motive; LOG q is then defined up to addition by /Ttp, as in the classical case. e)

More generally, let us consider a 1-motive [M

the universal extension splits canonically:

¢ . , T] , where T is a torus. In this case

G b = T x Hom(M,~a )v ; this induces a canonical

splitting of the Hodge filtration: H~)R[M , T] = F 1 ¢ Hom(M,K). On the other hand, let M ' denote the character group of T and q : M × M ' ...~. Gm the bilinear form induced by . Again the ~tale cohomology sits in an extension

0

~Hom(M,Qp)

1 [M ...~. Het

~T]

~ M' ®i/Qp(-1)

,0

Now assume that M and M ' are constant. Let ( m ] ) denote a basis of Hom(M,~) as well as its images in H 1 and H 1 resp.; let et R (#i). denote a basis of M ' , let d/~i/l+/~ j . be the corresponding basis in F 1 , and let ~j lift /~i/tp.

inside

Hlet" At last, let

(mi)

denote the basis of

qij = q(mi'#j) . Then in the bases of H1R r¢

M

(resp. H et I ) given by

dual to

( m ] ) , and set

v {d#j/l+/~j;mj}

(resp.

¥

{/~i,m i } ), the matrix of the comparison isomorphism takes the shape: tpI This completes the description of this isomorphism for

any Abelian variety with split multiplicative reduction.

32 II. Hodge classes.

1. The complex setting. a)

Let E be a field embeddable into ~: , and let A E be an Abelian variety over E . An

element ,eFOIHD1R(AE)~ha®HIFt(AE) von] = F 0 [ E n d H ~ R ( A E ) I On (for any n ) i s called a Hodge class if its image in

~1 @11 [End HB(A~:,q:)]

" " the lies m

i n al subspace rat'o

[End HI(A¢,~)] @n . By Deligne's theorem on absolute Hodge cycles [D2] , this definition does not depend on the chosen embedding E ¢ , ~ . Moreover, after a preliminary finite extension of E , one gets no more Hodge class by further extending E . It follows that the connected component of identity of the Hod~e group of A E (which is by definition the algebraic subgroup of GL [HD1R(AE) 1 which fixes the Hodge classes)is an E---form of the Mumford-Tate group of HI(A~:,~). It is known that the Hodge group is a classical reductive group. b)

Let us fix an embedding L : E ~

f . For any E-algebra E / , the E ~-hnear bijections

H1R(AE) ®E E ' ~~ H I ( A E ® ~ , Q ) ® 9 E ' which preserve Hodge classes form the set of E~-valued points of a E-torsor PL under the Hodge group; for E / = C , one has a canonical point ~ given by "integration of differential forms of second kind". Lemma 1: the torsor P is irreducible. Indeed, there exists a finite Galois extension E / of E such that the Hodge group of AEI is connected; hence the associated torsor P / is geometrically irreducible. But via the isomorphism L

HDR(AE) ®E Er = HDR(AE~) ' a Hodge class on A E is just a Hodge class on AE~ which is fixed by Gal(E~/E). Therefore PL is the Zariski closure of P~ over E , and is irreducible. Conjecture (Grothendieck): if E is algebraic over Q, 2, is a (Well) generic point of P (over

E). Thanks to the irreducibility lemma, this amounts to say that the transcendence degree over • of the periods equals the dimension of the Mumford-Tate group (here, "periods" means entries of a matrix of ~ w.r.t, bases of HD1R(AE , _ ) HI(Aq:,Q)) . _ [This deep problem is solved only for Abelian varieties isogeneous to some power of an elliptic curve with complex multiplication (Chudnovsky).

33 The conjecture can also be formulated as follows: every polynomial relation between periods, with coefficients in E , comes from Hodge classes. A major result in transcendence theory establishes this for linear relations (Wiistholz); the only Hodge classes which appear in this context are classes of endomorphisms.] 2. Behaviour under the p---adic comparison isomorphism Assume now that E is a number field; let v lp be a finite place of E , and K = E V be the completion of E w.r.t, v ; ]~ denotes the algebraic closure of E in l~. Let us choose an embedding 7 : 1~ ¢----,~ and denote by t its restriction to K . At last, let isomorphism:

%:

H1R(AE ) ®E BDR ~ ' HI(AE ®LE,9) @9 BDR

denote the composed

F.M.

HDR(AE) @E BDR

,%/

~ , H1R(AK) OK BDR

' H~t(AE) @gp BDR

~ ' Hlet(AET)®gp BDR ~'' HI(AE @7 ¢,9) ®9 BDR ~ , HI(AE

® t E,9) ®9 BDR

Blasius---Ogus [B1] and independently Wintenberger have recently proved the following striking result: Theorem 1. For every 7 above t, ~

is a BDR-Valued point of Pt"

[The Wintenberger proof uses the relative comparison isomorphism while the Blasius-Ogus proof uses Faltings's comparison theorem applied to smooth compactifications of "total spaces" of Abelian schemes]. With the notation of I 3, it follows formally that Hodge classes lie in (End H~)@n are Frobenius-invariant and killed by N In view of this theorem, it is natural to ask whether the p-adic analog of Grothendieck's conjecture holds, namely whether ~ is a (Well) generic point of P over K . [After I communicated the counterexample in prop. 3 to Fontaine, he suggested the following:] Conjecture 4: for "sufficiently general" 7 above t ,

~7 is a (Weil) generic point of P

over

34 See below, § 4. 3. Proof of proposition 1. In this example E = Q curve 0

{~p/qg , , gp(1) ~

, and AQ is the elliptic curve X0(ll ) . For p = 11 , AQp is a T a t e

q e pgp Tp(A )

With the notations of J qg @g gp

I 4b, consider the exact sequence

*0 , and let tp be a gp-generator of gp(1) such

2 2 that tp h Up is a g-generator of the image of h HI(A ~ % ~:,g) in h W p ( A ) for some fixed 3 ' : ~ ~ q: ; this determines tp up to sign. Let moreover v be a unit in gp such that w : = p1 ~ belongs to the rational subspace f ~ Q of fllAQp . According to I4b, we thenhave:

< Vtp,W> = "~tp. Now let g e Gal(~/Q) ; changing 7 into 7 o g modifies the Betti lattice inside Tp via the formula:

where g

denotes the image of g under the group homomorphism Gal(~/Q)

......, GL(Tp).

But in our case, this homomorphism is surjective, according to Serre [S 1]+ In particular, there . exists some g e G a t ( ~ / ~ ) , w i t h d e t g = 1 , and such that v t e T lies in the Betti lattice P P * ttl(A ®3`og I;,g) ; since det g = 1, changing 3' to 3' o g preserves t .

~

P

It then follows from the relation < ~tp,W > = ~- tp that the Zariski closure of ~7og over • is contained in a hypersurface of P . On the other hand, it follows from Serre's result and the next lemma that for some other 7 ' : ~ ¢'---+[ , the Zarisld closure of ~,~, over Q is the full torsor P. 4. Proof of proposition 2 (Abelian case). We prove the following variant for an Abelian variety A E over a number field E [Proposition 2 itself is proved in the same way with only minor modifications involving simple general facts

35 about absolute Hodge cycles contained in the beginning of H ] Let us fix 70 : ]~ ~

~ and denote by H 1

70

the rational structure H~(A]~ @70 ~:,Q) inside

H~t(A ,Qp) = H~t(A ,Qp) (for ]~ = algebraic closure of E in l~ where K = E vlp )" The ~g K ' v'

GaloisrepresentationHIt(A ,~p) isdescribedby a homomorphism : Gal(]~/E)

,GL(HI0)(QD).

Let us denote by G70 the Zariski closure of the image of Gal(Ig/E) over Q , which is the smallest algebraic subgroup of GL(H10) whose group of p-adic points contains the image of Ga~(g/E). Coniecture 5: the Mumford-Tate group of H170 is the connected component of identity in GT0 . [One easily checks that the truth of this conjecture does not depend on the choice of 70 ; on the other side, the fact that the Mumford-Tare group contains G O is a theorem of Borovoi [Bo] ]. 70 This conjecture is a weak form of the well-known conjecture of Mumford-Serre--Tate (replace by Qp in the statement). Proposition 2 ~: Conjecture 5 implies conjecture 4. Proof:

let

•E 70

denote the Zariski closure of

72 0

over

E , inside the torsor

P = Pt (~ = 701 E) " let GaT0 denote any connected component of GT0. For any ga • Ga 70(Qp ) ' let Cga : Spec BDR

~.. Spec E [ ~70 ] x Spec •p

, j ~ E × Ga

701E

be the composed morphism of affine schemes given by ( ~70,ga).

From lemma 1 and conjecture 5, it follows that GT01E = tJ Ga701E acts transitively on P , and that

Q ' G ~ 01E = P for any non-empty

E--subscheme Q of P . We can now make the

expression "sufficiently general 7" (in conjecture 4) precise: it means "any 7 of the form 7 = 70 o ga where ga e Im Gal(E/E) is such that ¢ga maps to the generic point"; indeed for

36 these embeddings 3',

~E= g----~E ~o" >--E .3`0 (~o)IE :--E.~o °";olE= e. It remains to prove the existence of (uncountably many) such ga ' To this aim, let us remark that there are only countably many subvarieties of G a70IE ( ~ 0 ) ; we denote them by Qn ' n e 84 . Hence there exist linear subspaces [ [ of End H 1 ® ~p , of codimension dim P - 1

3`o

such that

[ [ n G a70(~p) fl Qn ¢ I I n G~0(Qp)

for every

n . Any g a e ~ N G ~ 0 ( Q p )

being outside the countable subset Un ] [ N G~0(Qp) N Qn then satisfies the required property

_ E(~o )

g.

= a~olr (~o)

37 III. Covanishing cycles and th_...~emonodromy filtration.

1. Covanishing cycles. a)

Let again

A be an Abelian variety of dimension g over the

p-adic

field K , with

~rig semi--stable reduction. For any finite extension K ~ of K , let ~ K~ rigid analytic variety ("Abeloid variety") over K ~ .

denote the assodated

of the constant sheaf ~ on A ~ Kr i g/

can be interpreted as

i• g,~) ^rig

The (Cech) cohomology HI(A

the group of Galois covers of ~ K /

with group 27 [R1] [U].

For reasons which will soon be clear, we denote this group by M v (K ' ) . One defines this way an etale sheaf _M_Mon Spec K , described by the

~ - module M v := M v (K---) ; points of the lattice

M v will be called (integral) covanishing cycles.

b)

In order to understand the geometrical meaning of

M v , let us consider the Raynaud

extension G (resp. G / ) of A (resp. of the dual Abelian variety A ' ) :

T

~

M

M'

1

i

G

A

-'~

B

T/

--~ G '

---~ B f

A~

G is an extension of an Abelian variety B by an unramified torus T of dimension r < g (lifting the torus part of the semi---stable reduction of A ), and A (resp. ACp ) is the rigid analytic quotient of

G

(resp.

GE ) by the lattice M(K) of K---characters P M := M(K---) of characters) of T / ; and symmetrically for G / ...

This description of A~

(resp. the lattice

shows that M v is the dual of M ; in particular the (finite) ~ - action

is unramified (since T ~ is). [In Berkovich's astonishing theory of analytic spaces, one associates with A some pathwise connected locally simply connected topological space A an ; M(K) should

38 then appear as its fundamental group in the ordinary topological sense [Be] ]. c)

Composing the morphisms

M v ®~ Bp

) lim H1(A~:ig,~/pni/)GAGR,,,,,,,,, lim H~t(Af~p,~/pn~) +----n

p n

[where GAGR denotes the functor studied by Kiehl [K] ], yields a natural injection of ~p [ ~] -modules: : M v ®g g

Let

p

¢----,III.(A ,g_).

~L I( v

d) On the other side, the lattice M v (K) is naturally isomorphic to the group of rigid analytic homomorphisms from Gm to A' JR1] , see also [BL] for the variant over ~:p. Composing the morph/sms

iiOmrig(Cm,A,)pull-backHom(H1R(A,rig),H1R(~rig)) duality,Hl~R(A.rig) GAGR IIIR(A.) yidds a natural embedding: tDR: M v (K) ®~ K ~

tt~R(A ) .

[Le Stum [lS] interprets the image of I1)R as follows. By means of some compactification ~ of the semi-abelian group scheme A R over R extending A , there is the notion of strict neighborhood in

Trig,,K of the formal completion

A . For any

dArig-module

• , set

j+ ~ = li..x"~mJA. JA ~ ' where JA runs over all embeddings of strict neighborhoods of A inside Afig ; j+ is an exact functor, and there is a canonical epimorphism ~" the covanishing complex by ¢ := Ker(f~ rig ~ sequence , Rn(Arig,¢)

* ~j+ F [B]. Define

J+f/,~ rig) ' which gives rise to a long exact

, tt~)R(A)

......, ttrni.(~ ) 6

involving Berthelot's rigid cohomology of the special fiber ~ . The group Hl(Arig,¢) can then be

39 identified with Im ~I)R ; this justifies the label "covanishing cycles" by analogy with the complex case.] e) It turns out that the maps isomorphism; More precisely:

~et and

~DR are compatible with the Fontaine-Messing

Proposition 3: the following triangle is commutative: MY(K) @ Qp

/

~DR

H~R(A)

OK

B

Let

BD R

F.M.

1 A

Het( ~r) ®Qp BDR

Proof: let us introduce the Raynaud realization [M

~G] of the (rigid) 1-motive A.

The map

natural

Let

Hom(M(K),Qp) ~

can

be identified with

H~t [M

the

injection of

fi-modules

:

*G] .

On the other side, getting rid of double duality, one easily sees that ~DR can be identified with the natural embedding Hom(M(K),K) ~ ~ H1R [M ~ G] , see also [IS] 6.7. The required commutativity then follows from the fact that F.M. is tautological for the quotient 1-motive [M(K) ......... ,~1] (whose associated p--divisible group is ~ (~p/~p)n) . ~) An orientation of ~:p is an embedding of ~Upm(q:p)= ~p(1) into q:x ; this amounts to the choice of a generator tp of the Bp-module ~p(1) up to sign, [a further orientation of £ itself would fix the sign], or else to the choice of an embedding of Abelian groups X.(•m)

......~Wp(Gm)(= ~p(1)) .

By using an orientation of ~:p and duality, we get from c) an injection:

Jet: M'v (1) := M 'v ® X,(Gm)

Using the Raynaud 1-motive

[M ~ G]

pairing between H1 R and

M~V(1)

~. H~t(AL,~n) ® Tp(~m) ~ Tp(A .) . K ~" over 1~, it is then clear that the Fontaine-Messing

takes its values in

K't

for some finite unramified P extension K / of K (even in Ktp if the torus part of the semi---stable reduction ~ splits),

40 2. Ravnaud extensions and the q-matrix. Let f : A

J S be a semi-abelian scheme with proper generic fiber, S being an affine normal

connected noetherian scheme; we put S = Spec 9~, ~ = a)

Let us first assume that

Frac .~..

:~ is complete w.r.t, some ideal I (we set S O := Spec :~/I ), and

that the rank r of the toric part T O of A 0 = A x S S O is constant. One constructs the ttaynand extension over T

lifts

TO

0

~T ~

and

B

~

[CF] II, 0

*T

~G

....., ]3

~0 , where

is an Abelian scheme. There is also the Raynaud extension

~G ~ ....... ~B ~

* 0 attached to the dual Abelian scheme A ~ , and B '

is the dual

of B ; moreover rk T = rk T ' = r . These extensions arise via p u s h - o u t from morphisms of fppf sheaves

M

M'

.......~B , where M = X * ( T ' )

.....* B '

(character groups).

M ' = X (T)

The objects G , T , M ,B (resp. G ~ , ...) are functorial in _A (resp. A ' ) . b)

Replacing S by some open dense subset U , the Faltings construction (using an auxiliary

ample line bundle .~¢ on G ~

[CF] II 5.1), or methods of rigid analytic geometry ( [BL1] with

less generality), provide a t~ivialization

q

(independent of

.2'

[CF]

III 7.2) of the

Gm-biextension of M x M ~ obtained as inverse image of the Poincar~ biextension of B x B ' ; this amounts to giving a rifting [M

~bjG]

use some

on U ) . W h e n basis

q : M × MI ~

~G U

of

M

T O splits, so that M = M

((mi,/~j) } Gm,U

MU

of

M x M~

by a matrix with entries

in qij e

~B

(whence a smooth 1-motive

and M ' = M t are constant, one can order ~x

to

express

the

. [If moreover

A

bilinear form is principally

polarizable, such a polarization induces an isomorphism M = ~ M ~ , and then q : M ® M

, ~x

is symmetric. In the literature on Abeloid varieties, the associated q - m a t r i x is often referred to as the "period matrix"; however this terminology conflicts with the Fontaine-Messing theory, but some precise relation will be exhibited in I V ] .

c)

In order to understand the complex counterpart, we replace S by An , where ,~ denotes , the unit disk in ( . Assume that the restriction of f to the inverse image of S = A*n is proper, ,

where a

stands for the punctured a .

The kernel _h of the exponential map exp: Li..__eeA]S

, A is a sheaf of lattices extending the

41 local system

, . The (unique) extension in A of the fiber of h over 0 is a local {HI(As,B)}seS -_ system N of rank 2g - r . Via exp (which factorizes through N ), _A becomes a quotient of the semi-abelian family G = (Li....eeA_/S)/NN_ : A = G / M , where M denotes the sheaf of lattices A/_N_ (which degenerates at 0 ). This supplies us with a (complex analytic) smooth symmetrizable 1-motive *

[M ~

2]

over

1

S . Both the Betti realizations HI~_ and the De Rham realizations H~R_ (endowed with the Hodge filtration) of A and [ M - - - ~ G] are canonically isomorphic. However, one may not identify these "l--motives" because the weight filtrations differ, see below § 4. d)

We now start with the following global situation:

S 1 is an affine variety over a field l'~ of characteristic 0 ; 0 is a smooth rational point of S 1 , and Xl, ... ,xn are local coordinates around 0 ; fl : A1 ~S 1 is a semi-Abehan scheme, proper outside the divisor XlX2 ... x n = 0 , and the toric rank is constant on this divisor. Because

fl

~1 : ~1 ~

is of finite presentation, it arises by base change from a semi-abelian scheme ~1 (where ~ is a sub-Z/-algebra of E of finite presentation), with the same

properties as f l ' I f w e p u t

~=~[[Xl,...,Xn]]

, S = S p e c ~ (the completion of ~1 at 0),

I = (XlX2 ... Xn) , f = ~I/S ' we are in the situation a) b). Moreover, the open subscheme IJ may be defined by the condition XlX2 ... x n ~ 0 . It follows that the entries qij of the q-matrix belongto ~ [ [ X l , . . . , X n ] ] [XlX2 1-- Xn ] .

e)

Assume moreover that

E is a number field, with ring of integers

dE . Then ~ can be

chosen in the form dE [11 , where v is a product of distinct prime numbers. Thus for every finite place v of E not dividing u , the qij entries are meromorphic functions on Any' analytic on Av* n (Av , resp. av$ denotes the v-adic "open" unit disk, resp. punctured unit disk), and bounded away from 0 . On the other hand, one can also see (using construction c)) that the qij's define meromorphic functions on some complex polydisk centered at 0. [Remark: following [C], an element y of E[[Xl, ... ,Xn] ] l-

is said to be globally bounded if

-1

y • dE [ ~ ] [[Xl, ... ,Xn]] for some v , and if y has non-zero radius of convergence at every place of E . (Such series form a regular noetherian ring with residue field E , and the filtered

42 union of these rings over a~ finite extensions of E , is strictly henselian). One can show that the (x 1 ... xn)mqij's are globally bounded series (for suitable m) . The problem is to show that the v-adic radius of converge is not 0 for any v ] v . Using the compactification of Siegel modular stacks over ~ , one can find a semi-abelian extension of ~1 over an ~ - m o d e l of some covering of ~1 ' and afterwards, one has to use the 2-step construction of [CF] III 10 to keep track of the possible variation of the torus rank of the reduction, after replacing the divisor x 1 ... x n = 0 by vx 1 ... x n = 0] . b) Lemma 2. If v ~ v , then the entries of the q-matrix are units w.r.t, the v-adic Gauss norm.

Proof (sketch): let the v-adic

$ denote the completion of the quotient field of 2~ = ~ [ [Xl, ... Xn] ] w.r.t.

Gauss norm

I IGauss (= "sup" norm on

{ I Gauss by Gauss' lemma, hence characteristics.

$

~).

Because v is discrete, so is

is a complete discretely valued field of unequal

By construction of the Raynaud extension, the Barsotti-Tate groups associated to A / $ resp. to the 1-motive

[M/$

~P~G/$]

coincide. It follows that Grothendieck's monodromy pairing

associated to A / g is induced by the pairing M × M ~

+ $x

~~ given by the valuation of

the q-matrix w.r.t. { t Gauss ' Since A / $ has good reduction modulo the valuation ideal of $ (indeed its reduction is the generic fiber of the reduction of A modulo v , which is proper when v ~ v ) , this pairing has to be trivial:

{qij{ Gauss =

1.

g)

An example: let us consider the Legendre elliptic pencil with parameter x = A , given by the affine equation

v2=u(u-1)(u-x). Here one can choose ~ = ~[½] , and one has the explicit formulae: 16q = x(1 - x ) - l e - G / F

x = 16q(I ® ] (1 + q2m)(1 + q2m-1)-l)S m= 1

43 tD

1 12m ((~-)m/m.) x

where F = m=O

r0

G =2

{D

~ ((½)m/m!)2( ~ ~:)x m . m=l ~.=1

This example is studied thoroughly in [Dw].

3. Vanishing periods. a)

Let us take up the situation 2d again, and assume that E is contained in the p--adic field

K , with ~ C R . Assume also that the torus part of the semi-stable reduction splits. As before, we then have our constant sheaves of lattices M = M , M r = M on the v-adic unit polydisk t, nv ;let

{~}

b e a b a s i s of M t , and let {#~}

be the image of the dual basis of

M/V(1) under Jet (defined up to sign, see III lg). 1

*

On the other hand, we have the relative De Rham cohomology sheaf HDR(A/S ) which admits a canonical locally free extension to S (where the Gauss-Manin connection acquires a logarithmic singularity with nilpotent residue); in fact this extension is free because S is local, and we denote by {~j} a basis of global sections. We are aiming to give some analytic recipe to compute the 1 1 *n Fontalne-Messing "vanishing periods" Fp < #i ,wj(s) > of the fiber Al(S ) , s • a v , see III lf. b)

Let us express the composed morphism H1R(A/S*) can

~H R

S----ill

R T

"2_M , ® os

(roof = formal completion) in terms of the bases wj, d~/1 + ~ . We get a (2g,r)-matrix (~ij) with entries in 2~ = ]~ [ [Xl, ... ,Xn] ] . / Lemma 3. For any s e Av* n ' one has the relation wij(s ) = * 1_ tp < /~i ,wj(S) > • Moreover wij is

a bounded solution of the Gauss-Manin partial differential equations on An . Proof: the first assertion is easily proved by considering Raynaud's incarnation [M(s) ~G(s)] of the rigid 1-motive associated to Al(S ) , together with the trivial computation of Fontalne-Me~sing peno~ of the spUt torus T = -T(s): < " !i ' % / 1 +~j > = ~ ~ijtp The second

44 assertion follows from the horizontality of the map Hl~R(A/s*)Can" , HI~R(A/S ) ~^^ w.r.t, the Gauss-Manin connections v , and the fact that M ~ is formed of horizontal sections of H R(T

c)

(see also [vM]).

Let ~ denote a uniformizing parameter of R . We modify slightly the setting of 2. d) by

assuming that fl extends to a semi-abelian scheme ~ : _A ~ Spec R fl E

~ ~ ' proper

outside the divisor ~ x 1 ... x n = 0 , and with constant split toral part on this divisor. Again, the t~.ll.'8 converge on anv ' and for every point s e SI(E ) N a*nv ' the v-adic evaluation of wij at s may be interpreted as in lemma 3 (if furthermore E is a number field, the w..'s 1j are in fact globally bounded series). We next look for complex interpretation.

d) Let L: E

¢---4 •

be a complex embedding. We now assume that

following property: ~(~:) should contain the polydisk of radius convergence of the analytic solutions of Ganss-Manin in this polydisk).

By specializing to s , construction 2c provides an embedding: where A s := _Al(S) . Dually, we also have an embedding:

s e SI(E ) satisfies the Ixi(s) l

tB : M v ~

(to insure the

H~(A s ® ~:,~) ,

JB: M'V(1) = 2i~rM'V ¢'-4 HIB(As,(:'~) " In addition to the orientation of

~:p , we choose an orientation of

~: ; this eliminates all

ambiguities of signs, and allows to identify jB(#](1)) with #~ . Proposition 4. The following diagram is commutative:

eet ~ H~t(As

MV

,K,~p) ~

~ HI(As,¢,~ )

F'M'--I ]

I ~'11,

H 1 R ( A s ) @E BDR

v-adic ~

evaluation

H 1 R ( A s ) @ E ~: H~R ( A/s*)V

In particular (by duality), the complex evaluation of t

<

>

/~co m p l e x ~ij

at

s

evaluation gives the ')usual )' period

45 Mv

Proofi let us draw a middle vertical arrow

~

, defined by the obvious embedding

~DR ( -A/s*)V Mv = FM v ~

FH RIM

~ G]

/s

. =

1

,

rHDR(A/S )

(or equivalently, when n = 1 , by the

analog of ~DR in the rigid analytic category over the discretely valued field E((x))) . Then the commutativity on the L.H.S. is essentially the content of prop. 3; the commutativity on the R.H.S. follows immediately from the definition of ~B (details are left to the reader). This proposition suggests the following open question: assume that E is a number field, and denote by ]~ its algebraic closure of E inside Cp. Does there exist "~ : ]~ ~ that the following diagrams commute? Mv

•"

~B

,

~

H (A¢,~)

tet[

I

H 1et(AE, Qp)

~*

1 (A~:,Qp) , Her

M'V(1)

,"

I jet Tp(AE) ;

JB

, H1B(A¢,~)

• 7~,

C above ~ such

f Wp(A~:)

(We leave it as an exercise to answer positively, when AE is an elliptic curve, with help of [S2]). 4. The monodromv filtration. a)

In [G2] , Grothendieck constructs and studies thoroughly a 3-step filtration on Tp(A]~.) ,

the "monodromy filtration" (here, we turn back to the setting of section 1)). By duality, we get a 1 ,. it turns out that this filtration is the natural weight filtration on the filtration Wet on Het 1 of Raynaud's incarnation of the associated rigid 1-motive, loc. cit. § 14. Het b) According to the semi-stable philosophy (motivated by higher dimensional motives), it should be natural to handle the monodromy business on the De Rham realization. The monodromyfiltration W 1 = 0 ,

W0 = M v ( K ) ® K ,

W 2 = H ~ R ' Gr w ~ _ M ' ( K ) ® K , i s t h e canonical filtration associated with the nilpotent operator of level 2 defined by:

46 N:

HIR (A)

- - -

*

II

]~IR(A)

II [M_ --, C]

[M --, a] t

[o

,,iR [M

T]

II

O]

II

M'(K)

where the arrow at the bottom Grothendieck's monodromy pairing:

® K

M~(K)

-; M Y ( K )

~, M_V(K)

v(q) : M ® M r

q-*

® K

is the map induced by opposite of K nr

~~

(v = valuation), ibid

(we change the sign because we work on tIl~ R , not on the covariant H1DR). Assume moreover that M = M(K) . Then the cokernel of the map M ~ .....*M v inducing # is canonically isomorphic to the group of connected components of the special fiber of the N6ron model A , see [CF] III 8.1. The weight filtrations W and Wet are related via F.-M. :

Lemma4 (for M = M ( K ) ) : G r W q } G r W ( 1 )

(Gr:etoGr:et)®

K. Qp

In case A is a Jacobian variety, there is moreover a Picard-Lefschetz formula (loc. cit. § 12), where ~et(M v) appears once again as the module of covanishing cycles. c)

Like the Raynaud extension, the operator

N

admits a complex analog (which is

well-known). In the situation 2 c), let Dj = a j-1 x {0} x an-J C a n be the jth infinity". For any s • a *n

divisor "at

there is a monodromy action "around D i : M ~. e GL(HI(As,~))

which is unipotent of level 2. Set

N ~.j := ~ 1 logt ( M ~ ) - I • End HI(As,~:) . These nilpotent

operators are constant on a *n , and can be computed on the limit fiber by: N ~. j = - ReSDjV (the opposite of the residue at Dj of the Gauss-Manin connection). Under the identification

HI(As,~) _~ H~" [M__(s)

...). G(s)] ® Q , the "monodromy" filtration on

the L.H.S. associated with N ~. is just the standard weight filtration on the R.tt.S. [D1]. J d)

One can mimic the construction a) over any complete discretely valued ring instead of K ,

e.g. over

~=~[[X]]l,

I=(x)

, in the situation 3 d), with

n=l

Nf°r e End H 1 R I A / 5 ~ [ K ] I t he nilpotent endomorphism obtained this way.

; We denote by

47 Next, we wish to compare N , N f°r and N ° .

Let us consider a double embedding ~ r2 R and let s e SI(E ) . Assume that

] × (S) l v < 1 and

%c

that SI(~ ) contains the disk of radius I × (s) ] 2 " At last, set A s = A(s ) .

Proposition

5.

In

this

situation,

the

complex

evaluation

of

N ° e End H ~ R ( A s ® •) _~ End HI(As,c,~:) ; the v - a d i c evaluation of

N f°r N f°r

at

s at

is s

is

v(x(s))N e End H ~ R ( A s ® K ) . Proof:

the complex fact is well-known. The

v-adic

assertion relies on the equality

v(qij(s)) = (Valxqij) • v ( x ( s ) ) , which follows immediately from lemma 2.

[.Remarks:

dl)

if we only assume that

A~

_

is proper outside

~ 91

x = 0) , the monodromy filtrations corresponding to

Nfor (s)

limit. d2) A quite general definition of N is given in [CF] III

10.]

and

x = 0

(instead of

N = N s still coincide at the

48 IV. Frobenius and th__~e p---adic Betti lattice. 1. Semi---stable F.ro.b.enius. We take up again the situation I 3b), and explain a construction of the Frobenius semi-linear endomorphism ~/3 (due to Raynaud [R2] ). a)

Let

/3 denote a branch of the logarithm on

Kx . This amounts to the choice of some

uniformizing parameter of R , say ~ , characterized (up to a root of unity) by the fact that /3: Kx ,z_ w × (R/~R) x × (1 + ~R) ~

K factorizes through 1 + ~ R .

b) Let A be an Abelian variety over K with semi-stable reduction, and let [M ~/' ~G] the Raynaud realization of the associated rigid 1-motive (G sits in an extension 0

,T

*G .........;B

......* 0 , a n d ¢ is described by q : M × M '

Let us factorize q = ~:v(q) . q0 so that q0 : M × M ~

;@

.....*Gm). extends over R This amounts

m

to a factorization ~b= X~ • ¢0 , where )~~ : M

"

~T = Hom(M ~,Gm) is induced by ~:v(q)

and ¢0: M_M . _ ~G extends over R (we use the same notation ¢0 for this extension). Because T is a torus, the universal (M,x~)---equivariant vectorial extension of T splits canonically, ......

ld

which yields a canonical isomorphism of (Hodge) filtered K-vector spaces:

1

a/3: ttDR [M ~

0

G ] / R ®R K

.

~

R R[M_

=

R R(A)

.

For two uniformizing parameters w1 , w2 , the map z~/31 , a /32 are related by: (i)

a n n~ 1 = exp(- log w2/w 1 • N), where N is the operator defined in the previous section. ~'2 ~'1

[Note the similarity with the definition of the canonical extension in the theory of regular connections, and also with [CF] III 9]. c)

Let

[M - - ~

BT

denote the Barsotti-Tate group attached to the reduction

mod. ~

G ] / R (1), and let H 1 denote the K0-space obtained by inverting p in its crys/K 0

(1) Remember that the Barsotti-Tate group attached to [M ~ image of ¢0 under the connecting homomorphism Hom(M,pnG )

G ] / R is given by the ...~. Ext(M,pn G)

pn associated with the exact sequence 0

~ nG P

~G

~G .... .~0.

of

49 first crystalline cohomology group with coefficients in R 0 . Up to isogeny, BT splits into the sum of two Barsotti-Tate groups: the constant one M__(K)®g Qp///p , and t lira nG OR R / ~ R . [It P follows that tt 1 does not depend on ~ - in fact, it depends only on A R ® R / ~ 2 R , which crys/K 0 determines G ® R/~2R .] R The K0---structure H~ mentioned in I 3 bl) is just the image of H 1 under ~/~ inside crys/K 0 H1R(A ) ; the element ufl is ufl := - LOG ~a (defined up to translation by ~p(1) C B+ris) • By transport of structure, the

a---semi~inear

Frobenius on

H crys1 /K 0

provides the

a---semi-hnear endomorphism 9 = 9/3 on H 1 (a = Frobenius on K 0) . Using (i), one gets the following relation:

(ii)

~/~2 o ~ 1fll = exp(- P1 l°g(w2/Wl) ~ ~ p-a .N).

From the functoriality of Raynaud extensions

G

and of the rigid analytic isomorphisms

Grig/M = A rig , it follows that the semi-stable structure (ttl,9,N) is functorial in A . e)

That construction of Raynaud may be extended to the relative situation III 2, i.e. over = ~ - adically complete noetherian normal R0-algebra.

Let U C S p e c ~ be as in loc. cit., and let us choose a lifting ~ r e E n d U of the char. p Frobenius. By analogy with step c), we can construct, locally for the "loose" topology on U , a horizontal morphism ¢ / ~ a ) : a H R(A/U) ...~II . R(A__/U) ; furthermore, this morphism v "stabihzes" M U , and it can be globally defined there. [This is the "stability of vanishing cycles" mentioned in [Dw];indeed, when say analytic Dwork-Frobe~ius mapping].

~=R[~x]

, a:x,

~xp ,

eft is nothing but the

If A is the fiber A(s) of A at some point s e U fixed under g ,we recover ¢ ~ ( ~ ) = 9 f t . 2. Construction of H ~ A ) • From now onwards, we shall assume that A has multiplicative reduction. a)

With our previous notations, we then obtain the following consequences:

5O

al):

G=T, r=g,

a2):

the Hodge filtration splits canonically: H1 R = (MV(K) ®g K) $ F 1

a3):

the monodromy filtration consists of only two steps: GrWetH 1 et -~ (My ®g ~p) ~ (M' ®~ Qp(- 1)) (via ~et and Jet), GrWH1R -~ (MY(K) ®~ K) • (Mr(K) ®~ K ) , (these isomorphisms being compatible via F.M., by prop. 3 and its dual) MY(K) ®/~ K = Ker N , and

F 1 projects onto

Mr(K) ®g K

(this isomorphic

projection being given by F 1 =Colie Arig -~ Colie T rig -~., M / (K) ®~ K ) . a4): b)

the Fontaine-Messing isomorphism F.M. is described in 1 4 c).

The splitting of BT (up to isogeny) reflects on H~, and translates into an isomorphism:

~fl : u r (~o acts trivially on

W~I ~ ~1 n0~ no

Gr 0 = MY(K)®K 0 , and by multiplication by g

p

on the image of

Gr I = M r(K) ~ K 0 ) .

Let us now choose an orientation of ~p (see III 10: g ( - 1) := X • (Gm) ~-~ ~tp 1 C BDR ,and let us consider the etale lattice A_:= M v ~ M ~ ( - 1) , and let A := A_(I~) = A__(R---)= _A(Knr) , where K nr denotes the maJdmal subfield of K non ramified over K . ® Using ~/~ and the orientation, we can embed A into H~R K

[1_] c [tpj

® BDR ' and we K

call p-adic Betti lattice its image, which we denote by H~ [This is the dual of the lattice L~ mentioned in the introduction. The introduction of t p , the "p-adic 2id' , is motivated by the fact that the complex Betti lattice (in the setting III 4c) is stable under 2i~rNa0, not N®] . We thus get a tautological isomorphism:

51 5v/~: IIl~-lt K @ Knr'[tPJ"

~

II~ ~ K nr [tp]

where in fact K nr could be replaced by some finite extension of K , or else by K itself if T is split. From formulae (i) (ii), it follows:

(iii)

tt 1]32= exp(-log ~w2/~l ~ • N) • II1131.

From the very construction of II~ and the formula ~ p = p t p , we get: ® K nr [tp] . Lemma 5: The lattice II~ spans the Qp--Space of ~of-invariants in H1 R K

Remark: the image of 5~;1R~ under F.M. does no__Atlie in tt~t(A,Qp) ; compare with lemma 4. c) Let us now describe the complex analog of r./~ : A ~ II~ . So let A c be a complex Abelian variety in Jacobi form T~:/M (the quotient being alternatively described by q:M®M'

; ~:x ,where M/ = X * ( T c ) ) . L e t us orient ~ : , a n d c h o o s e a b r a n c h

complex logarithm, and compose with M ~

q : M OM /

floq.

We get an embedding

M / v 0//~: ~ Lie T c _~ H1B(A~:,~ ) ®~ ~ which factorizes through

turn provides an isomorphism

E~cv: A = M v • M ' ( - 1) = M v •

/ 3 of the

1 M'

H1B(Ac,Z/) . This in ~ ~tT (A¢,~)

(the

injectivity is a consequence of the Riemann condition Re fl~(q) < 0). [d) One can imitate the construction of the p-adic lattice in the case of an Abelian variety B with ordinary good reduction over

K = K 0 . Over

K nr

indeed, the Barsotti-Tate group

B(p) = lim n B becomes isomorphic to the B.-T. group associated to a 1-motive [M ¢ * T] , ,~ p where ¢ is given by the Serre--Tate parameters [K] . However, in contrast to the multiplicative reduction case, the lattice ~ M V e M/( - 1) obtained in this way is no.__Atfunctorial, as is easily seen from the case of complex multiplication (¢ = 1). e) The construction of Frobenius generalizes easily to the case of 1-motives. This allows to construct p-adic Betti lattices for 1-motives whose Abelian part has multiplicative reduction. We shall not pursue this generalization any further here.]

52 3. Computation of periods. a)

We shall compute the matrix of the restriction of

{d#i/l+#i} g -

"

in F 1 , {#~ = ~ . ~ ( - 1 ) ) ,

5~fl to

F1H~R

w.r.t, the bases

mV}g=l in H~, assuming that T splits over

j=l

K. In other words, we compute half of the (f0-p---adic period matrix. Proposition 6.

Let

qij = q(mi'#j) ' as in I 4 c). The following identity holds in

ItlR(AK) ®K K [tp] : g --t ~ d#j/1 + #j p#j + ~ fl(qij)miv . i=1 b)

Proof: it relies on a deformation argument. First of all, one may replace M by a sublattice

of finite index, such that q - ~;v(q)q0 with q0 = 1 rood ~; (in this situation BT splits actually, [ not only up to isogeny). Let us consider the analytic deformation M

~o

,T

over

8~ = R [ [~ij - ~ij] ] gi,j=l

" - - = X ~ ' - - ~0 M

~0

~ij = Kronecker symbol,

] ~T

~0

of being

given by the matrix ~ij (so that [M ~P*T] arise as the fiber at ~ i j = q i 0j ) ' F ° r t h e f i b e r a t )C~ ~ij = 6 i j : [M ~ i T ] , the F1H1 R coincides with r . ~ G r W H 1 R ) ; more precisely d#dl + #j = tp/~ i , at ~ij = ~ij • By definition of the Kodaira-Spencer mapping K.S. (see e.g. [CF] III. 9), one deduces that

d#j/1 + #j - t p/~j" + (~

q°.

1J K.S.)m v ' at ~ i j = q i 0j . i j =~ij

But in our bases, K.S. is expressed by the matrix d~ij/~i j (see [Ka], or [CF] ibid, where there is a minus sign because of a shghtly different convention). One concludes by noticing that log q0j. = c)

fl(qij) •

One could also argue as follows, using F.M.: it follows from 2 a 3) that d#j/1 + #j may be

expressed in the form t p#j" + ~'/3ijmV flij e K ; furthermore, these coefficients flij are uniquely ®0 Bss and is multiplied by p determined by the property that d#j/1 + #j - ~]flijmv lies in H~ K under ~ap. Let us show that flij = fl(qij ) satisfies this property: by 1 4 c), we have

53 d#j/1 + #j = tpFH-l(~j) + ~ LOG(qi,)m , j v so that d#j/1 + #j - ~ flijmv = E(LOG(qij ) - ~ q i j ) ) m v + tpFM-l(~j) . Because /~i £ H1et '

-

) ¢ (H ® Bss )~°-p , and we conclude by the foLlowing:

L e m m a 6: let c e K x . Then "the" element L O G c-tic of Bss is multiplied by p under the Frobenius ~aB.

Proof:

let us write c = ~v(C)c0 , so that

~ n LOG c0 - 1 o g c 0 = - l o g l i m ( c n ) p in B +c r.l s Let

C~n= ( ' " C n + l ' C n ) e l i m g

~

where ~ n is any lifting of cn =

, and let

x,---,x p We have

[ C n ] e W(lim k-)

LOG c - tic = - v(c)u/3 + LOG c O - log cO . Now

~n

(c0)p-n

e]~

be the Teichmiiller representative

[Cn] ~o _-_ [cnP ] = [C'n_l] = "¢Cn_1

,

whence

{ n

(lira ~npn) ~ = (lim ~P )P. It remains only to take logarithms and remind that 9/3u~ = pu/~. d)

Let

us

examine

the

(t~)-1

complex

counterpart,

as

in

2

c).

The

lattice

M ~ M/v(1) = h v ~~ H1B(A~:fl/) embeds into Lie T~ ; the subspace FOH1DR(A~: ) of HI(Ac,~ ) ® C - ttlDR(A~) is just the kernel of the complexification of this embedding. It follows that the canonical lifting

~ni

~ = mi - l~-i-~ 1 ,, mi P~(qij)#i

#i' = (t~flo )-l(

(we set

of

mi

inside

FOH1DR(Ac)

(1)) , and

#j

is given by

(1))) / 3 ( #j

" By

orthogona~ty ( F l i t 1 R = (FoH1DR) ±) , we obtain: Proposition 7: the following identity holds in tt~R(A~: ) :

d,j/1 + ,j =

+ SZ (qij)m .

[The compatibility (resp. analogy) between prop. 6 and formula (iii) resp. prop. 7., is a good test for having got the right signs. Although #jN is defined quite differently in the p-adic , resp. complex case, the exterior derivative of the coefficients of m~'s Kodaira-Spencer mapping.]

describes in both cases the

54 4. Periods in the relative case, and Dwork's p-adic cycles. a)

Let us consider the relative situation as in 1. d with r = g ; U being subject to be the

complement of divisor with normal crossings ~x 1 ... x n = 0. We set ~ = R [ [Xl, ... ,Xn] ] , and

[ -1 Xn I

we denote by ~Y the K-algebra generated by ~ Xl

1 Xn ] . The construction of H~ can be transposed to this relative setting: elements of :~ [ Xl ... We use "the" relative Frobenius ¢ ~ a ) to construct an embedding

A "

)

c_

[tp]),

such that ¢ a) I m A

H 1 (it is locally constant w.r.t, the loose topology), and we Because ¢~(a) is horizontal, so is --/3,a get:

Lemma 7: t t l R ( A / ~ [tp] )v = H~,a ®g K [tp] . H1 (for n = l , ¢ : x : ; x p) in terms of Dwork's b) In order to interpret the lattice --/3,a p---adic cycles [Dw], one forgets about tp (or better, one specializes tp to 1 : K [tp] ~.K , H1 a ~ M V S M ' ) ~fl,

. Let us for instance take back the example III 2g (Legendre). For

K = Q p ( ~ i ' ) (p ~ 2 ) , we have M = M(K), with base m . Setting v = u w , the period of the differential of the first kind w = du for the covanishing cycle m v at x = 0 is given by the residue of

du

= w -dw ~+l

at one of the two points above

u = 0

on the rational curve

w2 = u - 1 ; namely, this is ~ .

Let # be the basis of M / = M ' ( K ) lifted to H ~ , such that q = q ( m , / 0 is given by the formula displayed in III 2. g. Then after specializing tp to 1, the matrix of 5~/3 in terms of the bases x , w/ = v ( x ~i~)w d is F

xP

F log q =F logx-log16+

x(F log q) ....

(with determinant ( 4 x ( x - 1))-1).

1 + xF l o g x + . . .

Here "log" is standing for the branch /3, and 1~ for ~ F .

55 In fact, Dwork prefers to get rid of the constants log 16 and ~ , into

by changing the basis {#,m v}

-2vr2~{# + (log 16)mV,mv} . In this new basis, the entries of the period matrix lie in see

,

log 16 I - p

1

[Dw] 8. n . c) In section 3, we computed periods of one-forms of the first kind. The "horizontality temma" 7 then allows to obtain other periods by taking derivatives; still, we have to show that, in the multiplicative reduction case, any one-form of the second kind is the Gauss-Manin derivative of some one-form of the first kind. In other words: Lemma 8.

Let us consider a relative situation, as in III 2c or 2d. If r = g , then for any

k = 1, ... ,n , the smallest 1 * HDR(A/S ).

as [V(Xk ~ ] o~xk)] -submodule

1 * ) containing of HDR(_A/S

f 1 is

Indeed, this amounts to the surjectivity of K.S., which follows from the invertibility of its residue at x k = 0 ; this follows in turn from the fact that this residue (F1) .~o can _ ~M ' ( d d ) ® E -~( H1D R / F1) .... can ~ M._V(rid) ® E is induced by the non-degenerate pairing val(Xk) o q . In the situation of III 3 a) b), we can now complete the analytic description of the 1 * period matrix: take a basis wj of the canonical extension of HDR(_A]S ) in the form wj e F 1

j= 1,...g.

~j+g = V(XkO]SXk)Wj Lemma 9. The matrix of ~]5 w.r.t, the bases { ~ j } , {/~,mi} has the form:

itpWij(s)

*tp(XkO/dXkWij)(s)"

] (for A = A I ( S ) )

wij (s)log qij(s) (XkS/OXk( ~vi jlog qij))(s ) d) We are now in position to state the main result of this section IV, relating p-adic complex Betti lattices.

and

C Data:

dl):

a field E , doubly embedded E ~

; orientations of • and Cp. A branch K

/~ (resp. ~m) of the logarithm on K x (resp. on Cx) ; a uniformizing parameter ~v such that ~(7~) = 0 .

56 d2):

an affine curve

S 1 over

E ; a smooth point

0 • SI(E ) , and a local

parameter x around 0 ; a regular model 91 of S 1 over E fl R . N

d3): a semi-abelian scheme f : A ~91 , proper outside the divisor wx = 0 , and given by a split torus on this divisor. To f , one attaches as before the constant sheaf of lattices A=MV~M'(-1)

(outside x = 0 )

, and the bilinear form q : M ® M '

, Gm

(outside

~x = 0) . Taking bases of M , resp. M' , one may expand the entries of a matrix of q into Laurent series: qij =K~Ti'xnijg"+ h.o.t., and consider the double homomorphism from the E-algebra • E 1 .= E [log 7/ij,tj" ~~ C L~P] induced by /~, t ~ t p

d4):

(resp. ;8 , t ~ 2 i ~ ) .

a simply connected open neighborhood of 0 in S•, say ~; over

~\0,

A

is identified with the graded form (w.r.t. the local monodromy No ) of Rlia, nB. d5):

a point

seSI(E )

such that

s e ~ and

Ix( s )Iv < 1

(from this last

condition, it follows that the fiber A(s) has multiplicative reduction mod ~) . Combining the previous lemma with propositions 4, 6, and 7, we obtain: Theorem 2. The following diagram is commutative: A

t/ H~(A( s )) ®~ K [tp] p-adic e v a l u a t i o n at s ( v i a log x , , ~ ( x ( s ) ) % and E 1 , g[tp])

Hi(A(s) ®

¢,~) ®~ ¢

complex e v a l u a t i o n at s ( v i a log x ~ flm(x(s)) and H 1 R ( A / E I [ [ X ] ] [log x ] ) V

E1 - ~ •)

(In the example III 2g, E 1 is just Q(~-~ [t] , and the parameter x = A should be replaced by x = 16A) .

57 V. p-Adic

lattice an___ddHodse classes.

1. Rationality of Hodge classes. a) Let A E be an Abelian variety over a number fidd E . Let v be a finite place of E where A E has multiulicative reduction, and let K = E v denote the completion.

Conjecture 6. Let ~ e (End H1R(AE)) @n be some Hodge class (1). Then for every branch ~ of the logarithm on K x , the image of ~ under 5~ lies in the rational subspace (End H~,~) @n , where H~,~ := H~ ® ~ Q . (For instance, this holds if n = 1 just by functoriality of H~). b)

Let t : E ~

~: and let

Sh be the connected Shimura variety associated to the Hodge

structure HI(AE@tC,~) and to some (odd prime-to-p) N-level-structure; Sh descends to an algebraic variety over some finite extension E ~ of E , and AEr is the fiber of an Abelian scheme A , Sh at some point s e Sh(E r) . In terms of Siegel's modular schemes Ag,N [CF] IV, we have a commutative diagram Sh

"

* A

~®

n

rt

1E r

n -

~ X%N® I- 1

o

3E /

L ,¢NJ

where the superscript - denotes suitable projective toroidal compactificatious, see [H].

In fact A over

~

* Sh extends to a semi-abelian scheme over a normal projective model Sh of (namely

Sh = normalization of the schematic adherence of

~

(1) Some authors prefer to look at Hodge classes in the more general twisted tensor spaces @in

(%)~nl" ® ( s l y )

2(m3). however such spaces contain Hodge d~ses o~y if m 1 + m 2

is even (in fact if m 1 - m 2 = 2m3), and any polarization then provides an isomorphism of ml+m 2

/ oa these extra Hodge classes do not change the Hodge group.

. In particular,

in

58 W e considerthe followingcondition:

(*) There exists a zero--dimensional cusp in ~ , say 0 , such that 0 and s have the same reduction rood. the maximal ideal of R / . In fancy terms, this means that any Abelian variety with multiplicative reduction in characteristic p should also degenerate multiplicatively (in characteristic 0) inside the family "of Hodge type" that it defines [M]. Remark: condition (*) should follow from Gerritzen classification [Ge] of endomorphism rings of rigid analytic tori (which is the same in equal or unequal characteristics), in the special case of Shimura families of PEL-type [Sh] (i.e. characterized by endomorphisms).

Theorem 3. Conjecture 6 followsfrom (*). Proof: by definition of the Shimura variety, and by the theory of absolute Hodge classes [D2] , = ~(s) is the fiber at s of a global horizontal section ~ • r(End H1R(A/Sh)@n) v .

Let S 1 be an algebraic curve on ~ , j o i n i n g 0 and s , and smooth at 0 ;let x b e a l o c a l parameter around 0 , with Ix(s) I v < 1 . Then because 0 is a 0-dimensional cusp, A degenerates multiplicatively at 0 and we are in the situation where theorem 2 applies. n

The /~--periods of ~ admit an expansion in the form

~. a~. log P~x , with a 0 e E / [ Ix] ] , P~=0 af. • E~ [ [x] ] , whose complex evaluation (w.r.t e: E / ¢---* C) gives the corresponding complex period of ~ , according to theorem 2. Since ~ is a global horizontal section and a Hodge class at s , the complex periods are rational constants: a ~ = 0 for £ > 1 , and a 0 e Q . Thus the /~-periods of ~ = ~(s) are rational numbers. Remark: it follows (inconditionally) from theorem 1 and Fontaine' semi-stable theorem that the image of ~ under ~/~ lies in (End H1) @n /J

Qp.

2. p-Adic Hodge classes.

Let E ~ be some finitelygeneratedextensionof E. W e definea p--adic Hodge classon AE/ to be any element ~ of F0(end H~R(AEI) @ n ) -

such that for every E-embedding of E ~ intoany

finiteextension K / of K , and for every branch ~ of the logarithm on K ~x , the image of under 5~/~ liesin the rationalsubspace (End H~,Q)@n . Conjecture 6 predicts that any Hodge classis a p-adic Hodge class,and conjecture2 would identifythe two notions.

59 Proposition 8: if E is algebraically closed in E / , then any p-adic Hodge class ~ comes from 1 1 On] ~ b y F . M . (End HDR(AE) )On , and is sent into ['(End Het) Proof: the first assertion follows Deligne's proof in the complex case [D2]. To prove the second one, we remark that

~ • F 0 [(End H~)@hI ~=1 ; moreover, by changing ;5 continuously, the

lattice H~ is moved by exp(- log u.N), u • Rx . Since ~ has to remain rational w.r.t, all these lattices, we deduce that N~ = 0, and we conclude by Fontaine semi--stable theorem. Remark:

it is essential to take all

E-embedding

E ~ ¢-4 K

into account; for instance,

mVe FOHIR(AE, ) for E ' = K , and mVe H~ , FM(m v) e (H1.)~,et but it is highly probable that m v is not defined over ]~ fl K . 3. A p-adic period conjecture. For any E-algebra E ' , the E'-linear bijections H1R(AE) ®E E ' ~~. (H~,Q) ®Q E ' which preserve p-adic Hodge classes form the set of E'-valued points of an irreducible E-torsor Pfl under the "p-adic Hodge group" of A E (which is by definition the algebraic subgroup of GL H~R(AE) which fixes the p-adic Hodge classes; conjecture 2 would identify this group with the Hodge group). One has a canonical K [tp] -valued point of Pfl given by ~fl. A variant of conjecture 1 may be stated as follows: Conjecture 1 ' : for sufficiently general /~, ~

is a (Weil) generic point of P;3"

The next section will offer two partial positive answers. 4. Period relations of bounded degree. a)

We denote by E [ 5~Bv] _<~ the quotient of the polynomial ring: in 4g 2 indeterminates over

E by the ideal generated by relations of degree < * among (/~v)-~adic periods (v I P) • Hence for sufficiently large ~ , there is a natural embedding Spec E [5~/~v]_<~ C Pflv " The same construction

works

simultaneously

at

several

places

of

multiplicative

reduction:

b) Assume that A E is the fiber at s ¢ SI(E ) of a semi-abelian scheme A ) S 1 over an affine curve S1/Spec E , proper outside some smooth point 0 e SI(E ) , and degenerating to a split torus at this point. Let x be a local parameter around 0, and let ~ >> 0.

60 We lay down an extra normalization hypothesis:

(**)

the entries of the q-matrix expand qij = Yij xnij + "'" where Yij are roots of unity

(this is the case in example III 2g), if we set x = 16.~ and E = {~(~'l~). In these circumstances, we have the following two results: Theorem 4. Assume that Ix(S) lv is sufficiently small - w.r. to ~ - so that in particular A E = A(s) has multiplicative reduction at v. Let us choose ~ = ~ v such that ~x(s)) = 0. Then Spec E [ ~

_<~ = Pfl, and moreover any p---adic Hodge dass on A E is a Hodge class.

Theorem 5. Assume that A ...~. S 1 extends to a semi-abelian scheme over some regular model of SI over 0E , proper outside the divisor vx = 0 , v e ~ . Let V(s) denote the finite set of finite places v of E where lx(S)lv < [Vlv (so that A(s) has multiplicativereduction at v e V ) . Let us choose ~v such that ~ v ( X ( S ) ) = 0 , v • V ( s ) , and let E > 0 . If for every :E ~

~: , [ x(s) 1i. >- e , then the projections Spec E [ ( ~ v ) V e V ] _<, ~

P r y are surjective,

except possibly if s belong to a certain finite exceptional set (depending on ~,e). c)

In fact, the proof shows a little bit more: one can replace P / 3 in the statements by the y

1

,

specialization at s of the Sl-torsor formed of isomorphisms H D R ( A / S l ) ® ? ~tt ®? preserving global horizontal classes; this makes sense because any such class is automatically a 0 .-linear combination of relative Hodge classes, in virtue of: S1 Proposition 9

(Mustafin).

On an Abelian scheme

A

0 e S I\S~ , any element ~ of FiEnd H1R(A/S~)@n) v i s a cycles.

,S 1

degenerating to a torus at

linear combination of relative Hodge

See e.g. [A] IX 3.2. The argument given in the course of proving theorem 3 then shows that ~ is also a linear combination of relative p-adic Hodge cycles. d) We thus have to show that any relation (resp. "global relation" for theorem 5) of degree _< with coefficients in E between (fl)-periods of A(s) is the specialization at s of some relation of degree < ~ with coefficients in E [x]

between the relative /~v-periods (which belong to

E [tp,log x] [ Ix] ] in virtue of (**) and lemma 9). n..

Because tp is transcendental over E v , and /~v(~/ijx lJ(s)) = 0 , it suffices to replace in this

61 statement fly-periods by the v-adic evaluations of the G-functions wij , w l j ,

wijlogq~j,

1 / 1 1 -nij (wijlog qij) , where qij = ~ qij x = 1 ÷ ... This can be now deduced from standard results in G-function theory [A] VII thin. 4.3, resp. 5.2. See also, ibid IX for more details about the proof of a (complex) analogous statement.

62 References [A]

Andr6 Y.; G-functions and Geometry, Aspects of Mathematics E13, Vieweg 1988.

[Be]

Berkovich V.G.; Spectral theory and analytic geometry over non-archimedean fields II. Preprint IttES 1987.

['B] [Bo]

Berthelot P.; Cohomologie rigide, to appear in the "Ast6risque" series. Borovoi, M.; On the action of the Galois group on rational cohomology classes of type (p.p) of Abelian varieties, Mat. Sbornik 94 (1974) n°4, 649---652 (Russian).

[BL]

Bosch S., Liitkebohmert W.; stable reduction and Uniformization of Abelian varieties II, Inv. Math. 78, 257-297 (1984).

[B1 1]

Bosch S., Ltitkebohmert W.; Degenerating Abelian Varieties, preprint Mtinster 1988.

[CF]

Chai C.L., Faltings G.; Semiabelian degeneration and compactification of Siegel modular spaces. Forthcoming book (1990).

[C]

Christol G.; Globally bounded solutions of differential equations, to appear in Proc. of the Tokyo conference on analytic number theory.

[Cn]

Coleman R.; The universal vectorial bi-extension, preprint Berkeley 1989.

[Co]

Colmez P.; P6riodes des vari6t~s ab~liennes de type CM, preprint Bonn 1987.

[D1]

Deligne P.; Th~orie de Hodge III, IHES Publ. Math. N ° 44 (1974), 5-78.

[D 2]

DeLigne P.; Hodge cycles on Abelian varieties, in Lecture Notes in Math. 900 Springer-Verlag 1982 (notes by J. Milne).

[Dw]

Dwork B.; p---adic cycles, IHES Publ. Math. N° 37 (1969), 27-116.

[Fa]

Faltings G.; preprint Princeton on the "De Rham conjecture" 1989.

[F1]

Fontalne J.M.; Sur certains types de representations p-adiques du groupe de Galois d'un corps local, construction d'un anneau de Barsotti-Tate, Ann. of Math. 115, (1982), 59.9-677.

IF 2]

Fontaine J.M.; contribution to the proc. of IHES seminar 88 on "p-adic periods".

[F.M.]

Fontalne J.M.; Messing W.; p-adic periods and p-adic etale cohomology, Contemporary mathematics, Vol. 67 AMS (1985), 179-207.

[Gel

Gerritzen L.; On multiplication algebras of Riemann matrices, Math. Ann. 194 (1971), 109-122.

[%]

Grothendieck A.; On the De Rham cohomology of algebraic varieties, ItIES Publ. Math. 29 (1966) (especially footnote 10).

['G21

Grothendieck A.; ModUles de N6ron et monodromie, exp. IX, SGA 7, in Lecture Notes in Math. 288 Springer-Verlag 1970.

[%]

Grothendieck A.; Groupes de Barsotti-Tate et cristaux de Dieudonn~, S~minaire de Math6matiques Sup6rieures, n ° 45, Presses de l'Universit6 de Montr6al 1974.

63

[HI

Harris J.; Fonctorial properties of toroidial compactifications of locally symmetric varieties. Proc. London Math. Soc. (3) 59 (1989), 1-22.

[Ka]

Katz N.; Serre--Tate local moduli, in Surfaces Alg~briques, expos~ 5 bis, Springer LNM 868 (1981), 138-202.

[K]

Kiehl R.; Die De Rham Kohomologie algebraischer Mannigfaltigkeiten fiber einem bewerteten KSrper. IHES Publ. Math. 33 (1967), 5-20.

[1S] [M]

Le Stum B.; Cohomologie rigide et vari~t~s ab~liennes, t:h~se Univ. Rennes 1985. Mumford D.; Families of Abelian varieties, Algebraic groups and Discontinuous subgroup, Proc. Symp. Pure Math. vol. 9, AMS, Providence R.I. (1966), 347-351.

[Mu]

Mustafin G.A.; Families of Algebraic varieties and invariant cycles, Math. USSR Izvestiya Vol 27 (1986) n ° 2, 251-278.

[O]

Ogus A.; Contribution to this conference.

[RI]

Raynaud M.; Vari~t~s ab~liennes et g~om~trie rigide. Actes du congr~s international de Nice 1970, t.1, 473--477.

[m2]

Raynaud M.; Contribution to the proc. of IHES seminar 88 on "p-adic periods".

IS1]

Serre J.P.; Propri~t~s galoisiennes des points d'ordre fini des courbes elliptiques, Inv. Math., 15, 1972, 259-331.

IS2]

Serre J.P.; Abelian [-adic representations and elliptic curves. Benjamin, New-York, 1968.

[SGA4]

Artin M., Grothendieck A., Verdier J.-L.; Th~orie des topos et cohomologie ~tale des schemas, t. 3, LNM 269 Springer Verlag (1972).

[ski

Shimura G.; On analytic families of polarized abelian varieties and automorphic functions, Annals of Math. 78 (1963), 149.

[St] IT] IV]

Steenbrink J.; Limits of Hodge structures, Inv. Math. 31 (1975/76), 229-257.

[vM] [w]

Tate J.; p--divisible groups. Proc. of a Conf. on local fields. Springer Verlag (1967). Ullrich P.; Rigid analytic covering maps. Proc. of the Conf. on p-adic analysis, Hengelhoef (1986), 159-171. Van der Marel B.; thesis Rijksuniversiteit Groningen 1987. Wintenberger J.P.; p-adic Hodge theory for families of abelian varieties, preprint 1989.

THE NONARCHIMEDEAN BANACH-STONE THEOREM

Jestis ARAUJO, Universidad de Oviedo, ETSII, Castiello de Bemueces, 33204 GIJON, Spain J.MARTINEZ-MAURICA1, Universidad de Cantabria, Facultad de Ciencias, Avda. los Castros, 39071 SANTANDER, Spain

In 1987, E. Beckenstein and L. Narici ([BN]) introduced the Banach-Stone maps as those linear surjective isometries between two given spaces C(X) and C(Y) of K-valued continuous functions (X, Y are Hausdorff zerodimensional compact spaces and K is a nonarchimedean valued field) which derive in a natural way from some homeomorphism of Y onto X (see Definition 1 for details). In that paper they proved that, unlike the case of real or complex ground field, one can find linear surjective isometries T:C(X) --.) C(X) which are not Banach-Stone maps whenever X is not rind. In this paper, we extend this result to any X with more than one point (Theorem 2 and the closely related Theorems 3 and 7). Another evidence of the non existence of a nonarchimedean Banach-Stone theorem had been given by A. C. M. van Rooij ([vR], p.190). Also, we characterize the Banach-Stone maps in several different ways (Theorems 4, 5, 8 and Corollary 4). As a result some topological properties of the space of all Banach-Stone maps are given (Theorem 7 and Corollary 5). Throughout this paper K denotes a nonarchimedean commutative and complete valued field endowed with a nontrivial valuation. X, Y are always two separated zerodimensional compact spaces. By C(X) we indicate the Banach space of all continuous functions from X into K endowed with the sup-norm. C(X)' stands for the normed dual of C(X). A subset of a topological space is said to be clopen if it is both open and closed. We say that a clopen subset of X is proper if it is nonempty and different from X.

1 Research partially supported by the Spanish Direcci6n General de Investigaci6n Cientffica y T6cnica (DGICYT, PS87-0094).

65 1. BANACH-STONE MAPS.

D E F I N I T I O N 1 ([BN]) A linear isometry T o f C(X) onto C(Y) is a Banach-Stone map if there exists a homeomorphism h o f Y onto X and an a ~ C(Y), l a ( Y ) 1 - 1 , such that (Tf)(y) = a ( y ) f ( h ( y ) ) , f o r every y ~ Y , f ~ C(X).

REMARKS I If the ground field is R or C andX, Y are general (this is, zerodimensional or not) separated compact spaces, the Banach-Stone theorem allows us to conclude that every linear isometry of C(X) onto C(Y) is a Banach-Stone map. So the above definition of Banach-Stone map is of no use in the case of real or complex field. 2 There are non-homeomorphic spaces X and Y for which it is possible to find a linear surjective isometry T:C(X) ~ C (Y). In fact, letX~ Y be two non-homeomorphic compact ultrametrizable spaces

(e.g. let Y be the space which consists in adding an isolated point to the Cantor space X). Then C(X) and C(Y) are both linearly isometric to Co ([vR], p.74). F o r f ~ C(X), we define the cozero set o f f to be c(f) = {x E X:f(x) ;~ 0}. For U c X , ~u stands for the characteristic function on U.

D E F I N I T I O N 2 ([BN]) A map T o f C(X) into C(Y) has the disjoint cozero set property if c (f) n c (g ) = 0 implies c (Tf) n c (T g ) = f~ for f , g ~ C (X ).

R E M A R K We have c O O n c ( g ) = {x ~ X :f(x) ~ 0} n { x ~ X:g ( x ) ~ 0} = {x ~ X:f(x)g(x) ~ 0} =c(fg). Hence, the cozero set property means nothing else but: f g = 0 ~ (Tf) (Tg) = O. I f x ~ X, we define ex:C(X) --->K by e~(19=f(x) ( f ~ C(X)), and we shall say that e~, is the evaluation map at x. It is obvious that e~ ~ C(X)' and that ~ e~Jl = 1 for every x ~ X.

66 T H E O R E M 1 ([BN]) For a linear isometry T mapping C(X) onto C(Y), the following

statements are equivalent: (I) T has the disjoint cozero set property. (2) Its adjoint T' maps each norm-one multiple of an evaluation map into a norm-one multiple

of an evaluation map. (3) T is a Banach-Stone map.

C O R O L L A R Y I Let T be a linear isometry of C(X) onto C(Y). lf c ( T ~ u ) n c ( T ~ v ) = f ~

whenever U and V are two disjoint clopen subsets of X, then T is a Banach-Stone map. Proof. Let x e X. Let U, V be two disjoint clopen subsets of X. If c ( T ~ v ) n c (T~v) = 0 , then (T'e~) (~v) = (T~u) (x) = 0 or (T'e~) (~v) = (T~v) (x ) = O. Hence by ([BN], lemma 2), T'ex is a scalar multiple of an evaluation map. Also I1T'eJI = sup s,0 = sup

I (T'e~)C01 II./11 I e~(g)l

g,0 -I1~-

= II

I e~(Tf) I - s u p :,o tIAI e,Jl

=

1

•

Therefore, T is a Banach-Stone map because of Theorem 1. T H E O R E M 2 If #X >>2, . then there are linear surjective isometries T:C(X) ~ C(X) which

are not Banach-Stone maps. Proof. Let B c X

be a nonempty proper ctopen subset of X.

By ([vR], p, 188), every orthonormal set of characteristic functions of clopen sets can be extended to an orthonormal base of C(X) and by ([vR], p. 167) all the orthonormal bases of a Banach space have the same cardinality. Let Bt = {~} t.){g:i E I} and ~ = {~B} u{hi :i ~ I} be two orthonormal bases of C(X) extending {~} and {~} respectively. Now define T ( ~ ) = ~ and T(gl) = hi for every i E 1. We can extend T by linearity and continuity to a map T:C(X) ~ C(X). It is obvious that T is a surjective linear isornetry. Suppose now that T is a Banach-Stone map. Then there exists an a ~ C(X), I a(x) I-= 1, and a homeomorphism h :X --> X such that for every f in C(X) andx in X, (Tr~ (x) = a(x)f(h(x)). If we take xo ~ X - B , we have I (T~)(xo) l=t a(xo) I= 1 and I (T~)(xo)I=1 ~(x0) t=-0, a contradiction. Thus we can conclude that T is not a Banach-Stone map,

67

REMARKS 1 This result had been proved in ([BN]) when X is not rigid, this is, when there exists a homeomorphism h ~ ---~X different from the identity map. (In [KR] and [L] it is shown that there exist rigid spaces of arbitrary large cardinality). 2 IfX has only one point, every linear surjective T of C(X) onto C(Y) is a Banach-Stone map. Banach-Stone maps can be also considered as algebra almost-morphisms in the following sense: PROPOSITION 1 Let T be a linear isometry from C (X ) onto C (Y). Then, T is a Banach-Stone

map if and only i f T ~ is an invertible element of C(Y) and T(fg)=(T~x)~(Tf)(Tg)

(f,g ~ C(X)).

Proof. If T is a Banach-Stone map, it is obvious that T ~ is invertible in C(Y). Also

(T(fg)) (y) = a (y)f(h (y))g (h (y))

= ((T~x) (y))" (Tf) (y) (Tg) (y) for each y ~ Y, where a ~ C(Y) and h are as in Definition 1. Conversely, let ey ~ C(Y)' be an evaluation map. Let U, V be two disjoint clopen subsets of X. We have that

0 = e,(T(~v~v)), = (T(~.~))(y)

= ((T~x)(y))-;'(T~t:)(Y)(Tl:~v)(y). Then,

(T~v)(y) =

0 or (T~v)(y) = O. So T is a Banach-Stone map because of Corollary I.

The following result is a slight generalization of Theorem 2. T H E O R E M 3 If #X > 2, then there are linear surjective isometries T :C (X ) ---) C (X ) such that

T(~¢) = ~x which are not algebra morphisms. Proof. We shall distinguish two cases: i) IfX is not rigid ([KR],[L]) there exist two subsets of X, U and V, proper, clopen and disjoint, which are homeomorphic. Let ot E K, 0
88

f

(~f(x)+f(h(x)))

ifx e U

(T:3 (x ) =

ifx~ V otherwise

By using the same arguments as in ([BN], p.245) and applying Proposition 1, we arrive at the required conclusion. ii) IfX is rigid, letB be a proper clopen subset of X. By ([vR], Corollary 5.23), we can find an orthonormal base of C(X), B = {~B,~-B} t~{g~:i ~ I}, for some adequate g~ ~ C(X), i ~ I. Define now T(~a) = ~_R, T(~x_B) = ~ and T(g~) = g~ (i ~ I). We arrive at the the result by proceeding as in Theorem 2 and taking in account that the identity map is the only homeomorphism from X onto X.

2. S O M E C H A R A C T E R I Z A T I O N S .

F o r f ~ C(X), we denote v(f) = {If(x) I :x ~ X} - {0}. L E M M A 1 The following properties are equivalent for f ~ C(X): (1) cot) is clopen. (2) v(f) is finite. (3) infv(f) > 0. Proof. (1) ~ (2). Let, for each r > 0, A, = {x ~ X: If(x) I= r}. Obviously, A, is clopen for every r > 0 , and c(f) = u A , . r>0

Also, since c(]') is ctopen, then it is compact and there exist rl,r 2.... ,r, in R such that

c (f) = A,, uA:2.., u A r . Then v (f) is contained in { r l, r 2..... r,}. (2) =~ (3) is obvious. (3) ~ (1) Let ~ e K such that 0 <1 ct I< infv(f). Then, we have that

{x ~ X:f(x)= 0} =l f [ -t [0, let I) = {x e X: If(x) I
69 L E M M A 2 If T :C (X ) ~ C (Y ) is a linear surjective isometry, then thefollowing are equivalent: (1) T is a Banach-Stone map. (2) If c(]') = c(g), then c(Tf) = c(Tg) (f,g ~ C(X)) (3) If c (i") = c (g ) is a clopen subset of X, then c (Tf) = c (Tg ) (f , g ~ C (X )). Proof. (1) ~ (2),

(2) ~ (3) are obvious.

(3) ~ (1) Let U, V be proper clopen subsets of X such that U n V = f~. Let

f~,=~v+O~v

(~

K-{0}).

Clearly, we have that c(f~ = U u V for every o~ E K - {13}. Then c(TfD = c(T~vuv) because of (3). W e are going to prove that c (T~u) n c (T~v) = O. Otherwise, there exists so E c (T~v) n c (T~v).

Let

(T~u)(So) o~= Then

So~e(T~v+o~T~v)=c(Tf~o)

So~C(T~v+~T~v),

which

and

(T~v)(so)" hence

soc~c(T~w,v).

implies soE c(T~vuv) and

Then

if

13~K-{o.o,O},

we arrive at a contradiction.

So

c (T~v) n c (T~v) = 0 . Then, by Corollary 1, T is a Banach-Stone map. L E M M A 3 Let f, g ~ C(X). Then f +ctg ¢ C(X) is invertible for every tx ~ K, ~ ~ 0, if and

only if c ( D n c ( g ) = 0 and c ( f ) u c ( g ) = X. Proof.

First

assume

f + etg

is

invertible

for

every

ct E K - {(3}.

Since

X = c ( f + g ) c c f f ) w c ( g ) , then X = c ( f ) u c ( g ) . Let us see that c ( f ) n c ( g ) = 0 . Otherwise, take so ~ c ( f ) n c ( g ) and let

f(so) t~ =-g--~0) ¢ K - {0}. Since (/'+ ~ g ) (so) = O, then f + otog is not invertible. Hence c (D n c (g) = 0 . The converse is easy. L E M M A 4 Let tx, ~ ~ K - {t3}. Then, ira ~ R, a > 0, there exists'l ~ K such that I ~t t=1 o~ I / I ~ I

andO <1 o~+ 13"tl< a . P r o o f . L e t 5 E K - {(3} such thatt 5 I< min{l ot l,a} .Let ~'= ( 5 - ~)/13. Itis obvious that'/satisfies the requirements.

70 THEOREM 4 IfT :C (X ) --4 C (Y ) is a linear surjective isometry, then the following properties

are equivalent: (1) T is a Banach-Stone map. (2) infv(f) < infv(Tf) for every f e C(X). (3) lf v(f)= {1}, then v(Tf) = {1}. (4) v (f) c v (Tf) for e very f ~ C (X). (5) If f ~ C(X) is invertible, then so is Tf ~ C(Y). Proof. (1) ~ (2),

(2) ~ (3) are trivial.

(3) =~ (4) We infer from (3) that if v(f) = {r}, for some r > 0, then v(Tf) = {r}. Suppose that there exists f i n C (X) such that a ~ v (f) and a ~ v (Tf), a > 0. We can assume without loss of generality that 11311 -- 1. Then a < 1, because 1 ~ v(Tf). We can decompose f as g + h where g = f ~ , h =f~x-a ~ C(X)andA = { x ~ X: If(x) l>-a}. Obviously, a ~ v(g), Now, assume there exists y ~ Y such that I(Tg)(y)l=a. Since IIhll = IIThll < a, then I (Tf)(y)l= a. This proves that a ~ v(Tg). On

the

other

hand,

c(g)

is

clopen

because

infv(g)>0

(see

Lemma

1).

So

~ ~ , where ~ =j'~/f,{°,) for each i ~ {1,2 ..... n}. v ( g ) = { a = a l
W e obtain that Tg = ~ Teq.

n

Next, we are going to prove that c(Tcq) c u c(Tcq). Suppose that there exists an Xo ~ c(Tct~) i=2

such that ( T ~ ) (x0) = 0 for i = 2, 3 ..... n. Then I (Tg) (xo) M (Tcq) (xo) I- a~ = a, which is impossible. n

Hence c(Teq) c • c(T~i). i~2

So we know that there exists i0 ~ {2,3 ..... n} such that c(TctOc~c(Toqo) is not empty. Letx0 be an element of c(Tcq)nc(TCtio). Then we have that t (TctO(xo) I = a a n d l (T%o)(xo) t=ai, . Let bl, bl, ~ K such that

tb,o~ 1 .

lb~i = t , Let

us

consider

[51=blctl¢C(X),

now

v(131) = v(~io) = { 1}.

a~,

By

Lemma

0 < r =lb(T~O(xo)+(T~q)(xo)I< 1 and

~=b~,%oeC(X). 4

there

exists

Obviously, b ~ K

we

have such

that that

Ib l= 1.

Let f ' = hi31 + ~i,. Clearly we have that v(f') = {1} because c(~Onc(~q) = 0 . On the other hand r ~ v(Tr), with r ,

1. Now, we have arrived at a contradiction with our assumption.

71 Then v(D c v(Tf) for every f in C(X). (4) ~ (5) Suppose that there exists f ~ C(X) invertible such that T f ~ C(Y) is not. Then infv(f) > 0. Define A = {y ~ Y:JTf(y) 1< infv(f)}. Clearly A is clopen and nonempty because Tf is not invertible. Consider g = T - ~ . Since T -~ is an isometry, II gll = 1, and then there exists a point x0 ~ X in which g attains its norm, this is, J g(xo) I= 1. By Lemma 4, there exists a ~ K such that 0 < r = If(x0) + otg (x0)I < infv(f), where I o~ I=1f(xo) I> 0. Then r ~ v ( f + o~g) but r ~ v (T(f + ctg)), which is a contradiction. So T is invertible. (5) ~ (1) Let f, g in C(X) such that c(/) = c(g) is clopen and let A = X -cOO. We have that f + o ~ is invertible in C (X) for every ot ~ K - {0} because of Lemma 3. By (5), Tf+ c~T~ is invertible for every (x ~: 0. One can infer from Lemma 3 that c (Tf) n c ( T ~ ) = ~

and c (Tf) u c ( T ~ ) = Y. We

can obtain analogous results if we replace f by g. Hence c(Tf) = Y - c ( T ~ ) = c(Tg) and T is a Banach-Stone map by Lemma 2.

REMARK

It is easy to see that if T:C(X) --) C(Y) is a Banach-Stone map, then its inverse

T-I:C(Y) --) C(X) is also a Banach-Stone map. Indeed T -I is a linear surjective isometry. Since T is a Banach-Stone map, then there exist a ~ C(Y), l a(Y) l- 1, and a homeomorphism h:Y ~ X

such

that Tf = a (f o h) for every f ~ C (X). It is easy to see that

T-Ig = ~ 1

(g o h-')

(g

C(Y))

which proves that T -~ is a Banach-Stone map.

C O R O L L A R Y 2 If T:C(X) ~ C(Y) is a linear surjective isometry, then the following

properties are equivalent: (1) T is a Banach-Stone map. (2) i n f v ( T f ) < infv(f)for every f ~ C(X). (3) i n f v ( D = infv(Tf)for every f ~ C(X).

(4) If v(TjO = {1}, then v(f)= {1}. (5) v(T f) c v(l) for every f ~ C (X ). (6) v ( T f ) = v(I-) for every f ~ C(X). (7) I f T f ~ C(Y) is invertible, then so is f ~ C(X). T H E O R E M 5 Let T:C(X) --~ C(Y) be a linear surjective isometry. Then, the following are

equivalent: i) l f f ~ C(X) is a topological divisor of zero, then so is Tf ~ C(Y). ii) T is a Banach-Stone map.

72 Proof. i) =~ ii) Let us suppose that there i s f e

C(X) which is not inverfible but such that Tf

is invertible. Then there exists a point x0 ~ X in which f vanishes, and r = inf [ (Tf) (y) I> 0. yaY

Let

h(x)

ff(x)

if if

x~ U

x~U

where U = {x ~ X: If(x) l< r/2}. We have that h is a topological divisor of zero. Then so is Th. This implies the existence of a sequence (g,)~, N in C(Y) such that IIg~ll = 1 for each n ~ N and lim(Th)g, = 0 . Since llZf-Thll=llf-h[[ r

for each y ~ Y ,

we deduce that

l ( T h ) ( y ) f=l ( T f ) ( y ) l > r for every y e Y. For each n ~ N, let y~ ~ Y such that I g,(Y~) [= 1. Then, It (Th)g~ll >[ (Th)(y,) I [gn(Y~) [>-r for each n ~ N, and we arrive at a contradiction. So T is a Banach-Stone map.

ii) ~ i) is easy. R E M A R K With slight changes in the previous proof, it is possible to assure that the theorem also holds if we replace i) by i ' ) I f f ~ C(X) is a divisor of zero, then so is T f c C(Y).

COROLLARY 3 Let T :C (X ) --~ C (Y) be a linear surjective isometry. Then the following are

equivalent: (1) T is a Banach-Stone map. (2) T maps maximal ideals of C(X) onto maximal ideals of C(Y). (3) T maps closed ideals of C(X) onto closed ideals of C(Y). Proof. By ([vR], p. 209), we know that the maximal ideals of C(X) can be expressed in the form M={f~

C (X):./'(x~r) = O}

for some x,t e X and the map

M---) x M is a bijection from the set of all maximal ideals of C(X) onto X. Also, N is a closed ideal in C(X) if (and only if) there exists a subset BNofX such that

N= {f ~ C(X):f(x)=O '~x ~ BN}.

?3

(1) ~ (2) IfTis a Banach-Stone map, then there exists a homeomorphism h of Y onto X and

an a ~ C(Y),

l a(Y) I- 1, such that for each f i n C(X) and y in Y,

(Tf) (y) = a (y)f(h ~)). Let M be a maximal ideal of C(X). It is easy to check that T(9,0 = {g ~ C(Y):g(h-~(x~) = 13} which shows that T(~t) is a maximal ideal of C(Y). (2) ~ (3) Let Mbe a closed ideal in C(X). Then N is the intersection of all the maximal ideals containing it:

N=nM.. NC ~4

Since T maps maximal ideals onto maximal ideals, T(9~0= n T(gt0 Nc

is closed. (3) ~ (1) Let f e C (X) such that f(Xo) = 0 for some x0 e X. Let M be a maximal (and hence closed) ideal such that f e

~ We have that T(M) is a closed ideal of C(Y), and then there exists a

point y ~ e Y in which Tf vanishes. Then, by Corollary 2 ((7) ~ (1)), T is a Banach-Stone map. C O R O L L A R Y 4 If T:C(X) ~ C(Y) is a linear surjective isometry, then the following are

equivalent: (1) T is a Banach-Stone map. (2)Ifv(Dcv(g),thenv(Tf)cv(Tg) (f,g ~ C(X)). (3) Ifv(D = v(g), then v(Tf) = v(Tg) (f,g ~ C(X)). (4) lfinfv09 < infv(g), then infv(Tf) _
(f,g E C(X)).

( 6 ) l f l f ]<-Ig I, thenl Tf l<-ITg I (f,g ~ C(X)), (wherel f l<-Ig Imeanslf(x) I-<-Ig(x) Ifor every x~X). ( 7 ) l f l f l = l g l, thenlTfl=lTg I (f,g ~ C(X)). Proof. (1) ~ (2) ~ (3), (1) ~ (4) ~ (5), (1) ~ (6) ~ (7) are obvious. We shall prove simultaneously (3) ~ (1) and (5) ~ (1).

Let f ~ C (X ) such that v(f) = {1} and let g = T-~~r . Let A = {x ~ X :[g (x )[ = 1}. It is obvious that A ~ O because T -~ is an isometry and there exists some point in X where g attains its norm. Let us put g in the way g = g ~ + g ~ - A .

Clearly, i f x E X ,

[ g ~ ( x ) [ ~ {0,1}, this is ,

v ( g ~ ) = {1}, because there is at least one point xo ~ X with [ g~(xo) [= I. Also we have that foreach y ~ Y,

74 I (T(g{A)) ~') I =l (Tg) (y) - (T(g~x_a)) (y) I =1 1 -(T(g~x_a))(y)l =1 because IIT(g~_a]l = IIg~-all < 1. Thus v(Z(g~)) = {1}. Now, by using the hypothesis (3), v(Tf)= { 1} and by using the hypothesis (5), infv(Tf)= t, this is, v(Tf) = { 1}. By Theorem 4 we conclude that T is a Banach-Stone map. (7) ~ (1) Let f, g ~ C(X) such that c(f) = c ( g ) is a clopen subset of X. Then, by Lemma 1, there exists cc e K such that If(x) I<1 c~g(x) I for every x e c(f). Then If(x)+o~g(x) I=1 e~g(x) I for every x e X. By hypothesis, I(Tf)(y)+~(Tg)(y) I=[ (x(Tg) (y) I for every y E Y. Then if y ~ c (Tg), (Tg) (y) = 0 and then (Tf) (y) = 0. This implies c (Tf) c c (Tg). If we interchange the roles o f f and g we shall have the equality c(Tf) = c(Tg) which is equivalent, by Lemma 2, to being T a Banach-Stone map. R E M A R K We cannot replace (7) for the weaker form ( 7 ' ) l f l f l = 1, then lTfl= 1, f e C(X). For that, let B be a clopen proper subset of X. We know ([vR] p.167) that there exist gi ~ C(X) (i ~ I) such that {{~,~} u{g~:i ~ I} is an orthonormal base of C(X). Then, so is {ot~,~} u{gi:i ~ I}, where o~~ K,

o~I= 1 and 0
Define T:C(X) ~ C(X)

g~ "-) gi

(i ~ I)

and extend it by linearity and continmty to the whole space C(X). For every f ~ C(X),

I f 1= 1, there exist ~i ~ K (i ~ I), v ~ K and 13~ K such that 1131< 1

and f = Y. o~ig,+ v~x + ~8. iel

Then, for every x ~ X,

75

=l f ( x ) + 13(a- 1)~s(x) I =tf(x) l =1,

It is easy to see that T is a linear surjective isometry. However, T is not a Banach-Stone map. In order to see that, suppose there exist an a ~ C (X), I a (x) l-= I, and a homeomorphism h of X onto X, such that for each f in C(X) and x in X, (Tf)(x)=a(x)f(h(x)),. Then, for every x ~ X,

a(x) = ( T ~ ) ( x ) = 1. Hence ~n(h(x)) = ~(h(x))a(x) = (T~e) (x) = ct~B(x)which is impossible because o~;~ 0, 1. This completes the proof.

3. T O P O L O G I C A L PROPERTIES.

Let us introduce the following notation:

I (C (X ), C (Y ) ) = { T :C (X ) --) C (Y ):T is linear surjective isometry}. I (C (X), C (Y)) will be considered as a topological subspace of L (C (X), C (Y)). We shall write

I(C(X)) instead of I(C(X),C(X)). From now on, BS(C(X)), C(Y)) stands for the topological subspace ofl(C(X), C(Y)) consisting of all the Banach-Stone maps of C (X) onto C(Y). We shall write BS(C(X)) instead ofBS(C(X), C(X)). T I t E O R E M 6 I(C(X), C(Y)) is complete.

Proof. Let (T,),, N be a Cauchy sequence of elements of I(C(X),C(Y)). Then, (T,),,u converges to T ~ L(C(X), C(Y)). Obviously !1T.,'ll = lim n T,J]I = n)'][ for every f ~ C(X), and hence tl ,-4,~

T is an isometry. On the other hand, let us see that (T~1),, n is a Cauchy sequence in I(C(Y), C(X)). In order to see that, let e > 0. Then there exists no ~ N such that if n, m > no, ] 7", - T,,II < ~. So, for n, m -> no,

78 IIT2 ~- T2,~II <- I1T2~T,~-/11 UZ2,~l -< IIZ2'll UT~ - T~II tl ~ 1 = IIT,, - T, ll <E. Then (T2~),~ N is a Cauchy sequence in I(C(Y),C(X)) and, as above, it converges to an isometry S • L(C(Y),C(X)).

It is left to verify that T is surjective, For that, let us see that T o S is the identity map on C(Y). L e t e > 0 , g • C ( Y ) a n d n l e NsuchthatllT21g-Sgll < e f o r e v e r y n >nt. Also, letn0e N

(no>nO

such that IIZ,(Sg) - T(Sg ;II < e for every n _>no, Then, for n > no, we have IIZ(Sg) - gll -< max{ IIT(Sg) - T~(Sg ~1, I1T,(ag) - ell } -< max{ll T (S g ) - T , (S g ]I , IIT,II IIS g - Z 2~gll } <E. Since this inequality holds for all e > 0, we conclude that T(Sg) = g which implies that T is surjective. C O R O L L A R Y 5 BS(C(X), C(Y)) is a closed subset ofl(C(X), C(Y)). Proof. Let (T~)~ ~ s be a Cauchy sequence in BS (C (X), C (Y)). Let T be its limit in L (C (X), C (Y)). By Theorem 6, we know that T is a surjective isometry. Now, suppose that f, g • C(X) are such that If I=1 g I. Then l T f f I=t T,g I for every n • N and then, I (T.t3 (x) I=1 (Tg) (x) I for every x e X. So T is a Banach-Stone map (Corollary 4). THEOREM

7

I f #X > 2 , then

BS(C(X)) has empty interior in I(C(X)) (this is,

I(C(X)) - B S ( C ( X ) ) = I(C(X))).

Proof. Let T • BS(C(X)). Let e > 0. Consider ot • K such that 0
Let U be a proper clopen subset of X. Since T is a Banach-Stone map, c(T~v) is clopen and proper (Lemma 1 and Corollary 2) and hence ~_~fr~) belongs to C(X). Let {Iv} u{gi:i ~ 1} be an orthonormal base of C(X) extending {{tr}. Define S({v) = ~-~¢r~),

S(gi) = 0 for every i • I and

extend S by linearity and continuity to the whole space C(X). It is obvious that IISII -< 1, and hence T+~

belongs to the ball of center T and radius E, B~(T). It is also easy to see that T + ~

isometry.

is a linear

77 Let us see that T + ¢xS is surjective: Since {T~v} u{Tg~:i e 1} is an orthonormal base of C(X) and:

ll(Tg,)-(T+txS)(g,]l - 2

IlTg,]l (i ~ 1),

1 1 II(T~u)--(T+eS)(~v]l < ~ = ~ l t T ~ d l

then {(T + a S ) (~v)} u{(T + a S ) (gi):i E I} is another orthonormal base of C(X) ([vR], p. 183). Now it is easy to see that this last property implies that T + aS is surjective. Also, ~v is not invertible in C(X). However, (T + a S ) (~v) is invertible and we can deduce from Corollary 2 that T + aS is not a Banach-Stone map. REMARKS 1 If#X = 1, then BS(C(X))=I(C(X)). 2 Also, I(C(X)) is clopen in L(C(X)). We have proved in Theorem 6 that it is closed. Let us see that it is open. Let T E I(C(X)) and S e B,(T). Then, for every f ~ C(X), IIS ff/I = 11S( f ) - TO") + T(]'~I = [[T(fll = ll.fl which proves that S is a linear isometry. By using the same arguments as in Theorem 7, we can prove that S is surjective and hence B,_(T)c I(C(X)). Thus, I(C(X)) is clopen in L(C(X)). 2

4. A S P E C I A L CASE.

T H E O R E M 8 Suppose Y has not isolated points. Let T:C(X) --o C(Y) be a linear surjective isometry. Then, T is aBanach-Stone map if andonly if infv (Tf) > 0 wheneverinfv(D > 0 (f ~ C(X)). Proof. First, suppose that infv (Tf) > 0 whenever infv00 > 0 , f e C(X). Let us see that T maps invertible elements of C(X) onto invertible elements of C(Y) and then apply Theorem 4. In order to prove that, suppose there exists an invertible map f ~ C(X) such that Tf is not invertible in C (Y). Then, Y - c (Tf) is a proper clopen subset of Y (Lemma 1). Since Y has not isolated points, there exists a map h : Y - c (Tf) --~ K which is not locally constant at some x0 e Y - c (Tf) by ([GH] Corollary 5.4.). Without loss of generality we can suppose that h(xo) = 0 and then we extend h to the whole space Y by defining h (x) = 0 ifx e c (Tf). Obviously h ~ C (Y) because c (T.f) is clopen. If IIhll < infv(f), then infv(f+T-lh) = infv(f) > 0, because c(f) = X and IIT-lhU = IIhll < infv(f).

78 However infv(Tf+ h) = 0, because (Tf+ h) (x0) = 0 and Tf+ h is not locally constant at x0. So we arrive at a contradiction. The converse is trivial.

REMARK

Compare this result with Theorem 4, where some stronger conditions on v(f)

appear (e.g. infv(f) = infv (Tf) or v(f) = v(Tf)).

T H E O R E M 9 If #X > 2 and X has isolated points, then there exists a linear surjective isometry

T:C(X)--~C(X) such that i n f v ( T f ) > 0 whenever i n f v ( f ) > 0 Banach-Stone map.

( f ~ C(X))

which is not a

Proof. Let a e X be an isolated point. Choose b e X, b ~ a, and set

Tf:=f+f(b)~toi

( f ~ C(X))

T is a linear continuous map from C(X) onto C(X) and II hi ~ 1. One easily infers that T has an inverse, viz.

f --~ f - f ( b )~{.}. So, T is a surjective linear isometry. It is easily seen that v0 r) and v(Tf) differ by at most one point, so infv (f) > 0 ~ infv (Tf) > 0. Now assume T is a Banach-Stone map. Then, for every f e

C(X),

T f = c f f o h ) , where

c ~ C(X), I c l- 1 and h:X ---~X is a homeomorphism. So, in particular, (*)

c(a )f(h(a )) = f ( a ) + f(b ).

By taking f = {~,~ we find c(a){lai(h(a)) = 1, so that {~ai(h(a)) ~ O, i. e. h(a) = a and c(a) = 1. Then, from (*),

f(a)=f(a)+f(b)

( f ~ C(X)).

Hence f(b) = 0 for every f s C (X), a contradiction.

We are grateful to Dr. W. H. Schikhof for his valuable suggestions.

REFERENCES

[BN].- E. BECKENSTEIN and L. NARICI. A nonarchimedean Stone-Banach theorem, Proc. Amer. Math. Soc, 100 (1987) 242-246.

79 [GH].- L. GILLMAN, M. HENRIKSEN. Concerning rings of continuous functions, Trans. Amer. Math. Soc. 77 (1954) 340-362. [KR].- V. KANNAN and M. RAJAGOLAPAN. Rigid spaces.III, Canad. J. Math, 30 (1978), 926-932. [L].- F. W. LOZIER. A class of compact rigid O-dimensional spaces, Canad. J. Math. 21 (1969), 817-821. [vR].- A. C. M. VAN ROOtJ. Non-archimedean functional analysis. Marcel Dekker, New York, 1978.

Cohomologie

rigide

et th~orie

des ~-modules

Pierre BERTHELOT (IRMAR, Universit~ de Rennes)

Introduction 1. Op~rateurs diff~rentiels de niveau fin] 1.1. Structures partielles d'iddal ~tpuissances divis~es 1.2. Op~rateurs d'ordre fini 1.3. Opdrateurs d'ordre infini 2. La cat~gorie des ~ Q - m o d u l e s coh~rents 2.1. Les thdor~mes A et B 2.2. Ind~pendance par rapport au rel~vement 3. Isocristaux convergents 3.1. Cohdrence sur ~ Q des isocristaux convergents 3.2. Le complexe de de Rham d'un t ~ Q - m o d u l e 4. Isocristaux surconvergents : le cas constant 4.1. Image directe par sp~cialisation d~un isocristal surconvergent 4.2. Cohomologie locale ~t support dans une sous-varigtd tisse 4.3. Cohomologie locale a support dans un diviseur ~t croisements normaux 5. Isocristaux surconvergents associ~s aux caract~res 5.1. Isocristaux associds aux caract~res multiplicatifs 5.2. Isocristaux associds aux caract~res additifs

Introduction Soient k un corps parfait de caract~ristique p , W = W(k) l'anneau des vecteurs de Witt ~ coefficients dans k, K le corps des fractions de W. Si e est un nombre premier different de p, on salt associer a tout k-schema X une cat~gorie de faisceaux pour la topologie ~tale, la cat6gorie des faisceaux e-adiques constructibles, ~ partir de laquelle on peut construire une cat~gorie d~riv~e Dcb(X), la cat~gorie des complexes born6s de faisceaux e-adiques ~ cohomologie constructible, satisfaisant au "formalisme des six operations" d~gag~ par Grothendieck (notamment dans [11]

81

pour les faisceaux coh~rents et [SGA4] pour les faisceaux ~tales, et pour lequel nous renvoyons aussi ~ l'ouvrage de Mebkhout [17, II § 9]) : produit tensoriel, faisceau d'homomorphismes, image directe et image directe ~ support propre, image inverse et image inverse exceptionnelle. Dans le cas p-adique, la situation actuelle est tr~s loin d'etre aussi satisfaisante. On peut associer ~ tout k-schema s~par~ de type fini X des groupes de cohomologie rigide co~ncidant avec la cohomologie cristalline (tensoris~e par K) si X est propre et lisse, et avec la cohomologie de Monsky et Washnitzer si X est affine et lisse (cf. [2], [4]). On peut ~galement ~galement associer ~ X u n e cat~gorie de coefficients, la cat~gorie des F-isocristaux surconvergents, permettant notamment d'interpr~ter les espaces de cohomologie de Dwork comme groupes de cohomologie rigide ~ coefficients dans un isocristal [3]. Mais, outre le fait qu'on ne sait en g~n~ral d~montrer certaines propri~tSs fondamentales de ces groupes de cohomologie (notamment la finitude) qu'avec des hypotheses de r~solution des singularit6s, les F-isocristaux consid~r~s jusqu'~ present sont des analogues p-adiques des syst~mes Iocaux (ou faisceaux lisses) e-adiques, et ne peuvent donc fournir qu'une sous-cat~gorie de celle qui est n~cessaire pour d~velopper le formalisme des six operations de Grothendieck. On peut du reste penser que c'est pr~cis~ment faute d'une bonne cat~gorie de coefficients que la cohomologie p-adique n'a pas jusqu'ici connu un d~veloppement comparable ~ celui de la cohomologie e-adique.

Hirig(X/K),

Le d~veloppement, grace n o t a m m e n t aux travaux de Kashiwara et de Mebkhout, de la th~orie des ~x-modules holonomes sur une vari~t~ complexe X, et la possibilitY, qui en r~sulte par la correspondance de Riemann-Hilbert, de pouvoir interpreter alg~briquement la cat~gorie Db(x) des complexes born~s de faisceaux de C-vectoriels ~ cohomologie alg~briquement constructible, ainsi que les six operations sur cette cat~gorie (w~ir en particulier [6], [12], [13], [14], [15], [16], [17]), ont permis de remplir sur C le programme conjectur~ par Grothendieck concernant les "coefficients de de Rhara", programme qui apparaissait d~j~ dans [9] (voir aussi [10] p. 312, et l'introduction de [1], p. 22). Compte tenu des relations qui lient cohomologie rigide et cohomologie de de Rham, il semble doric naturel aujourd'hui de chercher ~ aborder le probl~me des coefficients de de Rham pour les vari~t~s de caract~ristique p au moyen de la th~orie des ~z-modules. C'est la d~marche que nous entreprenons dans le present article. Soient iv un anneau de valuation discrete complet d'in~gales caract~ristiques, de corps r~siduel k. Le point de d~part consiste ~ observer que, sur un ~V-sch~ma formel lisse ~r, de r~duction X sur k, il existe un faisceau d'op~rateurs diff~rentiels ~ qui op~re naturellement sur diff~rents types de ((9~r® Q)-modules d~finis par la cohomologie rigide (par l'interm~diaire du morphisme de sp~cialisation sp : ~VK --') ~ de la fibre g~n~rique de ~r dans ~r). De plus, la cat~gorie des ~ - m o d u l e s localement de presentation finie ne d~pend pas vraiment de ~r, mais seulement de sa r~duction X sur k.

82

La principale difficult~ de l'~tude de ~ vient alors de ce qu'il s'agit d'un faisceau d'op~rateurs d'ordre infini, qui peut ~tre vu comme un "compl6t6 faible" du faisceau usuel des op6rateurs diff6rentiels ~ r . En particulier, l'anneau des sections sur un ouvert affine du faisceau ~ Q := ~ r ® O n'est pas en g~n6ral un anneau noeth6rien. Pour contourner cette difficult6, n~us exposons dans la premiere partie une construction intrins~que du faisceau ~ comme limite inductive, pour m -> 0, des compl~t6s p-adiques ~(m) de sous-faisceaux ~m) de ~ r , appel6s faisceaux des op6rateurs de niveau < m, et qui poss~dent l'avantage d'etre des faisceaux d'anneaux nceth6riens. Ce point de vue permet en particulier de montrer que le faisceau ~ O est un faisceau d'anneaux coherent. La seconde partie est alors consacr6e aux propri~t~s des modules coh6rents sur des anneaux tels que ~(m), ~(m) Q, ~rQ- Pour ne pas alourdir le pr6sent article par trop de d6tails techniques, nous ne donnerons dans ces deux premieres sections que des indications sommaires sur les d~monstrations, en renvoyant a u n article ultSrieur [5] pour un expos6 plus syst~matique et d6taill6. Le reste de cet article est consacr5 a divers exemples de ~ Q - m o d u l e s coh6rents, permettant de faire le lien entre la th6orie des ~ Q - m o d u l e s et la cohomologie rigide. Le premier exemple est celui des isocristaux convergents, vus comme O~Q-modules munis d'une connexion int6grable. Nous montrons que la condition de convergence permet de les munir d'une structure canonique de ~ Q - m o d u l e , et qu'ils sont alors coh6rents sur ~ Q . Nous consid~rons ensuite quelques exemples d'isocristaux surconvergents le long d'un ferm6 de ~ : en prenant l'image directe d~riv6e d'un tel isocristal par le morphisme de sp6cialisation, on obtient un complexe de O~rQ-modules, et nous montrons que ce complexe poss~de une structure naturelle de complexe de ~ Q modules. Nous examinerons en particulier le cas de l'isocristal "constant" (~Y/Ks u r un ouvert Y de X, qui donne naissance/~ des ~ Q - m o d u l e s not6s R i v,t (q~rr, jouant le rSle des images directes sup6rieures pour l'immersion v : Y ~ X, ainsi que celui des faisceaux ~zi((2~rK ) d6finissant la cohomologie rigide ~ support dans le ferm~ compl6mentaire Z. Nous montrerons alors que, lorsque Z e s t lisse, ou est un diviseur ~ croisements normaux, ces ~ Q - m o d u l e s sont coh6rents. Cette propri6t6 reste conjecturale lorsque Z e s t une hypersurface quelconque, et devrait ~tre un point cl6 pour la compr6hension des propri~t6s de finitude de la cohomologie rigide. Nous examinerons enfin le cas des isocristaux surconvergents qui interviennent dans la th6orie p-adique des sommes exponentielles [3]. Le premier cas consid6r6, qui couvre en particulier le cas des caract~res multiplicatifs de IFq, q = pS, est celui de l'isocristal surconvergent X~ s u r ~mk = IP~ - {0, OO}d6fini, pour a • Zp, par le faisceau G ~ r sur ~ m g , muni de la connexion V telle que V(1) -- a d t / t : lorsque a n'est pas un nombre de Liouville, un calcul dfi a Laumon montre que le ~ O - m o d u l e sp.gffa obtenu sur ~ = IP~ est coh6rent. Nous terminerons avec l'6tude

83 des isocristaux s u r c o n v e r g e n t s sur A~ d6finis p a r la connexion V(1) = - s r d t s u r (0A~, avec ordp ~ = 1/(p - 1), qui donnent en particulier les F-isocristaux de Dwork . ~ associ6s aux caract~res additifs ~ de lFq : u n tel isocristal, qui poss~de une s i n g u l a r i t 6 irr6gulibre ~ l'infini, d6finit encore sur £ r = l P ~ u n ~ Q - m o d u l e coh6rent. Lors du colloque de L u m i n y "Systbmes diff6rentiels et singularit6s" en Juillet 1983, j'avais eu l'occasion d'avoir plusieurs discussions avec Ph. Robba concernant l'anneau ~ Q ; c'est avec 6motion que je lui d6die le pr6sent article. Les discussions et l'6change de correspondance que j'ai eu avec G. L a u m o n en 1985, dans lequel il a n o t a m m e n t mis en 6vidence la fa~on dont une partie de la th6orie des ~ x - m o d u l e s coh6rents s'6tend aux ~"~o' -modules (o~t ~(o) ~ r Q est le compl6t6 p - a d i q u e de l'anneau d'op6rateurs diff6rentiels engendr6 p a r les d6rivations), ont 6t6 une 6tape import a n t e pour m'aider ~ d6gager le rble que joue ici la notion de niveau d'un op6rateur diff6rentiel. J e r e m e r c i e enfin Z. M e b k h o u t pour les discussions que nous avons eues /~ plusieurs reprises dans les ann6es pass6es, qui ont 6t6 pr6cieuses pour m'aider ~ mieux comprendre la th6orie des ~ z - m o d u l e s . On d6signe par p u n nombre p r e m i e r fix6 dans t o u t l'article, et p a r ~(p) le localis6 de ~ p a r r a p p o r t ~ l'id6al p r e m i e r engendr6 par p. Tous les sch6mas formels consid6r6s dans cet article seront des sch6mas formels pour la topologie padique.

1. O p 6 r a t e u r s

d i f f 6 r e n t i e l s d e n i v e a u fini

(1.0.1) Soit f : X --+ S u n m o r p h i s m e lisse de sch6mas (resp. sch6mas formels). Rappelons [EGA IV, § 16] r a p i d e m e n t quelques dq~finitions c o n c e r n a n t le calcul diff6rentiel sur X ; pour s i m p l i f e r les notations, il sera sous-entendu dans ce qui suit que routes les notions consid6r6es sont relatives ~ S, que nous ne ferons pas figurer d a n s les n o t a t i o n s . P o u r tout e n t i e r n, on note ~ l'alg~bre du n - i ~ m e voisinage infinitesimal de X dans X x S X : si J est l'id6al de r i m m e r s i o n diagonale, on a doric ~ : := ( 2 z × z / J n+I. I1 existe sur :~: deux s t r u c t u r e s de (2z-alg~bre, corresp o n d a n t aux deux projections de X x S X sur X ; s a u f mention explicite du contraire, nous consid~rerons 9 ~ comme (2x-alg~bre par la s t r u c t u r e gauche, correspondant l'homomorvhisme a ~ a ® 1. S i t 1. . . . . t d sont des coordonn6es locales sur un ouvert U de X, r e l a t i v e m e n t ~ S, et si on pose ~i := 1 ® t i - t i ~, I e (OXx X, ~ est alors libre sur U, de base les produits _zk- pour Ikl < n, avec ~_k:= ~1kl... ,~dk~et lkl := k 1 + . . . + k d. Le faisceau ~ x n des op6rateurs diff~rentiels d'c, rdre <_ n sur X ( r e l a t i v e m e n t h S) est par d6finition le dual ~ O m o x ( ] ~ , (2x), et on d6finit le faisceau ~ x des op6ra-

84 teurs diff~rentiels par ~ z := Un ~Xn"Si on note (0 - [k-]~ "k la base de ~Zn duale de la base tk)k de 5~, les op~rateurs 0 [k-]v6rifient les relations suivantes : V k_', k" e_

lind, ~[k']~[k_"] = (k_'~,,_k")~,+_k,,],

Vk__e]Nd, V f e O z ,

~[klf =

~ k'+k"=k

_O[.k_'](.f)_~."].

(1.0.1.1) (1.0.1.2)

E n particulier, si on note ~i les d6rivations duales de la base dt i de ~J;, de sorte que, pour 1~ = (0 . . . . . 1. . . . . 0) avec 1 a la i-i~me place, on a 0_[M = 0i, on voit que les ~_[k_] j o u e n t le rSle de puissances divis6es des d6rivations ~i" Supposons que S soit un ~(p)-sch6ma. I1 r6sutte alors des propri6t6s g6n6rates des puissances divis6es que ~ z est engendr6 comme alg~bre sur (9z p a r les op6rateurs 0/[P"],pour n >_0. P a r suite, m6me si X est nceth6rien, les sections de ~ x s u r u n ouvert affine de X ne forment pas un a n n e a u noeth6rien en g6n6ral. P o u r tout entier m, nous allons c o n s t r u i r e un a u t r e faisceau d'op6rateurs diff6rentiels ~(xm) sur X, qui sera engendr6 par les 0/[P"], pour n :~ m, et dont les sections sur les ouverts affines seront des a n n e a u x nceth6riens. Cette construction repose sur la notion de structure

partielle d'iddal & puissances divisdes. 1.1. Structures p a r t i e l l e s d'id~al a puissances divisdes

Dans cet article, toutes les P D - s t r u c t u r e s consid6r6es sur les Z(p)-alg~bres seront suppos~es compatibles avec celles de l'id~al pTl(p) [1, I 2.2.1].

(1.1.1) D ~ f i n i t i o n s . Soient A une :~¢~)-alg~bre, I c A u n id6al, m u n e n t i e r positif. On appelle PD-structure partielle de niveau m s u r / , ou encore m-PD-structure, la donn6e d'un PD-id6al (J, 7) c I v~rifiant les conditions suivantes : (i) V x c I , xPm e J ; (ii) p I c J. Un id6al muni d'une m-PD-structure sera appel6 m-PD-idgal. Si (A, I, J, 7) et (A', I', J', 7') sont deux anneaux, munis de m-PD-id6aux, un m-PD-morphisme f : (A, I, J, 7) --~ (A', I', J', 7') est la donn6e d'un PD-morphisme f : (A, J, 7) --~ (A', J', 7') tel que f(I) c I'. P o u r m = 0, cette notion coYncide avec la notion usuelle d'id6al a puissances divis6es. Un exemple type d'id6al m u n i d'une m - P D - s t r u c t u r e est fourni p a r l'id6al m a x i m a l m d'un a n n e a u de valuation discr6te d'in6gales caract6ristiques, d'indice de ramification absolu e : si k est tel que e / ( p - 1 ) < k < e + 1, l'id6al m k est stable par les op6rations x, ~ xn/n!, et contient pm ; il d6finit alors une m-PD-structure sur m pour m > logp k.

85

S o i e n t A u n e W.(p)-alg~bre, I c A u n id6al. Le f o n c t e u r qui associe h t o u t e ~(p)a l g ~ b r e A ' m u n i e d ' u n m - P D - i d ~ a l (I', J', 1,') l ' e n s e m b l e d e s h o m o m o r p h i s m e s f : A -~ A ' tels que f ( / ) c I ' e s t r e p r d s e n t a b l e p a r u n e A - a l g ~ b r e A(rn)(/), m u n i e d ' u n mPD-id6al (/1, J v [-1) ; on d i r a que (5~m)(I), I1, J1, [-]) e s t la m-PD-enveloppe de l'iddal I. L ' a l g ~ b r e A(m)(/) e s t l ' e n v e l o p p e ~ p u i s s a n c e s divis~es u s u e l l e de l'id~al J = 1~ ) + p A de A e n g e n d r ~ p a r p e t p a r les xp~ p o u r x e I ; si on n o t e p a r u n - l e P D - i d 6 a l e n g e n d r ~ p a r u n ideal, on a J1 = ((I(P~) + P I)'A(m)(I)) " et

I1 = I'A(m)(I) + Jl"

(1.1.1.1)

(1.1.2) C e r t a i n s coefficients b i n S m i a u x m o d i f i e s j o u e n t u n rSle i m p o r t a n t d a n s ce qui suit. N o u s u t i l i s e r o n s les n o t a t i o n s s u i v a n t e s : s o i e n t k', k" d e u x e n t i e r s , et posons, p o u r u n e n t i e r m fix~, k ' = pmq, + r', k" = p m q . + r", et k' + k" = pmq + r, avec 0 <_r, r', r" < pro. On i n t r o d u i t alors les coefficients :

{k' +k"~ k'

._

J(m) "-

k" /(m) :=

q! q'lq"!'

'

(1.1.2.1)

t k" t(m)"

(1.1.2.2)

Le coefficient t k' J(m) e s t u n entier, et le coefficient k ' ; k " (m) u n e n t i e r p - a d i q u e . S'it n'y a p a s de confusion possible, nous o m e t t r o n s l'indice (m). Soit m a i n t e n a n t (A, I, J , ~,) u n e 2~(p)-alg~bre m u n i e d ' u n m - P D - i d 6 a l . P o u r t o u t e n t i e r k ~ IN, posons k =pmq + r, avec 0 <_r <pro. P o u r t o u t ~l~ment x ¢ I, et t o u t k _> 0, on pose

X {k} := Xr~q(XPm),

(1.1.2.3)

ce qui e s t b i e n d~fini p u i s q u e x pm~ J . Si n~cessaire, on p r d c i s e r a le n i v e a u m p a r la n o t a t i o n x [k}(m). L e s o p e r a t i o n s x , ) x {k} s e r o n t a p p e l ~ e s p u i s s a n c e s p a r t i e l l e m e n t divisdes de niveau m. E t l e s v~rifient des r e l a t i o n s a n a l o g u e s h celles des p u i s s a n c e s divis~es u s u e l l e s : (i) ~/x e I, x {°} = i, e t x Ik} e I pour tout k > 1 ; (ii) V x e I, V a e A, V k e IN, (ax) {k} = akx [k} ; (iii) V x , y e I ,

VkeiN,

(iv) V x e I , \ / k ' , k " e I N ,

( x + y ) [k} = x(k~x {k'l

(1.1.2.4)

k ~ (~,)(m)X{k'}y{k']; k'+k"=k ~k'+k"~

= t

(v) V x c I, V k', k" e IN, (x{k'})Ik'} =

k"

](m)

x(h'+k'].

q! x {k'k'], (q'!)a'q"!

a v e c ici k ' k " = p m q + r, 0 _< r < pro. L e s o p e r a t i o n s o p e r a t i o n s x ~ xk/q!.

(1.1.2.5)

(1.1.2.6) (1.1.2.7)

x ~-~ x {k}j o u e n t le rSle des

86 Supposons que A soit u n e alg~bre de polyn6mes, soit A : : R[t 1. . . . . td], et I := (tl, . . . . td). On m o n t r e a l o r s que A(m)(/) est u n R - m o d u l e libre a y a n t p o u r b a s e les ~l~ments t ®, k_• INd.

(1.1.3) Les op4rations pr~c~dentes p e r m e t t e n t de d~finir s u r t o u t ideal I d'une ~(p)alg~bre A m u n i d'une m - P D - s t r u c t u r e (J, 7) une filtration canonique, a n a l o g u e ~ la filtration I-adique, et ~ la filtration PD-adique sur un PD-id~al. P o u r tout entier n, soit I n l'id4al engendr~ p a r les 416ments de la forme ~1~{nl}. . . .~,{nr}~, avec x 1, ..., x~ • I, et Eini > n. On pose alors i{n} :: in + (i n ~ j ) Cette filtration s e r a appel~e filtration m - P D - a d i q u e de I. On dira que I est m - P D nilpotent s'il existe un entier n tel que I {n} = 0 , et topologiquement m - P D - n i l p o t e n t si la filtration m - P D - a d i q u e est plus fine que la filtration m-adique. P o u r que la m - P D s t r u c t u r e canonique d'un a n n e a u de v a l u a t i o n discrete d'in~gales caract4ristiques soit topologiquement m-PD-nilpotente, il thut et suffit que m > logp(e/(p -1)). Soient A une 7Z(p)-alg~bre, I un ideal de A, A(m)(I) la m-PD-enveloppe de I, I 1 le mPD-id~al de A(m)(I) d6fini en (1.1.1.]). On pose A~m)(/) := A(m)(I)/I~n+l}. L ' i m a g e de I 1 d a n s ~nm)(/) poss~de p a r p a s s a g e au quotient une m - P D - s t r u c t u r e canonique, et A~m)(I) r e p r ~ s e n t e le foncteur sur la cat~gorie des ~(p)-alg~bres A' m u n i e s d'un mPD-id6al (I', J ' , 7') tel que i,b+l} = 0, qui associe a (A', I', J ' , y') l ' e n s e m b l e des h o m o m o r p h i s m e s f : A ~ A' tels que f ( / ) c I'. Lorsque Fon p a r t d'une alg~bre de polyn6mes R[t I . . . . . td], et de l'id~al I := (tl, . . . . td) , on v~rifie que la filtration m - P D - a d i q u e s u r A(m)(I) est telle que i~n+l} soit t'id~al engendr~ p a r les t ® p o u r Ikl _> n + 1. I1 en r~sulte que A~m)(f) est alors une Ralg~bre finie libre de b a s e l e s t ® pour lk t _
(1.1.4) P o u r m" > m, t o u t e m - P D - s t r u c t u r e s u r u n id4al p e u t ~tre consid~r~e de m a n i ~ r e 4vidente c o m m e u n e m ' - P D - s t r u c t u r e . Si k =pmq + r =pro'q, + r', avec 0 _
(q!/q'!)x{~](m<

(1.1.4.1)

IIen r4sulte que, pour toute ~(p)-alg6bre A et tout ideal I c A, les m - P D -

enveloppes A(m)(I) f o r m e n t un syst~me projectif pour m variable. De plus, les morp h i s m e s de t r a n s i t i o n sont compatibles aux filtrations m-PD-adiques, de sorte que, p o u r n fix~ et m v a r i a b l e , les A~m)(I) f o r m e n t un syst~me projectif, qui est en fait constant et ~gal ~ A / I n+l pour m > logp (n) - 1 .

87

engendr~ par les d4rivations. Plus g~n4ralement, on d~duit de ces relations que ~(xm) est engendr~ comme Ox-alg~bre par les op~rateurs ~/~> p o u r j < m. (1.2.2) Pour m variable, les homomorphismes canoniques ~m+l)(I) -~ A~m)(I) d~finis en (1.1.4) fournissent par dualit4 des homomorphismes ~(zm)n--> t~(xmn +1), qui sont des isomorphismes pour m > logp (n) 1, le faisceau ~(m) ~tant alors ~gal ~ ~Zn" Soient , ~Xn m" > m, et k =pm~ + r =pm ~, + r', avec 0 _ cz(m') ~ X n envoie a (k-)(~) s u r ( g _ ! / q ' ! ) ~
~ m ~ ( X m)

>~X'

qui envoie ~_(k_>(~>sur q! ~[~], est un isomorphisme.

(1.2.3) P r o p o s i t i o n . Supposons que S soit na~thdrien, et soit m un entier. (i) Si S e t X sont affines, l'anneau D (m) := F(X, ~(m)) est ncethdrien. (ii) Le faisceau ~(xm) est un faisceau d'anneaux cohdrents. On dispose en effet sur D(xm) de la filtration par l'ordre, et le gradu~ associ4 grD(zm) est c o m m u t a t i f d'apr~s (1.2.1.2). I1 est nceth~rien, car il est engendr4, comme F(X, Oz)-alg~bre, par les griD~ n) pour i <_pro, qui sont des F(X, (3x)-modules de type fini. L'assertion (i) en r4sulte par un a r g u m e n t classique. Pour prouver (ii), il suffit alors de montrer que, lorsque X est affine assez petit, le foncteur qui associe ~ un D(xm)-module M le ~(zm)-module ~ ( m ) ~D~ ) M est exact. Comme ~(zm) est un (3~module libre pour X assez petit, et que, X ~tant nceth4rien, F(X, -) commute aux sommes directes, rhomomorphisme Ox ®r(z, Ox) D(xm) --->~(xm) est un isomorphisme, et r a s s e r t i o n r~sulte de r a s s e r t i o n analogue pour les (~xmodules, valable ~galement dans le cas formel. 1.3. O p d r a t e u r s d'ordre infini (1.3.1) Soient m a i n t e n a n t K un a n n e a u de valuation discrete complet d'in~gales caract~ristiques, d'id~at m a x i m a l m, et £r un V-schema formel lisse. Appliquant les constructions pr~c~dentes ~ S := SpfK et ~ £r, on obtient pour tout entier m > 0 un faiseeau d'op~rateurs diff6rentiels ~(m) sur ~. On notera ~(£r m) (resp. ~ ) le s~par~ compl4t4 de ~(~) (resp. ~ r ) pour la topologie m-adique ; on a donc par d~finition ~(rn)

:=

]iem_m n ~(m)/mn~(m )

li~mn~(m) Zn

88 1.2. O p ~ r a t e u r s

d'ordre fini

(1.2.1) Soit m a i n t e n a n t f : X ~ S u n m o r p h i s m e lisse de Z(p)-sch4mas (resp. s c h 6 m a s formels). A p p l i q u a n t les c o n s t r u c t i o n s qui pr6c~dent h l'id6al .¢ de ta diagonale d a n s X x s X, on obtient des faisceaux d ' a n n e a u x A~m)(.¢), qui seront not6s n A~lS(m), ou encore A~(m). Ces faisceaux sont m u n i s des d e u x s t r u c t u r e s de (9x alg~bre c o r r e s p o n d a n t a u x deux projections de X x S X sur X, et nous les consid~rerons g 6 n 4 r a l e m e n t c o m m e (gz-alg~bres p a r la s t r u c t u r e gauche. S i t 1. . . . . t d sont des coordonn~es locales s u r un ouvert U de X, et si ~ := 1 @ t i - t i ~ 1, A~(m)nest u n e (9x alg~bre finie libre sur U, de b a s e les _v® pour Ikl < - n : cela r~sulte de ce que A~r(m)nne d~pend que de (gx×z/,¢ n+i, qui e s t q u o t i e n t s u r U d ' u n e alg~bre de p o l y n 6 m e s coefficients d a n s (9z. On d6finit alors le faisceau des opdrateurs diffdrentiels de n i v e a u m e t d ' o r d r e n comme 4rant le dual de AX(m) n :

(m)rt

::

Je°mo (A (m),

Les h o m o m o r p h i s m e s surjectifs An+iX(m)--> A~(m) d6finissent des injections !~x(m)n ~

~(m) Y/JX n + l '

et on d~finit le faisceau ~(zm) des opdrateurs diffdrentiels de niveau m p a r ~(Xm) Pour

:---- U n

t o u s n , n', on dispose d ' a c c o u p l e m e n t s

c~(m)

"~X n"

"~XO~(m)nX ~(m)xn" --) ~(m)xn+n" O~llnlSa ~ '

comme

l e u r s a n a l o g u e s s u r les NZn d a n s [EGA IV, § 16]. On en d4duit u n e s t r u c t u r e d'alg~bre sur N(xm). D ' a u t r e part, N(xm) op~re sur (gz de la m a n i ~ r e s u i v a n t e : si f e (9x -

,~(m)

et t ' e ~ 2 n, la section P ( f ) e (9z est l'image de f p a r l'application compos~e OX

du ,

n

P

, ex,

avee d2(f) := 1 . f . Soient tl, ..., t d des coordonn~es locales sur un ouvert U c X, et vi := 1 . t i - t i ® 1. L a b a s e de -"x c~(m) de A~(m) n n duale de la b a s e (V®)k _ s e r a not4e (a_)k. Si n4cessaire, on pr4cisera le n i v e a u rn p a r la notationa_~-)(~). On d6duit des r e l a t i o n s de (1.1.2) les relations suivantes : V _k', _k" e INd,

o~O<-k') =\-k_'lk'+k'\lU--~-
(1.2.1.1) (1.2.1.2)

k" +k"=k Si k =pmcl + r, avec 0 -< r i < p m pour tout i, l'op4rateur a_~)joue donc le r61e de q!~k-/k!. P a r e x e m p l e , l o r s q u e m = 0, on a s i m p l e m e n t ~ 2 = _O~,et le f a i s c e a u ~(x°) e s t

89 en posant X n := g[" xs Spec~r/m n . Les faisceaux ~m) sont des faisceaux d'op6rateurs diff~rentiels d'ordre infini : si t 1. . . . . t d sont des coordonn~es locales sur un ouvert U c X, et ~l ..... ~d les d~rivations correspondantes, on a

F(U, ~(m))

__

{ ~ a~__~ [ ak- • U(U, (gir) ; a_k --) 0 pour k -) oo }. k

Comme ~(m)s'obtient en compl6tant ~(m)pour la topologie d6finie par un 616ment central, on d6duit de (1.2.3) :

(1.3.2) P r o p o s i t i o n . est ncethgrien.

Soit U c ~[ un ouvert affine. Alors l'anneau ~u~(m):= F(U, ~(irm))

(1.3.3) Puisque (~arest sans torsion, les homomorphismes ~ ) ~ ~(~') ---) t ~ sont injectifs, et il en est encore de m~me des homomorphismes ~m) _~ ~m') __~ ~ir. On d6finit un sous-faisceau d'anneaux ~ de ~ $ en posant

~ := u m ~ m) c~i~. Le faisceau ~ ainsi obtenu est ~troitement li~ ~ la notion de completion faible de Monsky et Washnitzer, comme le montre la caract6risation locale des op~rateurs de ~ . Soient en effet U un ouvert affine de £r sur lequel il existe des coordonn~es locales, a 1. . . . , ~d les d6rivations correspondantes, A = F(U, (gir), et It- I1 la norme spectrale sur A ® K. I1 est ators facile de voir que l'anneau D b := F(U, t ~ ) poss~de la description suivante : (1.3.4) P r o p o s i t i o n . Avec les notations prdcddentes, on a D*U : { ~ ak _~[k] ] ak • F(U, (gir) ; 3 c > 0, 77< 1 tels que V k_, IIak[[ < c77Ik-I}. k I1 r~sulte de la construction de ~ que l'on a ~ / w i n ~ -- ~Xn pour tout n. Si U est un ouvert affine de ~r, l'anneau D b n'est donc pas nceth6rien en g6n~ral, et on peut montrer que l'anneau D~UQ :-- Dtv ® Q ne rest pas non plus. On peut par contre montrer la coherence des faisceaux ~(m~ := ~(irm) ® O et ~ Q := ~ ® Q : (1.3.5) T h ~ o r ~ m e . cohdrents.

Les faisceaux c~m), ~(m)Q, ~ Q

son t des faisceaux d ' a n n e a u x

Le cas des faisceaux ~(m) et ~(m) -~r O se traite par les techniques consid~r~es dans la section suivante. Pour ce qui est de I ~ Q , nous nous limiterons ici ~ l'~nonc~ du point cl~ dont r~sulte la coh6rence, en renvoyant ~ [5] pour la d~monstration :

90 +1) (1.3.6) P r o p o s i t i o n . Pour tout m, les homomorphismes canoniques ~(m)Q _~ "~(m it Q sont plats (& gauche et & droite).

2. La cat6gorie des N)Q-modules coh~rents Dans toute la suite de cet article, on se place sous les hypotheses de (1.3.1). 2.1. Les thdor@mes A e t B

Nous explicitons ici les analogues des th~or~mes A et B pour les faisceaux coh~rents sur des a n n e a u x tels que ~(arm), ou N)Q. Les r~sultats qui suivent sont valables pour les ~-modules g gauche et ~ droite ; nous tes ~noncerons pour les 9modules ~ gauche.

(2,1.1) On suppose m a i n t e n a n t que le ~V-sch~ma formel £r est affine. On posera :

13(# ) := r( r,

£r /, ,,~-£rQ := ^

e

--

D~ := r(£r, e~), D~Q := D~r ® Q = r(£r, ~

e ) m) ® Q), ® Q),

les isomorphismes venant de ce que 95 est un espaee topologique noeth~rien. D~signons par ~ (resp. D) Fun des faisceaux ~(~m),~-~(=)Q,~ . , ~*~Q (resp. l'un A m des a n n e a u x ~ ), ~(m)Q, D~, D}Q). Si M est un D-module ~ gauche de presentation finie, on lui associe un Y-module ~ gauche M A de la fagon suivante : (i) Cas oh D = D(~m). Le D-module M est alors s~par~ et complet pour la topologie m-adique. Pour tout n _> 1, soit M le C0x -module quasi-coherent d~fini par M/mnM, qui est un ~ x -module de pr4sentation finie. On pose M a := limnM* n. ~(m)

""

o

(ii) Cas oh D = ~-~r Q. On peut trouver un D(m)-module de presentation finie M tel que M =/~/® Q. On pose alors M A := 3~/A®Q, ce qui ne d~pend pas du choix du r~seau/~/.

l~m/~(m)

(iii) Cas oiz D -- D). Comme D ) = et que M est de presentation finie, il existe un entier m, et un/~(m)-module de presentation finie M (m) tels que

M = D~ ®b}=)M (m). On d~finit al0rs M z~par

91

M A := D~ ®~xm)(M(m)) ~, et M ~ ne d6pend pas de la fa~on dont on a descendu M e n M (m). (iv) Cas oit D = D~Q. Comme on a encore D~Q = lim m D~ ^(m)Q, on proc~de comme dans le cas (iii) pour se ramener au cas (ii).

(2.1.2) T h 6 o r ~ m e (Th6or~me A). Soient ~ un ~Z-schdma formel affine et lisse, et D, D ddfinis comme en (2.1.1). Soit 3,( un D-module. Les propridtds suivantes sont dquivalentes : (i) ~ / e s t de prdsentation finie sur D ; (ii) Il existe un D-module de prdsentation finie M tel que d,t -- M A. (iii) Le D-module M := 17(5.r, ~ ) est de prdsentation finie, et l'homomorphisme canonique D ®D M --, ~t est un isomorphisme. De plus, si D est l'un des faiceaux ou D~Q, le foncteur M , >M A est exact.

~(m), "~(m)Q

Lorsque D = ~(m), ~(rn~ ou D~Q, il revient au m~me de dire que ~ est coherent. Dans le cas o~ D = t~(m), la d6monstration est analogue ~ celle de [EGA I, § 10]. Le cas off D = ~(m~ se ram~ne au precedent grace ~ l'existence, pour tout ~(m~_ module de pr6sentation finie ~ , d'un ~(m)-module de pr6sentation finie ~ tel que ® Q. Enfin, les cas o~ D = D~ ou D = D~Q se r a m ~ n e n t aux pr6c6dents par des a r g u m e n t s standards de passage ~ la limite inductive. Lorsque D = ~m), l'exactitude de M , > M ~ provient de ce que ~(m) est noeth6rien. Le cas de ~(m~ en r~sulte en p r e n a n t des modules entiers, et celui de D~Q vient de la platitude de D~Q sur les -~r Q, d'apr~s (1.3.6). Le m~me type d'argument fournit le th6or~me B :

(2.1.3) T h 6 o r ~ m e (Th6or~me B). Soient tY un ~r-schdma formel affine et lisse, D l'un des faisceaux ddfinis en (2.1.1), ~ un D-module de presentation finie. Alors, pour tout i > 1, on a Hi(g$, ~() = O.

2.2. I n d d p e n d a n c e p a r r a p p o r t a u r e l ~ v e m e n t

Soient respectivement C (m), t",(m),.,~ Q' C~, C~O" les categories des D(m), ~(m), D~, D~Q-modules ~ gauche coh6rents. On se propose de v6rifier que les cat6gories C (m), c(m~Q, pour m assez grand, et C~, C~Q, ne d6pendent que de la r6duction X de modulo m. Les 6nonc6s sont encore les m~mes pour les D-modules ~ droite.

92

(2.2.1) T h 6 o r ~ m e . Soient ~, ~" deux T'-schdmas [brmels lisses ayant m~me rdduction X modulo m. Si e est l'indice de ramification absolu de ~, et si m > logp(e/(p - 1)), les eatdgories C~m) et C(~m) (resp. ~'~(m)Q et C~)o) sont naturellement dquivalentes. Lorsque ~r et ~r, sont affines, il existe un isomorphisme ¢p : ~ ' ~ £r i n d u i s a n t l'identit6 sur X. On en d6duit pour tout m u n i s o m o r p h i s m e ~o*-semi-lin6aire ~ : ~(~n) - > ~(m), et l'extension des scalaires c o r r e s p o n d a n t e d6finit u n e 6quivalence de cat6gories e n t r e C(~m) et C~rm,). Si m > logp(e/(p -1)), cette 6quivalence est en fait i n d 6 p e n d a n t e du choix de l'isomorphisme (p. En effet, soient (p', ~0": :Y' ~ ~ ~ deux isomorphismes i n d u i s a n t l'identit6 sur X, et ~', ~" : ~(rn) ~ ; ~(m) les isomorphismes correspondants. Soient t 1. . . . , t d des coordonn6es locales sur ~ , t~ := ~'*(ti), t~' := ~o"*(ti), ~i les d6rivations correspondantes sur ~r, ~ := ~,(~i), ~, := ~,,(~i)" Comme t~'- t~ e mG~, et que cet id6at poss~de une m-PD-structure topologiquement m-PD-nilpotente gr~ice/i la condition v6rifi6e p a r m, la s6rie de Taylor T := ~ ( t " - t') ® 0ff(k> k

est d6finie et converge duns ~,n). On v~rifie facilement qu'elle est ind6pendante du choix des coordonn6es, et que, pour tout o p 6 r a t e u r P e ~(~), on a ~/"(P)T = Tv'(P) dans ~(m). On d6finit alors un isomorphisme e n t r e les deux foncteurs d'extension des scalaires d6finis par V' et V" :

en e n v o y a n t une section x de ag sur T ® x. De plus, les isomorphismes se,, e, v6rifient la condition de cocycle, ce qui p e r m e t d'6tendre la construction de l'~quivalence de cat6gories C(~m) - ) C(m,), au cas g6n6ral. Le cas des cat6gories ,~rn(m)Qet ~,~r'Q'~'(m)en r6sulte grace a rexistence de modules entiers pour les ~(~m~-modules coh6rents. P a r passage a la limite inductive, on en d6duit :

(2.2.2) C o r o l l a i r e . Soient ~, ~" deux ?/-schdmas formels lisses ayant m~me rdduction X modulo m. Les cat@ories C~ et C ~ (resp. C~Q et C ~ Q) sont naturellement dquivalentes.

(2.2.3) Nous renvoyons le lecteur a un article ult6rieur [5] pour la description des op6rations de fonctoriatit6 en ~r, et en X, des diverses cat6gories de ~ - m o d u l e s consid6r6es ici.

93 3. I s o c r i s t a u x

eonvergents

(3.0.1) Rappelons b r i b v e m e n t la d~finition d'un isocristal c o n v e r g e n t sur X (cf. [4, (2.3.2)], ou [2, (4.1)]). On note K le corps des fractions de 1/, et ~rgla fibre gdndrique du F - s c h d m a formel ~ : c'est un espace analytique rigide sur K, muni d'une application continue sp : ~ g ~ ~ , le m o r p h i s m e de sp~cialisation [4, (0.2.2)]. Lorsque ~ est affine, soit ~ = SpfA, on a ~ g = S p m (A ® K) ; l'application sp associe alors ~ un ideal maximal de A ® K, ddfinissant une extension finie K(x) de K, l'iddal maximal de A formd des ~ldments dont l'image dans K(x) est de valuation < 1 pour r u n i q u e v a l u a t i o n sur K(x) prolongeant celle de K. Pour tout ouvert U c ~, on a sp-l(U) = UK. Un isocristal c o n v e r g e n t E sur X est ddfini par la donn~e d'un (2~rK-module localement libre de r a n g fini E ~ sur ~g, muni d'une connexion int~grable V : E~---~E~®

~ ~r~, i

vdrifiant la condition suivante [4, (2.2.14)] : pour tout ouvert affine U = SpfA de ~ sur lequel il existe un systbme de coordonndes locales t 1. . . . . td, ddfinissant des ddrivations bl . . . . . bd, toute section e e F(UK, E~r), et tout 77< 1, on a [[_b[k]e [[~lk_l__>0 pour [k[ --->oo,

(3.0.1.1)

o£l b_®e est ddfini par raction des ddrivations b~ fournie par la connexion V, et [[ddsigne une norme de Banach sur F(UK, E~) [7, ch. 3, 3.7.3 prop. 3]. 3.1. Cohdrence sur ~ O des isocristaux convergents

(3.1.1) Soient E un (~%-module cohdrent, et $ = s p . E . U t i l i s a n t le fait qu'un faisceau c o h e r e n t sur u n espace affinoide est d~termin~ par ses sections globales, et que, pour tout ouvert affine U = S p f A c ~ , sp-l(U) = Spm A ® K est affinoide, on voit que sp*$ - ) E, de sorte que la donn~e de E ~quivaut/~ celle de $. De m~me, comme sp*~ - ) ~i~K' on a un isomorphisme canonique $ ® ~ r ) sp.(E ® D~rK), et la donn4e d'une connexion intdgrable sur le (9~rEmodule E dquivaut ~ celle d'une connexion intdgrable sur le (C0~ ® K)-module $. O n notera O ~ Q := O~r ® K. La donnde d'un isocristal convergent sur X dquivaut donc/i celle d'un O~rQ-module $ localement projectif de type fini, muni d'une connexion intdgrable V telle que la condition de convergence (3.0.1.1) soit satisfaite ; on dira encore qu'une telle connnexion est

convergente. Le G~rQ-module $ est alors muni d'une s t r u c t u r e canonique de ~ O - m o d u l e , p r o l o n g e a n t p a r continuit~ la s t r u c t u r e de ~ - Q - m o d u l e qui correspond ~ la connexion V. E n effet, il suffit pour la d~finir de construire pour tout ouvert affine assez

94

petit U de ~r un accouplement continu

> r(u, ~)

F(U, ~*~q) × F(U, ~)

prolongeant celui que d~finit V. On peut supposer qu'il existe s u r U des coordonn~es locales, d~finissant une base de d~rivations 01. . . . . 3d, et tout P e F(U, ~ O ) s'~crit p = ~ ak _~fk], oh les a k e F(U, G~rO) sont tels qu'il existe c > 0, 77< 1 tels que nak] < c~ I~-Ipour tout k. P o u r tout e e F(U, 8), la relation (3.0.1.1) entra~ne alors que ak~_[-k-]e-~ 0 pour ]k I -+ c~, de sorte qu'on peut d~finir P e comme somme de la s~rie P e = ~ ak_ ~_[-kle. k~ Nous allons m o n t r e r que, muni de la structure de ~ Q - m o d u l e ainsi d~finie, est coh6rent sur ~ Q . L a d~monstration s'appuie sur le r~sultat suivant d'Ogus [18, prop. 7.3], ~nonc~ dans [18] pour .@~), mais qui reste valable sur ~(m) :

(3.1.2) P r o p o s i t i o n . Soient ~ un schdma formel lisse de type fini, de rdduction X, E un isocristal convergent sur X, ddfini par un (9~rQ-module ~ localement projectif de type fini, m u n i d'une connexion intdgrable et convergente, et m e IN un entier. Il existe alors un ^ )-module ~, cohgrent sur C~p et un isomorphisme -lindaire

Supposons d'abord £r affine, avec des coordonn~es locales t I . . . . , t d ; posons zi = 1 ~ t i - t i ® 1. Soient A = F(~r, (2~r), M = F(~r, ~) ; notons AAAm)la m-PD-enveloppe de l'id~al diagonal de (A ®~ A) ® ~z/mn, et AA(m) : ~imn AAn(m) , AA(m) est donc le compl~t~ p-adique du A-module libre de base les _z®. Posons r/= ]p[p-,-a, et

B

=

(A~A{T,..

. )

pm+l

7~}/(pT~-, 1

..... PTd-,~'~+~))OQ.

Observons tout d'abord qu'il existe un h o m o m o r p h i s m e canonique B --~ AA(m) ® Q. pm+l E n effet, u n tel h o m o m o r p h i s m e envoie n~cessairement T i sur *i /p ; comme on a ^ ^ *ip~+t./10 = (P - 1)). z~P • ~+1} e AA(m) , et que AA(m) est s6par6 complet pour la topologie padique, l'assertion en r~sulte. C o m m e M est de type fini sur A ® Q, le produit tensoriel M ®A®Q B (pris ici pour la s t r u c t u r e gauche, d6finie par a ~ a ~ 1) est s~par6 et complet. P o u r tout e e M, la s6rie de Taylor O(e) = ~ O[k-]e® ,k = ~ 8e ®_~® k k

(3.1.2.1)

95 converge dans M ®A®Q B, car, si l'on pose k = prn+l~[ + r, avec 0 <_ri M ®A®Q B ainsi obtenue est A-lin6aire pour la structure droite de M ®A®Q B, i. e. que pour tout a e A, on a O(ae) = (l®a)O(e). P a r composition avec l'extension des scalaires p a r l ' h o m o m o r p h i s m e B --~ AA(m) ® Q d6fini plus haut, on trouve donc u n e application A-lin6aire pour la s t r u c t u r e droite, encore not6e 0,

O:M

>M®A®Q(~A(m)®Q).

Soit alors M' un sous-A-module de type fini de M tel que M = M' ® Q, et posons h~/= 0-1(M' ®A AA(m)) ; il est clair que ~ / ® Q -- M. Comme le compos6 de 0 avec l'extension des scalaires o par l'homomorphisme d'augmentation AA(m) -~ A est l'identit6 de M, on a M a M' et ~ / e s t de type fini sur A. Comme A est nceth6rien, M' de type fini sur A, et AA(m) le compI6t6 du A-module libre de base 2 ®, l'homomorphisme canonique

M" ®A £A(m) --->(Ok_M')^ est un isomorphisme, et t o u t 616ment x e M" ®A AA(m) s'6crit de mani~re unique comme somme d'une s6rie convergente x = ~ e_k® _v®, k

avec ek e M', et e~ -~ 0 si Ik_l -~ c~. P a r suite, d'apr~s (3.1.2.1), on a

~I = [eeM" [ Vk_, 3_~>eeM'}. D'apr~s (1.2.1.1), il en r6sulte que ~ / e s t un sous-D(~m)-module de M. Le O f m o d u l e coh6rent ~ d6fini par ~ / e s t alors muni d'une s t r u c t u r e de ~9(~m)-module ; comme le module de ses sections sur t o u t o u v e r t affine de ~r est s6par6 et complet, cette s t r u c t u r e s'6tend en une s t r u c t u r e de ~(~m)-module, et l'isomorphisme ~ ® Q - > est un isomorphisme de ~ m ~ - m o d u l e s . Dans le cas g6n6ral (que nous n'utiliserons en fait pas ici), on choisit u n r e c o u v r e m e n t fini (~r) de ~r du type pr6c6dent, et on recolle de proche en proche en r e p r e n a n t l ' a r g u m e n t d'Ogus [18, 7.5].

(3.1.3)

Proposition. Soit ~ un ~(rn)-rnodule, cohdrent en tant que (~-rnodule.

Alors : o

(i) $ est cohdrent sur ~(rn) ," (ii) l'homomorphisme canonique (3.1.3.1)

est un isomorphisme.

96 On peut supposer que ~r est affine. Soient A = F(~, (.9~r), D(~m) = F(~r, ~m)). Comme ~ est O~r-coh6rent, il existe un homomorphisme surjectif (2~ --~ ~, donc un h o m o m o r p h i s m e surjectif (~m))n __~ ~ ; soit aV son noyau. La filtration par l'ordre sur (~(m))n induit une filtration de aV par des sous-O~modules N/, qui sont O f c o h 6 rents puisque noyaux d'homomorphismes entre o ~ m o d u l e s coh6rents. Comme le foncteur F ( ~ , - ) c o m m u t e aux limites inductives, il en r6sulte que l'homomorphisme canonique (9~r ®A F(~, JO ~ N e s t un isomorphisme ; il en est de m6me de 0 ~ ®A D(m) --> ~(m). Comme, d'apr~s (1.2.3), D(~m) est nceth6rien, il existe un homomorphisme surjectif (D(~m))r -~ F(~r, ag), qui induit donc un morphisme surjectif (~m))r _~ N . La coh6rence de ~ sur ~ m ) en r6sulte d'apr~s (1.2.3). Pour prouver l'assertion (ii), observons tout d'abord qu'on dispose d'une action naturelle de ~m) prolongeant par continuit6 celle de ~(m), puisque, pour tout ouvert affine U de ~r, F(U, ~) est s6par6 et complet pour la topologie p-adique. On obtient ainsi un homomorphisme ~m) ®~i,~) ~ __>~ dont le compos6 avec (3.1.3.1) est l'identit6 de ~. I1 suffit donc de prouver que (3.1.3.1) est surjectif. Pour cela, on peut supposer de plus que ~r poss~de un syst~me de coordonn6es locales t 1. . . . . t d. Pour i = 1. . . . . d, notons ~ l'op6rateur 01V~). Pour tout entier r
(3.1.3.2)

k_

of1 les Bk_ • F(U, ~m)) sont des op6rateurs diff6rentiels d'ordre
(~i n i - 2 a i j ~ J ) e

= 0.

(3.1.3.3)

J
Par r6currence sur k, tout op6rateur ~!r)~,k of~ 0 < r <pro, peut s'6crire sous la forme

(r) ,k , ni ~i ~i = Qkr(~i - 2 a i j ~ [ j) + Rkr, j
(3.1.3.4)

o~t les Rkr sont d'ordre < pmn i par rapport/~ 0i" C'est en effet clair si k < n i, et, pour k > ni, on peut ~crire

~i)~i k = ~i(r)~i'k-ni (~i'ni_ •2 a i j ~ / ) J
+ ~(r)~,k-rti{V~ ui "i ~ t * i j v i ~,j~]' J
le deuxi~me terme 6tant un op6rateur d'ordre < pm(k - 1) + r. On en d6duit que, pour tout multi-indice k~ et tout op6rateur Bk • F(U, ~L)), d'ordre
97 chacun des ~i, il existe des op6rateurs Rk_ e F(U, t~m)), d'ordre < pm(n 1 + ... + nd), et Qik_e F(U, t~(~m)) tels que

Sk ~-'k- = ~. Qik (~n~ _ ~ aij0~j ) + Rk. j
De plus, les calculs s'effectuant dans F(U, ~m)), on a Q~k -* 0, Rk --~ 0 si Bk_ --~ 0. Appliquant ce qui precede ~ la suite B~_ 'k-de somme P, on peut alors r66crire P sous la forme

P = ~ ( ~ Q i k ) ( ~ [ ni- ~ a i j ~ ?) + E R k . k_

j
k_

Comme ~ k Rk est d'ordre fini, cela permet d'6crire dans ~m) ®~,~/~

P ® e = (~.Rk_)®e = l ®(~..Rk_)e , k_ k_ ce qui montre la surjectivit6 de (3.1.3.1). (3.1.4) P r o p o s i t i o n . Soit E un isocristal convergent sur X, ddfini par un (9~Qmodule ~ localement projectif de type fini, muni d'une connexion intdgrable et convergente. Alors : (i) Pour tout entier m, l'homomorphisme canonique >~(rn~ ®~Q $

(3.1.4.1)

est un isomorphisme, et il en est de m~me de l'homomorphisme canonique >~ Q ®~Q ~. (ii) Pour la structure de ~ O - m o d u l e coherent sur t ~ Q (resp. ~(rn)Q).

(3.1.4.2)

(resp. ~(m)Q-module) ddfinie en (3.1.1), ~ est

Soit m > 0, et choisissons, grace ~ (3.1.2), un ~m)-module ~, coh6rent sur (2~, et un isomorphisme ~ ® Q - > ~. L'isomorphisme (3.1.4.1) r6sulte alors de (3.1.3.1) pour ~ en tensorisant par Q. Passant ~ la limite inductive sur m, on en d6duit l'isomorphisme (3.1.4.2). D'autre part, ~ est ~rn)-coh~rent d'apr~s (3.1.3). Si on consid~re une pr6sentation de ~ sur ~m) : s

>

o,

on obtient par tensorisation avec Q, puis avec ~ Q (resp. ~(m) ~. Q~ sur ~rQ, une pr6sentation analogue de ~ Q ® ~ o ~ (resp. ~(~m)®~Q $), donc de $ d'apr~s (i). La coh6rence de $ r6sulte alors de (1.3.5).

98 3.2. L e c o m p l e x e d e d e R h a m d ' u n ~ Q - m o d u l e L'exemple le plus i m p o r t a n t d'isocristal convergent est l'isocristal "constant" (gz/g, d~fini par Cg~Q muni de la connexion trivale. Dans ce cas, on dispose comme dans le cas classique de la r6solution localement libre sur ~ Q fournie par le complexe de Spencer associ6 au faisceau tangent [17, I (2.1.17)]. Observons d'abord que la division par ~ est possible dans ~ m ~ et (3.2.1) P r o p o s i t i o n . Soient U un ouvert de ~ sur lequel il existe un syst~me de coordonndes locales t 1. . . . . td, ddfinissant des ddrivations 0 1. . . . , Dd" (i) Soit P = ~ k ak ~-~k-)• F(U, ~(m)) ; si a k = O pour tout k tel que k 1 = 0, il existe un unique opdrateur -Q e- F(U, ~m)) tel que pmp = QOl" (ii) Soit P = ~ k ak~ [k-] • F(U, ~ Q ) ; si ak_ = O pour tout k tel que k z = 0, il existe un unique opgrateur Q • F(U, ~ Q ) tel que P = QOl. L'unicit6 de Q r~sulte de ce que ~(rn) est sans torsion. Soit k tel que k 1 ¢ 0. Dans ~(m), on dispose de la relation (1.2.1.1)

I k sipm~k, (~11} = [ P msi Pm] k.

avec:

(3.2.1.1)

Quel que soit k 1, (kll~ divise donc pm dans :~p, ce qui permet de d~finir Q en posant Q = ~ ( . m / k l \ - l ~ ~{k_-!l) w \ 1 / f~k-, kl~.0

en notant 11 = (1, 0 . . . . ,0). L'assertion (ii) r6sulte de (i) par passage a la limite, mais peut aussi ~tre prouv~e directement : on a a k ~ k~] = (ak/kl)O~k~-l]~l, et il existe c > 0, et 71 < 1 tels que ]]akfl < c ~lk_l ; comme ]l/kl] = (9(k 1) = O(]kl), l'existence de Q en d6coule. (3.2.2) P r o p o s i t i o n . Soit Y-~ = £1~~ le faisceau tangent sur ~. L ' h o m o m o r p h i s m e ~ Q --~ (9~Q envoyant P • ~ Q sur P. 1 • (2~rQ fait du complexe de Spencer o

d

2

une rdsolution ~ Q - l i n d a i r e de (~rQ. E n partieulier, si Oz. . . . . Od sont les ddrivations correspondant it un syst~me de coordonndes locales, on a un isomorphisme

99 d

L'~noncd analogue sur ~(m)Q est vrai pour tout m e IN. k-1

k

Rappelons que la diff6rentielle d : N}Q ®of A ~-~r > ~ Q ®~r A :T~ est d6finie par k d(P ~ 3i^...AOk) = E ( - 1 ) i - l P 3 i ® 31A...A0~/A... A 3 k i=1

+ E (-1)i+JP ® [3i' 3j]A31A'''A~iA'''A~A'''A3k' i<j

pour P e ~ Q , 31. . . . . 3k e 5r~, On peut supposer qu'il existe sur ~ un syst~me de coordonn6es locales ; si 3i, .... 3d sont les d6rivations correspondantes, les sections 1 ® 3h^...A3ik forment une base de ~ Q ®o~ ASr~r• Comme la multiplication ~ droite par 3i e s t injective sur ~ Q , et que [0i, Oj] = 0, on volt facilement que le complexe

o -~ ~ Q %

d

^ ~ - ~ . -~ . . . -~

d

~ Q %~-~. -~ ~

-~

~+~Q/EfiQ.3~-~ o

est acyclique. I1 est clair d'autre part que le noyau de ~ Q --) C9~.Qest l'ensemble des op6rateurs de la forme Y'k~0 ak -~[!] e ~ Q , et un tel op6rateur appartient ~ Y'i ~Q'0~ d'apr~s (3.2.1). R e m a r q u e . On aurait aussi pu partir de la r6solution de Spencer de C9~csur ~}~), pals tensoriser sur t ~ ) avec ~(m~ et ~ O , qui sont plats sur t~}~), et appliquer les isomorphismes (3.1.4.1) et (3.1.4.2). Par contre, pour m > 0, l'~nonc6 est faux en g6n6ral pour C0~rsur 1~(~ m) et ~ n ) : par exemple, si dim(X) = 1, on a m

o~.

=

tn

~m>/E~m>o~> _ ~m>/E~m>~>, j=O

j=O

et, pourj > 0, les 3~j> ne sont pas divisib]es par 3. (3.2.3) C o r o l l a i r e . Soient ~ un ~ Q - m o d u l e it gauche, et ~t ® f2"~ le complexe de de R h a m de .At, ddfini p a r la connexion sous-jacente. Il existe un isomorphisme canonique dans la catdgorie ddrivde des faisceaux de K-vectoriels sur gf

I1 suffit en effet d'utiliser la r6solution de (3.2.2) pour calculer le complexe IRgfom~Q(C0~rQ, ,At).

100

(3.2.4) C o r o l l a i r e . Soient E un isocristal convergent sur X, ddfini p a r un Cq~rQmodule ~ localement projectif de type fini, m u n i d'une connexion intdgrable et convergente. Il existe des isomorphismes canoniques H/cony(X,E) -- Ext~ro((~£rQ, ~). En particulier, si X est propre sur k, il existe des isomorphismes canoniques H~g(X, E) = Ext~Q(O~Q, g).

Soit E~r le (2£rg-module ~ connexion int6grable d6finissant E. Par d6finition de la cohomologie convergente, on a H/cony(X,E) =/~(~rg, E~ ® D~r). Pour tout ouvert affine U ~ ~r, sp-l(U) est un ouvert affino~de de t~rK. Comme les faisceaux E£r ® ~ r sont coh6rents sur (2~rg, on a ]Rsp,(E£r ® g2~) -- sp,(E~ ® ~ K ) = 8 ® a ~ , et l'~nonc~ en r~sulte. Lorsque X est de plus propre sur k, la cohomologie rigide coincide par construction avec la cohomologie convergente.

4.

Isocristaux

surconvergents

: le cas constant

Soient Y un ouvert de X, Z le ferm~ compl~mentaire. Alors que les isocristaux convergents sur X consid~r6s dans la section pr~c6dente fournissent des ~ Q modules "sans singularit~s sur X", analogues des fibres ~ connexion int~grable en caract~ristique 0, les isocristaux convergents sur Y qui sont surconvergents le long de Z vont donner des exemples de ~ O - m o d u l e s avec des singularit6s le long de Z. Nous allons d'abord montrer que l'image directe d~riv~e d'un tel isocristal par le morphisme de sp~cialisation poss~de une structure canonique de complexe de la cat~gorie d~riv~e D b ( ~ Q ) . Nous montrerons ensuite, lorsque E est l'isocristal OYIK, et Z une sous-vari~t~ lisse, ou un diviseur ~ croisements normaux, que les faisceaux de cohomologie de ce complexe sont des ~ O - m o d u l e s coh~rents, et qu'il en est de m~me pour les faisceaux de cohomologie locale de (P~rKa support dans Z ; de plus, tous ces modules ont sur t ~ O la m~me presentation que les 5~z-modules analogues en caract6ristique 0. (4.0.1) Nous rappelerons d'abord rapidement ce qu'est un isocristal sur Y, surconvergent le long de Z [4, (2.3.2)]. Pour tout ferm6 Z c X, le tube de Z dans ~r est par d~finition le sous-ensemble ]Z[~r := sp-l(Z) c t~rg [4, (1ol.2)]. C'est un ouvert pour la topologie rigide de ~ g : s'il existe des sections f l . . . . . fr ~ F(~r, O~r), de r6ductions

101

j?l . . . . . 9~r, e F(X, OX) telles que Z = V ( f 1. . . . . 37r), alors ]Z[~- est l'ouvert de £rK d6fini par les conditions ]Z[r=

{xe~K

J Vi, Ifi(x)l
Plus g6n6ralement, si Z e s t l'ensemble des z6ros modulo m d'une famille de sections f l . . . . . fr de d)~r, on pose, pour tout ~. < 1,

]z[~ := { x ~ K

] v i , Lfi(x)l<x},

y~ := ~rK- ]z[~. Pour £ assez pros de 1, ]Z[x ne d6pend pas du choix des fi, ce qui permet de le d6finir par recollement lorsqu'il n'existe pas de sections globales d6finissant Z. De plus, le syst~me inductif des ]Z[x ne d6pend pas de la structure de sous-sch6ma ferm~ de X mise sur Z. On note v : Y c_~ ~, vx : Vx ~ ~ g les inclusions, et on pose, pour tout faisceau ab61ien F sur ~g,

v*F := li_m~_,l_ v~. v~* F,

(4.0.1.1)

_F_{z[(F) := Ker IF --~ vtF].

(4.0.1.2)

Les foncteur v t et _F]tz[ sont exacts, et, si F est un vtO~rK-module, l'homomorphisme canonique F ~ vtF est un isomorphisme [4, (2.1.3)]• Si v' : Y' c_~ £r' est l'inclusion d'un ouvert dans un second sch6ma formel, et f : £ r ' ~ ~ u n morphisme tel que f(Y') c Y, d6finissant f g : ~r~ ~ ~K, il existe, pour tout faisceau f sur ~g, un morphisme canonique

v t f --~ f g . v'tf}F.

(4.0.1.3)

Un isocristal sur Y, surconvergent le long de Z, est d6fini par la donn6e d'un vtG~ -module E~. localement libre de rang fini, muni d'une connexion int6grable V, telle que, si v 't est le foncteur analogue & v t sur ]X[~, et

Pl, P2 : (]X[~~z, v't(~} ) :~ (~g,

Vt('Q£r2

les deux projections du tube de la diagonale dans £rg x ~ g sur ~g, il existe un isomorphisme de v'tG~r}-modules t . • P2E~ -> p~E~induisant sur le compl6t6 * formel le long de la diagonale l'isomorphisme ~ que d6finit la s6rie de Taylor de V, donn6 en coordonn6es locales par : ~(1 ®e) = ~_[k-le ®_zk-. On dit alors que V e s t surconvergente. La surconvergence est une propri6t6 locale s u r ~. Soit E un vtG~r -module localement libre de r a n g fini, muni d'une connexion int6grable V. D'apr~s [4, (2.2.3)], il existe, pour X assez pros de 1, un ~ v - m o d u l e localement libre de rang fini Eo, muni d'une connexion int~grable Vo, et un isomor-

102

phisme horizontal E ) v?Eo . Supposons que Z soit un diviseur de X. Si U est un ouvert affine de ~ tel que Z = V(f), avec f e F(U, O£r), alors, pour tout X, Vx est d6fini p a r la condition [f(x)l > :t, et l'ouvert sp-~(U) c~ Vx = U g ~ V;~ est un ouvert affino~de de ~rg. Pour que V soit surconvergente, il faut et suffit, d'apr~s [4, (2.2.13)], que, pour tout ouvert affine U ~ 5g v6rifiant la condition pr6c6dente, et sur lequel il existe un syst~me de coordonn6es locales, et pour tout 77 < 1, il existe ).~ < 1 tel que l'on ait, pour tout e e F ( U K n V~., E0), avec £~ < ~. < 1, II0_[k-]e II~/t~_t~ 0 pour Ik_l -~ oo,

(4.0.1.4)

en n o t a n t ]- ] u n e norme de Banach sur F ( U K c~ V~,, Eo). 4.1. Image directe p a r spdcialisation d'un isocristal surconvergent

I1 est facile de voir qu'il n'existe pas en g6n6ral d'action du faisceau s p - l ( ~ Q ) sur (girK prolongeant par continuit6 celle de sp-l(t~ro) = ~£rg. Pour d6finir l'action de ~ O sur te complexe IRsp,Ezr, off E est un isocristal sur Y surconvergent le long de Z, nous utiliserons u n r e p r 6 s e n t a n t de ce complexe associ6 a des r e c o u v r e m e n t s de hr et Y. Commengons par un 6nonc6 g6n~ral f o u r n i s s a n t une r6solution a la Cech d'un faisceau de la forme v*E.

(4.1.1) Soit t = (~r/)i~! u n r e c o u v r e m e n t o u v e r t fini de ~r, et, pour tout i, soit ~i = (Yij)j~ji un r e c o u v r e m e n t ouvert fini de Yi := y ~ ~ " P o u r tout / = (io, " " , ih), on note ~ r = £rio c~... c~ ~ri~, et u~ l'inclusion de ~r~g darts ~rg ; on recouvre Y/:= Y c~ ~ r p a r le recouvrement 9i_ := (YiA)i, o~t j. = (Jo . . . . , jh ) e Jio x . . . x Jib' et :=

YioJo

" "

Yi.h.

L'intersection de k + 1 ouverts du recouvrement 9 / s e r a not6e Y/Z avec

\ J k o "'" Jkh k

h

Y _i_ = a=O Z=0 YiM,~"

On notera v~ (resp. v/j) l'inclusion de Y/(resp. I7/_/)dans ~:, et v[, v ~ les analogues du foncteur (4.0.1.1) pour les ouverts Y/, Y~ de ~r/. S u r ~r/g, on dispose du complexe de Cech pour les foncteurs v ~ d6fini en [4, (2.1.8)]. Appliqu6 ~ Ei := u*(E), il fournit un complexe

I-Iv[:/ i

> !-[ io~

>...

> I-I i_

>

....

103

qui est une r6solution de vZEi. Si / = (io, ..., ik) , et/-a := (io . . . . . ?a. . . . , ik), il existe un morphisme de complexes Pi~ : uia* ~t'(~ia, E/a)

>

u~, ~t-(~, E/)

prolongeant le morphisme u ~ , E h --> ui,Ei_, d~fini comme suit. Pour tout j -- (Jo. . . . , jk), notons _2a := (Jva,''' ,J~a). Pour toute section s de u/a, ~t'(O/a, E/a), on pose

pi~a(s)~_j__ pj~(s/a_~), :=

o5 p ~ est l'image par u/a, du morphisme de fonctorialit6 (4.0.1.3) d6fini par l'immersion £r/c___) £r/a, qui envoie Y/~ dans Y/aa" Pour / variable, ees morphismes donnent naissance & un bicomplexe q-ui sera not6 ~t. "(t, (Oi)i, E) ; en bidegr6 (h, k), il est donc d6fini par

=

I]

i=(io,'",ih)

m.(

1-I

J__=~o,'"J.h)

(4.1.2) L e m m e . Soient ~ ' un ouvert de ~, Y' un ouvert de ~ ' tel que Y" c Y, soient u : ~'g ~ ~K, v" : Y" c-o ~ ' les immersions correspondantes, et notons par l'exposant t les foncteurs (4.0.1.1) relatifs aux immersions ouvertes v e t v'. Alors : (i) Pour tout ~)~K-module F, l'homomorphisme canonique u,v'tF

>v t u , v ' t F

est un isomorphisme ;

(ii) Si Y" = Y (~ g$', l'homomorphisme canonique vtu,F

) vtu, v'tF

est dgalement un isomorphisme.

L'assertion (i) r6sulte de ce que u, v ' t F est un vtO~K-module , par l'interm6diaire de l'homomorphisme canonique vt(2~ ~ u , v't(2~rk. Si Y' = Y n £r', on a, pour tout ouvert affinoi'de W de ~K, F(W, v t u , v't F) = li_ma, z. F(W ~ Vz n ~

¢~ V[,, F)

= li~m~ F(W ~ ~ ~ £r~;, F) = F(W, v ~ u , F), car les ouverts V[ d6finissant le foncteur v't sur ~

sont 6gaux aux Vx ~ ~r~.

(4.1.3) P r o p o s i t i o n . Soit E un faisceau abdlien sur gSg" Avec les hypotheses et les notations de (4.1.1), le complexe simple associd au bicomplexe ~t..(t, (~i)i, E) est une rdsolution de vtE.

104

Pour tout k, il existe un morphisme canonique vtE -~ ~t0k(T, (~}i)i, E), d6fini par la famille des morphismes vtE --~ ui.(vti2Ei). Pour k variable, ces morphismes forment un morphisme de complexes du complexe vtE -+ vie ~ vtE -~ ... dont le terme g~n~ral est vtE, et la diff~rentielle 0 en degr~ pair, et IdvtE en degr~ impair, dans le complexe ~ t ° ' ( t , (~i)i, E). I1 suffit donc de prouver que, pour tout k, te complexe ~t0k(~, (~}i)i, E) "

) ~tlk(~, (~}i)i, E)

> ...

(4.1.3.1)

est une r~solution de vtE. Comme les ~'ig forment un recouvrement admissible de ~K, il suffit de prouver cette assertion apr~s localisation au-dessus des ~ i g ' on a alors u*(vtE) = vtiEi . S u r :~iK, te foncteur ~t'(~i, _ ) appliqu~ au complexe u*(~t'k(t, (~i)i, E)) donne un bicomplexe ~t.(~i, u*(¢t-k(t, (~i)i, E))). Pour tout faisceau F sur ~ig, le complexe ~t-(~i, F) est une r~solution de viiF d'apr~s [4, (2.1.8)] ; le complexe simple associ~ au bicomplexe ~t'(9i , u*(~t'k(t, (9i)i, 8))) est donc quasi-isomorphe ~ v~iu*(~t'k(t, (9i)i, E)). De m~me, le complexe ~t.(~i, vtiEi) est une r~solution de erie[El . Comme, pour tout v/t(9~g-module F, on a F - > vtiF [4, (2.1.3)], on voit qu'il suffit de prouver les deux assertions suivantes : a) pour tout h, on a

u*Cthk(t, (~i)i, E) - >vtiu*~thk(t, (gi)i, E) ; b) pour t o u t i = (Jo. . . . . jk ), le complexe

viii u* Ct°k(-T, (Vi)i, E)

>vii2 u~ Ct lk(t ' (Vi)i, E)

>...

est une r~solution de vtii viE i. Pour tout / = (i o. . . . . ih), soit/' = (i, i o. . . . . ih). Notons u i rinclusion de ~i_'g = ~ig ¢~ ~r/K dans ~rig, u~ celle de ~/'K dans ~r~K, Yi~ = ~ i g c~ Yt~, et vi~: Y~2 ~ ~EK" L'assertion a) r~sulte des isomorphismes

v:u*Cthk(~£,(~li)i ,E) =vii u*(

H

ui_.(

i-=(io,'",ih)

H

v[2Ei_))

J==(J-O,"'~)

i-=(io,...,ih) J=(iO,...~) =

=

H

H

i_=qo,...Jh) ;==(/o,...d~)

,t~, vit u /,. v/~./.

u /,. v./_2E/, 1-I H ,t ~=(io,...,iD ~=qo,...~)

= u* ~thk(t, (~/)i, E),

105

provenant des relations 6videntes u* u~. -- u~, u~* et ~'i . ,*. ~_i~ t = ~_ij " 'tu~*, appliqu6 aux ouverts iTEK c iTig et YiJ c Yi"

et de (4.1.2)

(i)

Pour prouver l'assertion b), on construit une homotopie sur le complexe

o~

v:~ viE,

>

vIi u~ o*o~(~, (v,),, ~)

>

Vt2 ~. u~C?lk(~, (~i)i, E)

) ....

On d6finit en effet un homomorphisme

~: : vli u* (

YI ui,( I ] vIi_Ei_)) i=(io,".,ih+1)- J__=(/-o,-.-J-k)

de la manibre suivante. Pour tout _i = (/o .....

i=(io,...,ih)

J = (io,.-.d.~)

avec Ja -- (Jao," -- • ,Jab), on pose

/k),

(JoJoo'"Joh

I

k, Jk Jko "'" Jkh On observe alors que, quels que soient i, 2, l'homomorphisme

_

tz ui uf*vi'j'Ei'

est un isomorphisme : en effet, en g a r d a n t les notations pr6c6dentes, on a comme plus h a u t

~:~ uT u,. vI:~- -_ : _

u;.<* v[:~ -- v:~ u,. v~:..

t l

_

_

- _

.

et d'autre part V ?ii

ui* ui_', vi'i_'Z ? ( -- v~ii ui_. vii'EEl'.

Comme Ybi_,= Yii r~ YiJ' on a vii,---v:? vi~ d'apr~s [4, (2.1.7)]. Le lemme (4.1.2) (ii) fournit donc les isomorphismes t£

v*tl u ; . v [ : .

_

-- v:iu'i . v i i'~v;~F. * " '* _ --- viiui_,vi_jEE,

d'oil l'assertion. Cet isomorphisme permet alors de d6finir ~ en posant, pour s •

V? ,,*(X,?h+1 k(~ i i ~i . . . .

(~}i)i ,

E), ~(s)i ~ := si,j,.

On v6rifie imm~diatement que ~ est bien une homotopie du complexe consid6r6. (4.1.4) L e m m e . Soient iT un V-schdma formel sdpard, de rdduction X sur k, iT' un ouvert affine de iT, u : iT'K c__+iTK l'immersion ouverte correspondante, f • F(iT', (gir), Y = D(f) c iT', v t le foncteur (4.0.1.1) sur iT~associd dt l'ouvert Y d e iT'. (i) Si E est un vtO~K-module de prdsentation finie, alors u . E est acyclique pour le foncteur sp,, ot~ sp : iTK --+ iT est te morphisme de spdcialisation. (ii) Si de plus E est muni d'une connexion intdgrable, surconvergente le long de X - Y, alors s p . u . E est muni d'une structure canonique de ~ Q - m o d u l e , foncto-

t06

rielle en E. Si ~'~ est un ouvert affine de ~', avec ul : ~'IK c._._>:~K' et si ]71 ~ ~'~ n Y est un ouvert de ~'~, de la forme Y~ = D ( f ~), avec f~ e F ( ~ , (2~), ddfinissant un foncteur v~ sur ~'~K, l'homomorphisme canonique sp.(u.E)

"~s p . ( u l . v ~ u ~ E )

est ~Q-lindaire. Comme E est de pr6sentation finie, il existe ~ < 1, et un Loaf[module coh6rent E o sur Vx tels que E = vtEo [4, (2.1.10)]. Pour tout ouvert affine U de ~r, et tout i, on a alors Hi(sp-l(U), u , E ) = Hi((U ~ ~r')K, vtEo). P u i s q u e ~ est s6par6, U n ~r, est affine, et (U r~ ~ ' ) g un o u v e r t affinoi'de de ~ . C o m m e Y est u n ouvert de ~ ' de la forme D(f), il r~sulte de [4, (3.1.7)] que les Hi(sp-l(U), u . E ) sont nuls pour i > 1, d'otl l'assertion (i). Supposons m a i n t e n a n t que E soit muni d'une connexion int6grable et surconvergente V. I1 existe alors ~'o < 1, et un O~K-module coh6rent E o sur V~o, muni d'une connexion int6grable Vo, tels q u e (E, V) = vt(Eo , Vo). Soient U u n ouvert affine de sur lequel il existe des coordonn6es locales, et P = Y~kak~ ~ c F(U, ~ O ) . Soient c > 0, 77< 1 tels que Ilak_ll < c~ tkl. D'apr~s (4.0.1.4), la surconvergence de V e n t r a i n e l'existence de ~tl, avec ~'0 < ~1 < 1, tel que, pour tout e e F((U n ~r')g (~ Vx, Eo), avec ~1 < )t < 1, on ait ] ~_~e IItl Ik-I --~ 0. On peut done d~finir Pe e F((U r~ ~')g n V~, E o) c o m m e somme de la s6rie convergente ~kak(~_[~]e). Puisque (U n ~r')g est affino~de, on a F((U ~ ~"')g, E) = lim~, F((U N ~ ' ) g ~ VA, Zo), et Faction de P sur F((U c~ ~')g, E) -- F(U, sp, u . E ) en r~sulte par passage a la limite inductive. I1 est facile de voir qu'elle ne d6pend pas des choix effectu6s, et, pour P variable, d6finit u n e s t r u c t u r e de F(U, ~ Q ) - m o d u l e sur F(U, s p . u . E ) . On obtient ainsi u n e s t r u c t u r e de ~ Q - m o d u l e sur s p . u , E . De plus, si (E', V') est un second vt(2~g-module localement libre de r a n g fini, et ~o : E --~ E ' u n h o m o m o r p h i s m e horizontal, il existe, pour ~o assez pros de 1, u n L0~rmodule coh6rent E~ sur Vxo, muni d'une connexion int6grable V~, tels que (E', V') --vt(E'o, V~), et un homomorphisme horizontal ~oo : E o --> E o tel que ~o= vt(~Oo).Pour tout ~t > ~'o, l'application F((U n ~¢")g ('h V M Eo) ~ F((U m ~')K ~ Y~, E~) est continue, done commute a Faction de P e F(U, ~ q ) qu'on vient de d6finir pour ~t assez prbs de 1. L a p r e m i t r e partie de l'assertion (ii) en r6sulte ; la deuxi~me partie vient de ce que, pour tout ~t, l'homomorphisme de restriction

F ( ( U n ~ " ) K n V~,E O)

>I-'((Ufh~["I)K(h VI~,Eo) ,

0it V 1~ = ~1/C- ]~/'1-}71 Ix, est continu, done compatible ~ l'action de P e F(U, ~ O ) .

107

Notant toujours v : Y ~ X I'inclusion d'un ouvert dans X, nous poserons, pour tout faisceau abdlien E sur ~K, ]Rv~,E := lRsp.(vtE),

R~v,Et

:= Risp.(vtE).

Si E est un vt(2~. -module, on a donc simplement lRv~,E = IRsp.E. Soit d'autre part T un ferm~ de X ; nous noterons

Comme le sugg~rent ces notations, les foncteurs lRv¢, et lR_F~Tjouent le r61e d'une image directe de Y dans X, et d'une cohomologie locale fi support dans T, pour les "fonctions h singularit~s surconvergentes'. (4.1.5) P r o p o s i t i o n . Soient ~r un ~-schgma formel sdpard, de type fini, de rdduction X sur k, Y un ouvert de X, de compldmentaire Z, T u n fermd de X, E un isocristat sur Y, surconvergent le long de Z. Il existe sur le complexe IRvC,E~; (resp. IRF_~TE£r) une structure naturelle de complexe de ~¢~Q-modules, qui en fait un complexe de Db(~¢~Q) fonctoriel en E. Si on note Y' = Y - (T r~ Y), et v" : Y' ~ X l ' i m m e r s i o n ouverte correspondante, la suite exacte de faisceaux sur ~ K 0

>Ft]T[E

>E

) v'tE

>0

d~finit un triangle distingud de Db(~Q), fonctoriel en E,

/'\ Soit ?~ = (g$i)i~i un recouvrement affine fini de £r, et, pour tout i, soit Oi = (Yu)jEJI un recouvrement fini de Yi := y (~ ~'i par des ouverts de la forme D(fij), off f i j e F(£ri, Gar). Notons v : Y ~ ~ ; d'apr~s (4.1.3), le complexe simple c t ' ( t , (Oi)i, E~) associ6 au bicomplexe ~t. "(t, (~)~, E~.) est une r6solution de vtE~, qui est 6gal ~ E~ puisque E ~ e s t un vtO~rK-module. D'autre part, avec les notations de (4.1.1), les ouverts ~_/ sont affines, et les YiA de la forme D(f/A) :, avec fi£ = l-[a,~ fi~J,K Le lemme (4.1.4) entra~ne alors que le complexe s p , ~ t ' ( ~ (~i)i, E~)-est un repr6sentant de 1Rsp,E£r. De plus, chacun des faisceaux sp.(u_i , v[iu*E ~) est muni d'une structure canonique de ~ Q - m o d u l e , fonctorielle en E, et lecomplexe sp, Ct'(t, (Oi)i, E~) est alors un complexe de ~ Q - m o d u l e s . Enfin, le passage h des recouvrements t', ~ . plus fins induit un morphisme de complexes sP, ~t'(t, (9i)i, E~) -~ sp, ~ t ' ( t ', (~')i', E~) qui est compatible h Faction de ~ Q

d'apr~s (4.1.4) (ii). En associant h E la classe

108

dans D ( ~ Q ) du complexe sp, ~t. (t, (~i)i, Eft), on obtient un foncteur qui redonne Ie foncteur lRv¢, par restriction des scalaires de ~ Q ~ (.9~rQ. Qu'il soit/~ valeurs dans Db(~rQ) r6sulte de la finitude des recouvrements employ6s, et de ce que l'on peut remplacer les complexes de cochaines par des complexes de cochaines altern6es. Pour traiter le cas du complexe IRF?TE~ on peut supposer que, pour tout i, le recouvrement ~i est choisi de telle sorte qu'il existe une sous-famille ~ de ~i formant un recouvrement de Y' ¢~ X i. Le faisceau F__~T[Ef admet pour r6solution le complexe E f ~ v'?Ef, qui admet lui-m6me pour r6solution le complexe simple associ6 au bicomplexe ~*" (T, (~¢)i, El)

>

~*" (~, (~i)i, El).

Par suite, le complexe IRF¢TE~radmet pour repr6sentant le complexe sp, (77"(~, (0i)i, E l ) --~ sp, C'*"(~, (0i)i, E~r). ,

Ce dernier est muni d u n e structure naturelle de complexe de ~ - Q - m o d u l e s d'apr~s ce qui pr6c~de, et on obtient encore ainsi un complexe de D b( ~ Q , ) ind6pendant du choix des recouvrements et fonctoriel en E. Comme c'est le c6ne du morphisme de complexes de ~ Q - m o d u l e s repr6sentant IRv~,E~ --~ tRv,?Ef, l'6nonc6 en d6eoule. (4.1.6) P r o p o s i t i o n .

Sous les hypotheses de (4.1.5), il existe un isomorphisme

canonique IRv,? ( E l ® ~ s ) (resp.

=

IRYom~Q((9~,Q, IRv¢,E~)

IRF_?T(Ef ® a'~2 = IR:/Com~Q(GfQ,

]RF__tTEf)).

Choisissons des recouvrements t = (~'i)iel de £r (resp. (resp. ~}i (Yij)jeJi de Yi:= Y ¢~ E'/), comme dans la d6monstration de (4.1.5). Soit E un isocristal sur Y, =

surconvergent le long de Z. D'apr~s le lemme (4.1.4) (i), le complexe simple ~t'(t, (~i)i, E f ® f ~ s ) associ6 au tricomplexe ~ t " ( t , (~i)i, E f ® f~g) est une r6solution du complexe E f ® f~K'/~ termes acycliques pour sp,, et fournissant doric un isomorphisme

]RvC,(Ef ® ~"2"~2 = IRsp,(Ef ® ~ ' 2 = sP*(~J"(~' (~i)i' E l ® f2"~K). D'autre part, le complexe IRsp,E, en rant que complexe de ~ Q - m o d u l e s , est repr6sent6 par construction par le complexe s p , ~ ? ' ( t , (~}i)i, E~). En utilisant la r6solution de OfQ construite en (3.2.2), on obtient l'isomorphisme

IP,Yom~Q(~9~Q, IRsp,Ef) = sp,~*'(t, (Oi)i, E£r) ®0~ ~rPour achever la d6monstration, il suffit donc de construire un isomorphisme sp, C?'(t, (~)i)i,E~) ® 0 ~

-

> s p , ~ ? ' ( t , (~i)i,E~,®fY~),

109

c'est h dire, avec les notations de (4.1.1), de construire pour t o u s / , j un isomorphisme c o m m u t a n t aux restrictions (sp.

¢

*

ce qu'on d~duit de ce que s p * ~ fini.

= ~ g , et de ce que ~

est localement libre de rang

De m~me, si l'on suppose, comme duns la d~monstration de (4.1.5), que les recouvrements Oi sont choisis de telle sorte qu'il existe des sous-recouvrements ~}~ formant des recouvrements des ouverts Y' n £ri, off Y' = Y - (T (~ Y), on obtient des isomorphismes IR_F~(E~ ® ~ r 2 ) = sp.[(~t.(~, (Vi)i,E~r®Fy~K)

IR~om~o(G~r e, IRFZTE~.) = sp. [C t" (t, (~9i)i,E~)

>

(~t.(~, (~}i,)i,E~®~'K)] '

) 5¢" (t, (9;)i, E~)] ®e~ ~ ,

et on conclut comme pr~c~demment.

(4.1.7) C o r o l l a i r e . Supposons X propre et lisse sur k, et soient v : Y ~-~ X une im-

mersion ouverte, T u n fermd de X, et E un isocristal surconvergent sur Y. Si l'on munit les complexes IRvt.E~et ]RF_?TE~des structures de complexes de D b ( ~ Q ) d d finies en (4.1.5), il existe des isomorphismes canoniques Hirig(Y/K, E) --- Ext~Q(($arQ, IRvt, E~), HIT(Y/K, E) = Ext~Q(~q~rQ, IR_FCTE£r). Comme on a par d~finition

Hing(Y/K, E) := Hi(£rK, E~ ® ~ 2

= Hi(~' IRv¢*(E~ ® ~K)),

HiT(Y/K, E) := H i ( ~ g , F_~T[(E~® g2"~K)) = Hi(~t', IRF¢T(E~® ~ 2 ) , le corollaire r~sulte imm~diatement de (4.1.6).

4.2. Cohomologie l ocal e £z s u p p o r t duns un d i v i s e u r lisse

On suppose ici que Z e s t un diviseur lisse duns X, et on garde les notations de (4.0.1). Nous nous limiterons au cas d'un isocristal constant sur Y, renvoyant h u n article ult~rieur pour des r~sultats plus g6n~raux, en liaison avec l'analogue pour les ~ Q - m o d u l e s du th~or~me de Kashiwara sur les ~-modules h support duns une sous-vari~t~ lisse [6, VI 7.11]. On observera que, Z ~tant un diviseur, les faisceaux Rivt. O~K sont nuls pour i > 0, et les faisceaux ~ ztz( ( ~ . K) sont nuls pour i ¢ 1 [4]. Si U = SpfA est un ouvert affine deX, e t f e F(U, G~) une ~quation de Z (~ U modulo m, on a

11o

F(U, sp. Lo~rK) = A[1/jq* ®K,

(4.2.0.1)

F(U, ]g'ztI(Lo~r~)) = (A[1/f]t/A) ® K.

(4.2.0.2)

Nous a u r o n s h utiliser le lemme suivant :

(4.2.1) L e m m e . Soient if" = SpfA un 1/-schdma formel affine rdduit, I1" HIune norme de Banach sur A ® K, f • A, B = Aoq ® K, oil AOO est le s@ard compldtd de Af, et (ak)k>O une suite d'dldments de A ® K vdrifiant les conditions (i) H existe des constantes c • IR, 7? < i telles que, pour tout k, IlakN < c ~k ; (ii) Dans B, on a

Z a k f -k = O. k>_O Si, pour tout j > 0, on pose J bj := ~ akfJ-k, k=O il existe c', et ~' < 1,pour lesquels on a ]ibjl] < c'~ 'J. Si l'~nonc~ est vrai pour une norme sur A ® K, il l'est pour toute norme ~quivalente. Comme A ® K est r~duit, la semi-norme spectrale sur Spm (A ® K) est une n o r m e de Banach, ~quivalente ~ toute norme de B a n a c h [7, 6.2.4, th.1], ce qui p e r m e t de supposer que I - il est la norme spectrale. Soit 77' • IR u n ~16ment dont une puissance a p p a r t i e n t au groupe de la valuation de K, et tel que 71 < ~' < 1. On consid~re le r e c o u v r e m e n t de ff'K par les deux ouverts affinoi'des V 1 :-- [XE.~K ] ] f ( x ) ] - ~ ' } ,

V2 := { X • ~ K ]

]f(x)l:>~'},

et il suffit de majorer les normes spectrales de bj sur V 1 et V2. P o u r x • V], on a J

tbj(x)t = ] Z ak(x)f(x)J-kl <- suPk_<j llakH~l"j-k <- c~'J" k=O Darts F(V2, (9~rg), on a ]ak(x)f-k(x)l < c(~/~') k. Comme A est r~duit, il en est de mSme de F(V2, (Q~K) [7, 7.3.2, cor. 10]. La norme spectrale est donc une norme de Banach sur F(V2, (9~K), si bien que la s~rie Y~k a k f -k converge vers b • F(V2, LO~-K).Comme b est d'image nulle dans B, on a If(x)] < 1 en tout point du support de b, et le principe du m a x i m u m e n t r a i n e que, quitte ~ a u g m e n t e r ~', on p e u t supposer que ~ k ak f - k = 0 dans F(V2, L0~rK).On obtient alors sur V2 la majoration

Ibj(x)] = i - ~_~ ak(x)f(x)J-k] <- suPk> j ilakH~'j-k < crfJ. k>j

111

(4.2.2) P r o p o s i t i o n . S u p p o s o n s que Z soit un d i v i s e u r lisse d a n s X, et soit v l'immersion de Y = X - Z d a n s X. Alors : (i) Le ~ Q - m o d u l e vt. (2~r~est coh@rent ; (ii) Si U est un ouvert de ~ sur lequel il existe des coordonn~es locales t 1. . . . . t d telles que Z = V(t 1) modulo m, et si ~ . . . . . ~d sont les ddrivations correspondantes, la suite (~Q)d

~

>~ Q

m , v,~O~rg

>O,

(4.2.2.1)

oft ~o(P) = P.(1/t~), et ~/ est dgfini p a r d

v(P1 . . . . , Pd)

=

P l a i t 1 + ~ PiOi,

(4.2.2.2)

i=2

est exacte. Comme ~ Q est coherent, la p r e m i e r e assertion r~sulte de la seconde, qu'il suffit de v~rifier lorsque U est affine. Posons A = F(U, O~r), et DtUQ = F(U, ~ Q ) . Observons d 'abord que l'homomorphisme ~ : D utQ --~ F(U, v .tO ~ ) = A [ 1 / t 1]t est surjectif : tout 61~ment f de A[1/tz]t s ~crit sous la forme ~,k>_o a k t l ~-1, pour une suite d'~l~ments a k e A telle q u e [[ak][ < C ~7k pour u n certain 7/ < 1 ; l'op~rateur P = ~k_>o (-1)kak D[k] a p p a r t i e n t alors &D?UQ et v~rifie P.(1/t 1) = f . Soit P e DtuQ tel que P.(1/t z) = O. Si P = ~,k ak ~-[k-], on peut e. c n. r e. 1-'. = .L .k akd_d-[k] - + ~,, Z,k_ ak~_[k-I, en r e g r o u p a n t dans ~ ' (resp. ~") l'ensemble des termes relatifs aux indices k tels que k i = 0 pour i > 1 (resp. tels qu'il existe i > 1 pour lequel k i ¢ 0). D ' a p r b s (3.2.1), l'op@rateur ~ k ak~[k-] a p p a r t i e n t ~ l'id@al & gauche engendr@ p a r 2 2 . . . . . ~d, donc& l'image de V. On peut ainsi supposer que P e s t de la forme P -- ~ k _ > 0 a k ~[k] 1" Puisque P e Ker(~), on a dans A[1/tl]t la relation

(-1)kakt~ k-1 = O.

(4.2.2.3)

k_>0 P o u r t o u t j >_0, soit bj e A l'~l~ment d~fini par j-1

bj := (-1) 7+I ~

(-l)kaktJl-k-l.

k=O

Comme P e DtUQ, il existe c, et ~ < 1, tels que IlakU < c~ k. D'apr~s (4.2.1), il existe donc c', et ~' < 1, tels que I[bj]l < c'~ 'j, et on peut d~finir un op~rateur Q • DtuQ en posant

V := ~ b j ~ ]. j>_O

On a alors Q t 1 = P : il suffit en effet de le v~rifier dans F(U (~ Y, ~ O ) d'apr~s (4.2.2.3), on a alors pour t o u t j > 0 l'~galit~ b = ( - 1 ~ E (-1)kak tj-k-1,

k>_j

D DtuQ , et,

112

ce qui permet d'4crire :

Qtl : E (~, (-1)k-JaktJl-k-1)D[~]tl j>_o k>-j k = E a k ( E (-1)k-Jt(-k-lo~'l)tl k >-O

j=O

: ~ ak(O~kltll)tl k>_O :P.

Comme b o = 0, il existe d'apr~s (3.2.1) Q • Dtuo tel que Q = Qi0i, d'ofi t'~nonc& (4.2.3) C o r o l l a i r e . Sous les hypoth@ses de (4.2.2) : (i) Le ~ Q - m o d u l e Yzt i(O£rg) est coh#rent ; (ii) Sur U, la suite > 0,

(4.2.3.1)

oh ~o(P) = P-(1/ti) , et Vest d#fini par d v(Pi . . . . . Pd ) = P l t i + ~ PiOi,

(4.2.3.2)

i=2

est exaete. Par image directe par le morphisme sp : ~"g -~ ~c, la suite exacte

fournit une suite exacte de ~ Q - m o d u l e s 0

> Oir Q

> v,t (glrK

>~tzi((.gir K) ........ > O.

La coherence de . t ~~$ .i ( o ~q), r~sulte donc de celle de G~rQ et v~ G~K. Si U = SpfA est un ouvert affine de ~ muni de coordonn6es ti, ..., td telles que Z = V(t i) modulo m, alors F(U, 9~*zi((#~r~) s'identifie ~ (A[1/ti]t/A) ® g. Comme, en t a n t que sous-D~Q-module de A[ 1/tl ]t ® K, on a A = (D~Q tl)-(1/ti), l'assertion (ii) r6sulte de (4.2.2) (ii).

4.3. C o h o m o l o g i e l o c a l e & s u p p o r t d a n s u n d i v i s e u r ~ c r o i s e m e n t s n o r m a u x

(4.3.0) Soient toujours Z c X un diviseur, Y l'ouvert compl~mentaire, v : Y ¢-+ X l'immersion correspondante. I1 parait raisonnable de conjecturer que, quel que soit Z, les faisceaux v.t (9~rK et 9Fztl((9~r2 sont coh6rents sur ~ Q . S o u s allons m o n t r e r q u e c'est le cas lorsque Z est un diviseur a croisements normaux, r6union de s o u s -

113

vari~t~s lisses se coupant transversalement. Cela permet aussi de traiter le cas des faisceaux Rive, GgrKet 9~zti(oirK) quand Z e s t une sous-vari6t6 lisse de codimension r.

(4.3.1) L e m m e , Soient A une V-alg~bre formellement lisse, possddant un syst~me de coordonndes locales t 1. . . . . td, ddfinissant des ddrivations 01. . . . . Od, et r,j, k trois entiers positifs, tels que r < d , j < k. On pose f = t l . . . t r, et, pour tout i > O, on note Qi l'opdrateur ddfini par r

r

Qi := 1-I (Oa [/-1] - ~ a[i]t a ) - I-Ioa[i-1]" a =1

(4.3.1.1)

a =1

On a alors r r H o g ] = (-1)r(k'j)fk-J l-I o~] a=l

a=l

k E (-1)r(i-J-1)fi-J-iQi"

(4.3.1.2)

i=j+l

Pour t o u t j > 0, et tout a, on a

t°Cll

=

On en ddduit la relation F

r

(-1)rf 1-I0g +il : a=l

F

,

1-[ (Ogl- 0g+i]ta ) : l ~ 0 g ] +Qj+t, a=l

a =1

et la relation (4.3.1.2) en r6sulte par r~currence.

(4.3.2) P r o p o s i t i o n . Supposons que Z vdrifie les hypotheses de (4.3.0). Alors : (i) Le ~ Q - m o d u l e vt, G~g est cohdrent ; (ii) Si U est un ouvert de Er sur lequel il existe des coordonndes locales t 1. . . . . t d telles que Z = V(ti...tr) modulo m, et si Oi. . . . . ~d sont les ddrivations correspondantes, la suite (~Q)d

~

>~ O

e

t ) v,~0~r K

) 0,

(4.3.2.1)

oit ~o(P) = P.(1/ ti...tr) , et ~/ est ddfini par r d v(P 1. . . . . Pd ) = ~,PiOit i + ~, PiOi, i=l

(4.3.2.2)

i=r+l

est exacte. L'assertion (i) r~sulte encore de (ii). Soit U = SpfA un ouvert affine de Er v~rifiant les conditions de (ii) ; posons f = t i . . . t r. Vn ~l~ment de F(U, v?, C0irK) s'~crit sous la forme g = Eak h~o

f-k-l,

114

la suite des ~l~ments a k e A K ~ t a n t telle qu'il existe c > 0, ~ < 1 tels que ]]ah[] _
k

k-k~ k-ka

( ~ ( i - k a - 1 ) t i - k ~ - l ~[i] t

[k] i:-k~+l

P o u r t o u t k • IN, soit I k l'ensemble des k_ tels que k a < k pour tout a, et qu'il existe a p o u r lequel k a = k. On r ~ c r i t P sous la forme r

P = E

E

k>O keIk

akll

((-1)

k-k~ tak-k~ 0a[kl +

-a=l

k E

(- 1)(G-k~-l)tG-k~-l o~.]ta ~

i~=ka+l

Posons alors P

Po

=

% I](-1) k-k,~

Z k>_0 k_eI~

k-ka

[k]

.

a=l

L ' o p ~ r a t e u r Po ainsi d~fini est bien d a n s DtUQ, car, pour tout k e Ik, on a k _< ]k], de sorte que ]ak_] <_c~ I~ < c~ k = c(~l/r) rk. D ' a u t r e p a r t , en d ~ s i g n a n t p a r J u n souse n s e m b l e v a r i a b l e de {1. . . . . r}, on p e u t ~crire P - P 0 c o m m e u n e c o m b i n a i s o n lin~aire & coefficients dans A d'op6rateurs de la forme

ak I-[0~ ] 1-[ o[i~lta = (ak H~[~ ] 11 ~[ia~-ll/ia) r I Data -a~J

aeJ

-a~J

aeJ

aeJ

avec k a < i a < k, J ~ {1. . . . . r} (car l'un des k s est 6gal ~ k), et J ~ 0. C e t t e derni~re condition m o n t r e que ces o p 6 r a t e u r s sont d a n s l'id6al engendr6 p a r ~1tl . . . . . ~rtr , donc d a n s I. L'ordre de chacun des o p 6 r a t e u r s

1-i

-a~J

1-I

aeJ

e s t c o m p r i s e n t r e k et rk, t a n d i s que le coefficient a k v~rifie c o m m e plus h a u t la m a j o r a t i o n ]Iak]] __Oak~[1k]'" "O[rk], avec

Z ( - 1 ) r k a k f -k-1 = P-(1/f) = 0. k>0

(4.3.2.3)

115

Pour tout i _> 1, soit Qi l'op6rateur diff~rentiel d6fini en (4.3.1.1). Pour tout k, la relation (4.3.1.2) permet d'6crire, en notant _~0] = i--jr ~[j] ,.la=l. a ,

k

k

k i-1

Eaj'O [/1 = ( E (-1)r(k-J)ajfk-J)o[kl + E ( E (-1)r(i-j-t)ajfi-j~l)qi. j=O

j=O

(4.3.2.4)

i=I j=O

D'apr~s (4.2.1), la relation (4.3.2.3) entra~ne qu'il existe c' et ~' < 1 tels que, quel que soit k, on ait h

H~-~(-1)r(h-J)ajf k-j II < c'71'k, j=o de sorte que, pour k -~ ¢¢, on a k

( ~ (_l)r(k-J)ajfk-j) ~_[k]___>0

j=O ~ ~(m) dans D~UQ, et m~me . . . . . . u O pour m assez grand. De m~me, on a pour tout i i-1

H2 (-1)r(i-j-1)ajf i-j-1 II <- C' T1'i-1. j=O Or l'op6rateur Qi est une combinaison h coefficients entiers, ind6pendants de i, d'op6rateurs du type I ] 0~ -~ I ] ~]t°, a~J aeJ off J c {1. . . . . r} est un sous-ensemble non vide. Pour tout J, la s6rie de terme g6n6ral i-1

( E (-1)r(i-j-I)ajf i-j-l) H O~ -11 H ~ o"[i]÷ a~ j=0 a~J aeJ converge donc dans D?UQ vers un op6rateur qui, d'apr~s (3.2.1), peut s'6crire sous la forme Qjl-[aeJOata, donc appartient h I. Par passage h la limite pour k ~ ~ , la relation (4.3.2.4) montre donc que P e I. (4.3.3) C o r o l l a i r e . Sous les hypotheses de (4.3.2) : (i) Le ~ Q - m o d u l e 9F~I(C0~rK) est cohdrent ; (ii) Sur U, la suite > O,

(4.3.3.1)

PiOiti + 2 PiOi ,

(4.3.3.2)

oh e(P) = P.(1/tl...tr), et ~/ est ddfini par d

V(Po . . . . . Pd ) = Potl""tr +

i=1

i=r+l

est exacte. La d6monstration est identique h celle de (4.2.3), en rempla~ant t 1 par tl... t r.

116

(4.3.4) C o r o l l a i r e . Soit Z une sous-vari@t4 lisse de c o d i m e n s i o n r d u n s X, et soit v l ' i m m e r s i o n de Y = X - Z d u n s X. Alors : (i) P o u r tout i, les ~ Q - m o d u l e s Rivt,(9~K et #f~zi((2~r~) sont cob@rents ; (ii) S i U est u n ouvert de #f sur lequel il existe des coordonndes locales t 1. . . . . t d telles que Z = V(t 1. . . . , t r) m o d u l o m, et si 31 . . . . . 3d sont les d@rivations correspondantes, la suite

(~Q)d

g

) ~}~Q

q~ >JCz#r((.Q~K)

> 0,

(4.3.4.1)

oft ~p(P) = P.(1/t r . .tr), et ~ est ddfini p a r

r d ~¢(P1. . . . . Pd ) = Y~ Pi ti + E Pi~i, i=l i=r+l

(4.3.4.2)

est exacte.

S i r = 1, l'6nonc6 a 6t6 montr6 en (4.2.2) et (4.2.3). Pour r > 2, la suite exacte longue des R i s p , et la nullit6 des yfti((qz~K) pour i ~ r m o n t r e n t que O~K = vt,(#~rx, 1" Rivt,(9~x=Opouri¢O,r1, e t R' r - i v,(2~ =Yf Z~ r ((9~rK). Rappelons d'abord le calcul de Yfztr((2~ra) sur un ouvert affine U v6rifiant les conditions de (ii) [4]. Posons Yi = D(ti) c X, et soient rio ...& : Yio r~ ... • Y& ¢--+ X les immersions ouvertes correspondantes. D'apr~s (4.1.3) et (4.1.4), on peut calculer les complexes IRvt, (ga~Ket IRF.~(9~g en utilisant le recouvrement de Y par les Yi, avec i = 1 . . . . . r : ]Rv~,(9~rx est repr6sent6 par le complexe commen~ant en degr~ 0 : d H sp*(v/?(9~2 > . ~ sp,(v~il (.9~K) >... > sp,(v~.., r (.PitK) > O, i=1 to
(PirQ

) H sp*(v/?('gKrK) ...... i=l

>

H sp*(V:oil(gKrK) to
>''"

) sp,(v~.., r (.P~K)

>

0.

Pour toute suite i o. . . . . ik, et tout ouvert affine U = SpfA c ~r, on a F(U, sp,(j:o...i~ OKrg)) = Ag[1/tio"" t&It, avec A K := A ® K, de sorte qu'on obtient un isomorphisme r Ag[1/tl"'" tr ]%/~Ag[1/tl'" ti'" tr]t > ~tzr(('~2" i=l D'apr6s (4.3.2), l'application DtUQ -+ A g [ 1 / t l . . . tr]? envoyant 1 sur 1 / t r . . t r e s t surjective, de sorte qu'il en est de m~me de ¢. De m~me, l'image de A g [ 1 / t r "ti" "tr]? duns A g [ 1 ] t l . . . t r It est le sous-Dtuo-module engendr6 par 1/t r "ti" "tr, soit encore le sous-D?vQ-module (D?uQti).lJtr..t r. La suite exacte (4.3.2.1) implique donc rassertion (ii).

117

5. I s o c r i s t a u x

surconvergents

associ~s aux caract~res

Nous d o n n e r o n s m a i n t e n a n t deux exemples de t ~ Q - m o d u l e s coh~rents d~finis par des isocristaux surconvergents non constants. Dans les cas les plus importants, ces isocristaux sont respectivement facteurs de l'image directe de l'isocristal constant par u n rev~tement de K u m m e r et d'Artin-Schreier [3], et sont A la base de la th~orie p - a d i q u e des sommes exponentielles d~velopp~e par Dwork, AdolphsonSperber, Robba, etc, dans laquelle ils correspondent aux caract~res multiplicatifs et additifs d'un corps fini.

5.1. I s o c r i s t a u x associ$s a u x c a r a c t ~ r e s m u l t i p l i c a t i f s ^1 On prend ici ~V= Zp, et le schema formel ~r est la droite projective formelle ]p~,. Nous d~signerons par t la coordonn~e canonique sur la droite affine formelle ~ = ~I

~I ^i - {oo} c ]p~., ~ ]a d~rivation par rapport ~ t ; sur l'ouvert ]p~. - {0}, nous utiliserons

la coordonn~e t ' = I/t, et nous noterons ~' la d6rivation par rapport ~ t'.

(5.1.1) On pose Y = IP~ - {0, oo}, et v : Y c-o X. Pour tout a ¢ Zp, on note X a risocristal s u r c o n v e r g e n t sur Y dont la r~alisation sur ~ g = ]plan est donn~e par vt~9~rn, m u n i de la connexion V d~finie sur A1Kan - {0} par V(1) = a t -l ®dt (cf. [3, (2.3)]). Rappelons que, si ]Fq est un corps fini de caract~ristique p, n un diviseur de q - 1, et X la puissance i-i~me du caract~re de Teichmttller sur le groupe pn(]Fq) des racines n-i~mes de l'unit~, X correspond dans la thSorie p - a d i q u e des sommes exponentielles/~ l'isocristal 9~x := X a , avec a z = i / n .

(5.1.2) P r o p o s i t i o n (Laumon). S u p p o s o n s que a e Z p soit non Liouville. Alors : (i) Le ~ Q - m o d u l e vt, J(a est eohdrent. (ii) S u p p o s o n s que a ~ IN. S u r ~ 1 , la suite de ~ Q - m o d u l e s

0

, v**X

0

(5.1.2.1)

ddfinie p a r qXP) = P.1, et v ( Q ) = Q ( t ~ - a), est exacte.

Lorsque a e Z, vt.9~'a est isomorphe en t a n t que ~ Q - m o d u l e ~ v,t (9~rn muni de la connexion triviale, par l'isomorphisme e n v o y a n t 1 sur t -~. On peut donc supposer que a ~ IN. Notons aussi qu'il suffit de prouver l'assertion (ii), car, avec la coordonn~e t', V e s t d~finie par V(1) = - a t "-1 ~ dt', et on dispose sur ~ 1 _ {0} d'une suite analogue/~ (5.1.2.1) obtenue en changeant a en - a , t en t', ~ en~'.

118

A1 Soit U = S p f A c A r un o u v e r t affine ; on a alors F(U, vt.Xa) = A ~ l / t ] t, et, p o u r tout n e ~ ,

~.t n = ntn-l+tnD.1 = ( n + a ) t n-1. Si P = ~ k ak ~[k], on a done

~o(P) :

a(a-1)...(a-k+l)

Z

k!

ak t-k =

k>_O

Z (k)a ak t-k, k>-O

of1 (~) e s t u n e n t i e r p - a d i q u e , p u i s q u e a e ~p. U n ~ l ~ m e n t f e A ~ l / t ] t p e u t s'~crire sous la f o r m e f = ~ k bk t-k, avec b k • A K e t ]lbk]] _< c ok p o u r u n 7] < 1. C o m m e a ~ IN, on peut poser a k := ( ; ) - l b k. Soit 77' tel que 77 < 9 ' < 1. C o m m e a est non Liouville, il existe c ' > 0 tel que I(~)l -> c'9 'k p o u r t o u t k [8, l e m m e 3.1], de sorte que

[]akH <- c?]kl(~)[ -1 <_ ( c / c ' ) ( ~ / 9 ' ) k. L ' o p ~ r a t e u r P = ~ h ak ~[k] a p p a r t i e n t done ~ D~Q, d'o~ la surjectivit~ de ~o. Soit Q • D ~ Q u n o p ~ r a t e u r tel que Q(t~ - a) = 0. Si Q = ~ k > o bk~[k]' on a

Q(t~ - a) = ~ ((k - a)b k + k t b k _ l ) ~ [k],

(5.1.2.2)

k>_O

d'ofi la nullit~ des b k p a r r ~ c u r r e n c e s u r k, et l'injectivit~ de ~. I1 est clair que Ira(V) c Ker(~o). Soit P • Ker(~o), avec P = ~ k ak~[k]" O n a done

Z (;)ak t-k = O. k>_O

P o u r t o u t j > 0, posons J

bj = E (;)aktJ-k,

bj = [ ( ; ) ( j - a ) ] - l b j .

k=O

D ' a p r ~ s (4.2.1), il existe c', 77' < 1, tels que HbjH <- c'9 'j. Soit 9" tel q u e 9' < 9 ''2 < 1. C o m m e a e s t n o n Liouville, il existe c I tel que, p o u r t o u t j, on a i t ~ - a ] > el9 ''j. D ' a u t r e p a r t , il existe c 2 tel que [(~)[ > c29 ''k. On o b t i e n t done u n e m a j o r a t i o n de la f o r m e IIbj[[ <- c"(9"/~?"2~, qui p e r m e t de d~finir u n o p ~ r a t e u r Q • D~uQ en p o s a n t Q = ~ j bj~ [j]. O n a alors

Q(t~ - a) = ~ ((] - a)bj +jtbj_l)OO'] j>_o avec •

(] _ a ) b j + j t b j

= (j)a -1 s2 ( ; ) a k t J - k + ( j a - 1 ) - l J ( ] - a - 1 ) - l t k=O

j-1

2

(;)aktJ-k-1,

k=O

d'ofl (] - a)bj + j t b j _ 1 = aj, et Q(t~ - a) = P, si bien que I m ( ~ ) = Ker(¢).

119

R e m a r q u e . On observera que la condition introduite sur a est pr~cisSment celle qui assure la finitude du groupe de cohomologie rigide H l g ( U / K , Xa).

5.2. I s o c r i s t a u x a s s o c i d s a u x c a r a c t ~ r e s a d d i t i f s

Soient q = pS, 1/o -- W(IFq) l'anneau des vecteurs de Witt & coefficients dans IFq, K o = Frac(Vo), g = K0(ttp), off pp c Qp est le groupe des racines p-i~mes de l'unit~, iV l'anneau des entiers de K. On notera encore ~ la droite projective formelle ~ , t la coordonn~e canonique sur la droite affine formelle ~ = ~{~} c ~ , t ' = 1/t, ~ la d~rivation par rapport ~ t, ~' la d~rivation par rapport a t'.

(5.2.1) On pose ici Y = A~, et on note encore v l'immersion de Y dans X. Rappelons [3] que, s u i v a n t Dwork, on associe & un caract~re additif ~, de IFq u n ~l~ment ~ e ?/, v~rifiant ordp z = 1/(p - 1).

(5.2.1.1)

Au caract~re ~ correspond alors dans la thSorie p - a d i q u e des sommes exponentielles un F-isocristal s u r c o n v e r g e n t sur Y, not~ ~ , , ou encore ~=, dont la r~alisation sur ~FK=]plan est donn~e par v t (2£rK, muni de la connexion V d~finie sur A~.(an par V(1) = - ~ v d t . P o u r ~tablir la coherence sur ~ r Q de v~.~=, la seule propri~t~ de z que nous aurons utiliser sera la relation (5.2.1.1). Rappelons tout d'abord le lemme suivant :

(5.2.2) L e m m e .

S o i e n t A u n e K-alg~bre de Tate, II" ~ u n e n o r m e de B a n a c h s u r A ,

(a n) u n e s u i t e d ~ l d m e n t de A , et ~ c K u n dldment tel que ordpr~ = 1/(p - 1). A l o r s les d e u x c o n d i t i o n s s u i v a n t e s sont dquivalentes : (i) Il existe c e IR, et ~ < 1 tels que, p o u r tout n, on ait Ilan]l < c / 7 n ; (ii) Il existe c" e IR, et ~' < i tels que, p o u r tout k < n, on air Ilan k ! / rck H<-C' rl 'n.

Si k = ~ i aiP i e s t le d~veloppement de k e n base p, et a(k) = ~ a i, on a ordp k! = (k - a ( k ) ) / ( p - 1). P a r suite, ordp ( z k / k ! ) = a ( k ) / ( p - 1) > 0, d'ofi (ii) ~ (i). Inversement, on a a(k) <_ (p - 1)(1 + logpk) < (p - 1)(1 + logpn),

d'o~ ]k!/z k] < p n , et (i) ~ (ii).

120

(5.2.3) P r o p o s i t i o n . ordp(z) = 1/(p - 1), et Alors : (i) Le ~ Q - m o d u l e (ii) S u r l'ouvert ~

Avec les notations de (5.2.1), soient ~ e ~ un dldment tel que . ~ = v*(2~rg, m u n i de la connexion V telle que V(1) = - ~ ® dt. vf,_~ est eohdrent. - {0}, la suite de ~ Q - m o d u l e s ,0

o

ddfinie p a r (p(P) = P. 1, et ~(Q) = Q (t '~ ~ ' - z), est exacte. S u r ~ 1 = ~ _ {oo}, le t ~ Q - m o d u l e v t , ~ est coh6rent d'apr~s (3.1.4), et il est en fait facile de voir qu'on a un isomorphisme

I1 suffit donc de prouver l'assertion (ii). Soient U c ~ - {0} un ouvert affine, avec U = SpfA, et U' = U - {oo} ; on a donc F ( U ' , O ~ ) = A { 1 / t ] = A [ t } ; o n n o t e r a D tuQ = F ( U , ~ Q ),D ut, Q = F ( U ' , ~ ~"Q t ),etonaune inclusion D*UQ c Dfu, Q. Comme ~ = vt (2~rKen t a n t que (_9~.K-module, on a aussi F(U, v**.~) = A[1/t'] t c A{1/t ~} = F(U', v * . ~ ) . Observons d'abord que, dans DtU.Q, les d~rivations ~ et 2' sont li~es p a r la relation t'~' = - t& E n ~crivant, pour k > 1, (k + 1)t 'k+l ~,[k+l] = (k + 1)t'(t'kO '[k])O' = - (k + 1)t'(t'k~ "[k])t2D, on en d~duit, par r~currence sur k, les relations

t,ka,[k ~

k

=

(_1) ~ ~

k-1 tia[i] '

(5.2.3.2)

(f--lt) t'icY[i]"

(5.2.3.3)

i=1

k

tk~[~] = (-1)k E i=1

Comme la connexion de ~ la formule

est telle que 2.1 = - ~, donc ~[i1.1 = (-~)i/i!, on en d~duit

~,[k]. 1

=

(_1) k

k ~ ( k-1 i - 1 ) t ' - k - i (- z)i/i!, i=1

de sorte que, p o u r P = ~ k >_ 0 "~*k ~,[k] ~' ~= r~t ~JUQ'

on a k

q~(P) = Z (-1)kak ( Z k>_O

(f--:) t ' - k - i ( - ~ ) i / i l ) "

(5.2.3.4)

i=l

Soit alors f e F(U, v * . ~ ) = A[1/t'] t. Grace au l e m m e (5.2.2), on p e u t ~crire f sous la forme

121

,f = ~ zkak/k!t'k, k>_O la suite d'~l~ments a k e A ~tant telle que IlakH <_c~ k, avec ~ < 1. Consid~rons alors l'op@rateur k

P = ~ (-1)kak t~'~[k] = ao + i~ ak ( ~ k>O

k~l =

(i-1)k-1t,i~,[i])

i=1

ao + ~ ( ~ k-1 t,i o,[i]. ( i - 1 ) ak) i>_l k>_i

C e t t e derni~re expression m o n t r e que P e D[tQ, et la p r e m i e r e m o n t r e que P.1 = f . P a r suite, (pest surjectif. P o u r p r o u v e r l'injectivit~ de V:, il suffit de m o n t r e r que la m u l t i p l i c a t i o n & droite p a r t'2~ ' - z = - ( 2 + ~r) est injective dans Dtu, Q. Or tout 616ment Q de Dtu, o p e u t s'6crire c o m m e s o m m e d ' u n e s6rie ~k_>0 bk ~k], o~i les b k f o r m e n t u n e s u i t e d'616m e n t s de A{1/t] telle qu'il existe c e IR, r; < 1, avec Hbkl[ < c~]k. On a alors Q(~ + ~) = ( ~ b~[k])(~ + ~) :: ~ (kbk_i + Zbk)~[k] , k_>O k_>O de sorte que la relation Q(O + z) = 0 e n t r a i n e que, pour t o u t k, b o = (-z)kbk/k!. O n d6duit alors de (5.2.2) la nullit6 de I[boll, donc de t o u s l e s b k. Soit I = I m ( v ) l'id6al & gauche e~gendr6 dans DtUQ p a r t'2O ' - z ; il est clair que I c Ker(~o). P o u r p r o u v e r :['inclusion r6ciproque, nous allons d'abord o b s e r v e r que tout o p 6 r a t e u r P = ~k_>0 ak O,[k] e D b Q est congru modulo I & un o p 6 r a t e u r de la forme P' = ~k>_O Ck t'k~'[k], o~i les % e A scnt tels qu'il existe c e JR, ~ < 1, avec IIckl[ < c~ k. Pour cela, on m o n t r e que, d a n s D~UQ, on p e u t 6crire P = P ' + P", avec P ' du type voulu et P" e / , en p o s a n t :

p' = ~ k!ak~'['~]~[k]/(--~) k, k>_O

p,, = ~ ak~'[k](1 _ k!~[kl/(-~)k). k>_l

En effet, on a p o u r t o u t k > 1, ~,[k] ~[k] = ~},[k]t,kt~[k] = (-1)k ~

( ki --ll ]~ ~,[k]t,k+i~,[i],

l<_i<_k d'o(~ r o n d@duit que b,[k]~[k] est u n opOrateur de la forme ~,[kl~)[k] =

~

tbkt,J~,[j] '

l~j<_2k avec ajk e ~ . On p e u t alors ~crire, en p o s a n t a~ = k ! a k / ( - z ) k,

P" := ao + j>_l : Z ( 2k>_j ~ a jk a'~t t,1 "j~'[j] v , ce qui m o n t r e , compte t e n u de (5.2.2), que P ' est un o p ~ r a t e u r de DtuQ de la forme voulue. D ' a u t r e part, on a pour to~at k > 1

122

k-1

1 -k!O[k]/(-z) k = 1 - ( - O I z ) k = ( ~

(-0/z)i)(l+0/z),

i=0 de sorte que, pour m o n t r e r que P" e I, il suffit de prouver que la s6rie k-1 Q- = ~ a k ~ , [ k ] ( ~ (_O/~)i) = ~ ak(i!/(_z)i)o,[kl~[i] k>_l i=O O<_i
~,[k] ~[i] =

2 °tjkit'J-k+i~'[J], l<_j<_k+i

o~ ajk i c Z, et ajk i = 0 p o u r j < k - i. On obtient donc Q " = ~,( ~

ajki(i!/(-z)i)akt'J-k+i)o 'b].

j>-i i,:k j<-k+i

D'apr~s (5.2.2), il existe c" e IR, et ~' < i tels que, pour tous i _
H(i!/zi)akH < c'~ 'k. Comme, dans la sommation d6fic.issant le coefficient de 0'[J], on a j < k + i < 2k, il en r6sulte que Q" ~ DtuQ. I1 reste alors & d~montrer que si un op6rateur de la forme P = ~k_>o ak t'k~'[k], avec UakU < c~ k, ~ < 1, est dans Ker(~), il peut s'6crire sous la forme Q(t'20 "- z), avec Q e DtuQ. D'apr~s (5.2.3.4), on a dans A{1/t'} la relation k 2 ( - 1 ) k a k ( 2 ( ik-1 - 1 ) t,_ i (-z)i/i!) = O. (5.2.3.5) k>O i=l Posons alors c o = ao, et, pour i > 1,

c i : ((-~)i/i!)~(-1)kak(k:~), k>_i

de sorte que la relation (5.2.3.5) s'6crit Ci t'-i = O. i>_O

(5.2.3.6)

On observera que, pour tout i, [c~] < c Hi. Si on pose, pour t o u t j _>0,

J

bj = 2 c i t 'j-i, i=o

il r6sulte de (4.2.1) qu'il existe c' e IR, et 77' < 1, tels que l'on ait [bj[ <_c'~ 'j pour t o u t j . D'autre part, on p e u t grace & (5.2.3.2) 6crire dans Du, o 1 o p e r a t e u r P sous la fbrme k P = ~, ak t'k~'[k] = ao+ ~- ( - 1 ) k a k ( ~ (i-1)k-1ti~[i]) k>-O k>l i=1

123

k

E (-1)kak(E ( ik-1 - 1 ) ti(~[il - (-It)i/i!)), kkl

i=l

la deuxi~me ~galit~ r~sultant de (5.2.3.5). Comme on a i-1 ~[i] (-z)i /i ! = ((-z)i/i!)

(i~(-O/g).")(l+

-

_

0/~1,

j=O il suffit de m o n t r e r que l'op~rateur k i-1 Q = ~ ( - 1 ) k a k ( ~ (f-_l) ti((-z)i/i!)(~,(-O/~)J)) k>_l i=l j=0

appartient ~ DtUQ. Or on peut encore ~crire Q sous la forme

Q = ~ ( ~ ((_ ~)i/i!)(~ (-1)kak (k-~))t,_i)(_~/z)j, j>O i>j

k>_i

c'est & dire

Q = ~ ( ~ c i t ' - i ) ( - ~ / z ) j. j>-O i>j

Compte t e n u de (5.2.3.6), on a donc aussi

Q = _ ~(~cit'-~i)(-~/z)J. j>-O i<j

P o u r j _> 1, on a d'aprbs (5.2.3.3) (_~lrr)j = (]!ll~J)t, j ~ ( jh--1l ) t'h ~ ,[hi h=l

On en d~duit donc

Q -- -co- ~ (~ c~t J ,)o~,,~J)(~ (i -1) t h~ ~) j>_l i<_j

h=l

= "-Co - Z ( E (Jh-11)(J!/nJ)bj) t'ho'[h]" h>_lj>_h

D'apr~s l'estimation vue plus h a u t pour ]IbjHet le lemme (5.2.2), il en r6sulte que Q e Dtuo, ce qui ach~ve la d6monstration. R e m a r q u e . Si on prend un ~l~ment zr tel qu.e ordpz < 1/(p - 1), la connexion correspondante sur ~ ne v~rifie plus la condition de convergence (3.0.1.1), de sorte que s p . ~ n'a plus de structure de ~ Q - m o d u l e . D'autre part, si ordpz > 1/(p - 1), le rayon de convergence de e x p z t est > 1, et exp~:t d~finit un isomorphisme . ~ = Ox/g.

124

Bibliographie [EGA] A. Grothendieck (avec la collaboration de J. Dieudonn6), Elements de G~om6trie Alg~brique, Publ. Math. I.H.E.S. 4, 8, 11, 17, 20, 24, 28, 32. [SGA4] M. Artin, A. Grothendieck, J.-L. Verdier, Thdorie des topos et cohomoiogie ~tale des schdmas, Lecture Notes in Math. 269, 270,305, Springer-Verlag (1972). [1] P. Berthelot, Cohomologie cristalline des schemas de caract~ristique p > 0, Lecture Notes in Math. 407, Springer Verlag (1974). [2] P. Berthelot, Gdomdtrie rigide et cohomologie des vari~t~s algdbriques de caract~ristique p, Journ6es d'analyse p-adique (1982), in Introduction aux cohomologies p-adiques, Bull. Soc. Math. France, M6moire 23, p. 7-32 (1986). [3] P. Berthelot, Cohomologie rigide et thtorie de Dwork : le cas des sommes exponentielles, Ast~risque 119-120, p. 17-49 (1984). [4] P. Berthelot, Cohomologie rigide et cohomologie rigide h support propre, ~ paraltre dans Ast6risque. [5] P. Berthelot, ~-modules cohdrents, en preparation. [6] A. Borel et at., Algebraic D-modules, Perspectives in Math. 2, Academic Press (1987). [7] S. Bosch, U. Gtintzer, R. Remmert, Non-archimedean analysis, Grundlehren des math. Wissenschaften 261, Springer-Verlag (1984). [8] G. Christol, Un thdor~me de transfert pour les disques singuliers r~guliers, As~risque 119-120, p. 151-168 (1984). [9] A. Grothendieck, On the de Rham cohomology of algebraic varieties, Publ. Math. I.H.E.S. 29, p. 351-359 (1966). [10] A. Grothendieck, Crystals and the de Rham cohomology of schemes, in Dix exposes sur la cohomologie des schdmas, p. 306-358, North-Holland (1968). [11] R. Hartshorne, Residues and Duality, Lecture Notes in Math. 20, Springer Verlag (1966). [12] M. Kashiwara, Faisceaux constructibles et syst~mes holonomes d'dquations aux ddrivdes partielles ~ points singuliers r~guliers, S6m. Goulaouic-Schwarz, 1979-80, exp. 19, Ecole Polytechnique (1981). [13] M. Kashiwara, The Riemann-Hilbert problem for holonomic systems, Publ. R.I.M.S. 437, Kyoto University (1983). [14] Z. Mebkhout, Cohomologie locale des espaces analytiques complexes, Th~se Universit6 Paris VII (1979). [15] Z. Mebkhout, Une dquivalence de catdgories, Comp. Math. 51, p. 51-62 (1984). [16] Z. Mebkhout, Une autre dquivalence de catdgories, Comp. Math. 51, p. 63-88 (1984). [17] Z. Mebkhout, Le formalisme des six opdrations de Grothendieck pour les ~X-mOdules cohdrents, Travaux en cours 35, Hermarm (1989). [18] A. Ogus, The convergent topos in characteristic p, Grothendieck Festschrift, Progress in Math., Birkh~iuser (1990).

EXTENSIONS DE D-MODULES ET GROUPES DE GALOIS DIFFERENTIELS par D. BERTRAND Univ. Paris Vl, Maths, T. 46, 4, Place Jussieu, F-75252 Paris Cedex 05. La theorie des equations differentielles lineaires & second membre peut 6tre vue comme un analogue differentiel de la theorie de Kummer. Pour tout entier n donne, celle-ci decrit les extensions d'un corps K engendrees par les racines n-iemes des elements de K ; elle repose sur une bonne connaissance de I'extension cyclotomique d'ordre n de K . Celle-l& etudie les equations de la forme Ly = g , 05 L e s t un operateur differentiel donne; on sait (methode de variations des constantes) le rele qu'y joue I'equation homog~ne correspondante. Inspire par le travail de Bashmakov et de Ribet [23] en theorie de Kummer sur les groupes algebriques, je montre ici que cette analogie formelle se transporte sans peine & I'etude des groupes de Galois non r~ductifs associes & ces deux situations. La premiere partie traite de la notion d'extension de D-modules, qui englobe celle d'equation & second membre, et dont on decrit les proprietes elementaires. On determine dans la deuxieme partie I'image des representations galoisiennes definies par de telles extensions. La troisi~me partie passe en revue quelques applications, en particulier aux equations hypergeometriques et & des questions de transcendance. Je remercie R. Coleman et N. Katz, dont les travaux [11] et [17] m'ont fourni le cadre de la presentation qui suit, F. Beukers, qui m'en a indique le lien avec la theorie de Shidlovsky, ainsi que Z. Mebkhout et J-P. Ramis pour d'utiles discussions.

{}1. Extensions Soient

de D-modules. X

un ouvert propre de la droite projective

P I ( C ) , et O X

126 son alg6bre affine. La

Ox-alg~bre

est engendr~e par une d~rivation

D = DX oq

des op6rateurs diff~rentiels sur

de O X . Par

X

D-module, on entend ici un

fibr6 ~ connexion

V sur X . Vu la forme de X , il revient au meme de Ox-module libre de type fini V ~ muni d'une action ~q(V) de c3

consid6rer un v~rifiantles sur

propri6t~s usuelles des d~rivations.

O x , o~(V)

Dans une base

est repr~sent6e par un op~rateur

matrice c a r r i e

& coefficients

dans

OX

o~ + A(B ) , o i

On notera

B

de

V~

A(B ) est une

H*(V) les groupes de

cohomologie du complexe de de Rham alg6brique 0 --> V---> V -> 0 d~fini par 8 ( V ) - En d'autres termes, HO(v) est form~ des sections horizontales (globales) de autres

V , tandis que

groupes

dimensions

sont

h*(V)

nuls).

HI(v) Ce

sont

des

espaces

de 8(V)

vectoriels

(les

sur C ,

de

finies.

S o i e n t maintenant D-module

s'identifie au co-noyau

V , V'

deux

D - m o d u l e s . On note

V*

le

Hom(V,Ox) (muni de la connexion duale) , de sorte qu'un morphisme

de V vers V' est une section horizontale (globale) du D-module V'®V* (voir par exemple [13], [2]) . Une extension de V par V' est une suite exacte 0 -~ V' -~ E --> V -~ 0 darts ta cat~gorie des D-modules; par abus de language, on la notera encore E Le proc6d6 usuel de Baer (ou des t e c h n i q u e s plus fonctorielles, voir [22]), permet de munir t'ensemble ExtD(V,V' ) des classes d'isomorphismes v6rifie Lemme

d'extensions de

V

par

V'd'une

structure de groupe, qui

: 1 (Coleman

[11]):

les groupes

I--XtD(V,V' )

et H I ( v ' ® v *)

sont

canoniquement isomorphes. D6monstration:

soit

E

un (repr6sentant d'un) 616ment de

ExtD(V,V' )

Puisque X est affine, la suite exacte de fibr6s associ6e & I'extension E ~ est scindabte. Choisissons-en une section s : V ~ -~ E ~ . Alors, l'application O~(E)S - soq(V) est un O x - h o m o m o r p h i s m e dans H I ( V ' ® v * ) L'application

de

ne d~pend pas du choix de

V ~ vers V '~ , dont la classe s , ni du repr~sentant de

~E E .

: EXtD(V,V') -~ H I(V'®V*) : E-~ ~(E) = ~E d6finit I'isomorphisme escompt6, c o m m e on le v6rifie ais6ment bases convenables -et si I'on convient d'~crire les 616ments de colonne -, la transpos6e de la matrice d'un repr6sentant celle de

de

{E

(dans des (Ox) n en

apparaTt dans

~(E) sous la forme d'un rectangle en position inf6rieure gauche ). On peut pr6ciser cet 6nonc~ lorsque

connexion canonique), universelle du D-module

V'

--

Ox

(muni de sa

en c o n v e n a n t d ' a p p e l e r e x t e n s i o n v e c t o r i e l l e V la classe d'isomorphisme des extensions E de V

127 par une somme directe finie

U

de copies du

D-module

O X , telles toute

extension 0 -~ U' ~ E' -~ V -~ 0 de ce type se d~duise de E par image directe sous un unique morphisme de U dans U' (voir [12] pour une g6n~ralisation de cette d6finition, et [15] pour un analogue dans la cat~gorie des modules de Drin'feld). Une telle extension existe : c'est I'extension de V par U = H o m ( H I ( V * ) , O X ) qui correspond, dans I'isomorphisme fourni par le lemme 1, & I'~l~ment identit6 de H I ( U ® V * ) = Hom(H I ( V * ) , H I(V*)) . Le rang h I(V*) de U est donn~ par le lemme suivant, o5 S d6signe I'ensemble des points & I'infini de X , et Irr(V) la somme des irr6gularit6s de V aux diff6rents points de S .

Lemme 2 : soit V un D-module de rang h I(V*) = ( c a r d ( S ) - 2) n + En particutier, si V e s t

n . Alors,

Irr(V) + h0(V *)

un module irr~ductible non isomorphe ~

V = O x ) , il existe exactement

(card(S) - 2) n + Irr(V)

extensions de V p a r O X lin&airement ind~pendantes sur

(resp. C .

D ~ m o n s t r ~ t i o n : le premier ~nonc6 resulte imm~diatement comparaison de Deligne ([13], p. 111; voir aussi [17]'.) : h0(V) - h I ( V ) - -

O X (resp. si card(S) - 1)

du th6or~me de

n x(X) - Irr(V) ,

appliqu6e au D-module V* , dont l'irr6gularit~ vaut celle de V . Pour le second, noter que si V est irr6ductible, V* I'est aussi, et n'admet donc pas de section horizontale globale, & moins d'6tre isomorphe & O X (de fa~on g~n~rale, tout morphisme non nul entre deux D-modules irr~ductibles est un isomorphisme, les fonctions rationnelles dont la d ~ r i v ~ e Iogarithmique appartient & O x 6tant des unit6 de O x ). Renlarque 1 : Une extension E de V par V' fournit par dualit~ une extension E* de V'* par V* . Le lemme 2 permet donc ~galement de d~crire les extensions de O X par V • eNes sont param~tr6es par H I(V) . Pour les applications, de type rationnel, que nous avons en vue, il s'av~re commode de travailler dans la cat6gorie des D-modules, o5 D = D K = K[o~] corps

d~signe l'anneau des op~rateurs diff~rentiels & coefficients dans le K = C ( P 1 ) des fonctions rationnelles sur X , et o5 les notations

pr~c~dentes (et suivantes) s'~tendent de fa~;on naturelle . Tout D-module V d~finit par extensions des scalaires un D-module, qu'on notera V K , ou encore V . La remarque concluant la d~monstration pr~c~dente, jointe & un calcul local aux diff~rentes places de X , montre en effet que si deux D-modules V, V' sont isomorphes en tant que D-modules, il le sont d~j& en tant que D-modules. De plus :

Lemme

3 : - / ' application naturelle

(reals non surjective) .

: ExtD(V,V' ) ~

ExtD(V,V')

est injective

128 : posant W = V'®V* , on se ram~ne & montrer que I'application H I ( w ) darts H I ( W K ) = WK/oq(w)W K est injective, c'est-&-dire

naturelle de qu'un 616ment

f

de

WK

dont I'image par

oq(W) appartient &

W

est

n~cessairement dans W . Cela se v6rifie par un calcul local (noter qu'en toute place de O x , de param6tre local t , le g6n6rateur ~3 de D est 6gal & d/dt & une unit~ locale pros). Explicitons ces 6nonc6s dans le cadre des equations diff~rentielles. Si M est un 616ment unitaire de D -- OX[~ ] , le Ox-module D/DM est naturellement muni d'une action (& gauche) de 8 , qui en fait un D-module V(M) , et d'une base canonique B(M), form~e par les classes des premieres puissances _> O de oq . Le dual de V(M) est isomorphe & D/DM* , o5 M* d~signe I'op~rateur diff6rentiel adjoint de M (voir [16], 1.5, ainsi que [17], 2.9 et [18] pour une pr6sentation plus intrins~que), et les solutions de M s'identifient aux vecteurs horizontaux de ce dual De m6me, I'application t(o ..... g) ~ g 6tablit un C-isomorphisme de Hi(V(M)) sur le conoyau Ox/M*O X de M* dans O X . Soit alors M -- L'L un op~rateur d6compos~, o~ L , L' 616ments unitaires de D d'ordres > 0 . II d6finit une extension : E(L,L') : 0 - , V ( L ' ) ~ V ( M ) -~V(L) -~0 ,

sont des

o~ la secende fl~che d6signe la multiplication & droite par L , et la troisi6me la surjection canonique. L'~l~ment de ExtD(V(L),V(L')) associ~ correspond & Ia classe darts H I(Hom(V(L),V(L')))

du

Ox-homomorphisme

~L,L' qui envoie

le dernier vecteur de B(L) sur le premier de B(L') et annule les autres vecteurs de B(L) (pour s'en convaincre, choisir {B(L')L, B(L)} pour base de M ). Lorsque L' -- Gq, le compos6 E des isomorphismes EXtD(V(L),Ox) ~ H I(v(L*)) -~ O X / L O X

fournit le lien annonc~ darts I'introduction avec les ~quations &

second membre. Plus pr6cis6ment, on a : Lernme 4 : i) si

g

est un #l#ment non nul de O x , le D-module

est isomorphe ~ un #l#ment Eg(L) de cide avec la classe de g dans

ii) supposons que Alors, I'extension

E(L,oq-g-l~)g)

ExtD(V(L),Ox) dont I'image par ~ coin-

Ox/LO X . V(L) soit irr#ductible et non isomorphe ~

E(L,~)) est triviafe dans

si

~L est divisible ~ droite dans

o#

f appartient ~ O X .

EXtD(V(L),Ox) si et

D par un op~rateur de la

Ox .

seulement

forme oq.f-lGqf,

D 6 m o n s t r a t i o n : i) la multiplication & droite par g d6finit un isomorphisme de D/D(o~-g'l~g) sur D/Dcq= ( O x ) K , et I'imagedirecte E'g(L)de la D-extension

E(L, oq-g'loqg ) de

V(L)

par O X

sous cet isomorphisme est

donn6e dans

H I(HomK(VK(L),K ) par le compos~ par la multiplication & gauche

129 par

g

du

K-homomorphisme

~L, o~-o~g/g de VK(L) dans

compos6 provient d'un morphisme de d'une

D-extension

de cisaillement repr6sentative de si t'6quation

V(L) ~ dans O X , E'g(L)

K.

Comme ce

provient en fait

Eg(L) . (En terme concret, on effectuera la transformation

de matrice

diag(g,

~E(L,~-Sg/g)

Idn)

sur la transpos6e de la matrice

dans la base {L,B(L)} ) .

ii) D'apr~s ce qui pr~c~.de, E(L,~) est triviale si et seulement Ly = 1 admet une solution f dans O x . Alors, ~)L appartient &

I'id~al & gauche engendr6 dans D par son annulateur minimal ~-f-lo~f . Invers6ment, si cette condition est v6rifi~e, f est une solution de ~L et Lf est une constante. De I'hypoth6se faite sur V(L) , on d~duit comme au lemme 2 que cette constante n'est pas nulle, et 1'6quation Lf = 1 est bien r~soluble dans O x . R e m a r q u e 2 : On prendra garde au fait qu'un ~l~ment L de D peut 6tre irr~ductible dans D sans 1'6tre dans D (il n'y a pas ici de lemme de Gauss !). C'est par exemple le cas de I'op~rateur hyperg~om~trique sur X = P1- {O,l,oo} associ6, pour des param~tres a et c g6n~riques, et un entier n > 1 , & la fonction z l - C 2 F l ( l + a - c , l - n - c , 2 - c ; z ) . En revanche, V(L) est irr6ductible si et seulement si

VK(L )

l'est (l'intersection avec V

d'un D-sous-module

W

de

V K est un r6seau W stable sous ~(V) ) Convenons enfin d'appeler ~quivalents deux op~rateurs unitaires L , N de D tels les D-modules VK(L ), VK(N ) soient isomorphes. Le crit6re de trivialit6 ci-dessus se g~n~ralise de la fa~on suivante :

Lemme 5 " s o i e n t L , L' deux 6,!~ments unitaires de D tels qu'aucun sousmodule non nul de V(L) ne soit isomorphe ~ un sous-module de V(L') . Alors, I'extension E(L,L') est triviale si et seulement s'il existe deux ~l~ments N , N' de D , respectivement dquivalents ~ L , L' , tels que L'L = N N ' . D~monstr~t!Qn

: si

E = E(L,L')

est triviale,

E admet

un sous-D-module

V'

isomorphe & V(L') , et tel que E/V' soit un D-module isomorphe & V(L) . On conclut en ~tendant les scalaires & K , et en notant comme dans [24], 2.3, que I'image de 1 dans E/V' en est un vecteur cyclique. Invers~ment, les suites exactes de D-modules fournies par les d~compositions L'L , NN' permettent de construire, comme dans [19] et [24], 2.2, un D - h o m o m o r p h i s m e du sous-module VK(N ) de E K dans VK(L ) , qui ne peut ~viter d'6tre injectif (donc bijectif, et de fournir alors une section de ~l~ment non nul de

VK(L' ) . I1 existe alors un

E K ) que s'il contient un

D-morphisme,

donc aussi un

D-morphisme, non trivial d'un sous-module de V(L) darts V(L') , ce que notre hypoth~se interdit. Par consequent, E(L,L') est triviale dans ExtD(V(L),V(L')), et donc dans ExtD(V(L),V(L')) .

130

Remarque 3 : la premiere partie du raisonnement ci-dessus entrafne la version relative suivante du lemme du vecteur cyclique : si L (resp. L' ) est donn6, les 616ments de E D ( V K ( L ), VK(L')) sont t o u s repr~sent~s par des extensions de la forme E(N,N') , o~J N (resp. N' ) parcourt I'ensemble des 61~ments de D ~quivatents & L (resp. L' ) . On peut donc se limiter dans ces questions #, l'6tude des op6rateurs d6compos6s dans D .

§2. Representations

galoisiennes.

On reprend les notations du paragraphe pr6c6dent. On fixe de plus une extension diff6rentielle minimale K' de K , de corps de constantes C , dans laquelle t o u s l e s 616ments de D admettent un syst6me fondamental de solutions, et on note G = Gal;)(K'/K) le groupe de Galois diff6rentiel de cette extension. C'est un groupe pro-alg~brique, dont on n'aura en fait & consid6rer que des quotients de dimension finie (et en nombre fini) du type suivant. Soit V est un D-module de rang n . On note V °q le C - e s p a c e vectoriel de dimension n de V(K') form~ des vecteurs horizontaux de V Q K ' . L'extension minimale K V de K o~ sont d~finis les ~16ments de V °~ est une extension

de Picard-Vessiot,

dont

le groupe

de Galois

Galoq(Kv/K) s'identifie & I'image de la repr6sentation on d6signera par

H V = Galoq(K'/Kv)

diff~rentiel

naturelle de

G

GV = sur

V ~) ;

le noyau de cette repr6sentation.

Les

g r o u p e s de c o h o m o l o g i e que nous c o n s i d ~ r e r o n s seront relatifs & ces repr6sentations, et & des cochaines continues. Le dual V* de V fournit la m6me extension de Picard-Vessiot K V que V , et une repr6sentation de G V sur (V*) °q = (V~))* duale de celle de V . De m6me, pour un second D-module V' , la representation (V'®V*) °q = V'~3®V °q* de G annule H V c~ H V, -- HV@ V, , et KV,®V* est contenu dans le compositum

K v . K V, = KV~)V, des corps

KV

et K V , .

sentation

Soient alors E une extension de V par V' . Comme la repreE °q de G est une extension de V °~ par V '°q , le corps K E contient

K V ~)V'

Le

groupe

de

Galois

diff6rentiel

GU(E) = Galo~(KE/Kv@v,) de l'extension

K E / K v . K V,

s'identifie & I'image de

est nul si

H V c~ H V,

dans

E

est triviale. De fa£~on g~n~rale, il

Aut(E °~) , et donc & un sous-espace

vectoriel de Hom(V°q,V *~) . C'est cette image qu'on cherche ici & d~crire. Pour all6ger, on d6signe par W le D-module V'®V* , par ~ les projections canoniques E -~ V e t E °~ ~ V °q , et par GSS(E) = G V e V, ,, Galoq(Kvev,/K ) le

131 quotient de G E par GU(E). Soit g un repr6sentant dans W de I'image de E dans H I(W) . Trouver le corps de d6finition K E d'une base de E °q (i.e. d'un "syst6me fondamental" de vecteurs horizontaux de E ) revient & rechercher la plus petite extension diff6rentielle de KVE)V, oQ I'extension 0 --->V' ®Kv@ v, --> E ®Kv@ v, --> V ® K v ~ V, --> 0 se trivialise; en effet, V ®KvE)v,

et V' ®Kv@ V, sont d6j&, par d6finition de

KV@ V, , isomorphes & des sommes directes de copies du trivial

KV~ V, . C'est donc le corps

une solution

f

KVSv,®D - module

K E = KV®V,(f ) engendr6 sur

KV® V, par

de 1'6quation oqw(f) = g .

Comme en th~orie de Kummer (voir [23], inspir6), ce corps est bien ind6pendant de la solutions different d'un 616ment de W °q , et K w de W e s t contenue dans KV@V,. Dans

dont ce qui suit est fortement solution f choisie, puisque deux que I'extension de Picard-Vessiot ces conditions, I'homomorphisme

VE : HV~V' --> W°q : c /--> WE(~) = off- f ne d6pend que de E . II fournit I'injection canonique, mentionn6e plus haut, de son image GU(E) dans Hom(V°q,V'°) . Quant & I'application C-lin6aire W : EXtD(V,V') --> Hom(Hv~v',W°q) : E /--> w(E) -- WE , on I'obtient par construction en composant I'isomorphisme ~ avec I'application : H I(W) ---> H 1 (G,W ~)) qui, & la classe de g , associe la classe de cohomologie de I'homomorphisme crois6 : ~ /-> c g - g de G dans W °q, suivi de la restriction H I ( G , w Gq) ~ H I ( H v ~ ) v , , W ° ) ) de G & HV@ V, Quitte & remplacer K' par une cl6ture diff6rentielle de K , on peut encore d6crire W de la fa~on suivante. La suite exacte de G-modules: 0 ~W~)--> W(K')~W(K')-> 0, o5 la troisi6me fl6che est ~)(V) , donne naissance & un op6rateur cobord : W(K')G/~(w)(W(K')G ) = H 1 (WK) --> H 1 (G,W c3) , et W se d6duit de cette fl6che par les homomorphismes d6crits aux lemmes 1 et 3 d'une part, et par la suite d'inflation-restriction : 0 --> H 1(GSS(E), W cq) -~ H 1 (G,W °q) --> HomGss(E)(HvE)v,,WCq ) d'autre part. En particulier, I'image stable sous I'action de

Gw

GU(E) de WE est un sous-espace de W ~)

, et I'application

W

est injective d~s que

H I(GV@V,, W ct) est nul. Le lemme suivant permet, dans de nombreux cas, de v6rifier cette derni~re hypoth~se, et joue ainsi le r~le du th6or~me 90 de Hilbert (voir [26] pour un analogue p-adique plus souple). Lemme 6 " s o i t G un groupe alg#brique r#ductif sur representation rationnelle de G . Alors, H 1 (G,V) = 0 .

C , et

V une

132 D6monstratiQn : la cohomologie de G s'injectant darts celle de sa composantre neutre, on peut supposer G connexe. Par r6currence sur la dimension de V , on se ram~ne au cas o~ V est une representation triviale ou irr~ductible. Dans le premier cas , on note que G n'admet pas de quotient unipotent. Dans le second, le centre de G est form~ d'homoth~ties (Schur). S'il n'est pas r~duit & I'identit~, le lemme de Sah montre que H I(G,V) est nul. Sinon, G , et son alg~bre de Lie g , sont semi-simples. En particulier (voir [8] §6, exercice l.b), Hl ( g , v ) = 0 Mais la cohomologie (par cochaines continues) de G s'injecte darts darts la cohomologie de g , et H I(G,V) est encore nul. (Lorsque G est semi-simple, on peut 6galement ~tablir ce lemme en consid6rant le produit semi-direct de G par V associ6 & la representation V , et en notant que d'apr~s le th6or~me de Levi- Malcev - voir [8], 6.8, corollaire 3; [28], th6or6me 3.1.18 -, il n'admet qu'une classe de conjugaisons de sections de sa projection sur G .) Les remarques pr6c6dentes ram6nent 1'6tude de GU(L) & celle des sous-D-modules de W . Plut6t qu'un ~nonc6 g~n~ral, nous donnons maintenant quelques exemples simples d'appNcations de ce principe. Th6oreme

1 : s o i e n t V , V' deux D-modules tels que V ® V ' * soit D-module irr#ductible, et soit E une extension non triviale de V p a r Alors, Galcq(KE/Kv~v, ) est isomorphe ~ Hom(V~,V '°q)

un

V'.

D6monstration • puisque W = V®V'* est irr6ductible, V , V' et les repr6sentations W c3 , V8 , V '°~ le sont 6galement. Par cons6quent, le groupe G V ¢ V , , dont V@V' est une representation fid61e compl~tement r6ductible , est r~ductif, et les remarques pr~c~dentes, jointes & I'hypoth6se faite sur I'extension E , entrai'nent que I'homomorphisme "PE n'est pas nul. Son image GU(E) est ainsi un sous-Gw-module de W °~ non nul, et remplit donc tout W ~3. II est facile d'6tendre cette proposition & des situations semi-simples, en consid~rant les images directes ou inverses de E associ~es aux diff~rents facteurs de V e t de V' . Voici une illustration typique de cette d6marche. Th(~or~me 2 • soient V un D-module irr~ductible , et par une somme directe V ' = (Ox) r de copies de images directes de

E sous les projections de

E une extension de V

O X . On suppose que les

V' sur ses diff#rents facteurs

O X sont des #l#ments de ExtD(V,Ox) lin#airement ind#pendants sur C . Alors, Galo~(KE/Kv) est isomorphe ~ (V*°~)r. D6monstratiQn • soient =i les

, i = 1 ..... r ,

les projections en question (pour

D-modules, et pour leurs espaces de sections), et

La construction de I'homomorphisme

E i l'extension

~ 6tant fonctorielle, ~(Ei)

(=i),E. coi'ncide

133 avec HV

~iO~E , et I'injectivit~ de

~

entra?ne que ces

dans V *Gq sont lin6airement ind6pendants sur

I'image GU(E) de HV dans Mais puisque d'apr~s Schur)

C

Consid~rons alors

(V*°q)r par leur produit. C'en est un Gv-module.

V *{) est par hypoth6se

de (V*{)) r que s'il existe des

r homomorphismes de

Gv-irr6ductible,

GU(E) ne peut diff~rer

G V- endomorphismes (c'est & dire des scalaires,

a I ..... a r non tous nuls tels que al~(Er) + ...+ ar~'(E r) annule

H V . Ceci contredit l'ind~pendance lin~aire des En termes concrets, premi6re partie •

~(Ei).

on a d6montre,

avec le notations de la

C o r o l l a i r e : s o i t L un op#rateur diff6rentiel ~ coefficients dans rang n , et irr#ductible. Notons K L I'extension de Picard-Vessiot de K , et soient

fl ..... fr des #l~ments de

K' tels que

K, de L sur

Lf 1 ..... Lf r soient des

#l#ments de K lin#airement ind#pendants sur C modulo LK. Alors, le degr# de transcendance de I'extensk)n KL(c3ifj ; j = l ..... r, i=0 ..... n-l) de K L est #gal

,~ hr. Terminons par une version tordue du th6or6me 2 •

Th6or#.me

3 : s o i e n t V un D-module irr#ductible , L(1) . . . . . L(r) des Dmodules de rang 1 deux a deux non isomorphes, et E une extension de V par V' = L(1)e)... e ) L ( r ) . On suppose que I'image directe E(i) de E sous la facteurs

L(i) est un #l#ment non nul de

projection de

V' sur chacun des

ExtD(V,L(i)).

Alors, le groupe de Galois diff#rentiel du corps

KVe)L(1 )~... e) L(r) est isomorp~e a la somme directe des groupes

K E sur V*°~®L(i) °~ .

D6monstration :les representations de G associ~es aux D-modules Horn(V, L(i)) sont les produits de sa representation V *~) par des caract~res :Zi distincts. Elles sont donc irr6ductibles et deux & deux non isomorphes. Si la conclusion du th6or~me ~tait incorrecte, la projection du groupe de Galois 6tudi6 sur run des facteurs V*C3®L(i){) serait donc nulie, et l'extension E(i) correspondante serait triviale. Le lecteur averti rapprochera 1'6nonc~ du th6or~me 2 ou de son corollaire, et plus encore sa d~monstration, du th6or6me 1.2 de [23] . Les hypotheses de semi-simplicit~ jouent dans les deux cas un r61e fondamental. Mais Ribet ~tend dans [23] sa m6thode & des 1-motifs de type "doublement"mixtes. Cela correspond, dans la situation diff6rentielle, & supposer que V est lui-m~me une extension non triviale de D-modules semi-simples, et ron peut encore obtenir des analogues des r~sultats de [23] dans ce cas (voir la remarque 5 de la derni6re partie). En fait, le cadre d'6tude naturel des ~quations diff~rentielles semble 6tre celui des motifs & n poids

134 (voir par exemple [25]). Nous reviendrons sur c e s r e m a r q u e s ult6rieurement.

§3.

Applications.

Les 6nonces precedents permettent d'~tablir de facon unifi~e divers r~sultats (pour I'essentiel bien connus) d'ind6pendance alg~brique de fonctions classiques, en choisissant convenablement le D-module V . Nous en donnons ici quatre exemples. Les deux premiers concernent des ~quations fuchsiennes, et ne refl~tent doric que des propriM6s standard de la monodromie. Des singularit~s irr6guli~res apparaissent dans les deux derniers.

a) Le?a~ V-- O x . D'apr~s le lemme 2 , le dimension de rang

C-espace vectoriel

card(S) - 1 . Par ailleurs, le groupe

EXtD(OX,Ox)

est de

Ox*/C* est un Z-module libre

card(S) - 1 (consid~rer I'application R6s S : OX* ---> Hom(S,Z) : u / ~ {s/--> r~sidu en s de du/u} ,

et appliquer le th~or~me des r6sidus). Un lien direct entre ce groupe et cet espace vectoriel est fourni par l'application qui, & une unit~ u de O X , associe I'extension

Edu(o~)/u(8) de O X par O X (notations du lemme 4); celle-

ci d6finit en effet un homomorphisme de groupes X : Ox*/C* -~ EXtD(Ox,Ox), dont on v6rifie ais6ment te caract~re injectif. Mais mieux : Lemme 7 : l'homomorphisme ~ s'6tend en un homomorphisme injectif (donc un isomorphisme) ;L®I de (Ox*/C*) ®C dans ExtD(Ox,Ox) . [L'injectivit6 de d'~l~ments de

X exprime que les classes modulo 2i~Z des Iogarithmes OX* multiplicativement ind6pendants sont lin6airement

ind~pendantes sur Z , et le lemme 7 qu'elles le sont m~me sur C .] D~monstratiQrl : 1'6nonc~ revient & affirmer que I'image de (Ox*/C*) par

R6s S

est un r~seau du C-espace vectoriel qu'elle engendre dans Hom(S,C) , ce qui est clair. Dans I'esprit du point b) ci-dessous, on peut encore interpreter cet ~nonc~ de la fagon suivante . Soit X h I'espace analytique associ6 & X . La suite de cohomologie d~duite de la suite exacte de l'exponentielle sur X h fournit un isomorphisme du groupe ( O x h ) * / e x p ( O x h) darts H l ( X h , 2 i ~ z ) . En second lieu, l'intersection de OX* avec exp(Ox h) ind6finiment divisibles dans

est form~e d'~i6ments

OX* , d'apr~s (par exemple !) l'argument de

monodromie de Shafarevich-Manin (cf. [20], p.215), et est doric r~duite aux constantes, Enfin, comme V -- O X n'a que des singularit~s r~guli~res, H I ( x h , c ) = Hl(xh,(vh)°q) -- H I ( x h , v h) est isomorphe & H I ( v ) (cf. [22], §0) ,

135 donc & E x t D ( O x , O x ) H I(Xh,2i=z)®C

. II reste & joindre & ces remarques I'isomorphisme de

avec H l ( X h , c )

pour obtenir rapplication

X , et le lemme 7 .

En combinant le lemme 7 au th~or~me 2 (ou plutSt, la traduction de I'un au corollaire de I'autre), et en les ~tendant pour plus de g~n6ralit6 au cas des rev~tements finis de X , on a finalement 6tabli : T h ~ o r ~ m e 4 : soient C , telles que

61 ..... ~r des fonctions holomorphes sur un ouvert

de

exp(~.l) . . . . . exp(~r) soient des fonctions alg#briques multipli-

cativement ind6pendantes modulo

C* . Alors,

61 ..... ~r

sont alg#briquement

ind6pendantes sur C ( x ) . Bien st3r, il ne s'agit I& que d'un cas particulier du th6or6me d'Ax [2] sur I'analogue fonctionnel de la conjecture de Schanuel. Mais il est int~ressant de noter que I'essentiel de sa preuve s'est concentr6e au lemme 7, qu'on peut voir c o m m e un analogue fonctionnel de la conjecture de Leopoldt. b) Le cas des ~ouations de Picard-Fuoh$. Pour all6ger, nous nous limitons ici & d6crire la th~orie de Manin [20], telle que I'a r6cemment interpr6t6e Coleman [11], dans le cas d'un schema elliptique A sur X , qu'on suppose ne devenir constant sur aucun rev6tement fini de X (autrement dit, en notant encore A la courbe elliptique obtenue par extension des scalaires & K , I'invariant j(A/K) n'est pas constant ) . La connexion de Gauss-Manin munit alors le fibr6 H l d R ( A / X ) , de rang 2 sur X , d'une structure de D-module irr6ductible. En particulier, son groupe de Galois diff6rentiel est isomorphe & SL2(C ) (Picard-Lefschetz, ou Gauss), et toute section non nulle Soient

co de D1A/X

en est un vecteur cyclique.

L -- LA/X,co 1'6quation de Picard-Fuchs correspondant &

et x/-->{co 1(x), co2(x)}

une base de solutions de

p6riodes fondamentales de

L

co,

form6e d'un couple de

cox sur A x . Pour tout point rationnel

P

sur A(K)

(c'est-&-dire pour pour toute section P de A/X ), I'int6gration sur un chemin de A x allant de 0 & Px permet de d6finir une section de classe de Nilsson (au sens de [13], p.122) du fibr6 tangent relatif

LieA/X . D a n s la base duale de

co , elle s'identifie & une fonction up , bien d~finie & I'addition d'un ~l~ment du groupe

£2= Z coI + Z co2 pros . Dans ces conditions,

uniforme sur X , & croissance et

Lup

est une fonction

mod~r~e ~. I'infini, donc un ~l~ment

mp

de O X ,

I'application p:

A(K)/Ator(K)

qui associe & la classe de

--> EXtD(V(L),Ox) P

I'extension de

, V(L) par

O X correspondant & la

classe de mp dans O x / L O x (lemme 4), est un homomorphisme de groupe. Le

136 thdor~me du noyau de Manin ([20], §6; [11], Th~or~me 1.4.3) en affirme le caract6re injectif. (Combin~ avec la g6n~ralisation du lemme 2 & une base X quelconque, ce th6or~me fo~rnit d'ailleurs une majoration bien connue du rang des groupes de MordelI-Weil des courbes elliptiques sur les corps de fonctions.) Mais mieux :

Lemme 8 : I ' h o m o m o r p h i s m e

A(K) ®C

~®1 de

dans

E x t D ( V ( L ) , O x ) est

injectif [Les classes modulo ~ des Iogarithmes elliptiques d'61~ments de A(K) lin6airement ind~pendants sur Z le sont aussi. L'injectivit~ de tx exprime que leurs classes modulo ~2®C+ O X fe sont encore, et le lemme 7 qu'elles le

sont m6me sur C Rappelons & ce propos qu'un schema isoconstant n'a pas de multiplication complexe.]

elliptique

non

D~monstration : pour I'injecltivit~ de IX lui-m6me, on note que si Ix(P) est nul dans H I(V(L), OX) sans qlue P le soit dans A(K)®Q , il existe un 61~ment f de

OX

tel que up - f

s0it une combinaison

C-lin~aire de

(°1, (02 , et le

th~or~me de Picard- Lefsc!hetz montre que c'en est alors une cembinaison Q-lin~aire (voir [19], p.192) ; par l'argument rappel6 au lemme 7 , un multiple non nul de P serait alors ind~finiment divisible dans A(K) , ce que la th~orie des hauteurs interdit. (Voir [11] pour une pr6sentation plus intrins~que de ces arguments, et pour une d6monstration de nature alg6brique.) Quant & Ix®l , on peut le traiter de la fagon 61~mentaire suivante. Soient a 1 ..... a r des nombres complexes lin~airements ind~pendants sur Q , et P(1) ..... P(r) des points d'ordre infini de A(K) tels que a 1 u P ( l ) + ... + arUp(r) appartienne ~. ~®C + Ox

. Le th~or~me de Picard-Lefschetz montre de nouveau que quitte &

translater les

up(i)

par des ~l~ments de

~ ® Q , cette expression appartient

& O X . Mais d'apr~s le th60r6me 1 et I'injectivit~ de

I~, chacun des groupes

de Galois GU(ix(P(i))) est isomorphe & ~ ® C . Les D-modules Ix(P(i)) ~tant fuchsiens, I'action d'une monodromie convenalement choisie dans GaI~(K'/K) fournit alors une relation de d6pendance C-lin~aire absurde entre

(01 et (02-

En combinant le lemme 8 au th~or~me 2 (~tendus une fois encore aux rev~,tements finis de X ), on a finalement ~tabli :

TheorY.me 5 : consid~rons deux fonctions alg~briques que

( g 2 ) 3 / ( g 3 ) 2 ne soit pas constante . Soient

strass d'invariants

g2(x) et g3(x) telles

to x la fonction de Weier-

{g2(x),g3(x)}, {(0I(X), (02(x)} une base Iocalement holomor-

phe de son r~seau de p~riodes, et u 1 ..... u r des fonctions holomorphes sur un ouvert de

C , telles que

ind~pendantes sur

les fonctions

u 1 ..... Ur, (01, (02 soient lin~airement

Z, et que to x(Ul (x)) ..... to x(Ur(X)) soient des fonctions

137

alg~briques. Alors, les 2r fonctions

u l(x),

(d/dx)u l(x) . . . . . Ur(X), (d/dx)ur(X)

sont alg~briquement ind~pendantes sur le corps C(x,e~ 1 (x),co2(x),(d/dx)co 1 (x)) . Noter que cet ~nonc~ West pas de m6me nature que les r~sultats de [10] . Ceux-ci concernent en effet une fonction de Weierstrass ~o fix6e, c'est-&-dire un schema elliptique A/X constant. Remarque

4 : une extension d'un D-module du type

V(LA/x,co) par

OX

apparaTt 6galement dans I'interpr~tation donn~e par Beukers ([4], prop.l) de la m6thode d'Ap~ry pour I'irrationalit6 de ~(2) . Idem pour I'~tude de Log2 , avec une extension par O X d'un D-module de rang 1 , trivial sur un rev~tement fini de X . Existe-t-il un lien direct entre ces extensions et ces hombres irrationnels ? c) Le cas hvDera6om~trioue. Les groupes de Galois des op~rateurs hyperg~om~triques L irr6ductibles sont enti~rement connus, gr&ce & I'extension faite par Gabber et Katz [17] des r~sultats de [6] . Ces articles reposent sur des arguments purement alg~briques (ou formels, b. I'infini). Pour L confluent, une mMhode analytique, due par Ramis et d(~velopp~e dons [21], consiste & 6tudier le groupe de Galois de (K~) h ® V(L) sur le corps (K,~) h des fonctions

m~romorphes

& l'infini, en interpr6tant les matrices de Stokes (d~finies au moyen d'un proc~d~ de resommation canonique) comme des automorphismes diff~rentiels. Elle permet en principe de traiter 6galement les op(~rateurs hyperg~om~triques r~ductibles (voir [14] pour les op6rateurs de rang petit). Mais il est plus commode de faire appel dans ce c a s & la m6thode (purement alg~brique) du §2 . Voici comment.

choisir

Pour all6ger, nous nous bornerons au cos confluent. On peut alors X = Gm pour base, sans modifier la convention du §1 sur tes

D-modules (voir [17] o~ =

pour une presentation plus adequate).

On pose :

x(d/dx) , Ox(cc ) = V(o~ - ( z ) ,

o0 c~ est un nombre complexe quelconque, et on rappelle que I'application c~ /-e Ox(c~ ) ~tablit un isomorphisme de C/Z sur le groupe des classes d'isomorphisme de Soient

D-modules fuchsiens de rang 1 sur n > m

X .

deux entiers > 1 , et {c( 1 ..... C(m}, {151 ..... 15n} deux

families de nombres complexes. On note L = Lm,n( {c(i} i, {13j}j) = Nj=I ..... n(~ + 15j-1) - z r[i= 1 ..... m( ~ + (~i) t'op6rateur hyperg~om~trique g6n6ralis~ d'ordre n correspondant. I1 admet une singularit~ r~guli~re en 0 , o0 les exposants valent {1- 13j}j, et une singularit~

irr6guli~re

en

~

, o~ I'irr~gularit~ vaut

1 , la partie de pente

138 nulle a une Iongueur m , et les exposants correspondants valent {~i}i • Par ailleurs,

le

D-module

V(L)

West congru & I'un des

est irr~ductible si et seulement si

13j modulo

Z (voir [6], lemme

aucun des

c~i

4.2; [17], corollaire

3.2.1) , et h I(V(L)) est alors 6gal & 1 (lemme 2). En particulier, il n'existe e s s e n t i e l l e m e n t qu'une extension non triviale d'un D-module hyperg~om~trique irr~ductible V(Lm,n({O~i}i,{13j}j) ) par O X Elle se trouve 6tre encore donn6e

par un op6rateur

{13j}j=l .... n + l ) ,

ot]

hyperg6om6trique

M -- L m + l , n + l ( { O t i } i = 1 ..... m + l ,

C~m+1 et ~n+l d6signe deux entiers

blement choisis. Plus pr6cis6ment (volt [9], proposition (a) si O~m+ 1 >_ 13n+1 extension de

OX

1

rationnels

, V(M) d~finit une extension de V(L) par

par V(L)

convena-

: OX

(et une

dans le cas contraire) ;

(b) pour L irr6ductible, cette extension est donc triviale (en vertu du lemme 4 (ii)) si et seulement s'il existe un param~tre entier L3j de L majorant

strictement

strictement

C~m+1 , ou un param6tre entier

c~i de

L minorant

13n+1

Le th~or~me 1, joint aux r~sultats de [17] (et & une torsion ~ventuelle par un module du type OX(-o~)) fournit donc le groupe de Galois de t o u s l e s op~rateurs hyperg6om~triques facteurs (voir [9], th6or6me 2).

M

dont

le semi-simplifi6

poss~de

deux

Plus g6n6ralement, soit M un op~rateur hyperg~om6trique d'ordre q , r6ductible dans I'anneau D . Puisque card(S) vaut ici 2 , on d~duit du lemme 2 et de I'additivit6 de h 0 - h 1 sur les suites exactes de D-modules que le semi-simplifi6 de V(M) est une somme directe d'un module hyperg~om6trique irr6ductible V(L) de rang n < q , et de r = q - n modules de type OX(O~) (voir [17], cor. 3.7.5.2). Supposons de plus que V(L) soit un quotient de V(M) , et que ces r hombres o~ ne soient pas congrus modulo Z . Comme Ext(Ox(c~),Ox(c~')) -- Ext(Ox(c~-oQ, OX) = 0 si ( z - o~' n'est pas entier , V(M) est alors une extension de V(L) par une somme directe de D-modules de rang 1 , pour laquelle on peut 6noncer :

Th6or6me

6 (K. Boussel [9]): soient q > p > r des entiers >_ 1, et c~1 ..... O~p,

~1 ..... ~q des nombres complexes v~rifiant les conditions suivantes :

i)

c~i- ~ i e s t un entier > 0

ii) aucune

des differences

n'est enti~re . Alors, I'extension contient

le

de

de

i-- 1 ..... r;

c~i- ~ i' (l_
Picard-Vessiot

compositum

Picard-Vessiot

pour

du

de

~j-~j, (l<j<j'
c~i-~j

(i,j > r)

M = Lp,q({CCi}l
corps C(x,xC~i

,l
et de I'extension de

L = Lp_r,q_ r ({c~i}i> r, {~j}j>r), et le groupe de Galois

139

diff6rentiel

relatif correspondant est isomorphe ~ c ( q - r ) r .

D6m0q~.trati0n : les hypoth6ses faites sur les r premiers param~tres de M entrafnent, d'apr6s le point (a) ci-dessus, que V(M) est une extension de V(L) par Ox(c~ 1)¢ ...¢ Ox(c~r) . On tire de ii) et du point (b) que V(L) est irr6ductible, et que l'extensions de

V(L)

par

Ox(c~i) d~duites de V(M)

projection canonique n'est triviale pour aucun indice permet de conclure.

par la

i < r . Le th~or~me 3

R e m a r q u e 5 : On trouvera dans [9] une analyse exhaustive des types de d6composition possibles des op6rateurs hyperg~om~triques r6ductibles, et de leurs groupes de Galois. Par exemple, si rhypoth~se de positivit6 de i) est mise en d6faut par les r' < r premiers couples de param6tres, V(M) est une extension par V' = Ox(c~r,+l ) ~ ...¢ Ox(c~r) d'une extension E' de V" = Ox(C~l) ...¢ Ox(c~r, ) par V(L) : c'est la situation mentionn~e & la fin du

§ 2 . Sous

I'hypoth~se ii), la dimension du groupe de Galois relatif y croft de (r-r')r' unit6s .~le point clef ~tant, comme dans [23], 3.3, que le seul sous-G-module de E '° se projetant surjectivement sur (V") °~ est E '°q lui-m6me). En revanche, des d6g6nerescences apparaissent Iorsque ii) West plus satisfaite. d) Lien avec la m6thode de Shidlovskv. Soient E un D - m o d u l e , et f une section horizontale de l'extension de E & I'espace analytique X h . Supposons que S soit d6fini sur la clSture alg6brique Q a de Q dans C , et que E et f v~rifient les hypoth6ses d'arithm6ticit~ de la th6orie des E-fonctions de Siegel, en un point base alg6brique (z0 de X u S . En particulier, E et ses fibres Ec~ aux points cz de X(Q a) sont munies bien connue des r~sultats de puissance sym6trique stf de entier t > 1 , il en est de m6me,

d'une Qa-structure. Dans sa g6n6ralisation Siegel, Shidlovsky a d~montr6 que si la f engendre le D-module StE pour tout en tout point alg6brique c~¢c~ 0 de X , des

puissances sym6triques de f(cQ Qa-espace vectoriel E~ (voir [27]

vis & vis des puissances sym~triques du pour des raffinements de cette assertion).

En d'autres termes, apr~s choix d'une base de E sur Qa[x] , les composantes de f(~) sont 'homog~nement' alg6briquement ind6pendantes sur Qa d~s que celles de f le sont sur Qa(x) - ou sur C(X) , cela revient au m6me en vertu de la d6finition des E-fonctions . Cette derni6re hypoth6se ~quivaut encore & dire que I'orbite de stf sous I'action naturelle du groupe G E -- GaI~)(KE/K) sur (StE) C3= St(E ~) engendre St(E c3) sur C , pour tout t >1 . Dans [6], Beukers, Brownawell et Heckman ont montr6 qu'& condition de supposer G E r#ductif, le r6sultat de Shidlovsky peut se d6duire de la m~thode originale de Siegel, et retrouve de ce fait un caract~re effectif. Cette hypoth~se fimite n~anmoins beaucoup le champ d'application de teur m6thode. La d6marche du §2 permet dans de nombreux cas de rem6dier

140 & cette difficult6, gr&ce & 1'6nonc~ suivant, ol) on reprend les notations du {}2 relativement & une extension 0 -~ V' --> E ~ V ~ 0 de D-modules. T h ~ o r ~ m e 7 (Beukers [5]): s o i e n t t un entier > 1 , et de projection v =~(f) sur V ~ , tel que • i) I'orbite de

Stv sous I'action de

ii)rorbite de

v sous ['action de

Alors, rorbite de

GV engendre

#l&ment de E ~ ,

s t ( v °q) •

GU(E) cofncide avec

stf sous raction de

fun

GE engendre

(V') ~3 . St(E{)) .

La d6monstration du th6or~me 7 repose sur la filtration naturelle de StE par les D-modules S i E ® s t - i v ' , et sur une r~currence sur i . Son application au th6or~me de Shidlovsky est claire sous l'hypoth~se du th6or~me 1 , o~ sa condition ii) est satisfaite d~s que v n'est pas nulle. Dans des cas plus g6n6raux, on consid6rera une filtration {E J} de E de gradu6s irr6ductibles, et on proc~dera par r6currence sur j , en combinant le th6or~me 7 & I'argum,ent mentionn6 & la remarque 5 ci-dessus. Signalons, pour conclure sur un ton plus arithm~tique, que les r6sultats de B6zivin et Robba sur les op~rateurs de Polya L (et leur application & I'ind6pendance lin6aire sur Q de valeurs de fonctions classiques) s'interpr6tent 6galement en termes d'extensions de D -modules. Le th~or~me principal de [7] (voir aussi [1]) peut ainsi se lire comme un crit~re local-global de trivialit6 pour certains 616ments de H 1 (V(L)*) = E x t D ( V ( L ) , O x ) , t'expression 'local' se r~f~rant ici aux ~quations diff6rentielle p-adiques associ6es & L . II serait tr~s int~ressant de d6velopper 1'6tude 'kumm6rienne' du {}2 dans ce cadre, en recherchant un analogue diff6rentiel des groupes de Tate.-Shafarevich.

R~f6rences [1] [2] [3] [4] [5] [6] [7] [8] [9]

Y. ANDRE : Crit~res de rationalit~ et d'alg~bricit~; expos6 & Luminy, Mai 88. J. AX : On Schanuel's conjecture; Ann. Maths., 93, 1971, 252-268. D. BABBITT - V. VARADARAJAN : Local m o d u l i for m e r o m o r p h i c differential equations ; Ast6risque; 169-170, 1989. F. BEUKERS • Irrationality of =2, periods of an elliptic curve, and F1(5); Birkh&user Prog. Maths, 31, 1983, 47-66. F. BEUKERS : communication orale, Juin 1989. F. BEUKERS-D. BROWNAWELL-G HECKMAN : Siegel normality ; Ann. Maths, 127, 1988, 279-308. J-P. BEZIVIN - P. ROBBA : Rational solutions of linear differential equations; J. Austr. Math. Soc., 46, 1989, 184-196. N. BOURBAKI : Groupes et alg#bres de L i e , Chap. 1 , Paris, 1971. K. BOUSSEL " Groupes de Galois des 6quations hyperg6om~triques ...; CRAS Paris, 309,1989, 587-589 (et article en preparation).

141 [10] D. BROWNAWELL - K. KUBOTA : Algebraic independence of Weierstrass functions; Acta Arith., 33, 1977, 111-149. [11] R. COLEMAN : Manin's proof of the Mordell-Weil conjecture over function fields; manuscrit, Berkeley, Mars 1989. [12] R. COLEMAN : lettre & B. Mazur, Mars 1987. [13] P. DEL1GNE : Equations diff~rentielles a points singuliers r~guliers ; Springer LN. 163, 1970 . [14] A. DUVAL - C. MITSCHI : Matrices de Stokes et groupes de Galois des ~quations hyperg6om~triques ...; Pacific J. Math., 135, 1988 . [15] E. GEKELER : De Rham cohomology and the Gauss-Manin connection for Drinfeld modules; ce volume . [16] N. KATZ : On the calculation of some differential Galois groups; Inv. math., 87, 1987, 13-61. [17] N. KATZ : Exponential sums and differential equations ; Princeton UP, & paraftre. [18] L E D . T. - Z. MEBKHOUT : Introduction to linear differential systems; Proc. Symp. Pure Maths, 40, 1983, 2, 31-63. [19] B. MALGRANGE : Sur la r6duction formelle des ~quations diff6rentielles & singularit6s irr6guli~res; manuscrit, Grenoble, 1979. [20] Y. MANIN : Rational points of algebraic curves over function fields;Izv. 27, 1963, 1395-1440 -- AMS Transl., 37, 1966, 189-234. [21] J. MARTINET - J-P. RAMIS : Elementary acceleration and multisummability; manuscrit, Luminy, Septembre 1989 . [22] Z. MEBKHOUT : Le th~or~me de comparaison ...; Publ. math. 69, 1989, 47-89. [23] K. RIBET : Kummer theory on extensions of abelian arieties by tori; Duke Math. J., 46, 1979, 745-761. [24] P. ROBBA : Lemmes de Hensel pour les op6rateurs diff6rentiels ... ; Ens. math., 26, 1980, 279-311. [25] T. SCHOLL : Remarks on special values of L-functions; manuscrit. [26] J-P. SERRE : Sur les groupes de congruence des vari~t6s ab~liennes; Izv. AN SSSR, 28, 1964, 3-18. [27] A. SHIDLOVSKt : Nombres transcendants [en russe]; Nauka, 1987. [28] V. VARADARAJAN : Lie groups, Lie algebras, and their representations ; Springer GTM, 102, 1984.

D u a l i t y in Rigid A n a l y s i s Bruno Chiarellotto * Dipartimento di Matematica Pura e Applicata Universita' di Padova Via Belzoni 7 , 35131 Padova Italy

Throughout the article K denotes an algebraically closed field, complete for an ultrametric valuation (We wii1 call such a field an ultrametric field ).

In this paper we want to give a partial answer to the question:

is it poaaible to have a Serre-type duality in the rigid analytic netting ? i.e. Given X a regular rigid analytic variety of dimension n a~d M a coherent sheaf of Ox-modules is it possible to define topological pairings:

H ' ( X , M ) × E x t ' ( X , • , fl~) ---~K H'(X,M) × zx~:(x,M,n?~) --.t( in such a way that each space is the strong duM of the other (via the involved pairing) ([SED], [m~l, [SVO])? We need some definitions and tools to solve this problem: a notion of cohomology with "compact" support, a "canonical" topology on the cohomology groups and a notion of "residue". In this paper we will construct the duality for a rigid Stein space X [LU] and a coherent sheaf .~[ on it. Here is an outline of the contents. In §1 we introduce the notion of a family of compact (or, better, co-compact) supports. In §2 we will prove the duality for X = A~( and the structural sheaf by introducing a residue map (2.8) (Note 2.12 deMs with the problem of the unicity of this map). In §3 we will provides the Serre duality for a coherent sheaf on A ~ . In §4, theorem (4.21) gives the duality for a coherent sheaf on a Stein space. The key point in the proof is the existence of a closed immersion for a Stein space in an a~ine rigid space [LU]. Finally in §5 we will prove that the notion of residue for a Stein space arising from §4 is independent of the closed immersion. Of course there are still some gaps in the theory. One is the fact that we deal, here, only with non-singular spaces. On the other hand little is known about a rigid analytic variety viewed as a topos. Furthemore our notion of compact support can not work for m~noid spaces (see (1.6)). It does, however, seen to work for " rigid spaces without boundary" (see [LUT]) and we plan further investigation along these lines. Some interesting results on cohomology with supports in the rigid analytic setting can be found in Y. Morita's article in this volume (see also [MOS]). * Supported by C.N.R. Italy.

143

The most important part of this work was prepared during our visit to the "Equipe L.A.212" of CNRS (Universit~ Paris VII): we thank that institution. We must also thank Professor Z. Mebkhout because he gave us the basic ideas of this paper and much assistance during its preparation. We should also mention Professor J. Fresnel, Professor M. van der Put and Professor P. Berthelot for much help and many suggestions we had from them.

§1 We refer to [BGR], [FRE], [F-vdP] for the basic notion about rigid analytic varieties. Throughout the article we will employ the "strong" (saturated) topology on them. It's natural to associate a situs to a rigid analytic variety X (based on the strong Grothendieck topology) and finally to consider the associated topos, which we will indicate by R i g ( X ) ( as in the case of the fitale topos, one can find a topologically generating family ( [SGAIV]VII1.7, 3.1, III4.1: the admissible affinoids)). All the morphisms of the situs are injections: it follows that the elements of the topos associated with an admissible open set are open objects of R i g ( X ) ([SGAIV]IV2.1). We want to introduce a family of supports which will play the role of the family of compact sets in the classical case. In Rig(X) it is, perhaps, easier to define a "dual" notion i.e. a family of co-supports ( this was in the mind of Grothendieck in tile first edition of [SGAIV]): D e f i n i t i o n 1.1. A faznily of co-supports is a subset "¢ of the family of open sets of Rig(X) (i.e. subobjects of the final object), subject to the rules: L The intersection ( in the sense of R i g ( X ) ) of a finite number of dements in ¢ is still in

ft. Any open set which contains an element of ¢ belongs to ¢. Our point is that a finite union of admissible attinoid subdomains of X should be considered a "compact" (note that a finite union of admissible at~noids is an admissible open

set([BGR]9.1.4)). We assert: P r o p o s i t i o n 1.2. (Fresnei) I[ X is a separated rigid analytic variety and Y is a finite union of admissible afilnoids of X then X \ Y (as set) is an admissible open subset of X (thus associated to an open object of Rig(X)). Proof. i. Suppose X = S p m A is an affinoid space and Y is a rational subset:

Y = {z e X ;

[ fi(x) ] <_ [ fo(z) l,

i = l...,n}

" 0 f i A = A. It is possible to findaTr E K* such that I fo(x) 1> Ir~ I for w h e r e f i C A, ~ i = every x E Y. If X+ = { x e X ; Ifo(x) l > l w t }

x_ =-- { x e X ;

If0(=)l
then { X + , X_} is an admissible covering of X and fo(x) is invertible in X+. We put

x~ == { ~ e x + ;

f,(=)

I~1>

11.

n Then Ui=l X+i is an admissible open of X+ ([BGR]9.t.4) , and of X . On the other hand

(x\Y)nx_

= x_

144 n

(x \ Y) n x+

U x;

:

and it follows that X \ Y is an admissible of X ([BGR]9.1.2).

ii. Suppose X = SprnA and Y C_ X is a finite union of admissible affinoids .

Y=0

Then

, i=1

is a finite union of Y/ rational subdomains in X ([BGR]7.3.5), so

x Y : N(x\y,) i=I

By i. ,we have that each X \ Y, is adanissibie, and thus, so too is the finite intersection.

iii. In the generic case take {Xi}iel an admissible alTinoid coveting of X. For each i, (x \ Y).n x, : x, \ (Y n xd. But Y MXi is a finite union of admissible affinoids of Xi ([BGR] p.393 , since X is separated ) : Xi \ (Y f3 Xi) is an admissible of Xi ( by ii.) and, of X ([BGR]p.354). Thus being true for each Xi of the admissible covering {Xi} it follows, from the definition of strong topology, that X \ Y is admissible ([BGR]9.1.2) Q.E.D. So far we can talk about the open object of Rig(X) associated with an admissible open which is the complement of a finite union of admissible affinoids. P r o p o s i t i o n ( D e f i n i t l o n ) l . 3 . In Rig(X), with X a separated rigid analytic variety, the family of open objects given by "the open objects of Rig(X) containing an open set which is the complement of a t~nite union of admissible at~noids" is a family of co-supports. We will indicate it by [c. P r o o f . We apply the previous proposition and we note that in a separated rigid analytic variety the intersection of two admissible affinoids is still an admissible affmoid ([BGR]9.6.1).Q.E.D. R e m a r k . We don't know if each open object in Rig(X) is associated with an admissible open of X. We don't know if the family [c is a local family. Naturally we are going to define the sections with compact support in the complement of a co-compact using "the exact sequence of eohomology with supports" ([SGAIV]V6.5). We recall that to each open object is associated a closed one ([SGAIV]IV). Finally for £ E obRig(X) we have H.~°(£) -- 1hrn___.H~c\u(£) ue[c (if [/1 is an open object of Rig(X), £([]1) = Hora(U,,£)) and the sections

H°(C) = n m g~\v(C) UE[c ([SGAIV]V.6).

145

It is, then, possible to define the derived functors (cfr. [ARI,[BER],[SGAIV]V.6). Using the fact that the limit is filtering, for 5r • obRig(X)~b (category of abelian sheaves) we get: H__~P(.T) =

liLn __H%\u(Sr) UE[c

H~(7) :

lim H ~ W ( J : ) u~[c

and

R e m a r k 1.4. In the same way, we can define Ezt~x,c and $ ~ o x , c : they too satisfy the direct limit construction. R e m a r k 1.5. The set given by the open objects which axe the complement of a finite union of a~noids are a co-final family for the direct limit. R e m a r k 1.6. If X is a compact (proper over K) rigid analytic variety, the compact support cohomology is the usual one. On the other hand, for X = SpmA, with A an a~noid algebra, the compact support cohomotogy and the usual one coincide. As we will see, afflnoid spaces won't appear in the duality theorems (2.11). One of the tools we need is the Yoneda-Cartier pairing for the Ext* in the rigid setting ([CA],[SUO], [HARD]). To this end, we state the following extension lemma ([SUO]2.1]): L e m m a 1.7. I f Z is an injective Ox-module and

H--+M an injective map of Ox-modules, if V is an open in Rig(X) and

¢ • E~t°~,x\v(x,N,z) then it is possible to tlnd an extension

-~ • Ezt°ox,x\v(X,M,Z). P r o o f . We need only note that ([SGAIV]V6.8.6)

Ezt°o x,xx v( X, "P, ~ ) = Ext°o x (X, P , H___°x\v( :F) ) for 7~, 9v Ox-modules and use the fact that if 2" is an injective then tI°x\v(Z)) is injective ([SGAIV]V4.11). Q.E.D. Finally we can define the Yoneda-Cartier pairing

E~tPox(X, M, 7) × E~t~o~,~(X,~', G) ---,

E~tv,:,~(X, '+~

34, ~)

R e m a r k . The same argument works for ~cxtox.

§2 This paragraph deals with the simplest case of S e r r e - t y p e duality : the structural sheaf of the atone rigid space A ~ .

146

To accomplish this goal, we need to define topologies and a residue formula in such a way that the pairing

gives a perfect duality (each space is the strong dual of the other). We recall that A ~ is a quasi-Stein space [KI] (for the notion of Stein space in rigid azmlysis see [LU]). D e f i n i t i o n . A rigid separated variety is a quasi-Stein space if there exists an admissible affinoid covering { Ui }ieN (countable) such that

Ui c_ Ui+l and

ox ( u , + ~ )

--,

Ox(UO

is dense. R e m a r k . For A~( such an admissible covering is given, for example , by {B d }icN

B~, = {(*D c A~,- Ixj [_< e'} with e C] K* l,e > 1. Cartan's theorems A and B hold [KI] , since OA~c is a coherent sheaf

H (i A Kn , O A ~ ) = 0

i>0.

We recall the defimtion of coherent sheaf in the rigid setting: D e f i n i t i o n 2.0. [BGR] In a rigid analytic variety X a sheaf of Ox-modules M is coherent if there exists an admissible affinoid covering {UI}Iu such that the restriction of M to each Ui = SpmAi is a sheaf associated to a finite Ai-module Mi

R e m a r k . One can prove that for a coherent Ox-module M and for every U = SpmA admissible attlnoid of X, the restriction Ad W is a sheaf associated to a finite A-module Mu : MIv

= My.

Each affinoid algebra A is a Banach space: every submodule of a finite A-module is closed and every A-linear map between finite A-modules is continuous ([BGR]3.7.3,6.1.1). It follows that A" My

-

.4

-

~4(V)

has a canonical (by the open mapping theorem [BGR],[BOU]) topology as Banach space. On the other hand an o~moid algebra A is of countable type : it follows that Mu is of countable type, again by [BGR]. We wish to endow H ° ( A ~ , OA~) with a canonical topology. First of all we notice that for A ~ and, in general, for a quasi-Stein space, the admissible countable affinoid coverings form a co-final system for the set of all the coverings. (Each admissible countable at~inoid covering of an a~noid space admits a refinement which is a finite admissible at~noid covering.

147

A quasi-Stein space has a countable affinoid covering) . Take an admissible countable affinoid covering ,/2, and consider its Cech-complex in OA?~ :

c'(u, o.;~) From the acyclicity of the affinoid subdomains, its cohomology will be H ° ( A ~ , Oh~). On the other hand the sections of the structural sheaf on each attlnoid form a Banach space: the Cech-comptex C'(b/, O A ) ) is a complex of Fr6chet spaces ([BOU]I3.2). We can put the induced topology (of Pr~chet Space) on H°(A~(, Oh~ ). If P is an admissible countable affinoid covering which is a refinement of L/, there will be continuous maps between Pr6chet spaces:

c'(u, OA~:) ~ C'(V, OA~) which induce isomorphisms in cohomology. But the open mapping theorem (still valid in the ultrametric setting [BOU]) allows us to conclude that this isomorphism in cohomology is, actually, a homeomorphism ([RR] temmal) for the induced topologies : ~0 : H°(U, OA}) ---4 H°('l), OA~.). This gives a canonical topology as Fr6chet space of countable type to H°(A~:,OA~.) (the sections of the stuctural sheaf on each atilnoid subdomain constitute an affinoid algebra which is a Banach algebra of countable type. Every subspace of a space of countable type is of countable type) . In particular H ° ( A ~ ' O A ~ : ) = { E a°~xC~ do, e K a6N -

la.l,l~l=0

lim

eelK*l

}

(2.1)

We now treat the space H2(A~c , f/~} ). Once we have fixed a system of coordinates on A~:, x a , . . . , x,, , a co-final system of compacts is given by: B = {B,={xeA~(,

Ix~l<,

i=1,...

,n}

eelK*l

}

(2.2)

(In fact on every affinoid the maximum modulus principle holds ([BGR]6.2.1)). We can write

H cp ( A Kr*,

A~) =

~n

lim

HB,(AK,fiA~) p

n

n

B, 6 B

P r o p o s i t i o n 2.3. Under the previous hyphothesis n H cp ( A K ,f~)

For

p ~

= 0

p#n

n

H c"( A K" ' f ~ A " ~ )={

E xa a--z~ d x l h . . . Adz,, ~e(N*)-

aa E K 3e El K* I

lim laol~l~t=0} I~1--'+¢~

Furthemore, we can endow this space with a canonical topology. P r o o f . Once we have fixed a global system of coordinates we can use the results of Morita [MOS] for the vanishing of the eohomology. In fact the eohomology will be the direct limit of n

n

n

~

n--1

r~

148

(A~¢ is a quasi-Stein and ~ is coherent). On ,, ~ K \ ~, A~ J there exists a canonical topology. Indeed F / ~ is, in particular, a coherent sheaf: the countable admissible ai~noid coverings of A ~ \ B ~ form a co-final system for the coverings (even if A ~ \ B ~ is not quasi-Stein, but it is finite union of quasi-Stein ...) : the Cech-complex of f t ~ of each such a covering consists of Fr~chet spaces.The same argument we used for OA~ shows that there exists a canonical topology (not a priori separated) for H n-1 ( A ~ \ B~, ~ ) i.e. independent of the covering ([RR]lemmal). For H ~ ' ( A ~ , ~ , ~ ) we will take the direct limit of these canonical topologies. The fact that B (2.2) is co-final is independent of the coordinates: the resultant topology is canonical Q.E.D. Among all the coverings which one can use in order to calculate H'~-~(A~ \ B e , g t ~ ) there is the following ld = { U ~ , i = { x e A ~ g ;

IxiI>e}

i=1,...

,n}

(2.4)

Every U~,i is quasi-Stein and the sections of a coherent sheaf on a quasi-Stein form a Fr~chet space (If .7- is a coherent sheaf, on each affinoid its sections form a Banaeh space. Taking ml admissible countable affinoid covering V, the global sections will form a closed subspace of C'(I),.7-) which is a Frdchet space. The topology is canonical (see the argument for H°(A~-, OA~:) [RR]).). On the other hand the intersections of the elements of/d are still quasi-Stein spaces: the Cdch-complex C°(/d, fl~.~: ) is a complex of Fr6chet spaces, tt~e induced topology in cohomology is the canonical topology in H n - t ( A ~ \ B~, 12~.~.). In particular this topology on H ~ ( A ~ , f/~?~) is given as the direct limit of the Banach spaces a~ = {

E ctC:(b f N ' )

n

a~ dx, A . . . A d x , ; a~ e K xa ,

lira l a , ~ l e l ~ l - - - 0 [M~+o¢

with ~ CI K* IFrom (2.3) it is possible to give a residue formula for H{(A~c , ~ } resx :

H~"(A~,fl~.~c ) ----* K

}

) : (2.6)

aa

E x--J dxi A . . . A dxn ---+ a(l,_ .1) ~c(N*)" "A priori" such a continuous map depends on the choice of coordinates. We will see later that it is unique up to a constant (2.8). W'e can prove: T h e o r e m 2.7. The spaces H:(ALa~

)

and

H°(ALOT,~)

endowed with the canonical topologies are in perfect duMity (one is the strong dual of the other) using the pairing: H cn( A Kn , ~ An~ : ) X

HO¢A

~ nK,

on

n A ~ c~ ) - " ~ H en( A Kn , flA~:) ~K

In particular H°(A~., O ~ c) is a reflex/re Prdchet space of countable type and H~(A~:, f~.~ ) is a strong dual of a Fr6chet space, complete, reflex/ve and Hausdor~. P r o o f . This is an adaptation of some results about duality between direct and inverse limits of Banach and F%ehet spaces in the ultrametric setting ([MO-SCH], [MOH] Note

149

the different techniques used in the case that the field is, or is not , spherically complete). The only remark consists in noticing that the canonical topology on H 2 ( A ~ , g~?~) can be (formally) defined as the direct limit of bounded series: A,,b = {

aa x"'g d x l A . " . A dxn ac, E K

E c~E(bfN,)n

sup [ a~ [ ~lol < + ~ } I~1~+~

(e e I K* ]). Q.E.D. Using the previous theorem we can prove that our notion of residue is essentially unique

[AK]. P r o p o s i t i o n 2.8. On H~(A~-, F t ~ . ) and H°(A~:, O , ~ ), endo~'ed with the canonical topologies, suppose there exists res

:

c t

K,

) ~

K

which gives a perfect duality between the two spaces. T h e n ~ a multiplication by a constant in K

is equM to res~ (2.6) up to

P r o o f . In particular res= will be an element of the dual of H2(A~,~VXI ) with the n ' OnA~:) such canonical topology. By the perfect duality for ~:-e-s,there will be a E HO~A ~ K that: n n n a n n n g c ( A K , aA~() ~ He ( A K , ~ A ~ )

K

will be commutative. On the other hand the same argument works for ~ using a b E H°(A~(, O 1 } ). F i n a l l y , there is the commutative diagram:

respect to res~ ,

K Using again the fact that res~ gives a perfect duality, we conclude that ab=l

in H ° ( A ~ : , O ~ b ) . But for entire analytic functions (2.1) this is true if and only if a, b are

const=t ([BGRIS.1.3,[AMD. Q.E.D. We will call the generic residue (up to a constant) in A ~ : f e z , . R e m a r k 2.9. The same duality works in the case of a free sheaf O A~, p P E N, or if we substitute the whole affine space with: o

Be = { x e A ~ ¢

I

Ixil <e i=1 .... ,n

}

(2.10)

R e m a r k 2.11. If we take X = S p m K < z >; [BGR], H o ( x , O x ) = K < x > and this is not self-dual. This shows that our definition is not valid for rigid affinoid spaces.

150

N o t e 2.12, We want to check the compatibility of res. in A ~ and the algebraic residue in H ~ ( P ~ - , Q ~ ) (via G.A.G.A. in the rigid setting [KO]). This will show the invariance of res~ for automorphisms of P ~ . By lemma 3.7 one has: i

n

n

i

n

TI

HB,(AK,~A,K) ~-- HB,(PK,~'Ip~). Consider the maps: n

n

n

7 : H ' - I ( A ~ ( \ B~,f~7~)-----~HB~(AK,fIA"K) and n

n

rt

~

tz

n

n

n

n

n

To construct these maps one takes an injective resolution:

0~

~n

i

~

d

~

d

~tp~K----~go---~dl-----~g2 . . . . . . .

its restriction to A~. will be an injective resolution of Q ~ . the following exact sequence:

0 -+ tI;,(A~(, J.) ~

(2.13)

For the sheaf f~-~7~ one obtains

H'(A~c , J . ) ----+ H ' ( A ~ \ B,, d.) ---* 0.

For 7 we proceed as follows : take l E H " - I ( A ~ ( \ B,, f / ~ ), it can be considered as a section of d~-I on A ~ \ Be, then there exists [ E H°(A~(, d ~ - l ) whose restriction to A~- \ B, is I. Applying d of (2.13), d i e H°(A~(, d , ) and, in particular, d l e g °Be\r A "K, J , ) : 7(class of l) = class of all.

The extension with zero (dr), ¢ H°(P~., d.) gives:

r(elass of d ) = cI~ss of (d), e H " ( P L a ~ ) . On the other hand we can associate to the following quasi-Stein admissible covering of P~: (we will indicate by x 0 , . . . . , z , the coordinates of P~(; A~( is viewed into P~( as the set

u0 = {(~) I~0 ¢ 0}): U0 = {Uo,Ui.,

i = 1,. ....

,n}

(Ui,, = {(xk) I xi # O, I~-~I < ! } ) a quasi-Stein admissible covering of A ~ \ Be:

U={V~=UonU~,,

i = 1 , ..... ,n}

(V, = U0 n U~,, = {(x~) I x0 ¢ 0, I~1 > e}. For the associate alternating Cech-complex

c"-~(u, a2~, ) = c"(Uo, a ~ ) and for each k E [0, n -- 1] there exists:

C~(u, f~,~ ) I~ ,ok+, (u0, a~,~) { gao...a~,O (g ......... ) ~

0 otherwize

151

(ai ~ [1,n]), which associates the same section if the open set U0 appears in the nerve ofg/o

(in this case we will always impose 0 in the last position), 0 in the other cases. With these definitions the diagrams (k e [0, n - 1t)

(2.14)

c~+~(u,,f~,7, ) A%

c~+~(u,,ftr,~,)

are commutative i.e. : f~+~o8 = ~0of~. It is then possible to define a map:

Finally tile main point of this note consists in showing that the following diagramm: H " - I ( A ~ \ B~, f~.~)

K

rest/

~'1

1

f

H"(U0, a ~ )

,

H " ( P ~ , n~Tc)

is commutative, too. The symbol res represents the algebraic residue in P~( ([HARA]): in the rigid setting the G.A.G.A type theorems hold [KO]. Take ~ C / : / n - l ( L / , f ~ ) ( we will denote in the same way its representative ~ E z n - l ( ~ ,kl , f i nA ~ ) -~,- __cn--l(~.,~ V~. k , f i nA~l:.itsresidue, r e s . ~ , i s g i v e n b y thec°efficient°f -x t .-,x. z k " The same value is assumed by res on f(~) i.e. for ~ viewed in Z n-1 (b/0, f~5~) = cn( L/0, f/~'~ )" This shows that the E1 part of the diagramm is commutative, We now study the part ~2. Consider E c n ( l ~ o , ~ ' ~ ) = cn-l(l'~k , f~nA~:~ (i.e. we are taking an element of Hn-I(A~( \ B ~ , f ~ c ) and the element of H " ( L / o , f ~ ) associated to it by f ), we want to show that the related elements of H"(P~f, ~2~,~) and H '~-1 ( A ~ \ B,, f/~c ) are in the relationship expressed by .~-2Take i(~) E e " - a ( H , J0) = Z"(//0, J0) (2.13). The injective sheaf 3"o is acyclie [SGAIV] : there exists ~l E Cn-~rLt ~ , fl"A~] such that ~(~,) = i(~) (2.14). Using the map f,,_2 (which is , actually, an injection) by (2.14):

~0(/.-~(~1)) = i(O e

z"(Uo, Jo),

this means that :n-2({1) realizes i(~) as a bord in C'(/J0, J0). By (2.13) : 8( d~x) = d ~ l = di~ = 0

6o( d.f.-K1) = d*of.-2~l = di~ = O. It follows that d~l E z n - 2 ( U , J1)

dfn-2~l = f n - 2 d ~ l E Zn--I(u0, J1).

We can repeat the same argument for J1 : there will be (2 E Cn-3(L/, J~) such that ~ 2 = d(t. For P ~ we take fn-a((2) e Cn-2(L(0, J1): ~Ofn--3~2 = fn-2~2

: fn-2d~l : dfn-2~1,

152

as before d~: E z~-a(/g, J2) and df~-a~2 = f , - a d ( : e Z"-2(U0, J~). At the end we obtain d(,,_l E Z°(U, g~_~) and we put d~,_ ~ = l C H°(A~,-\B~, J~-~); of course dI = d2(,_~ = 0 so n n l E H ' ~ - I ( A ~ \ B e , QA~). For P~; we c a n represent d{n-t = ( /~i)i=l,/~i G F ( U o r'iUi,, & - - l ) , then fod(n-1 = dfo~n-a = ()~1,o,... ,)~,,,o,013,.. ..O,,,n) E Zl(blo,J,~-l) (-~i,0 = )'i)- By the acyelicity of Jn-~ there will exist ~,~ 6 C°(U0, J . - 1 ) such that 8o((,,) = dfo~,~-I = fod~n-,. To construct (~ one takes an extension [ E H°(A~-, J . - 1 ) of I C H°(A~c \ B,, J,,_~) and by our choice of coordinates [ ~ F(U0, J._~ ) ; thus z,,

=(l-,o,,.

....

,o.)

~c°(Uo,&_,).

Using (2.13)

d~ = (d~;0,,. .... ,0~) c e°(U0,&) whence we obtain (d[,O~, ... , 0 , ) = (di), e H'(PiZ a~,7,) as in the construction of r and VThis concludes the proof of the commutativity of .E2. As we said, this note gives a " unicity statement " to our notion of residue. In fact by the compatibility we can derive the invariance for automorphisms of P ~ .

§3 We want to extend the duality to a coherent (gA~-module (cf. (2.0) and [BGR], [FRE D. One of the most outstanding features of coherent OA~-modules on a rigid analytic variety X is the fact that for this thick category the rigid variety has sufficiently many points i.e. if .Ad is a coherent OA~-module .M,: = 0 V x C X then .a.4 = 0 ([FRE],[BGR]9.4). As we said A ~ is a quasi-Stein space, it follows that for a coherent OA~-module .M : HP(A~-,34) = 0

ifp¢O

[KI]. In the first part of this paragraph we will study the pairing for 3d: Ho(A~(, 3d) x E x t ; t , . ~ ,c(A~, M , f~17,.) ----+ K

(3.1)

In H°(A~:, 2,4) (for a coherent OA~.-module Ad) it is possible to introduce a canonical topology. Once we have noted that the sections over ~n atfinoid admissible open of a coherent sheaf form a Banach space of countable type (see after (2.0)) the same arguments we used for OA~: (§2) allow us to conclude that H°(A~¢,.£4) is a Fr~chet space of countable type [RR]. We don't know if our family of supports is a local family: for this reason we cannot give a canonical topology on E x t ~ . ¢ ( A } , A~I,f ~ , ) A K ~

K

"

Later, (3.14), we will consider the pairing (f14 ia a coherent OA~-module ) " ,.M) X Ext~-~(A~,.M, HeP( A K

fl~)----*K

p=O .... , n

(3.2)

A K

We want to study the topologies on these spaces. For H~(A~., .M) as usual (§2) we have : •

rt

Hc(AK,Ad ) =

1hm H ~ , ( A ~ , . M ) . ~elK'l

153

Since the space A~- is quasi Stein we get H~,(A~c,.M ) -~ H P - I ( A ~ c \ B , , A 4 )

p>_2

H b , ( A ~ , M ) _~ H ° ( A ~ \ B * , M ) H°(A~.,.M) and H ° (A~'¢, M ) is a closed subspace of H°(A~:,,M). For each H ' ( A ~ c \ B~, M ) and H ° ( A ~ , M4) there is a canonical topology (see proofs of (2.3), (3.1)), which will induce a topology on H ~ , ( A ~ ¢ , M ) . The direct limit of these topological spaces H"B , kt A "K~ 34) will give the canonical topology on H~(A~c,.M)~ We want now to introduce a canonical topology on E x t o A ~ ( K, Ad, A~:)- We need some results: P r o p o s i t i o n 3.3. If ~, .T are coherent Ox-modules in a r/g/d analytic variety X , then

are coherent Ox-modules for each i and

&tb,, (.% ~), = E=to,,.,(.ry,Gy). P r o o f . For the first claim ta~e U = S p m A an admissible affinoid open of X . The fact that the restriction of an injective Ox-module 27 to U, 271u, is still an injective O r - m o d u l e ([KO13.8), implies: i

i

EXtox(.~,~)l v = gXtoxlv(.~lU,~lU ). But 9~lU = ff~ where F is a finite A-module. It is possible to write O ----* FI

----*A m --+ F ----~ O

and/;'1 is a finite A-module (A is noetherian). The relative sequence of associated Ou-modules is still exact: 0 ~ ~ ~ "-'~ A - - ~ F = 5flu ~ 0 (3.4) [BGR]. In particular F1 is a coherent Ou-module. F i n a l l y , by induction on the cohomology, using the fact that

g Xt°x w ( J:lU, ~lU ) is coherent if ~" , G are coherent [LIU] and the "thickness" of the category of the coherent Ou-modules we ~an reach the desired conclusion. For the second statement the point is, as in the classical case, that the fiber Z~, y E X , of an injective Ox-module 27, is an injective Ox,y-module (the proof is as in ([GROT]IV) using th~ fact that an affinoid algebra is noetherian [BGR]). Q . E . D . R e m a r k 3.5. The second statement is true for every O x -module ~. C o r o l l a r y 3.6. l.f X = A~¢ and .T, ~ are coherent OA~-modules, then i SXtb^ ~ (2 r , ~) = 0

i f i > n.

P r o o f . From the proposition (3.4), we know that

Sxt~ ~^~vc~ , G)

154

are coherent sheaves : it follows that the points are sufficient to separate these sheaves. On the other hand (3.4): ti i Exo,.~ (.r ,G)~, = gXto,~,,(.r,, ,~,,)

for every y E A~-. But OA~,y is a regular, noetherian local ring of dimension n and 9t-~ , Gy are finite COA~:,y-modules. This implies that V y E A~.: i g Xto A,K,~(.T'y , ~ ) = 0

if i > n Q.E.D. To conclude the study of (3.2) we note that, since A ~ is quasi Stein and EXtrA (.hd, f ~ . ) are coherent OAT -modules (3.4) :

ExtPoA.(A~K,Jtd,a~.K) = H°(A~c, gx~o^7 ( A d , f t ~ ¢ ) ) and thus it is possible to give a canonical topology to ExtPo,~ (A~(, 2M, ft~,~ ) (global sections of a coherent sheaf): Fr~chet spaces of countable type (3.1) (for the spectral sequence see

([SGAIV]V4.10)). We have defined canonical topologies on the objects appearing in the pairing (3.2) and (3.1). Before stating the duality theorems we need an "excision lemma" in the rigid setting. L e m m a 3.7. Under the previous hypotheses t'or .h4 , OA~,-module H~, (/~,,, 3d) = H°B,(/~q , .M) = H~,(A~c,.Ad)

with e < ei < ej ; e, ei, ej E[ K* I • P r o o f . For notations see (2.2), (2.10). For the proof see ([MOS]lemmal). Q.E.D. R e m a r k 3.8. One can prove that the extentions of the previous isomorphisms (3.7) to the cohomological groups are homeomorphisms for the canonical topologies when .M is a coherent OA~-modnle. It is only necessary to observe that a countable admissible affinoid o

covering of A ~ \ Be will induce a countable admissible covering of B,~ \ B~ consisting of quasi-Stein spaces (an affinoid is quasi-Stein). R e m a r k 3.9.t'3.2 We recall that by our definition:

EXtPoA. c(A~:,.M, f ~ )

=

lim E z t S ^

B,(A~-,.A.4,~,, )

for every p E N. By (3.1) we can, now, state: 3.10. Under the previous hypotheses, the space H ° ( A ~ , .h,4) endowed with the natural topology given after (3.1) is a reflexive Fr~chet space of countable type. The pairing:

Theorem

H°(A~,.h4) x Ext~9^~,c(A~,.M,f~)

----* t (

m ~ e s E~t~ A~ ,~(A "~, n4, f ~ ~ ) the dual o/. the previous space, rUahemore

ExtPo^~,c(A"g,.h4,~2~7~) = 0

l i p ¢ n.

(3.1)

155

P r o o f . We know that H°(A~,.M) is a Fr~chet space of countable type. What we need to show is that it is reflexive, that there is a bijection (as sets) given by the pairing:

EXt~A~: ,¢(A~, Ai, ~ )

----* ( H ° ( A ~ , .M))',

• n and the vanishing of the other groups of cohomology for ExtoA~,c(AK, 34, ~

(3.11)

).

T o calculate H°(A~., 34) one can take the admissible afl]noid covering {B~, }ieN, e e I K* I, e > 1 and H°(A~c,.M) = fimH°(B,,,34). Each B~ is an affinoid admissible open and .A41B,~ is an associated module. We note that

Oh.u(Be) = { ~_~ a~,x'~ a , ~ e K

lira

l a - l e ilk'i=0}

is a Banach space (in particular it is an afl~inoid K-algebra) under the norm

It ~ - . ~ °

II, = ~ p I a. I~ 'l"l.

In the previous situation (2.0)

.M(B,,) = OA~(BE')" A where A is a sub Oh~ (Bd)-module. We consider two cases. If the field K is spherically complete [MOH], then one can prove that the maps of Banach spaces

OA~(B~,) --~ OA~(B,,-~)

i EN

axe c-compact (i.e. the image of the unit ball is relatively c-compact : the proof is given using the analogous result for bounded functions [MOH],[MOS]). From the facts that

H°(B~,,3/[)- OA"K(B,')" A and that .Ad is coherent: H°(B~,-,, .h4) _

OA~,(B,,)" ¢~ O A ~ ( B , , - , )

--

OA~(B,,-,)"

As in the classical case (i.e. over C) , the image of a e-compact set by a continuous map is still e-compact. We can conclude that all the maps of Banach spaces

H°(B,,, M) ---* H°(B,,-,,.M) axe c-compact. On the other hand by [KI] (the covering {B~,}ieN gives the structure of a quasi-Stein space to A ~ ) :

H°(A~, 34) --~ H°(B,,, 34)

156

has dense image for every i. It follows from ([MOH]3.4) that H ° ( A ~ , 2 , 4 ) is a reflexive Fr6dmt space of countable type and its dual is

lirnH°(Bd,3d) '

H°(A~.,A4) ' =

K the field K is not spherically complete we can reach the same conclusion. In fact, in this case each Fr&het space of countable type is reflexive ([SCH]9.9) : in particular this is true for H°(A~c,~4). Again , the fact that each H ° ( A ~ , . A 4 ) - - ~ H°(Bd,.A4) is dense, allows us to write

limH°(Bd,Ad) '.

H°(A~,Ad) ' =

So, for general K ultrametric field, H°(A~c, M ) is reflexive and H°(A~,.M)'

h m H ° ( B d , M ) '.

=

IEN

One can have a homeomorphism of topological spaces (2.10) o

H ° ( A , ~ , . M ) = lira H°(Bd,

f14)

iEN

and the fact that

o

H°(A~c, 3/[)' = lim

H°(Bd, 3.4)'.

iEN o

o

(In fact each H ° ( A ~ , 3 A ) --~ H°(Bd,~d) is dense. The set B e, (2.10) is quasi-Stein : to show this fact one can choose a covering {B~. } . ~ s , e. ~ e', e. C] K* 1. In general every sequence {rl.}.¢N , r/. el K* 1, qn -* +oc can give an admissible affinoid covering of A~(, o

{ B ~ . } . e N , which makes A ~ quasi-Stein). We prove now that the dual of

H°(Bd,.Ad)

is

o

given by

Ezt~ A ~ , ~(Bd, A4, Ft~.K ) via

the pairing

o

o

H°(Bd,.M) x

ExtrA. ,,(Bd,2Uf,a~,~c ) --* K

(3.n)~

K

and that

o

ExtPo^~:,~(Bd,M,a~,~c ) = 0 With

o

( Be,,

f/

~,~ )

i f p # n. o

=

lim _ .

(Bo,.M , " )

tt<~'

(In fact we don't know if the notion of compact support we gave is a local one. Morally we o

are working with the compacts inside B d). A n d from (3.7) o

lim. ExtoA~.,B, P " M , f~AK ". ) = (AK,

lira -- . ExtPo^~.,B , ( B e , , M , ~ : ) .

rt<e'

r/<:e*

_

(a.7)i

157 o

To prove (3.11)i one should first note that on Be, (open polydisk) there exists a finite free o

o

resolution of ~4 (on BE,, ~4 is associated to a finite OA.K(B,,)-module ) [GRU]: 0--+

O".'

. . . . . . . .

Bj

Ono '

Bj

~

M

.

tB,~

---* 0.

(3.12)

We will prove the two assertions in (3.11)i by induction on the length l ( M ) of the free o

resolution (3.12). If/(A4) = 1, M is free (of course in B E, ) and one can see the duality as in (2.9). Suppose all assertions true for length < n - 1 and take M , l ( M ) = n. T h e n

O ---~ A ---~ O q

B#

--~ M o

IB,~

o

--~ 0.

o

where .A is coherent in B,~ and 1(,4) = n - t. But since B ~ is a quasi-Stein space, the short sequence of reflexive Fr6chet spaces of countable type : o

0~

o

H°(B,,,A)

o

---* H ° ( B , , , O q. ) -----* H ° ( B , , , M )

--~ 0

is exact. Take the dual sequence : it is still exact (if K is spherically complete the argument is as in [SED] using the Hahn-Banach theorem, if K is not spherically complete the "countable type" hypothesis allows us to have a Hahn-Banach type extention theorem [SCH]): o

o

0 +--- H ° ( B , , , A ) ' ~

o

H(B,,Oo°

q )' ~

HO(Bd,M) ' ~

O.

B#

By induction the sequence o

0

o

ExtoA~,¢(B~.,A

o

" ~'/A~) ~-- ExtoA),~(B6,0q.

,~'~A~J~' ~-- E x t ~ A ~ , c ( B ~ ' , M , 1 2 ~ ) '

~--O

is isomorphic to the previous one and still exact. We have proved (3.11)i. Using (3.7)i we find that: o

lira(EXtrA" ,c(B~, j t 4 , ~ ) ) i

= h m ( h m E x t o.

K

t

s-(K,-h'4,~'/.~))

=

rl<e,

= ExtbA~,c(A~,.A/l,fl~:). By the compatibility of the pairing (3.11)i at each i, the dual of H°(A~,.M) is

o

lim( Ext~A~,c( B d , ]v[, ~

)) --- Ext~A~,c( A ~ , . h 4 , ~ c ) "

i

Q.E.D.

R e m a r k 3.13. We d i d n ' t p u t a topology on Ext~)A~ ¢ ,c- Of course, imposing the strong dual topology deriving from (3.1) on it, the pairing will become a perfect pairing: one space is the strong dual of the other. We now study (3.2).

158

Theorem

3.14.

Under the previous hypotheses the spaces Exto-A~ (A~K,]M,fl2~ ) and

Hep ( A Kn , 34) endowed with the canonical topologies are, respectively, a ref/exive Fr~chet countable space and a complete ret~exive, Hausdorf[ strong dual of a Fr~chet space. The pairing : H~V(A~:," M ) x Extg-vAT, ( A ~ , M , a ; , ~ ) -----, K p = 0, . . . , ,~ (3.2)

is a perfect pairing (each one is the strong dual of the other) . P r o o f . The spaces

E x t r a ~ (A~c, M , 3 2 ~ ) endowed with the canonical topology are Frfichet spaces of countable type : see after corollary ¢)

(3.6). The first step, as usual, is to show the duality (3.2) in the case B d (e E[ K* [, e > 1)(this time with given topology on both sides). By [GRU] there exists a finite free resolution of 34 o . We make our induction on the length I(3/[) of this resolution. For l ( M ) = 1 , M is free o n / ~ d and we use (2.9). Suppose

O ----~ A ----~ O~ ---+34 o ---*0. Bel IBd and I(2t4) - 1 = I(.A) . By induction we may suppose the result known for .A (i.e. theorem o

(3.14) at level B d ). ~,Ve obtain the long exact sequence

0~

n

o

q~

0

q

o

Ext°A~(Bd,M,f~AK, )--4Ext o^~. ( B ~,,O°B,, 'fl2~)~-~

(3.15) 0

Ezt o. and

o

.

n

-

$

1

( B e , A, flA" )---~Ezto.

o

,

E x t oi ^ } ( B e , , A d , f l ~ } )

o

,

n

(Be,.M, g/A" ) ~

0

o

n ~- Extto-2} ( B d , - 4 , f t A })

i _> 2

(3.15)i

o

(We know that ExtPo,~ ( B d , 3,4,5r ) = 0 for p >_ n + 1 and for every A d , ~- c o h e r e n t ) . By induction, we can write the strong dual sequence for the middle part in (3.15):

H:(b,,, o~ )2-H:(b,,, ~)

(3.17)

B#

o

with the canonical topologies (As in the proof of (3.10) for a 9v, O A ~ - m o d u l e : H~(B~,, ~ ) = o

lim H ~ , ( B d , Y=).). The map (3.17) is a part of a long exact sequence:

o --~ H:-,(b,,, M)22H:(b,,, ~)~--2 o

t

o

) ~--2H:(B,,,34) ---. 0

--. H 2 ( B , , , O :

(3.18)

B d

and •

o

.

o

H~(Bd,3d)~_H~+I(Bd,A)

for

i<_n-2

(3.18)i

159 o

(by the inductive hypothesis HPc(Bd, .4) = 0 if p _> n + 1). We need to know that (3.18), and (3.18)~ endowed with the canonical topologies are the strong dual sequences of (3.15) mad

(3.15),. We divide the proof of this fact in two parts: a) we will prove that taking the transposed sequence of (3.15) and (3.15)i we get (3.18) and (3.18)i as sets but the topologies in o

o

o

H : - I ( Bd, .£4 ),Hc( B d , 34) and H~( B d , .M) ( i < n - 2) will be the induced by (respectively) o

(3.17) and H~+~(Bo, A); /3) we will prove that Ct ,St in (3.18) and the various isomorphisms in (3.18)i are homeomorphisms in the images for the canonical topologies. Then, using the fact that, by induction, we know that the strong dual of o

o

(

,QA

(Bd, A, f~A~)

z )

and o

o

are realized by (3.17) and H2-P(B,, , A) with canonical topologies, we conclude that the o

o

strong dual of E x t ~

(B,,, 34, ft~. ) is H cn - - . ( B ,.," 3//) via the pairing. A7¢ K For ce), one can take a sequence {r/,~}neN, r/, --~ ei, r/, e[ K* [ and to obtain (3.10): o

0 . q f~ 0 q f~. EztoAyc(B,,,O° , A~) = h-m E x t o ATe( B ."" , O .n~ , a ~ ) = l i m A . Bd r/n and each element of the projective limit is a Banach space of countable type. We will denote by {B.} the analogous Banach sequence relative to the space o

E~t°o.~ ( B.,,.4, aJ,,k ) and by ~ , the maps of Banach spaces

A,-~B.

(3.19)

(continuous because OA k (Bn.)-linear and A~, B~ are finite O a ~ (Bn.)-modules). Since the ~an of (3.19) are Oh~.(B,.)-linear, they have closed images [BGR]. It follows that

0~

kerqan ~

An-~-~B~ ~

Cokerqan --~ 0

(3.20).

is an exact sequence of Banach spaces of countable type which are finite OA~(Bu. )-modules. If the field K is spherically complete ( as in the proof of (3.10)) the maps in the projective limit {qan} are c-compact : it follows that all the relative maps

Ker~p, ~

Ker~o,_a

,

Coker~,an ~

Coker~on_1

are c-compact.One can write the transpose of (3.20)n with the Banach dual topology:

0~

ker~Ont

e----

An*----jO - ' ~ " ' " n e--- C o k e r ~ . ' ~

0

(3.20)~

By the Hahn-Banach theorem, which is valid because K is spherically complete [IN], (3.20)~n is still exact and continuous. If K is not spherically complete, the fact that (3.20)n is made

160

up of Banach spaces of countable type allows us to conclude that (3.20)" is still exact (there is, in fact, an extension theorem without identity for the norm [SCH]) and A ~. / ~ t B ., ~Ker~p,/, Ker~ t ~ Coker~, ~ with the induced topologies ( in the spherically complete case this isomorphism will be isometric. Here, by the open mapping theorem [BGR], we can only say that it is a homeomorphism). On the other hand the countability hypothesis implies that each element of (3.20), is a reflexive space [SCHI,[vRO]. We note also (independently of whether or not the field is spherically complete) that each map which forms the projective limits {Ker~,~}, {Coker~,} , {A,}, {B,} has dense image, and the same is true for every map from the projective limit to the spaces which form it. All the hypotheses required to prove the duality theorems for inductive and projective limits ([MO-SCtt], [MOH] ) are verified for (3.20)" and (3.20),. We get a perfect duality between (3.15) (which is the projective limit of (3.20),) and (3.18) (which is the inductive limit of (3.20)') i.e. each space in (3.15) is reflexive and has its strong dual in (3.18) (this last sequence is exact by the exactness of the direct limit). R e m a r k . For (3.15)i and (3.18)i the arguments are the same. But the topologies we found are the induced from ..4, and coq. , and we don't know if they B~i o

coincide with the canonical ones which can be defined in H$(B~, f14). fl) We will prove that the maps e t ,St in (3.18) are homeomorphisms to the image using the canonical topologies: this will give the proof that the topology which gave the perfect o

o

duality in c~) (i.e.induced from the canonical topologies in H$(Bc,,.A) and H¢(B~,,Oo " " q ),by B~ o

¢)

induction) coincides with the canonical topology on H2(B,,, .hi) and H 2 - I ( B d , .h/[). For ~) one can study the following exact sequence for every rI < ei o

H"-:(b,, \ B,, M)

\ B,,A)

H " - I ( B , , \ B , , O q ) ¢---~H"-I(B,, \B,,2t4) --, O.

(3.21)

B,i

The limit of (3.21) for the various q < e~ gives (3.18) and the canonical topologies on it. It is enough to show that the maps in (3.2t) are homeomorphisms for the canonical topologies. o

Taking ,then, a countable afflnoid covering/4 of B d \ B, s one can calculate the previous maps using the sequence of Fr4chet spaces:

0 ---*C'(/4,A) ----~C'(LI, O q ) ---.C'(bt, A4) --~ O.

(3.22) o

Using the open mapDing theorem [BGR], [BOU] and the facts that H"-a(B~, \ B~, 0 q ) is a Hausdorff space and that all the maps are continuous we obtain the desired conclusion. So, we know the theorem for •

o

o

Hc(B¢,,A4 ) and

Ext" ( B~, , .Ml, ft~,~ )

o

o

where

H ' ( B d , 2,4) = lira H~, (BE,, A4) rt<e i

with the canonical topologies. One notes that: o

Hc (B~,, M) = lira/-/B. (AK, M)

161

and the isomorphism is, actually a horneomorphism (3.8). For (3.14) we now need to prove that o

limExt°( Bd,A4,~2k ) = Ext°(A"K,./t4,912b ) o

ca

li) H:-'(B~,, M) ; lim(lim H~'(B~,, M)) = H : - ' ( A L M),

(3.23)

r/<e ~

endowed with the canonical topologies, are in perfect duality (each one the strong dual of the other) via the pairing (3.2) and that Ext°(A~g,M,12~:) is reflexive. The canonical topologies coincide with the topology given by the two limits in (3.23), once we have imposed the canonical topologies for each "U-step". We write o

o

Ext°(Bd, 3d, Ft~: ) ~

Ext*(Bd-,, A4, ft2~: ) ~

Ext°(Bd-,, .M, a ~ )

and the space in the middle, which we will denote by Di-1, is a Banach space of countable type. All maps in the sequence {Di} are continuous and with dense image (as those in the inverse limit in (3.23). Use proof of (3.10)) : if the field K is spherically complete the maps axe c-compact, if not each Di is reflexive [SCH],[vRO]. Take the strong dual sequence o n~°

!~rc

o

(Be,,M) e--, Di-l' ~ H en - - o (Be,-,, IvY)

and get an injective limit (of injective maps) of Banach spaces. For the two limits we have the hypotheses of the theorems of duality ([MO-SCH],[MOH]): it follows that the inverse limit of {Di}, E x t ' ( A ~ , M , ~ , ~ ) , with inverse limit topology (which is the canonical) is reflexive and in perfect duality (via the pairing) with the direct limit of {D~'} , H ~ - ° ( A ~ , M ) , with the direct limit topology (which is the canonical topology) Q.E.D.

§4 We want to extend the Serre type duality to the case of a rigid Stein space. We recall the definition (note the difference from that for quasi-Stein [KI],[LIU]). D e f i n i t i o n 4.1.[LU] A rigid analytic variety X is a Stein space if it is separated and if there'exists an admissible countable afflnoid covering {Ui}ieN , Ui = S p m A i , such that

U~ c Ui+~

and Ox(V~+,) --~ Ox(Ui) is dense, and such that the Ui are defined by

v~ = {~ ~ vi+~

I/.~+~(~)f-
with ai+1 E K * , [ ai+l [< 1 and ( f l +l) a topologically generating system of Ai+l. As in the classical ease it was shown by [LU] that there exists a closed immersion of X in an affine rigid space i : X --~ A~.

162

By means of this map we want to transfer the duality for X in A ~ . generalities about closed immersions and the relative adjoint functors.

We first need some

D e f i n i t i o n 4.2. A morphism i

: X 1 --~

X 2

of analytic rigid varieties is a closed immersion if there exists an admissible affinoid covering {Ui}iez of X2 such that the induced map, for each i,

i-' (udA-,u~ is a morphism of affinoid spaces, i - l ( U i ) = SpmBi, Ui = S p m A i and the inherent map

is sudeetive ([BOa]9.5.3). R e m a r k . In general it can be proved that in a closed immersion the inverse image of an affinoid is an affinoid. We want to show how a generic closed immersion i

: Xl ~

-¥2

(4.3)

represents Rig(X1) as a closed subtopos of Rig(X2) [SGAIV]. By definition (4.2), i(X1) is a closed analytic subset of X2 ([BGR],[FRE]). In the saturated topology of X2, U = X2 \ i(X1) is an open admissible:

j

: U=X2\i(X1)~X2

By the fact all the morphisms in the situs axe injections, U will be an open object in Rig(X2) and will characterize an open subtopos. From i (4.3) we have

i.

: Rig(X1) ~

Rig(X2)

P r o p o s i t i o n 4.4. The functor i, is a £ully £aitb_full functor. An object jr E obRig(X2) is a sheaf of the form i.G, Q 6 obRig(Xt), if and only if j ' J : (restriction) is the ~nal object in

nit(u). P r o o f . From i of (4.3) we get a morphism of topos which we write in the same way. By general theory i, asad i* (inverse image as sheaves of sets) axe adjoint. The functor i, will he fully faithfull if ([SGAIV]VIII.6):

i*i,~ ---* G is an isomorphism for every ~ E obRig(X O. It is enough to prove this (for example) on each affinoid which defines the immersion (4.3): ({Ui}iEz), i - l ( U i ) in X1. On each such affinoid we show the isomorphism of sheaves for the weak rational topology [FRE] (The weak rational topology is a Grothendieck topology where the admissible opens are the rationals and admissible coverings axe the finite rational ones. The relative strong topology is the usual one (§1. and [BGR]). The usual "unicity" theorems connecting sheaves for the weak rational topology and the relative extension to the strong still hold). In this case if we take W C i - l ( U i ) rational, there exists W ' C_ Ui such that W ' Cl i - l ( U i ) = W ([FRElpag.232). Then:

i*i.G(W) = 9 ( W )

163

as presheaves. But, in particular, ~ is a sheaf for the weak rational topology, too: it follows that i*i,~ and G are isomorphic as sheaves for the weak rational topology (see [BGR]). Since the extention to the strong topology is unique, they will be isomorphic for the strong topology, too. For the second part of the proposition we will work as in ([SGAIV]VIII.6). It remains to show that if f" E obRig(X2) has a restriction to U which is the final object, then there exists a G E obRig(X1) such that i . ~ = f . This is equivalent to the fact that the canonical homomorphism .T ---+ i . i *.T (4.5) is an isomorphism on X2. In order to prove this isomorphism we restrict , as before, to an affinoid Ui of a covering {Ui} related to the closed immersion. On each such Ui it is enough to prove that there is an isomorphism in (4.5) at the level of presheaves for the weak topology ([BGR]. Here the weak topology on affinoid space is the topology in which the admissibles are tile afilnoids and the admissible coverings the finite affinoid ones). Take W C_ Ui affinoid and consider at level of presheaves i,i*J:(W) ~ - - ~:(W). What we need to prove is the following fact : take an admissible open T of Ui such that T n i - l ( U i ) = W Cl i - l ( U i ) then, in the previous hypothesis on .%', 5r(T) : a~(W). Of course one can suppose T _C W and Ui = W. The new situation is i-~(Ui) C T c W

i-~(Ui) N W _c T.

For our purposes it fs enough to show that there exists an admissible affinoid covering {Xa} of W such that is an isomorphism for every c~. n u t W = S p m A and i -1 (U/) cl W = V(A) ,A = (f~,... , fk), fi E A ( A affinoid algebra is noetherian), by definition of closed immersion. Then there exists r Et K* I such that T _DV,(A) where

V,(A)={xESpmA; Put Zi = {x E W = S p m A

]fi(x) l < %

iE[1,k]}

[ fi(x) I > _ ~r } i = 1 , . . . ,n. Then

ff = { u d A ) , z l , . . .

,z, }

is an admissible attinoid covering of W and Zi f) i-a(Ui) = @and (Zi n T) C) i - l ( U i ) = @. Q.E.D. R e m a r k . The idea of the covering is from Fresnel's book [FRE]. In particular (4.4) shows how to a closed immersion i

: X---'A~

there corresponds a closed subtopos of R i g ( A ~ ) complementary of the open Rig(U), U = A~( \ i ( X ) . From the previous results, as in ([SGAIV]VIII) :

i, : Rig(X)ab - - * Rig(AnK)"b i* : Rig(A~)ab ~ Rig(X)~b

(4.6)

164

induces an equivalence of the categories of the abelian sheaves on X and the abelian sheaves on A ~ whose restriction to U = A~c \ i(X) is zero. In particular i, is an exact functor [SGAIV]. Using the structural sheaf OA k on Rig(A~) and the induced one i - I O A } ( this is the inverse image as set. For visual reasons we decided for this notation instead of i*OA} : i-aOA} is the n o t a t i o n for topological spaces) on X ([SGAtXqIV.14), the functors of (4.6) specialize to

i,

: Rig(X)i-~o,.

i"

: Rig(A~)o..~

~

Rig(A~)o^~¢

---, Rig(X)~-,o.~

(4.6)

which maintains the same properties as (4.6). In particular for i. there exists a right adjoint ([SGAIV]IV.14):

: Rzg(AK)o,) ~

Rig(X)i-~oA~

i.e. if .7-E Rzg(X)i-~OA~ and 9 C Rog(AK)oA.K:

Horno.~ (i..7:, 9) = Homi-~o^~¢ (.F,i:9)

(4.7)

Using the definition of ([SGAIV]IV.10.5.1) one can show that (4.7) can be localized in (4.8) (~om(5r, 9) is the sheaf which represents V ~ Horn(J r" x 17,9). The functors i , and i* are exact : i,(~" x i'V) = i , ~ x i,i*V = i,2:" x V because i,.7" has restriction to the complement of i(X) which is the final object ([SGAIV]IV.14.3)). In particular i ~ is left exact and it sends injective objects to injective ones. It should be noted that if we restrict i, to

i.

: Rig(X)o x ~

Rig(Ang)o.~

(4.9)

the adjoint i: can be defined " n : Rzg(Ag)o.k ----* Rig(X)ox

i !

maintaining the same properties [SGAIV]IV.12):

([SGAIV]IV.14.5).

In fact from the previous results (see

Horno^k (i.F,9) = Horni-~o^k (.U, i ' 9 ) = Horni-~o,~: (F, Homi-~o,~: (Ox, i ' 9 ) ) = = Horni-*o,~: (.U, i*Homo,~ ( i . O x , 9)) = Homox (.F, i*Homo^~ (i.Ox, 9)). So, in the case (4.9),

i'g = i'~o,,o^~, (i.Ox, g).

(4.10)

(i*, image inverse as sheaves of sets : it is exact) This agrees with the notion of i ~ in ([HARD]II.6). R e m a r k . These arguments are valid for every closed immersion. We want to specialize the closed immersion (4.3) to the case in which X is a Stein space, irreducible and regular of dimension q. In this case there exists a closed immersion i

: X ~

A~¢

(4.11)

165 and all the properties we found still hold. particular case.

We study i ! and its derived functors in this

If I is the coherent sheaf of ideals of A~¢ defining the closed immersion (4.11) we can write 1 Z/Z2 ~ ~A~ ® O x ~ f~lx/K -----* 0 (4.12) which is an exact sequence of coherent sheaves ( f~lA~c and ~2~;/K are locally free by hypothesis of regularity [F-vdP], [BKKN]). One can show exactly as in the algebraic case ([HAA]II.8), using some results obtained by Fresnel ([FRE]pag.222) on the dimension of irreducible rigid varieties, that (4.12) is an exact sequence of locally free sheaves on the left ,too :

0 - - * Z I Z 2 --* f~lAk ® O x ----* :~21X/K ----+ 0

(4.13)

and Z is locally generated by n - q elements (locally in the rigid sense . . . ), if q = d i m X . From (4.13) we can write [HARD]:

A,,-q 5~1i 2 ® ^q~2~XlK

AncI 1

~OAT~ ® OX-m~

Putting A

n~'~l

n

A~: = ~A~

and

^q~x/u = ~qx/K (invertible sheaves), it is possible to rewrite the previous isomorphism:

i*~

®LOA~¢/X '~ ~qX/K

where WA~lX ~ (A'*-qz/Er2) ' and i*, this time, indicates the inverse image as sheaves of '! n modules. We will now study *'flAk and its derived functors. To this end we consider (4o10):

sx¢~,,~

(Ox, f12~, ).

(i* the inverse image as sheaves of sets is exact, O x is understood in A ~ i . e . I . O x ) . By previous considerations (3.3) , these sheaves are coherent ( i . of a coherent sheaf is coherent [BGR]) : we can study them locally (using global sections on affinoid domains). By choosing an admissible atEnoid covering {Ui} of A~c such that on each Ui = SpinAl

ox, o,

Ai =

with {f~i)) an A-sequence, ZIv , "~ ( f ( 0 ) : one can work exactly as in [HARD] by mean of t h e Koszul resolution associated to {f~i)}. We conclude i

n

~gXltti(OXIoi,~-~A~clUi) = 0

If i ~ n - q ,

and then by local arguments i

n

E~bx(OX,OA~ ) = 0

i T C h - - q.

166

For the case i = n - q , always from the complex, on each Ui there is an isomorphism, which will depend on the choice of {f~i)}: fin v.

~

K Iu~

ZIu" f~A"

*

*

Ui

K

All these local (on Ui) isomorphisms can be glued using the morphism

A~< [U~ Z l U ' f l ~ : IU~

det(c,~ )

tu~nu~

AnK [Uj Z l ~ fl'~: IUj I~:,nuj

in Ui n Uj, where (cii) E OA~(UI ClUj) is the matrix of the map between: (f~) and (f~). We can then define a global map which is an isomorphism

: &gTJ(Ox,a2~)

:+

~o,nox(/, - ' T

(on each Ui, ~o composed with the evaluation on f~i) A . . . . . . A the following isomorphism

V

: gXto-xq(Ox,aX~)

_T_+

' ze2~

f(~2~ is ~ ' d

a2~®Ox®iA

e

n

)

We conclude with

~- "!

-'-~)

(4.14)

Applying the inverse image (as sheaves of sets) which is an exact functor:

i*&t~-,,q(Ox,f~2 K. ) ~-

i*

n ~A~ ®~A~,/x ~- f~x/~c

and i*gxtb~(Ox,fl~)

= 0

if i ¢ n -- q- In the language of the derived categories we state n Rz'! (f~A~) -~ ftqx/K[q - n].

(4.15)

We recall our situation : X is Stein , irreducible, regular of dimX = q and we have [LU] a closed immersion i : X --+ A~ (4.11). P r o p o s i t i o n 4.16. Under the previous hypothesis on X, if M is a coherent Ox-module, there exist the isomorpMsms:

gxtb,,.~ (i.M, ft~.,, ) = i . g4-("-q)t~o~w-,,'A a ~ ) P r o o f . The closed immersion i. is, in particular, proper : i . M is coherent [KIE]. To prove the proposition it is enough to take an injective resolution: 0 ----~ ft~,k - - * Z ° and apply

Horn(i,M,Z °) = Hom(M,i~'Z °) using the fact that i ! sends injective objects to injectives [SGAIV] (the same argument for 7from being i, exact). Q . E . D .

167

every sheaf.hi of Ox-rnoduIes:

P r o p o s i t i o n 4.17. For

H~(A~c,i.34 ) ~

H2(X,.M ).

P r o o f . We need to show that each affinoid V C_ X is the intersection of a finite union of afflnoids of A~: and X i.e. there exists I1,...,... Im admissible atllnoids in A ~ such that u~I~ n X

=

V.

Take {Ui}iEN an affinoid countable covering of A~- and study {Ui n X}iEN which is an admissible affinoid covering of X and {Ui fl V}iEN which is an affinoid covering of V (X is separated). This implies that there will be a finite subcover : k

U ( v , n v ) = v. j=l

For each j = 1 , . . . , . . . , k

uj n v c u j n x c Uj where U] n V is a closed affinoid space of Uj. As in ([FREI,[BGR]) one can prove that there exists a finite number of rationals {Ij,t}'~=l of Uj (and so admissible affinoids of A ~ ) such that (Uzi,t)nWjnx)

=

vinv.

I

By the fact that i. (exact functor) sends injective objects into flasques which are acyclie for the functor "sections with support" ([SGAIVIV), we conclude. Q.E.D. R e m a r k 4.18. If .T is an Ox-module then

~o,~ox (~, -) of an injective object is a flasque sheaf ([SGAIV]V.4.10). In particular it is Pc(X, - ) acyclic. From (4.16) and (4.17) it follows that:

E=~o.~,o(A~, i.M, a3,~) i

n

= E=6-~(?~-') (M, a~).

R e m a r k 4.19. Trivially one has

H'(A~,i.Z4) = H'(X,M) (i. is exact, and sends injective in flasque ([SGAIV]V.4)). It follows that the only group different from zero is H ° . On the other hand the two canonical topologies coincide (§3. eft. remark (4.20)). R e m a r k 4.20. In X, for a coherent sheaf A4, it is possible to define a canonical topology on H°(X, M ) . By (4.1) X admits an admissible countable affinoid covering and the sections of a coherent sheaf on an afflnoid form a Banach space (2.0). The same argument can be repeated for tt~(X,M) since it can be written as the direct limit of spaces of the type ttn-I(X\U, M) where U is an afllnoid. For the relationship between the canonical topologies H°(A~c,i.M) and H°(X,M) it is enough to notice that i-l(U) of an a~noid tr C A ~ is an affinoid : the two topologies are the same. Finally we can state

168

T h e o r e m 4.21. If X is a regular, irreducible Stein space (4.1) of dimension q, the spaces Extqo-~( Jv4,flqx) and H I ( X , . M ) endowed with the canonical topologies (4.20) are, respectively, a reflexive Frdchet space of countabIe type, and a compIete, reflexive Hausdorff space There exists a pairing H;(X,M)

× Extq-x(.M,~lqx) - - ~ K

(4.22)

which is perfect (making one the strong dual of the other). Again for the canonical topology, H°(X, M ) is a reflexive, Fr~chet space of countable type, and there exists a pairing × E z t ~ X~ ¢(M,12qx) ~

H°(X,M)

K

(4.23)

making E:t~,~(M, ~ ) its du~. P r o o f . By [LU] there exists a closed immersion i

: X ~

A~-.

(4.11)

Then by (4.17) we have H:(A~,i,M)

-~ H ~ ( X , M )

and this is a homeomorphism for the canonical topologies (4.20). On the other hand by (4.16)

e~;:'.K (i.M,a;o)K = i.&4,-2(M,a~) and the isomorphism is a homeomorphism for the canonical topologies. In fact from (4.16) they axe the global sections of two isomorphic coherent sheaves on A ~ (4.20) (and a A-llnear map between two finite A-modules is automatically continuous if A is an a ~ n o i d algebra ([BGR]3.7.3)). We use the duality in A ~ to conclude. Same argument for (4.23). By the way, one can note that, putting the strong dual topology, arising from the pairing, on Extqox,¢(~4, flqx) the pairing (4.23) becomes a perfect duality (one space is the strong dual of the other) Q.E.D.

§5 Implicitly we have found a notion of residue for a regular, irreducible Stein space. One fixes a closed immersion i : X ~ A ~ . It is, in fact, possible to define a map ~

i

: Hqc(X,~qX/K) ~

n:(A~,fl,~)

in the following way: q

q

g~¢ ( X , ~'~X/K)

q

-- H eq( A Kn, ~ ,". ~ Xq / K )

~-- H ~ ( A ~ c , $ X t " - q ( i . O x , g t ~ ) [ n

n

-

'~

rt

~ H ¢ ( A K , t.RY~A~c[n - q]) --~

n ,~t~) - q]) ~_ E x t r a ~ ( A K" , ~ •. O X , ft~.~) ----+ H cn ( A K

(5.t) ( E x t n - q ( i , O x , ~%~ ) is the only cohomological sheaf different than zero). And finally

H~(X, '

.

.

.

.

One can ask if this notion is independent of the closed immersion we axe using. We will answer this question in the affirmative (up to a multiplication by a constant).

169

Consider another closed immersion for X

j

: X---,A~

we obtain the commutative diagram

h~

X

(i,j)

I.

A~:+M

(5.2)

AT~ where (i,j) is a closed immersion. It is possible to construct k., k., closed immersions such that the diagram

A~

x

("J)-~

\/:

A ~ +m

(5.3)

A~ is commutative. In order to see the existence of k, ( the method for km is similar) consider the sheaf of ideals 17 defining the dosed immersion i in A ~ . The facts that A~¢ is a Stein space and Z is a coherent sheaf (the affinoid algebras are noethcrian) imply that the sequence

OA~ (A~-] is exact. If we take ~- : i ( X ) ~ A T and compose with the various projections of A T we get ~i C z (i (X) ) = -o' ^~.( A ~ c ) " By the previous observation each 3, can be extented to a global section Ji. Finally we can define kn(x) = (x, J1,... , jm). From the upper part of (5.3) we obtain the commutative diagram:

170

q

q

g~(X,~x/K)

-

(i,j)

O n+rn h

ggn+m/An+m

--c

~

wK

,°~A~+~'J

(For showing the commutativity one can use the fact that i ~, (i,j) ~, k" transform injective objects into injective objects §4, and work as in ([HARD]III.7). Note that Ri' ( R j ' ( a " . + T , ) [ ( n + AK = m) -- n])[n --p] ~ aPXlK "" R(i,j)'(a~+2,, )In + m - p ] ) (analogous situation for the lower part of (5.3)) Finally the problem of unicity is simply to compare H~(A~, ~)

~"

rz,+m;A,+-~ On+m -c ~ K ,~oAT+~j

(5.5)

~n+rn

K and we expect that the two residues ;

reSn

r e s n + m O ~n

are equal up to a constant. The point is to show that res.+m o k . satisfies a duality theory as resn did with the same canonical topologies on the various spaces : it will then sufficient to apply proposition (2.8). Applying §4 to kn there is an isomorphism (5.1):

HntAn+m ¢~m k n+m c t K , ,t.OA~+ta( n. OA~,~"~A~+m)) ~

n+m rt+m Exto^.K+~,c(kn,OA~.,~A~+~ )

(5.6)

(degeneracy of the spectral sequence). There exists the "dual" sequence:

E x t 3 x . K (gXt~x~+" (k..OAT,,

.+m ), ~A~+~ .+m ) ~ - ~A~+~

H°(A~:+m, k . . O A ~ )

(5.7)

By mean of the Koszul complex one can prove that m

trt

[L

¢r~

~n+m

~ (~n+trt

and so (5.7) is a homeomorphism for the canonical topologies, too. But (5.7) is in fact the transposed situation of (5.6) using the duality theory in °~+m From the reflexivity of the spaces which appear one concludes that (5.6) is a homeomorphism for the strong dual topology via the pairing in on+"* Note that the strong dual topology for

is the canonical one §3. On the other hand

Hc. (AK,n f/AK ) n. ~Hc" ( A~-~+m.., k n . ~ A ) ""K ~ H~ (A~:+m, CXt~ . +,. (kn*OA~, °"+m"~A~+m ~ H AK

171

are all homeomorphisms ((4.20) §3). it follows that reSn+m

o'kn

n n n : H e (AK,~A~c)

n-Fm

----+ E x t o ^ ~ + ~ , c ( k n . O A ~ g , ~ A ~ + ~ -"-*

nq-m

ir2fn+m[An+ra "~ t"K

)

c}nTrrt x, resn+,n ,OOA,,+,,,/ ----~ K 1¢

is continuous for the canonical topology and it satisfies the duality for this topology. We can apply (2.8). N o t e 5.8. Professor Berthelot pointed out to us the possibility of giving a "unicity statement" for our residue. Consider X a Stein space and fix (possibly after an extension of the ground field K) a K-rational point. Consider local coordinates and impose that in those coordinates the residue is defined as in §2. This notion is unique (note 2.12), at least, for projective automorphisms of the coordinates.

Bibliography [AM] Amice Y. : "Les nombres p-adiques". Presses Universitaires de France, Paris (1975).

[AIq Altman

A.; Eleiman S. : "Introduction to Grothendieck duality theory" . Lecture Notes in Mathematics 146, Springer-Verlag, Berlin (1970).

[AR] Artin

M. : "Grothendieck topolog!ies". Notes on a seminar by M. Artin, Harvard University (t962).

[BER] Berthelot P. : "Cohomologie rigide et cohomologie rigide £ support propre". A paraitre dans Ast~risque. [BGR] Bosch S.; G/intzer U.; Remmert R. : "Non archimedean Analysis". Grundlehren der Math. Wissenschaften 261, Springer-Verlag, Berlin (1984). [BEEN] Berger R.; Kiehl R.; Eunz E.; Nastoid N.J. : "Differentialrechnung in der anMytischen Geometrie". Lecture Notes in Mathematics 38, Springer-Verlag, Berlin (1967).

[BOU] Bourbaki N. : "Espaces vectorie}s topologiques". [CA] Cartier P. : "Les groupes E x t ~ ( A , B ) ''. S~minaire

Hermann, Paris (1958). Grothendieck t.1, 1957 n.3 Secretariat

Math., Paris (1958). [FRE] Fresnel J. : "Cours de g~om~trie m~alytique rigide'. Universit~ de Bordeaux I (1984). [F-vdP] Fresnel J.; Van der Put M. : "G~om&rie analytique rigide et applications". Progress in Math. 18, Birkhauser (1981). [GROW] Grothendieck A. : "Sur quelques points d'alg~bre homologique'. Tohoku Math. J. 9, n9-221 (1957).

[GaUl Gruson

L. : "Fibres vectoriels sur un polydisque ultrara~trique'. Norm. Sup. 4 ¢ s~rie, 1, 45-89 (1968).

[HARA] Harthshorne [HARD] Harthshorne

Ann. Scient. Ec.

R. : "Algebraic Geometry". Springer-Verlag , Berlin (1977).

R. : "Residues and Duality". Lecture Notes in Mathematics 20, SpringerVerlag, Berlin (1966).

[IN] Ingleton

A.W. : "The Hahn-Banach theorem for non-Archimedean valued fields". Proc. Cambridge Phil. Soc. 48, 41-45 (1952).

172

[KIE] Kiehl R.

: "Der Endlichkeitssatz fur eigentliche Abbildungen in der nichtarchimedischen Funktionentheorie". Inventiones Math. 2, 191-214 (1967).

[KI] Kiehl R.

: "Theorem A und Theorem B in der nichtarchimedischen Funktionentheorie". Inventiones Math. 2, 256-273 (1967).

[KO] Kopf U.

: "Uber eigentliche Familien algebraischer Varietaten uber affinoiden Raumen'. Sehriftenreihe Math. Inst. Univ. Miinster 2 Serie, Heft 7 (1974).

[LIUt Liu

Q. : "Un contre example au crit~re cohomotogique d'att:inoidit6". C.R. Acad. Sci. Paris t.307 S6rie 1, 83-86 (t988).

[LU]

Lfitkebohmert W. : "Steinsche R£ume in der nichtarchimedischen Funktionentheorie'. Schriftenreihe Math. Inst. Univ. Miinster 2 Serie, Heft 6 (1973).

[LUTI L{itkebohmert

W. : " Formal-algebraic and rigid-analytic geometry". Math. Ann. 286,

341-371 (1990).

[MOH] Morita

Y. : " A p-adic theory of hyperfunctions I ' . Publ. Res. Inst. Math. Sci. 17, 1-24 (1981).

[MOSt Morita

Y. : "A p-adie theory of hyperfunctions II". in "Algebraic Analysis", vol.I, 457-472, Academic Press (1988).

[MO-SCH] Morita Y.; Schikhof W.H.

: "Duality of proiective limit spaces and inductive limit spaces over a non-spherically complete non arehimedean field". Tohoku Math. J. 38, 387-397 (1986).

[R.R.] Ramls

J.P.; Ruget G. : "Complexe dualisant et th~or~mes de dualitfi en g~omfitrie analytique complexe". Publ. Math. I.H. ES 38, 77-91 (1970)

[vaOl van

ROOIJ A.C.M. : "Nonarchimedean functional analysis". Monographs and Textbooks in Pure and Applied Math. 51, Marcel Dekker, Inc. Basel (1978).

[SCHI Schikhof W.H.

: "Locally convex spaces over non spherically complete valued field I, II". Tijdschrift van her Belgisch Wiskundig Genootschap Serie B, fasc. I, 28 187-224 (1986).

[SED] Serre J.P. : [SGAIV] "S~minaire

"Un thfior~me de dualitY". Comm. Math. Helv. 29, 9-26 (1955).

de G~om~trie Alg~brique. Th~orie des Topos et Cohomologie ~tale des Schemas". M. Artin ; A. Grothendieck L.N. in Math.~ 269-270 (1972).

[svot

Suominen K. : "Duality for coherent sheaves on analytic manifolds". Annales Ac. Scient. Peunicae, 424 (1968).

On the Frobcnius Matrices of Format Carves Robert F. Coleman Deparurent of Mathematics University of California Berkeley, CA 94720

Let p be a rational prime. If A is an abelian variety with good reduction over an tmmmified extension K of Qp, then there is a natural Frobenius-linear endomorphism of H~)R(A,K). The Jacobian of a Fermat curve F m : xm+ym = 1 has good reduction over O~ so long as (m,p) = 1, and in [C-GK] using ideas of Katz, Gross, Koblitz and Dwork we computed the matrix of the corresponding endomorphism with respect to the the basis represented by the differentials of the second kind on F m corn,i,j = xiyJ(y/x)d(x~) where {0 < i < m, 0 < j < m, i+j ~ m} in terms of special values of the p-adic gamma function at rational arguments. Here, as we shall do henceforth, we identify the first de Rham cohomology group of F m with that of its jacobian. In general, the jacobian of F m 'has good reduction over an algebraic closure Qp of Qp but not over an unramified extension of Qp (see [CM]). As a result we no longer have a canonical Frobenius-linear endomorphism. Instead we have a canonical action of the crystalline Weil Group (see [BO] or [CM §2]) on H~R~m,K). We call the corresponding matrices Frobenius matrices. In this article, we use the results of [CM] on the stable model o f F m as well as those of Ogus [O] on CM motives, in this volume, to compute these rnalrices when p > 3. We were not able to complete the computation using only what we knew about the stable models. Gratifyingly, we were able to obtain just enough information about these matrices to be able to plug into Ogus' theor3r and finish the job. "Fnese matrices involve special values of an extension of the Morita gamma function to all of Qp and bear a striking resemblance to the formula for the complex periods in terms of the classical gamma function. This is explained, in a large part, by an observation of Anderson [A] elucidated in this context by Ogus [O], which asserts roughly that the "Fermat motive" factors naturally into the product of three versions of the same "ulterior motive." In Section I, the problem is precisely stated. We define the extended gamma function in Section II and describe our main results in Section IIL In Section IV, we extend the methods of Katz [K] for computing Frobenius matrices to the case of bad reduction. The relevant results of [CM] are recalled in Section V and we complete the computation of the man'ices by Sections VI and VII. Finally we note that from the results proven in this paper, one may deduce the formula for the 1 ~ 1 co~ts of Jacobi sum Hecke characters proven in [CM]. Thus this formula now has this Crystalline proof, an l -adic ~tale proof given in [CM] and a p-adic &ale proof sketched in [C-AI]. We are indebted to Ogus for suggesting the problem solved in this paper as well as for explaining to us how to apply his recent results [O].

L

'I~

General Set Up.

Fix an algebraic closure (~p of % a non-trivial non-archimedean absolute value I Ion ~ and let v be the valuation on Qp such that v(p) = 1. Let I = {(r,s) ~ O f Z ~ O f L r ~ O, s $ O, r+s $ 0}. To each element_ (r,s) in I

and each element a in the Weil group of Qp (seebelow) we shallassociatean element ~c~r,s)in Qp. For x e O j Z or Q, we let (x) denote its unique representative mod Z in (0,1]c'~ ((x) = x-[x], if x ~ Q - Z

174

and 1 otherwise). For g, a funcdon defined on Q , and r e Q r/,, we will let g(r) denote g((r)). Let (r,s) e I. Set e(r,s) = (r~-(s)-(r+s) and L(r,s) = (r+s)e(r's) (this equals (-1)~r's)K(r,s) in the notation of [C-GK]). •

1

Let m --

be a positive integer such that m.(r,s) = 0 and let Vm,r,~ denote the class of mL(r,s)o~,ij m HDR(Fm,Qp) where i = re-(r) and j = re-(s). If n is a positive integer divisible by m and f : Fn -4 Fm is the obvious map, then (1.I)

f*Vm,r,s = Vn,r,r

Remark. Note that Olin,i,j = -xi'myJdy/y = m'lxi'myj'mdxm.

(1.2)

Suppose K is a fmite extension of Qp. The Well group of K is the subgroup W(K) of Gal(0p/K) consisting of elements o whose restriction to the maximal unramified extension of K is an integral power of Frobenius. This integer will be called the degree of o and denoted deg(o). We set W = W(Qp). Now if X is a curve over K with stable model over a Galois extension K' of K in ~p and ~ • W(K) such that deg(~) > 0 there is a canonical element q)(c) e E n d ( ' ) (see [CM] §2). If ~ is arboreal (which is equivalent to the Jacobian of X having good reduction), this endomorphism induces an endomorphism **(o) of HIR(X,K ') and this map extends to an anti-homomorphism from W(K) into EndK,(HIR(x,K')). Moreover, if Y is another curve over K with arboreal reduction over K' and f : X -4 Y is a morphiun defined over K', then as morphisms from H~R(Y,K,) to HDR(X,K 1 ,), (1.3)

~,(o)of °*

=

f*o~*(o).

In [BO], a semi-linear action Peds of the crystalline Weil group Wefts(K) on HIR(X,K ') is defined. When K = t ~ Weds(K) is just W. Moreover, G a l ( ~ K ) acts on H~)R(X,K') ~ HIR(X,K)®K ' and if o • W(K), peds(o) = ~*(o)oo (see Theorem 4.8 and the remark following its proof in [BO]). We let Gal(Qp/Qp) -act on ( O / Z )u, u >_ 0 e Z , by setting or = or, for o E Gal(0p/Qp) and r e (O/Z) u, where P is the finite id~le of Q which describes the action of o on roots of unity, (i.e., the inverse image of

o[~

under the

~ d n map).

Proposition 1.4. Let o • W. Suppose q e I and m is a positive integer such that rnq = 0.

¢,*(a)Vm,o.~ q = 13dq)Vm,q for some [~o'(q) e Qp which does not depend on m (because of (1.1)). We now fix m and drop it from the notation unless there is danger of confusion. Suppose q • I. Using the fact that ~ * is an anti-homomorphism we deduce, (1.5.1)

~ ( q ) = ~dx'tq)13x(q).

Using the fact that Peds is a homomort~sm, we deduce

(1.5.2)

~o~(q)

=

~g°kO°~dq) •

175 Combining 0.5.1) and 0.5.2) v,,e see that 13~q)a = ~o.(o3" For q • I, t.t(o,r) = (olr)o-(r} • QIW]. If 0t • Qp, ¥ • Q[W] we let ctv =(ctlth) nv for any positive integer n such that n-~ ~ Z[W] and any n-th root in Qt,' ctl~" of ct. It will be implicit in use of such an expression, without further qualification, that the resulting element of (~p is independent of the choices of n 1/n and ~x . ~nm

1.6. Let (r,s) • L t = r+s and o ~ W. Then 13o(s,r) = ~o(r,s) and 13o.(r,-t)= (-1)~°'s)-~°'t)[3o(r,s).

Remark. (-1) p'(°'s)~(°'t) = +1 and if p is odd, u ~ O f / a n d deg(o) = 1, (-1) gt(o'u) = (-1)p(°'lu)'(u) regarding p(o'lu)-(u) as a 2-adic integer and this latter equals [p(olu)] when (u) • Zp. Proof. This follows from (1.4) using the automorphisms defined over Qp, f : (x,y) ---) (y,x) and g : (x,y) (~t/y,1/y) of Fm where ~ is an m-th root of -1 and the facts that f*vr,s = Vs,r and g *Vr,s = (" 1)¢(r's)c(r)vr,-(r+s)" [] The following result is Theorem 19 of [C-GK]. Theorem 1.7. Suppose q = (r,s) • I. If p does not divide the order of (r,s), and deg(c) = 1, then

13o(o0 = (- 1)r(q)p¢('°" ~q)rp/Fp(r>rp(s>. Note. The sign here is different from that found in [C-GK] when p = 2. "Dais discrepancy is explained in the section at the end of this paper, 'T_.rram to [C-GK]:' Both proofs of this theorem presented in [C-GK] use the fact ttat Fm has good reduction over Qp when p does not divide m. In ttlis paper we will apply the results of [CM] on the stable reduction of Fm to investigate the case in which p does divides m. It is also worthwhile to record the relationship between 13o(r,s) and the analogous numbers corresponding to the curve Fm : um+v m+1 = 0. For q • I, let rlq denote the cohornology class of the differential L(q)uivJ(v/u)d(u/v) where (i,j) = m(q). Defme To(q) by the formula ~*(O)rl o- lq = "/o(q)'qq 1.8. If q = (r,s) • I, ~/o(q) = (-1)~°'r)+~°'s)13o(q). Remark. (-1) g°'r)+'a(°'s) = +1. I f m is prime to p and ~ is an m-th root of -1 then ~o = ~p when p is odd, • . k(o,r,s)~, . . and (-1) m = -1 when p = 2. In these ~ , yo(r,s) = t-l) po(r,s), where k(o,r,s) = [p(olr)] + [p(o-ls)] when p is odd and (o'lr)-(r) + (o'is)-(s) when p = 2' Proof. This follows from (1.,1) using the morphism f: (u,v) ~ (~u,~v) from F m to F m and the fact that mf*v

- rm((r~s)~ r,s - ~ r,s ' []

Prolmsition 1.9. S u p p o s e m i s p r i m e t o p a n d q = ( r , s )

~ I, mq=O. T b e n i f d e g ( o ) = 1,

176

To(q) = (.p)e(-a-'q)r,p
1rp<s>lFp<-(r+s)>"1.

Proof. This follows from Lemtrm t .8 and the remark following it, using: If t e Q/Z, mt = 0 and t ~ 0 , / ( t ) = p-[p~oIt)] and [
IT.

An extension of the Morka G a m m a

Function to Qp.

If x • • , we let (X)p= (x mod Zp) identifying ~ with the subgroup Z[1/p]/Z of O fL in the natural way and if x e Qp, we set [X]p = x if x e Z p and x-<X)potherwise. In addition, if x • Q/Z, we let Xp denote the image of x in Qp/Zp under the natural projection. If a is a non-zero element of (~p with integral valuation, c0(a) will denote the unique root of unity of order prime to p in Qp such that Ic0(a)-a/pV(a)l < 1. In this case, we also set ~(a) = pV(a)c0(a) and a* = a/~a). Note that ~c(r)= K(rp) when r • Q/Z, rp ~ 0. The natural ring homomorphism from Z into Ro =: Z/(p-1)xZp extends naturally to an injective ring homomorphism from Z[1/p] into R =: Z/(p-1)XQp and we identify Z[1/p] with its image. For a • Qp, we set {a} = 1 if ~ e pZp and pV(a)a, otherwise. For ct e Z[l/P]>o we set

(2.1)

t'p(o~a) =

I-I{+k}

where the product runs over integers k, 0 <_k <_[ct]- 1. If z = (x,a) e R, we set s(z) = a. Then Fp extends uniquely to a continuous funcuon from R into Zp. It is determined by the functional equation (2.2)

['p(Z+l) = {s(z)}['p(Z)

and the normalization Fp(z) = 1 for z e Z[1/p], 0 < z <_ 1. For z = (x,s) E R, set [z] o = (x+l-(-s)p,[s]p) which lies in Ro. (Note: For a e Z[1/p], [(ct, a)] o = ([a],[a]).) For z = (x,s) ~ R o and a • Z ~ we set aZ= c0(a)Xa*s. Then f'p also satisfies (2.3)

f'p(Z)['p(1-z ) = (-1) i(z)

where i(z) = (z-1)-[(z-1)/p] for z in Roand i(z) = [z]o otherwise. Moreover ~p(Z) =- {s(z)} |z]° modulo s l Z p . For s e Qp w e set Fp(s) = f:'p((S),S) when p , 2 and F2(z) = e(z)['2(0,z) where ~(z) = -1 when Izt _> 1 and 1 when tzl < 1. The restriction of Fp to Zp is the Morita gamma function, q'hen when p ~e 2, and Izl > 0, (2.4)

['p(Z) = c0(s(z))[Zl°Fp(s(z)),

It follows that, (2.5)

Fp(S) -= (-1)P'Is*[SIP rood s l Z p ,

177

S'Up(S) when. Isl > 1 (2.6)

Fp(s+l) =

-{Slrp(S) when Is~ _< 1. Also,

Fp(S)Fp(1-s) = (- 1)/ (s)

(2.7)

where

g (s) =

0 [s]2

Ist> l , p > 2 Isl > 1 p = 2

(s/p)p I+[s/212

Isl < i, p > 2 Isl _< i, p = 2

In the above and the following, we regard an element a of Z[1/p] as contained in Z 2 when p is odd and a appears in the exponent of -t. We have the following Gauss multiplication formulas: If n > 0, (n,p) = I then, n-1

,-[sl

n-1

(2.8) n-1

1~ rp((s+i)/n)

=

rp(s)n n('~) nl llF_Ci/n)

if Isl -< 1.

where B(s) = z-[z/Plo for any z ~ R such that s(z) = s. F'mally,

"~Fp((s+i)/p) = Fp(s) if Isl > 1 and (2.9)

We state the duplication formula when p > 2 and Isl > 1 which is a special case of (2.8): (2.10)

rp(s)rp(s+l~) = rv(2S)rp((S)p+(-1)v/2)(1/2*)I2~lp.

(Note that Fp((2S}p,r2)Fp((2s}p/2+l/2) = Ip((S)p+(-1)(2S)p/2) = Fp((S}p+l/2){(S)p -1/2}*e(sP'@.) The proofs of the above formulas follow the same lines as the proofs of the corresponding formulas in [C-GK]. For future reference, we record: Suppose (r,s) e I and (r+S)p~: 0. Then (2.11)

Fp((r)+(s)) = Fp((r+s))L(r,s)~(r+sff (r's).

178

III.

St~m~

of th~ Main Ram~Im.

ff A is a group acting on an abelian group M on both the right and the left, we say these actions commute if (om)x = a(rnx) for a, "~• A and m • M. f f g a is a one-cocycle on A with values in M for the left action, we say that go is a left cocycle and if for the right action, we say go is a right cocycle, ff it is a cocycle for both these actions, we call go a bi-cocycle. The unmodified expression "cocycle" will mean left cocycle. Suppose S is a set on which W acts through its abelian quotient (like Q/Z) and T is a left W module like t)~p. Let M(S,T) denote the group of functions from S into T. We let W act on M(S,T) on the left via (aH)(r) = H ( a t r ) a and also on the right via (Hc)(r) = H(o4r) for H e M(S). We set M(S) = M(S,Qp). These two actions commute and by formulas (1.5.1) and (1.5.2), 13a is a bi-c.ocycle with values in M(I). We now reformulate some results of [O] in our context. Let ~.~ denote the group of roots of unity in Qp. Let Fl(O)(r) = pF~ (c)(-r) in the notation of [O] (rt is a prime of Q above p in [O] but its choice turns out not to affect Frl(o)(r)). Then F t ( a ) is a left cocycle on W with values in M(Q]Z,0p/g**) and it follows from Lemma 1.8, [O; Theorem 2.5] and [O; Theorem 3.13] that (3.1)

rl(a)(r)rl(a)(s)/Fl(a)(t) = I]o(r,s) mod ll~.

for (r,s) • I, t = r+s. Remark 3.7 of [O], after applying a Tate twist, asserts in our language, l ~ m m a 3.2. Let S be a subset of Q ~ stable under the action of the finite id~les and multiplication by 2. Let I(S) = Ic'~S2. Suppose T a is a cocycle on W with values in M ( S , Q p / ~ such that To(0 ) = 1 if 0 • S and To(l/2) = pdeg(a)~ if 1/2 e S. Define a cocycle B a on W with values in M(I(S),~,/Iz~ by Ba(q) = TO(r)TO(s)/Ta(t) for q = (r,s) • I(S), t = r+s. Suppose for every r e S-{0,1/2) and o e W, Bo(r,r) = [3o(r,r) mod tx~. Then, BO(q) = ~a(q) mod ~t~, for all o • W, q • I(S). Moreover, TO(r) = Fl(C)(r). l.,eanma 3.3 Let A be a group and d a homomorphLsm A onto Z

Suppose M is a left A module and for

each c • A such that d(a) = 1 we have a go • M such that ol(gox-go) = gr~-Xgo, for a, x • WU(Qp) such that deg(a) = 1 and deg(x) = 0, is independent of a defines a cocycle on the kernel of d. Then go extends uniquely to a left cocycle on A with values in M. Proof. Uniqueness is clear. We will def'me gx for x • A, d(x) -> 0 by induction on d(x). For d(x) = 1, it is already defined. For d(x) = 0, we set g~ = a'l(gox-go). Suppo~ n > 1 and ~ is def'med for 0 < d(x) -< n and satisfies P(n) :

gox = agx+go, for a, x • A, d(a), d(x) > 0, d(ox) < n.

This is true f o r n = 1 by hypothesis. We will define g~ for d(x) -< n+l so that P(n+l) is satisfied. Suppose a, A such that ct~ = x. If deg(oO, deg(~) > 0 set g~ = agl3+ga. We must check that this is independent of the choice of o~ and 13. Suppose 0 < d(p) < d(13). Let y = ~p and ~ = p-l]]. Then d(~) > 0 and "1:= 75 and 0tgl~+g~x= 0~(pg~5+g~+got = (~og~5+Ctgp+ga = yg~5+gy, by induction. By reversing the roles of ct and 13 in this argument we deduce that gx is well defined and satisfies P(n+l) for a, x ~ A, d(c), d(x) > O, d(tr~) < n. Suppose therefore that d(x) = n + l and x = t ~ with d ~ ) = 0. Write a = op, y = pl~ with d(p) = 1. Then, g.~ = ogy+go. = o(pgl~+gp)+g O = opgff,,-ogp+ga = ctg~+go, by

179

induction. Now for d(x) < 0, we set gx = -W,x-1 and one can easily check that go is a cocycle, o 3.4. Let A, M and d be as in the previous lemma but also suppose that A acts on M on the right and these two actions commute. Suppose that go is a left cocycle which satisfies the following property P'(n) :

gox = go~+g,: arid (Igo = goo for d(o), d(l:) _>0 and d(ox) <- n

for n = 1. Then gx is a fight cocycle as well. Proof. Suppose go satisfies P'(n) for n >_ 1. We will show it satisfies P'(n) for n+l. Suppose d(z) >_d(c) > 0 and d(ox) = n+l. Let x = cO. Then, as d(cp) __O, go = goP"gop = cgP+go-goO so cgp = and

gp+go(p-l)

2

gox = o gp+Ogo+g o = cgp+°(go(p-1))+°go+g o = ogp+go+goop = gx+gox, as og o = goo. One can make similar arguments when d(o) _>d(x) > 0 or either is in the kemel of d. Finally, OXgox = Ol:(goX+~) = o(Xgo+gx)x = Og.:ox = C(gxo+go)x = (ogn+go)OX = goxo'x. It is now easy to show that go is a cocycle for the right action.

Let O o denote the left cocycle on W with values in M ( Q p / / ~ such that Go(r) = Fp(r)"1 ifrp = 0 and

deg(o) = I and Gc(r) =: Fp(O'Ir)/Fp(r>if rp :~0. That this existsfollowsfromLemma3.3 and one can show it is a bi-cocycleusing Lemma3.4. Let X : W -~ Z~ denote the cyclotomiccharacter, Wx the kernel of Xand W%°the subgroup of W~ consistingof elementsdegreezero._ For o E W, we let o~a) = co(X(o))and Wo~the kernel of coin W. Let r~be a fixed(p-l)-st root of -p in Qp. For c of degree one, let (-1){'t(a'r)tY/l:<°'lrX°'P)Go.(r)if rp = 0, Fo(r) = -rp(lf2)tm(°~(2(r))(°-l)/2~Kr)~°'r)Go(r) if rp # 0.

Remark. Fo(r) is of the form ~Df(ct'r)Go(r), where ~ is a (p-1)m-th root of unity and fro,r) = (-o-lr) if deg(o) = 1, rp = 0 and fro,r) = deg(c)/2 + v(r)(
~(~) (~ Da ~(~) " ((2u(r))) " lc(r) ' ,

if rp = 0, ffrp~O,

180

from which it is obvious that Fix(r) =: (Fgt(pr)/Fp(pr)) p'l for p e W ~ deg(p) = l, is independent of the choice of p. Moreover, H x is a cocycle on W ~ with values in M(QfZ) (m fact, a coboundary ) whose restriction to 0 W x is a bi-cocycle. Now, since p is odd, 0)('t;)l't(°'r)K (rX't-l), Fm.(r)/Fa(z r)

=

if rp = 0.

o)(z)gt(a'vb)((2x(r))(~lg2~(r) ~t(x'r), if rp * 0.

It follows from I.emma 3.3 that F o extends to a left cxx:ycle on Wo~. It also follows easily from this and I.,enmaa 3.3 that B o extends to a left cocycle on W with values in M(I). Finally, we must show that the restriction of F o to W z is a right cocycle. Ftrst it is clear that for o of degree 1 in WX, o F a = Fo(I. For degree zero this same identity follows from the fact that F a is a bi-cocycle on W~. Next, if deg(a) = 1, deg('O = 0 and rp = O, r~(o'r)/Fo(CV)

= (x(r)~) (~'1) = F~(r) since x fixes all roots

of unity and Fox(r)/Fo(r) = n <°'lr>(m-°) = Fz(r). As we can also prove similar formulas when rp ¢ 0, the proposition follows from l_emma 3.4. [] L e m m a 3.6. Suppose, o E W, deg(o) = 1, t e Q / Z and t:~ 0. T h e n i f p is odd or tp = 0 Fo.(t)Fo(-t ) = (-1) ~t(a't) 6o(a)p. Proof. Suppose now that t # 0 and deg(o) = 1. First, if tp = 0, Fo(-t) =

(- 1)}~(a"t)lyg(1-(aqt)XaP)Fp(1-(t))-

1 = (. 1)l (tkg(a,t)pr a-px 4a'ttXa-P)Fp(t}.

From this the lemroa follows, in this case (see the proof of Proposition 1.9). Second if tp # 0, p is odd and o e W x then (2~1-(t))) (a-l)/2 = (-2~(t)) (al)~,

g(o,-t) = (a-1)-I.t(o,t),

~ - 0 = -~(t), Fp(1-(olt))/Fp(1-{t)) = Go(t) "1, (o.p)/2 = _o)(O)/l:m(o-p)/2 and Fp(l]2) = (-1)(P+I)/2Fp(1/2)'I,

The lemma follows, a

Let (OjZ)p, = {r e Oj'Z : rp = 0} = ~ and let I0 = {(r,s) e I : {r, s, r+s } ~ (Q/Z)p, }, 12 = {(r,s) ~ I : {r, s, r+s}n(Ofl)p, = ~ } , I 1 = I-(Iou12). Also let 12 = {(r,s) ~ 12 : v(r) = v(s) = v(r+s)} and

r = IouI1uI ~. L e m m a 3.7. There exists a bi-cocycle H o on W z with values in M(I) such that if a e W x mad q = (r,s) e L r+s = t then Ho(q) = 1 ff q e IouI 1 and Ho(oq) = ({(r)}(r){(s)}(s>/{(r)+(s)}(r~(s))* ° x if q ~ 12 and deg(o) = 1. Proof. Since Ii, i a {0,1,2}, is stable under the action of W we may assume q e 12. We will use Lemn-ms 3.3 and 3.4. T h e ~ for x ~ W ~

181

0-Io~(oq)/Ho(oq))°'t = ({(r)}f0{(s)}0>/{(rD-(s)}(r>+0>) "c-I which depends only on x and is a cocycle on W~. On the other hand, if we let (o-lr)-(r) = a, (o'ls)-(s) = b, then a and b are elements of Z p and so as x is inertial

H.m(q)/1-I~(-c'lq) = ({(o'Ir)}(r){(O'Is)j(s>/{(o-Ir)+(O'Is)}(r>+(s)) x'l. Now, since none of r, s or t lie in (OjZ)p,,

({(o-~r)}(0) ~-I = ((r×l+pa)I/P) ~q, ({(o~s)}('>) ~q = ((s)(l+ph)I/P) ~-I

and ({(o-lr)+(o'ls)}(r)+(s)) x'l = (((r)+(s))(l+p(a+b))l/P) "c'l

Hence,(Ho~(oq)/Ho(oq))°-1 = H~q)/Ho(xlq)' The computations necessary to show I-Io is a bi-cocycle are similar, o We note that I-Io(q) = I-Io(qp) and Ho(oq) = (((1-~(q))l'~{(r)}(r){(s)}(s)/{(t)}(t))°'l)* when q = (r,s) 12. Using this one can show that the restriction of Ho to the fixer of (l+p)

1/p

in Wz factors naturally into the

product of three bi-cocycles with values in M(OJZ). We may now state, Proposition 3.8. Let q = (r,s) e L t = r + s . S u p p o s e p > 3 o r p = 3 a n d q e r. Then f o r o ~ W z, (3.9)

~o(q) = r'~(~ro(r)ro(s)/ro(t)-

One can show that this formula, in the case in which oq = q, is equivalent to the formula for the local co~ts of the Jacobi sum Hecke characters, Theorem 523 of [CM]. In fact, this theorem implies that (3.9) is false when p = 3 in the excluded cases (although, it true up to roots of unity.) (See [P] for the formula corresponding to [CM; Theorem 5.3] in the case p = 2.) We deduce from ~ r e m 1.7 and Proposition 3.5, the following theorem when q ~ I0.

PrOlX~ition 3.10. Suppose o e W, deg(o) = 1, q = (r,s) e Ioul 1 and t = r+s. Then if p is odd or tp # 0

~o(q)= °(°)-c¢o~'0ro(r)ro(s)/r'o(t) When q e I l, this will be proven in Section VI. In the excluded case, it is true up to a sign. The conclusion of Proposition 3.8, in the case q e IowI I is an immediate corollary. We also state the corresponding result for the symmetric Fermat curve. Let Vo(r) = (-1)~(°'r)Fo.(r) ifr OJ'Z and o e W. 'Iben, it follows from Proposition 3.5 and Proposition 3.10 that trader the same hypotheses as in Proposition 3.10,

(3.10

To(q) = P&g(o)c°(°)r(-°a q)ro(r)Fo(S)Fo(-0-

As was pointed out in [O], it follows immediately from this and Lemma 3.2 that Fl(O)(r) = Fo(r) mod g**wbenrp = 0. We also obtain a fomaula for Fl(O)(r ) when rp # 0 as Corollary 6.5 below.

182 Silppose now that p is odd. To completely determine ~o(q), at least when p > 3, we define -~(IJ,2 r

"~

r)q.(. ] )(2~ .l r),r2).

A i r ) = ( 2 ) ) * F p ( ( r } + ( - 1 ) ( 2 r ' / 2 ) / I ' p ( ( a -1

for r ~ ~

and, if f is the order of 2 mod p-V(r) we define an element DO(r) in ~r,/Ix by f

DO(r)= llJ(A (r/2 ) "' i

where a i = 2i'1/(2f-1). (Here r/2 denotes the element s of ~

such that 2as -- r.) Then D O is the unique

cocycle, on W with values in M(QpF/p; Q ~ / g ~ such that, if Do(r)2/Do(2r) = AO(r) mod }.t.. (See [O; Lemma 3.3].) Also note that when a e W x, A a = 1 and D a = 1 mod It,,.. We extend ~

to all of W and

Proposition 3.12. There exists a unique cocycle I-Iaon W with values in M(I), ff p > 3, and on M(r), if p = 3, with the following properties: Suppose q • I, q • !", if p = 3 and q ~ IouI 1, if p = 2. Let qp = (r,s), t = r+s and~J e W. l f q • I2, H~(q) = DO(r)Do.(s)/DO(t) mod go, and H~(q) -~ (X(o)

1/2-e(¢ 1 ~cs,r) ~a,s) p4a,t) qP))*({(-r)} {(-s)) /{(-r)+(-s)} )*

modulo (a+pn'C~), and H(c0 = 1, otherwise. Moreover, when r ~ (~/_~, Ho(r,r) = AdO. It is clear that ~ must be the same cocycle as that defined above when (s ~ W l. This proposition and the following theorem will be proved in Section VII. The determination of [3o, when p > 3, is completed by, Ttw.a~em 3.13. Let q = (r,s) ~ I, t = r+s. Supixssep>3 or, p = 3 a n d q Then if a c W and deg(a) = 1,

e r, o r p = 2 , q • lowI 1 and ~ e 0 .

lBo(q) = O(O)e(~lq)I~q)Fdr)Fo(s)/Fo(t ). Propositions 3.8 and 3.10 are immediate corollaries of tiffs theorem. For ~ ~ W and r e O fL set Fo.(r) = Ic(rp)~(cs'r)4t(a'rp)Go(r). Since HO(q) only depends on qp and Go(u) -- 1 for u e 1 ~ / / ~ , u ~ 0, the following is a corollary of Theorem 3.13, although we actually prove it In'st. Proposition 3.14, Let q = (r,s) e ½, t =r+s. S u ~ p > 3

o r p = 3 a n d q e r. T h e n i f o e W,

~do)/130(,~ = o(o)~('r~~("-h)Fo(~)Fo(s)~o(0. It is interesting to observe that the restriction of a .-o F a to Wo~ is a one-cocycle and, using the Gauss multiplication formula (2.9), that

(3.15)

Fo((r+i)/p) = Fo(r)

Also, using the above results (me can prove:

if rp ~ 0.

183

(3.16)

13o(S,t)13o(r+s,t) = 13o(r,s+t)13o(r,s)

when all the terms in this equation are defined. (One should also be able to give a geometric proof of this equation using the ideas in [A]. See [O;, Remark 3.51.)

IV.

Computing Using Formal Expansions

In this section we generalize [C-GK; Theorem 17] which was a generalization of [K; Theorem 6.2]. Let K be a finite extension of O.p, with ring of integers A and residue field k. Suppose X and Y are complete curves over A with arboreal reductions ~ and ~7. Suppose f : ~ ~ ~f is a morphism of curves over k. Because the Jacobians of X and Y have good reduction, f induces a homomorphism f from H1R(Y,K) to •

.

*

H1R(X,K). Suppose P and Q are smooth points on X and x7 over k such that f(P) = Q. Let s and t be rational functions on X and Y which are uniformizing parameters on the residue disks above P and Q respectively (i.e. map these disks onto the open unit disk), E an element of A* and q is a positive integer such that f~I = gsq . Suppose o is a differential of the second kind on Y regular on the disk Q and v is a differential of the second kind on X regular on the disk P. Let tm

n

~

n

= ~n=la(n)t dt/t and v = ~n=lb(n)s ds/s be the e ~ i o n s

o f o in tand v in s.

4.1. Suppose f * o = txv in cohomology and {hi} is a sequence of positive integers such that la(ni)/nil --~ ,:,,,, Then (x = Limi_.)~ qEnia(ni)/b(qni). Proof. Let U be the dagger lifting of ~

in X and V the dagger lifting on ~fns in y. Let S be the connected

component of U above the congxment of)~'ns containing P and T the connected component of V above the cong~onent of ~fns containing Q. The main thing to observe is that the de Rham cohomology of X (resp. Y) injects into the dagger cohomology of U (resp. V) and the morphism f may be lifted to a morphism F from U to V such that on S, (F't) I T = £sq" The rest of the proof follows the same lines as the proof of [C-GK; Theorem 17]. [] Unforttmately, one does not know too much, a priori, about the seque~:e {ni} in the above theorem. Since the set {a(n) : n > 0} is Ixntrgt~ it follows that ni ~ 0 as i ---) 0*. The following result guided in us during our computations. Proposition 4.2. With notation as in the above theorem, suppose there exists an automorphism ct of T of order e such that ct(Q) = Q, the quotient T/(ct) has genus zero and ct*I = 1I for some % e k* of order e. Then n i is not congruent to zero m o d e for large i. Proof. Let ~ be the Teichmfiller lifting of %in K, The automorphism ct lifts to an automorphism a of order e of T, such that fit(Q) = Q and ~*t = ¢t. It follows that T/(a-) is a dagger space of genus zero. From this and the

184

fact that ~ is of the second kind we see that ~ = ~l(tl*)it.o is exact on T. As 0 n = qf~-=0a(qn)tqndt/t, we see that the set {a(qn)/qn} is bounded. D

V.

Recollections of the Stable Reduction of F e m a t Quofienm Fix m = pnd, (p,d) = 1 and let a and b be integers such that (m,a,b) = 1. Let Fro,a,b denote the quotient of

F m whose function field is generated by w = xayb and u = xm. Then Fro.a,b has affine equation w m = ua(1-u) b. •

.

m

a

b

Let C = a+b. In [CM], we calculated the stable model of the curve with affme equauon w = u (I-u) (-1)

c

over Qp when p was greater than 3 or when p = 3 and either 3n divides a (or b or c) or (3,abe) = 1. This curve is isomorphic to Fm,s.b over an unramified extension of Op so long as p > 2 and therefore has the same theory of stable reduction. We will recall and slightly generalize some of the results of [CM]. Suppose n > 1. If pn divides b, then the map from Fm to Fm,a,b factors through the curve with aft'me equation m

(5.1)

d

x +t = 1.

It turns out that the corntmtation of the stable model of this curve is very similar to that of Fm,~b. n

Suppose either (i) Xm is the curve with affine equation (5.1) or (ii) X m = Fro,a,b, p does not divide a or b, (p,c) = 1 and either p > 3 or p = 3 and (3,abe) = 1. Because of the similarities between these two cases we will describe what we need to know about the stable models of these curves simultaneously. Let m denote the stable model for Xm over Qp. Then by [CM Theorem 3.4], ~m may be defined over Up,

m = AuB, where A and B are each closed curves contained in

m such that Ac'tB is finite and

the natural morphism from - m to -mlp collapses each comtxment of B to a smooth point. The curve A is isomorphic to -m/p in such a way that the map from A to it induced by the natural map from X m to Xm/p is the absolute Frobenius morphism. Moreover B consists of the rn/p disjoint irreducible components each isomorphic to the Artin-Schreier curve zP-z = v q (where q = d in case (i) and 2 in case (ii)) and each is attached to A at its "point at infinity." We will call the components of B the new components. We will now describe the affmoid subdomains of Xm corresponding to new components (i.e., if Y is an irreducible component of -m, the set of points of X m which reduce to a point in the interior of Y is an affmoid subdornain). Recall, x p I = -p. For a positive integer k let ~ denote the group of k-th roots of unity in Q-p. Case(i). Fix a't in ~ such that xd = -pn~ and suppose "f E t~m. D e f ' m e v a n d z - t b y t h e e q u a d o n s t = x v a n d x =

~O-r~'lz9. "n~ (pn/~)'l(xm - (1-td)) -= ¢-z. t - v d rood rc

and for each 'y ~ lain, X~t= Xm,.t =: {(x,y) e X : lyl _< IpnTtlTM and Ix-~ _< Ircl) is an affmoid subdomain of Xm which corresponds to a new component By with affme e q u a t i o n ~ - ~ = V2. Moreover, B~,= By iff ybf lap (these are the m/pnew components). Inthis case,let A = gm/p.pand for 5 e A, s e t X s = X - t a n d BS= B./for any representativey a gm ofS. Inthis case, z~./= d(1-(l)/~ + z.tand d(1-t~'l)/x is an integer when ~ is a p-th

185

root of unity. Case(ii).

LetC(a,b)=a~bb/~andF={y•~_p: n+¢

Fixaxin~pSUChthatx2= -p n+e

(-p

Tin= C(a,b)} andlete=v(ab). Setp=p(a,b) =:-c3/2ab.

,~ D e f m e v b y u = a / c + w a n d ~ f o r y • -I

m

a

h

r~byw=~l+~-~z~.

Then

2

~OC(a,b)) (w -u (l-u)-) -= ~-zT-v mod re.

In this case, the atTmoid subdomain X y = Xm, Y =: {(tt, v) • Fa, b : lu-a]cl <

Ip~+~lIn, Iw-M < IC(a,b)Jl~l,~l},

y • F of Fro,a,b corresponds to a new component By with affine equation ~-2¢ = y2; By = By iff I~P/' • %. In this case, let A = F/gp and for ;5 ~ A, set X 8 = Xy and B 8 = By for any representative 7 e F of ;5. In this case, zt7 = d(1-~-l)/pep'g + z7 and d(1-~'l)~epTt is an integer when ~ is a p-th root of unity. By functoriatity (see (1.3)) the elements of o of W of positive degree act on B. By [CM; Theorem 4.3] this action is described by

(5.2)

¢(o)* : '¢

¢y_t pde-6'(~) ---> % V" .

In particular, ¢(o) takes B8 to Bso. Supl xyse ~ • P'm" Let a t denote the automorphism of X such that at(x,w ) = (~x,w), in case (i), and the automorphism such that a{(u,w) = (u,~w), in case (ii), then zty~ (5.3)

zy

at* : V ~

V.

The following proposition will also follow from our explicit computations, nevertheless we include a conceptual proof by way of motivation. Proposition 5.4. Suppose h = (r,s) e I and tot,s is the pullback of a differential rl on X. Suptxrse ;5 • A. Then the restriction of rl to XS is exact iff (m/p)w = 0. Note: The hypotheses imply Sp = 0, in case (i), and mh ---k(a,b) mcxl m, for integer k, in case (ii). Proof. First, (m/p)h = 0 iff "q is the pullback of a differential ~' on Xm/p. Let P denote the smooth point on "m/p to which B8 maps. Suppose (m/p)h = 0. Then as rl' is regular on the residue class above P and hence exact on this residue class since every differential on a disk is exact it follows that r I is exact on X& Now suppose that TI is exact on X 8. Let V 8 denote the wide open above B 8. It follows that ~i is exact on V 8. It follows from (5.3) that a;(V~) = Vts. Since rl is an eigenvector for the action of {ag : ~ • la~n}, we see that r I is exact on wseaV8 = V the wide otx'n above B. Let W denote the wide open ~ in Xm above A. Since the ~ t s of B are each attached at only I

-

I

-

I

-

.

one point to A, it follows that H~(Xm,Qp) is isomorphic to HDR(W,Qp)~BHDR(V,Qp) vm the natural map (see [C-RLC {}IV]). Next, the statement that the map from A to -m/p is finite, radicial and surjective implies

186

that the composition I -I -I -HDRCXm/p.Q p) ~ HDR(Xm,Qp --->HDR(W,Qp) 1

--

I

--

is an isomorphism. This implies that the class of rl in HDR(Xm,Qp) must lie in the image of HDR(Xm~p,Qp) and so (m/p)w

=

O. t~

Caa'ollary 5.5. Let notation be as in the proposition. Suppose 8 e A and (m/p)h ~ 0. Let U be a residue class of X~ and F a rigid analytic function on U such that dF = rl. Then F is unbounded on U. Proof. By [K], ff c0 is an overconvergent differential on X& ~ it is the exterior derivative of a bounded function on U iffits de Rb.am cohomology class lies in the unit root subspace. Hence the corollary follows immediately from the proposition and the fact that B~ is an Anin-Schreier curve (as the unit root subspace of the first crystalline cohomology group of such a curve is zero). D

VI. A C o z ~ m f i o m

In this section we will prove Proposition 3.10 when q c I 1 and deduce from this a formula for 13o.(r,s) d when r = s. Let notation be as in §V, case (i). In particular, x = -pnrc. In this section, ffj e Z , we let (-pnr0j/d denote ~J. Suppose q = (r,s) e L mr = ds = 0, re(r) = i and d(s) = j. Let rlr,~ denote the differential x~-mvJdv/v on , X m. Using (1.2), we see that the pullback of ~rstOFm via(x,y) ~ ( x , / ) i s - m p n.~-pnre).-(s) o~j. W e may expand ~r,sat the point w = O, x = l (which is a point on the affmoid subdomain X m j ) in v (which is a parameter on the residue class of this point on Xm,1). We Find

TIr,s = ~

( ~ ) ~ ((r~l)vd~+Jdv/%, := ~

I=0

ar,s(k)vkdv/%'

k=O

Suppose that (m~)q ~ 0, which means p does not divide i. Now ~*(C)'qr, s = ¢x'qo,r.as in cohomology for some a i n Qp and ]3~(cq) = (-pn~z)(s)'(as)a. It is easy to see that lar,s(k)/kl--> 00 as lid ~ 0, k -=j mod d, k > 0. (This gives mother proof of Proposition 5.4 in this case.) It then follows from Theorem 4.1 and (5.2) that •

=

c-I k

.~-I.1., .tAr)-Ix/(, )g+v(o~)- Ix

where t = p~s)-(ps) = [p~s)] which is less than p. If.~ -=0 rood (p-l), the g-th term of this limit equals

t

n

Taking limits in R, we have

p

t

187 Lim/__,(O.(s,)~'~ {-(r)+k} = Lim,~(O ,.(s))Limlv_~((.r)p,(.r))~-~{h-l+k} where Z ~ Z>o and h ~ Z [ I ~ ] ~

As

~'~{h-l+k} = l - ~ l]'l {(h)+g } / 1 ~h]-I {(h)+g}

for h close to ((-r)p,(-r)) by (2.4), Liml_Ko_(sDk~ {-~r)+k} =

Limz-,-(,>Limb~ lmFp (h+/)/FpO0 = rp(1-(r)-(s))/Fp(1-(r)) = (-1)fFp(r)/Fp((r)+(s))

where f equals (s) when p = 2 and 0 otherwise, since (r}p = (r+s)V Similarly, L i m ~ : o ~(s)

{--{cr}+k}=

Lim~_._+(O,.(s)~(<-o'r>p,(-Or))Fp('h+1~÷L),/Fp(h) =

(-D ro(-(or>)trp((or>)/Fp((or)-Kos)),

by (2.4) where f equals (os) when p = 2 and 0 otherwise. Noting that os = ps, [p(s)] = p(s)-(ps), (-(ps)/p)p =

[p(s)l/p and [(2s)/2] 2 =-(s)-(2s) rood 2 when p = 2, we see that .(.1)f'f(_l)/(°s)(7-~s~(_(or))) [p(s)l = x[P(s)lc0(Or)(°s)-(s)~

and so ~(oq) equals (6.1)

px(')(°'v)(-~:(or>)(m>~sH Go(o'rlGgoS)rp((O'r>+(os>)rp((rFKs)) -1.

Hence using (2.11) we conclude that J3o(Cl)equals Go(r)Go(s)/Gg(r+s) times (.1)~(°,s) ¢~(O)'e(°'lq)N(°'lsXo-P)k~r)~°,0"i't(°,t)" This combined with Lemmas 1.6 and 3.6 implies Proposition 3.10, in the case q E I v

Inparticalar,

Proposition 6.2. Suppose q = (r,s) e 1I. t = r+s and Sp = 0. Then v(~o(oq)) = deg(o)(-s) + v(rX(r)..(oI) -((t)-(m))). Observe now that the quofiem of Fro, Y = Fro,l,1 :

win=u(1-u) is isomorphic to the curve X :

xm+y2 = 1,

via f*(x,y) = (41/row, 1-2u) (where 4 ltm will be a fixed m-th root of 4 for this section and by 4 x/m we will he.an (41/m)x.) Hence, we can determine the Frobenius matrix of Y from that of X.

188

More precisely,ff t0i = xi'mdy ("=" -m'Ico2m2.i.m), v i = wi-mdu ("=" m-lo)m.i.i), then f'c0i = -2-1-4(r)vi.

Letabe anelernentWofdcgreeone. Letr=i/mands= 1/2. Ttxmincohomology • *(o)f°*c0i = f*
-1 (or) -4

~o(r,s))

v

i'

where i' = m.(or} and using (1.3). Hence, if q = (r,r),

[3~(oc0 = 4"~'(°'or)~(o(r,s)) By (6.1), ~o(r,s)) equals Go(crr)Go(1/2)(Fp((Or)+l/2)[Fp((r)+l[2))

times tm(°'P)'r2(-(Orp))(1-°)t2and

~o(q) = (L(q)/L(oq))~3~q). Now Go(l/2) = (-1)(P+l)t2Fp(l[2), Oo(r)(Fp((Or)+l/2)/Fp((r)+l/2) = (Oo(r))2.Fp(or) Fp((Or)+l/2)(Fp
and by (2.10), rp(or)rp((OV)+l/2)/Fp(r)Fp((r)+l/2)

=

(2 *)[2(r)]p-t2(or)]p.I"p((OT)p+(-1)(2°r)P/2)Fp(2(Or)). (Fp((r)p+(- 1)(2~)p/2))Fp(2(r)))"1. Using (2.11), we see that (L(o-lq)/IA:q))(Fp(2(r))/Fp(2(olr))) = o(o)e(atq)~(2r)e(°l~W'(q). Moreover, (2*)12(°'tr)lp'[2(r)lP = (4~°'r)2-~°~) *, and O}(4-~(o'r))k~2~)~°'tq)'c'(q) = z((r)/(2r))2~(°'r)K(2r¢(°'lq)'e(q) = K(r)2O(°'r)/K(2r)~(°'2r). as

E(o-lo)-e(q) = 2~(o,r)-g(o,2r) + E(oqq)(l~). I-~,w,~e,we have,

Proposition 6.3. Suppose r E ~

rp *0, q = (r,r), o e W, and deg(o) = 1. Then J3o(q) equals the

product of GO(r)2/GO(2a'),¢J0(o)e(°'lq)z(r)74K°'r)/l¢(2r)~°'2r) and

-rp(1/2~(°*>~<rp>(°~A.(qS. Corollary 6.4. The conclusion of Proposition 3.14 in is true when r = s and true modulo g® in general. Proof. The first IXn'tfollows immediately from the proposition. Let n he a posidvc integer. To prove the second part we will apply Lcmma 3.2 with S = {u e O f / : v(u) = -n) and To(r) = ~r_~°'r)-tt(°'rr)Go(r)F1(o)(rp)for o e W and r e S. Then T o is clearlya cocycle on W with values in M(S,Qp/g**). The firstpart of the corollarycombined with (3.1)implies thatT o fulfilsthe other requirements of Lcrnma 3.2. Itfollows from thatIcrnma thatTo(r)TO(s)/TO(t)= ~o(r,s)rood p.~ for (r,s)e I(S), t = r+s. Applying (3.1)again wc dcducc the corollary, o Also from Lemma 3.2 we deduce,

Corollary 6.5. Suppose ¢J e W, r e Of/and rp * 0. Then,

189

Fl(o)(r)

=

p'aeg(°)/2p(°'tr)'(r)Do(rp)Go(r ) mod U~o.

VII. A Consruence. In this section we will complete the proofs of Proposition 3.14 and ~ e m 3.13. Let p be an odd prime and suppose q = (r,s) ~ 12, t = r+s and v(s) = v(t). Let (a,b) = m((r)-l,(s)-1 ), (a',b') = m((or)-l,(cs)-l), c = a+b, c' = a'+b'. Then (p,be) = 1. Let e = v(a). We will use the notation, where appropriate, of case (ii) §V. In particular, x will be a square root of -pn+Cx and T an m-th root of C(a,b) (7 "=" C(-l,(s>-l)). Then C(a',b') = C(a,b)f(aibJ/ck)m(l+miffa)*'(l+mj/fb)b'/(l+mk/fc) c' where f, i, j and k are integers such that (a',b') = f(a,b) + m(ij) and k = i+j. As (l+mi/fa) a' -~ l+mi rood (m2/a)Zp etc., it follows that there exists a unique m-th root 7' of C(a',b') in Tf(aibJ/ck)(l+pn'eZp) and 7' is independent of the choices of f, i and j. We note that the functions u and w'= wfui(1-u)j are also co-ordinates on Fro,a,b = Fm,a,,b,. If Q is the point (a/c,ya) in (u,w)-coordinates and P is the point (a'/b',7') in (u, w')coordinates, then Q lies on X ~ P lies on X~ and ~I~(o)(P)= (~. Define v and v' by u = a/c + xv = a'/c' + xv'. Then ~a+m,b+m = wdu = ..'t'wdv and O)a,+m,b,+m= -'t'w'dv'. Thus we may expand (0a+m,b+m at Q in v (which is a parameter on the residue class (~),

%.m,~m = "W ~ A ( g + l ) v

dv

where

T__~A ( / + 1)vg = (( 1-(c't/a)v)'( 1+(c't/b)v)b) lhn. Similarly, we may expand 0)a,+m,b,+mat Pin v', oo

~a'+m,b'+m = "~7'~0 A'(/+l)v'/dv'" In fact, g

u -u -v ahn bhn

with a similar formula for A'(g + 1). The above formula for A(g + 1) was applied by David (5~az-Uribe, using Maxsyrlm, to obtain some computati~aal evidence supporting Proposition 3.8 (before it was proved). However, it is difficult to use this formula to get congruences. Therefore, define the coefficients BOa) by the equation: (l-(e~/a)v)a(l+(e~/b)v)b = l+.~__~lBOl)~hvh.

Then it follows that B( 1) = 0 and B(2) = .¢3f2ab = 9(a,b) and

(7.1)

v(B0a)) _> ~fan{v(~)-ie : 0<_i.gh} _> -he + ( e - Max(e,v0a))).

In particular, v(B(h)'th) ~ n + l / ( p - 1 ) s i n c e p > 3 o r p = 3 a n d e = 0 .

Moreover,

190

'~

"~

~

h h

"

((1-(c'r./a)v)'(l+(c'r./b)v)b) l h n - 'V ('m')(Y__, B0a)x v y.

In the following the expression x ~ y modulo (l+pU~p) will mean y ~ 0 and x/y a (l+pUZp). I,emma 7.2. Suppose deg(o) = ] and u is a positive integer. Then l/m n+e (pU-l)/2 (1 +pn~..l@ A(p u) -= ((pu lV2)(-p r~o(a,b)) ) modulo

(7.3)

and v(A(pU)) = u/2. Moreover the same is true for A'(pu) with (a,b) replaced by (a',b). e~

Proof. Fix a posidve integer u. Now A(p u) = ~

o~j~here

and the sum rims over sequences of positive integers (i2,i3,...) such that ~ i h = j and ~9-~hih = pU-1. For a positive integer x, let S(x) denote the sum of its p-adic digits. Let K = (pU-1)/2. Now ctj = 0 for j > K. Moreover, o~(pu.l)/2 equals the right hand side of the congruence (7.3) which has valuation u/2, since S(K) = u(p- 1)/2. Hence all we have to do is show v(aj) > (n-e) + u/2 for j < K. Suppose j < K and let i = K - j. Then =

Kv(x)- i n - v(,j0 = (n +e+l/(p-1))K- j n - (j-S(j))/(p-1) = in + eK+ S(j)/(p-1) + i/(p-1). Now S(x-i)>S(x) -i. Therefore, v(xPU-l~li...)),hn, -> in + eK + u/2.

When p = 3, e = 0 and so v(o~j) >_v(, p 3. In general we have,

u-I 1/m ( j )) which concludes the proof in this case. Assume now that p >

and

~v(B(h))i h >_~((1-h)e - v0a))ih = (j-K)e - ~v0a)i h (using (7.1) so v(ctj) >_i(n-e) + u]2 - ~v0a)i h. Since n > e, it is enough to show i > ~v(h)i h or what is the same that ~(v0a)+l)i h < K/2. But ~(v(h)+l)i h = ~ ( t + l ) D t where D t = v ( ~ tih" Now since ~ h J h = 2K and h -> Max(2,pV00), we see that 2D 0 + pD 1 + . . . + ptD t + . . . < 2K. Thus, ~ ( t + l ) D t < K - ~ ( p t / 2 - (t+l))D t < K,

191 since p > 5 unless D t = 0 for t > 0. But in this case, Y'.(v(h)+l)i h = D O = j < K. This concludes the proof.

13

tl

As this lemma implies IAa,b(pU)/p I --)o. as u --* o. we may use Theorem 4.1, (5.2) and the fact that V = ~" to deduce: Proposition 7.4. Suppose o e W, and deg(o) = 1.

~3~(oh) = (~,a/,{) Lira tK't°'l)PUA(pU)/A'(pU+ 1). Remark. That a limit formula of this rdmpe must exist follows from Proposition 5-5. Also one can predict that the formula must involve A(k) for k odd using Proposition 4.2, since the hyperelliptic involution of X.t fixes and takes V to -V. Proposition 7.5. Suppose q = (r,s) e 12, t = r+s and o e W such that deg(o) = 1. Then ~o'(q) is congruent -1

tO B0)(o) "e(° q)ro(r)ro.(s)/Fo(t) modulO (l+pn'eZp), where -1

s

B = (:c(o)~:2-~(" qP)*({(-rv~}~":P{(-sp}~" P/{<-~p-~-@~"%*. Proof. Using Lemma 1.6 we may suppose that v(s) = v(t). Let notation be as above. Fu-sl~ 2(

= (-d)

n .(pU-pU+l)/2f I/m ~,/'

-prO

l / m ~-1

k(pU.1)/2.)~.(pU+1.i)/2/

pU(p-1)/2

p((p

u+l

-D/2)!/r~

for some m-adic integer x and ((p-+l-1)/2)!B¢-~

u(p-1)/2 u

((p -1)/2)!).O+mx)

U(p.1

)/2((pU-1)/2)!) =

~(P-1)/2((pU+l-1)/2)!B~(P'lXpUly2((pU-1)/2)!) = r¢(!0-1)/"2((p u + l -1)/2)!/(-p) (pu- 1)/2 ((p u -1)/2)!)

=

-(.~)'(Pq ¢2rp( 1/2+p~÷ I/2).

Since (a,b,c) - f(a',b',d) mod pn, p(a',b')/p(a,b) e f(l+pn'eZp). Hence setting p = pep(a,b) and using Lemma 7.2 we observe that modulo pn-e+If2

pACpU)/A,(pU. 1) _= .p;,l:-(p-t )/2( d/o)pUfp -1y2f( 1-pu+1):2Fp( 1/2+pU+ 1/2). Now ('~°Â)pU~l'pu+l)/2 =-(-(0(o)pn+e~)(°q)/20~(o)l/2)* modulo (l+pUZp) and (20~(o)abc'3m) (°-ly2= -I

u

(2o(o)(r){s)/(t)) (°-l)f2. Hence, ff u >__n-e, (x° )P pA(pU)/A'(pu+l) is congruent to -p~(°P)/2(2o)(o)(r){s)/(t))(°l )/2Fp(1/2)(Z(o) 1/2)* modulo (1 +pn'ezp). On the other hand: Let ~ = p'~a, ff = p a, A = p~C(a,b), A' = p'ea'c(a',b'). Fix an m-th root ~. of A and

192

let~.' be the m-th root of A' in kf(~Jl~/ck)(l+pn'e~p), ofp.

" n ~ t ¢ =p~°'~r)x°/X'and

where B = {(r)-l}g(~'m){(s)-I }N°'°0/{(r)+(s)-2} ~(o'°'r)+g(°'°O. Let 8 = e(q)-e(crq). Now, as No,or)+g(o,as)-g(o,o't) = E(q)o-e(oq),

c0(B) = ~t)-8~(r)~a'°'r)(s)~°'°s)/(t) ~°'ea)) and, peta(o,or) = p-V(t~pV(r)~(o,Or)pV(S~(o,oS)/p(t~(o,o~). Also, B * = ({(-r)}l(r)]pl(m)]P{(-s)}[(S)]P-[(°S)lP/{(-r)+(-s)}[(t)lpt(m)]P+~)

*

•(((-r)}~(°'°~P{(-s)}~(°'°'~/{(-r~-s)}~(~'°9)*. Moreover, ({ (_r) }[(r)lp'[(°'r)lp{(_S)}[(S)lp'[(°S)lp/{(_r)_~_s) }[(t)]p-[(°'t)]p+8). =({ (t)} -,5)*({ (r)} [(r)lp-[(m)]p{(s)} [(s)Ip-[(os)]p/{ (t)} [(t)lp-[(m)lp), and

({(.r)}~(°'°~ {(.s)} ~(°'%)/{(_r~_s)} ~(°'°~P)* =

rood (l+pn'ezp). (Note, here we used ({(-u)+x}P(c'°u)) * - {(-u)}~°'au)((l+pnx)l/Pa)°-~(°) mod (l+pfZp), if x e Zp, u ~ O.gtZp and n _
Go(oOGo(as)/%(ot)

~-

z(o)*~(~qP)~(~)({(r)]t(°~'-[(°~>lr'{(s)]I<'>]Pq<°s>~'/[(0}[<%-[(°%)*

modulo (l+pn'eZp) by (2.5), the proposition follows, t~ Corollary 7.6. With the above hypotheses, v(13o(r,s)) = deg(o)/'2 + v(r)((fflr)-(r))+v(s)((ols)-(s))-v(t)((olt)-(t)). We see that the two sides of the equation in Proposition 3.14 are congruent modulo (l+pn-eZp) when q 12. This combined with Corollary 6.4 implies that they are equal. Let J = I2t'~OTf/p) 2. By Len3ma 3.5 the ratio, do(q), of 13o(q) and 0)(o)~(°lq)l"o(l')l"o(s)/T'o(t) extends to cocycle on W with values in M(J). Moreover, it follows from the proposition that do(q) is congruent B

193

modulo (l+pn-eZp). Since, I]o(q) = r't(o)(r)Fl(o)(s)/Fl(o)(t) rood ~ . by (3.1), it follows from Corollary 6.5 that do(r,s) = Dcr(r)Do(s)/Do(t) mod It**. Finally, from Proposition 6.3, we see that do(r,r ) = Aq(r) when r • QJZp. By taking Ha = do, we complete the proof of Proposition 3.12 and Theorem 3.13 in the case q • J. Theorem 3.13, in general, follows from this and Proposition 3.14. n References. [A]

Anderson, G., cyclotomy and a Covering of the Taniyama group. Comtxrsitio Math., 57 (1985), 153-217.

[BO]

BertheloL P., and A. Ogus, F-isocrystals and de Rham cohomology. I, Invent. Math. 72, 2 (1983), 159-199.

[C-GK] Coleman, 1L, On the Gross Koblitz Formula, Advanced Studies in Pure Math. 12 (1987), 21-52. [C-RL]

• Reciprocity Laws on Curves, Comtx~itio, 72, (1989) 205-235.

[C-AI]

, Anderson-lhara Theory: Gauss Sums and Circular Units, Advan. in Math., 17, (1989) 5572.

[CM]

_ _ and W. McCallun% Stable Reduction of Fennat Curves and Local Components of Jacobi Sum Hecke Characters, J. reine angew. Math. 385 (1988) 41-101.

[K]

Katz, N., Crystalline cohomology, Dieudonn~ modules and Jacobi sums, in Automorphic Forms, Representation Theory and Arithmetic, (Tata Institute of Fundamental Research. Bombay, India, 1979), 165-245.

[O] [P]

Ogus, A., A p-adic Analogue of the (3aowla-Selberg Formula, this volume. Prapavessi, D., On Jacobi Sum Hecke Characters Ramified only at 2, to appear in The Journal of Algebra.

Errata for [C-GK] On page 24, Proposition 2 requires the hypothesis r >_Ipl. On page 44, line 12, I= {(r,s) • (1/m)Z/Zx(1/ha)Z/Z, r ¢ 0, s # 0 and r+s # 0}. On page 45, line -7, the term "(p- 1)/~Ja/m)" should be replaced by "(p-lXakn)". On pages 46-47, in lines 4 and 7 on page 46 and line t2 on page 47, h and k should be switched On page 47, lines 1, 2, and 10 on page 47, the f'h-sth in the exponent should be replaced by a k; lines 4 and 6 switch h and k; in the corollary, t should be -1 for all p; in Theorem 19, tr,s = (-1)~:(r's)for all p. On page 48, line 10, "l+K(pr,ps)" should be replaced by "l-K(pr,ps)" (note that ~(pr,ps) must be 1 in this case); lines -10, -12 and -13, the values of Fp on the left hand side of the equations should be replaced by their irlv~'se.,s. On page 49, line 1, insert "let x • Zp"; line 2, replace with "(i) l(1-x) ~ l ( x ) mod 2".

Supercongruences

Matthijs J. Coster Centre f o r Mathematics and Computer Science P.O. B o x 4079, 1009 AB Amsterdam, the Netherlands

A b s t r a c t . In this report we will discuss special congruences. We explain how the

congruences arise from formal groups and then we gNe some examples. Key Words & phrases: (Super-)Congruences,Formal Groups, Conjecture of Attdn and Swinnerton-Dyer.

1. Introduction. This paper deals with so called "supercongruences". Before we will explain this term, we give some definitions which we need in the explanation. Let K be an algebraic extension of Q. Let p be a prime which splits in K as p = zr~. Let I • lp be the valuation on Q in such a way that [~ [p = 1 and Ire Ip =p-1. We will consider Jr as an element of Zp. Let {un } n=~ be a sequence of rational or p-adic integers. In this paper we will consider the congruences

u(mp r) =-a.u(mp r-l) mod p2r,

(1A)

u(mp r) = ct.u(mp r-l) mod zcJ~r,

(1B)

and

where ~, m and r are positive integers and a is an integer and zc is an p-adic integer. Therefore (1A) is a congruence in Z and (1B) is a congruence in Zp. In Section 2 we will give an introduction in formal groups. We will show that congruences (1A) and (1B) with -- 1 arise in a natural way from formal groups. Especially, we will give a sketch of the Conjecture of Atkin and Swinnerton-Dyer. In Section 3, 4 and 5 we give some examples of congruences (1A) and (1B) with coefficient )~ > 1. In such cases we call the congruence

supercongruence. At the moment supercongruences cannot be proved by use of formal groups. In each case a separated proof has to be given. A lot of proofs will be omitted in this paper. For these proofs we refer to [12]. In Section 6 some conjectures are given. 2. T h e

conjecture

of Atkin

and

Swinnerton-Dyer.

Let K be a commutative field with char(K) = 0 and let R be a subring. (In our case we will choose R = Zp). We denote by R[[T]] the set of power series in the variable T with coefficients in R.

195

Let F(X,Y) ~ R[[X,Y]]. We call F(X,Y) a commutative formal group law if F(X,Y) satisfies the following properties.

F(X, Y) = X + Y + (terms of degree > 2), F(X, F(Y, Z)) = F(F(X, Y), Z), F(X, Y) = F(Y, X).

(2)

We derive from (2) that F(X,Y) satisfies moreover the following properties.

F(X, O) = X, there is a unique i(T) e R[[T]] such that F(T, i(T)) = O. Let ~

= {X(T) e R[[TI] : X(0) = 01. We define a formal addition +~ron ~ r b Y

X(73 +~-Y(T) = F(X(T), Y(T)). It turns out that/J'@ with +sris a group. This group is called a formal commutative group in one variable over R. From now on let ,~rbe a formal group over R (i.e.

~r: ( ~ r ,+at)). We define the logarithmf(T) of the formal group ~" by

f(T) e K[[TI], f(T) = T + (terms of degree > 2), f(F(X, Y)) =f(X) +f(Y).

(3)

The last condition can be replaced by

F(X, Y) =f-10c(X ) + f(Y)), where f - l ( T ) ~ K[[T]] is the power series such t h a t f - l ( f ( T ) ) fiT) satisfies the property

f(T) = ~

u(n).Tn/n with u(n) e R.

= T. We find that

(4)

n=l

We call (.o =f(T) dT

(5)

the differential form related to the formal group ~. We consider the formal Dirichlet series

L(s, ~') = ~ , un/nS

(6)

n=l

where ~r(T) = ~

unTn/n is the logarithm of the formal group J .

n=l

Two formal groups ~ ' ( w i t h f and L(s, ~r)) and ~ (with g and L(s, ~ )) are isomor-

196

phic over R if there is a formal group homomorphism h: ~r._._) ~ with h(T) ~ R[[T]] and h(F(X,Y)) = G(h(X),h(Y)). In our case (that char(K) = 0) we have 0o

h(T) = g-1(I(73 ) and L(s, ~)/L(s, ~ )= ~ -v(n) -7, n=l

with [ v(n) [p <[ nip.

n

We have a theorem due to Honda which says that for each formal group ~r there exists a formal group ~ such that the Dirichlet series related to ~ has p-adic numbers as coefficients. In formula: T h e o r e m 1. ( H o n d a ) . Let ~rbe a formal group over Zp. Then there exists a formal group ~, isomorphic to 5rover Zp such that oo

L(s, ~ )

=

(1-Zpj-l-Jsb(I))

-1

(7)

j=l

where b(1) are p-adic integers. Proof. See [20, pp. 441-445] or [12, pp. 18-23].11 Corollary

2. Let ~rbe a formal group over Zp. L e t f(T) = ~ u(n) . T"/n be the n=l

related formal group. Then there exists p-adic numbers b(1) such that u(mp r) - b(1).u(mp r-l) - ... - pr-l.b(r).u(m) =- 0 mod pr,

(8)

for m,r positive integers and p is not a divisor of m. Proof. See [20, pp. 441-445].11 We will apply Theorem 1 and Corollary 2 on formal groups related to elliptic curves. Let be an elliptic curve over Z and let co be a holomorphic differential form on N. Let ,~rm be the formal group related to co. Suppose that 6rm is a formal group over Z. Let

Lp(s) = ( 1 - a p .p-S +p.bp .p-2S) -1 be the Dirichlet series related to the Hasse-Weil zeta-function of the elliptic curve. Then a theorem of Honda Cartier and Hill says that the formal series related to the Dirichlet series is isomorphic to the formal group .~r0~. C o r o l l a r y 3. ( C o n j e c t u r e of A t k i n a n d S w i n n e r t o n - D y e r ) . Let o~ro~be the formal group as defined above. (i) We have

u(mp r) - ap. u(mp r-l) + p . bp.u(mp r- 2) - 0 mod pr. (ii)

(9)

If ~ is ordinary over Zp (i.e. bp = 1 and ap~ O) then we have u(mp r) = ~.u(mp r-l) m o d p r.

(10)

"197

where ~z suchthat [ 7rl e = 1 and ~ i s a r o o t o f X 2 - a p X +

l =0.

Proof. (i) Corollary 2 says that (9) must be a congruence of form (8). Then we use the theorem of Honda Cartier and Hill, which was mentioned above. This theorem says that the coefficients b(1) and b(2) coincide with the integers ap and bp of the Dirichlet series and that the other coefficients b(n) equal zero.

(ii) Notice that -I

It is not difficult to see that the formal group related to Lp(s) is isomorphic to the formal group which is related to the Dirichlet series ( 1 - ~.p-S~ -1 . (See t20a>.,, \

/

3. Generalized Ap6ry numbers. n

n

2

The numbersb(n)=k~=l(k)

" ( n k k ) a n d d ( n ) = k=l~(k)z'(nkk)2 were introduced by

Roger Ap6ry and played a role in the proof of the irrationality of 5(2) and ~'(3) respectively. Many papers deal with congruences on these numbers. We mention Chowla, cowles and Cowles [9] and Gessel [16]. For these numbers Mimura [21] proved some congruences of the form up_ 1 = 1 mod p3, where p is a prime, p > 5. F. Beukers [4] generalized these congruences to

u(mp r - 1) - u(mp "-I - 1) modp 3r, where m and r are any positive integers. Now we consider the so called generalized Ap~ry numbers, which are defined by WAB~(n) =

'

(11)

,I: =

where A,B ~ Z~O and e = +1. We have for the generalized Ap6ry numbers the following theorem. Theorem

4. Let w(n) be as defined above. Let p > 5 be a prime. Then f o r any

m, r ~ Z>_I we have

w ( mp r) =_ w ( m p r - t ) modp 3r j f A > 2 f°r~A = 1 and B > 1,e = - 1 and w ( mp r - 1) = w(mp r-1

3r .f B > 2 1) modp f°r~B = 1 a n d A > 1,e= (-1) A.

Proof. The proof is very technical. See [12, pp. 49-55]. I

198

4. Binomial coefficients. Since the work of Fermat it is known that every prime p -= 1 mod 4 can be written as p = a2+b 2 for integers a and b in an essentially unique way. Without loss o f generality we may assume that a ~ 1 mod 4. Gauss proved by counting the number of solutions of the elliptic curve ~: y2 = X4+l rood p in two, essentially different ways, that

e@_) - 2a mod p

(12)

By applying Corollary 3 on the elliptic curve ~, congruence (12) can be generalized to

J mod ' where

m,r

are positive integers and m -- I mod 4. Here i denotes a p-adic integer such

that i2 = -1 and

bi = -a

mod p. Beukers conjectured in [4] the congruence

=-(a+bi)

modp 2

(14)

This was proved by Chowla, Dwork and Evans [10]. Van Hamme [18] generalized (14) to

__

= (a+ bi) .

modp r

(15)

for any positive integer r. This congruence can be generalized to the supercongruence

~)-(a+bi)

~)modp

2r

(16)

We get another example by considering primes p - 1 mod 3. Then 4 p = e 2 + 3 f 2 for certain values e andf. Without loss of generality we may assume that e - - 1 mod 3. Choose the p-adic number ~: = (e + 3fq/-3)/2 such that I ~ Ip = 1 . Starting from the elliptic curve N: y2 =

1_4X 3, Corollary 3 implies the congruence (~(mp'-l))

~(mpr-

1)

-

_ (~(mp'-'-l))

~'~(mp'-'

1)

mod pr

(17)

199

for any positive integers m,r with m - 1 mod 3. H o w e v e r congruence (17) can be improved to the supercongruence

(~(mff-1) [½(mff-1)

i2. m

_ j i /gt

P

r-t

-

:'t.~(mpr-'-

1"" ;l

(18)

1)) m ° d p 2 '

In the general case we define for a , ]3 positive integers with ~z+]3 _< d the binomial coefficient ifn - 1 mod d

(19)

else. We have for these coefficients the congruence

v(mff) =- ~.v(mp r-l) m o d p ~ where

-

g(

(20)

)

This result can be found using formal group theory ( n a m e l y f ( T ) = ~ n=l

v(n)

• T n is a

/'/

formal logarithm over Zp for p = 1 mod d) or the p-adic F-function (cf. [22, pp. 111114]). In the case that d = 2, 3, 4 or 6 we can improve congruence (20). The following theorem deals with the improvement. T h e o r e m 5. Let d be 2, 3, 4 or 6. Let p be a prime with p = 1 mod d. Let m and r be

positive integers with m = 1 m o d d. Let ~z, ]3 ~ Z> 1 with a+]3 < d. Then the binominal coefficient v(n) satisfies the supercongruence v(mpr) =--g(p),,,V "-1" ~.v(rnpr-1) modp2r

(21)

where g(p) ~ Zp with g(p) -= 1 mod p and fr - Fp ( ~-)Fp ( ~) d P r o o f . We prove congruence (21) using the p-adic F-function. The proof is based on a formula of Gross and Koblitz [17] which expresses the p-adic F-function in terms o f Gauss sums and on a formula of Diamond [14] which expresses the logarithmic derivative in terms of the p-adic logarithm. See [11].11

200

5. Values of the Legendre polynomials. This section contains joined work with L. van Hamme. Nice supercongruences exist for the values of some Legendre polynomials. These polynomials can be defined by (22)

¢n+k) ¢t-l) k k=O

andthey satisfy F(X) =

&

1

I 1 - 2tX + X

= ~.a Pn(t)X" ,=0

(23)

Let K be an algebraic extension of Q. Let p be a prime which splits in K as p = sr~. I-et t ~ K with I tip < 1 and consider the differential form (24)

dX = ~_~ P , ( t ) X " d X d l - 2tX2 + X4 n=0

on the elliptic curve g : y2 = x(x 2 + Ax + B). The theory of formal groups predicts a congruence of the form as described in Corollary 3 P-~(,,,,,-'-t) (t) = ~'P~f,,p.-'-l) (t) modp"

(25)

for any positive integer r and positive odd integer m. It turns out that if g has complex multiplication, congruence (25) can be changed into a congruence rood ~2r. We have the following theorem. T h e o r e m 6. Let K = Q ( ~ , elliptic curve

t/7 ) with d a square-free positive integer. Consider the

: y2 = x(x 2 + Ax + B) with A, B ~ K

(26)

Let A = A 2 - 4 B . Let oa and oo' be a basis of periods of ~

~r= o0'/co ~ Q ( ~

and suppose that

) (which implies that the curve has complex multiplication), ~ has

positive imaginary part and A = 3 go (co/2), ~ - = go (09/2 + ¢o'/2) - go (co72), where go (z) is the Weierstrass go-function. Let p be an odd prime which does not divide d and p = srsr, where st, ~ ~ Q ( ~ ). Suppose that u, v, x, y integers and v even. Then we have

Pii_(mpr_l)(~--) = era/¢-1. -~.P,, where e = i y(I -x)+p-2 . Here i = tTf.

sr = u + v ~

and ~lr = x + y ~ with

r-i ..(_a_.._'~mod rc2r

(27)

201

We

first

g:y2=

give

an

example

x ( x 2 + 3 x +2).

We

in

which

Theorem

can

choose

periods

6

can

be

applied.

Let

co and co' in such a way

that ~o(co/2) = 1 and co'/co = v = i. Let p - 1 mod 4 be a prime. Let i be a p - a d i c number such that i 2 = -1. Fix the sign of bi such that a - bi mod p. Let zc= a-bi. Then we have ~1: = zci = b+ai. Hence e = i y(1-x)+p-2= i - b = denote a(n)

=

(_l)(P-1)/4. W e

k~=l\kJ'k, k ) " The numbers a(n) have been used for proving that log 2

is irrational with measure of irrationality 4.622 [1]. Carlitz proved that the numbers a(n) satisfy f o r p - 1 mod 4 the congruence a

- ( - 1)

(28)

• za moa p.

Since a(n) = Pn(3), we have for those primes the supercongruence a

mp 2

_ (_l)~-1).~.a

modp2~.

(29)

Another proof o f this supercongruence in the case m = r = 1 has been given by van Hamme in [ 18]. Sketch of the p r o o f of T h e o r e m 6. Let L = Q ( ~ ,

q-d) and

R={ cz ~ L: ord~(a) > 0}. In this proof we will denote = v/g

(30)

•

dx

x

We consider the holomorphic differential form co = - - - . Let t = - be a local parameter at 2y y infinity. We express co in terms of t and we get

co =

dt

4 1 - 2At2 + At 4

= ~ c(n). tZndt.

(31)

,=0

Then we define the local parameter z at infinity by d z = cO

Hence z can be expressed as a function of t by

(32)

202

z = ~-' c(k)

2k+1

(33)

k~=oZk+ i t and t can be expressed as a function of z by ~(z) - ~(~)

t= z+...=-2.

(34)

#(z)

Notice that t(z) is an elliptic function. Since g has complex multiplication we have zc e End(~). More specified we have t(7~z)

= F(t(z))

1 + 7ca2t-e(z) + ga4t-4(z) + ... +7~ap_l t I -P(z) (35) = rltP(z).

1-

7~dz

t-Z(z)

-

red4 t -4

(2) + . . . -

gd.p_ 1 t l - P ( z )

where 13, ai, dj ~ R . This formula is due to Weber (cf. [23]). Formula (33) imply the formulas r~z =/..~ 0= 2/+c(1)t2l+l(rCZl )

(36)

and e~

r~z= ~ r c . c(k) t2k+l

(37)

k=o 2 k + l Substitute (35) for t(rcz) in (36). Consider in equations (36) and (37) the coefficient of

7~p2r tmP r mod

mp

r" We get the coefficients 1

mff "1 c ( ~ ( m p r - l - l ) ) . ~ ] mp'-

1

(38)

and

mP r c ( l ( m p r - 1 ) )

(39)

respectively from formulas (36) and (37) respectively. This implies the congruence of the 1

theorem. We can calculate that T1 = i y (1 - ~) +p - 2. (V/-~) ~(P- x). See a more detailed proof in [13] R

203

There are only 8 values t with these nice supercongruences over Z (cf. [12, pp. 87-89]).

6. Conclusion. In Section 5 we introduced the numbers a(n), which are generalised Ap6ry numbers. They satisfy supercongruence (29). The numbers a(n) are related to an elliptic curve. The Ap6ry numbers b(n) and d(n) as defined in Section 4 are related to K3 surfaces (cf. [7]). They satisfy other congruences which are comparable to congruence (29), namely r-1

,40 and (41)

where a+bi is as defined in section 4 and ~ is a root of some polynomial of degree 3 (see [6] or [24]). Beukers and Stienstra conjectured in [6] and [7] the supercongruences

b

=(a+bi)Z.b mpr

1-1

modp 2"

(42)

and

d('~

t -- ~ "d( rap'21- 1~1m°dpZ'"

(43)

Van Hamme [19] proved (42) in the case that m = r = 1. Recently Young [24] proved (43) in the case that m = r = 1. The rest of the conjectures is at the moment unproved. Perhaps, the proof of Theorem 4 gives a good possibility to prove the rest of the conjectures. 7. REFERENCES. [1] [2] [3] [4] [5]

K. Alladi and M.L. Robinson: On certain values of the logarithm, Lecture Notes 751, 1-9. R. Ap6ry: Irrationalit6 de 4(2) et ~(3), Ast6risque 61 (1979), 11-13. A.O.L. Atkin and H.P.F. Swinnerton-Dyer: Modular forms on noncongruence subgroups, Proc. of Symposia in Pure Math., A.M.S. 19 (1971), 1-25. F.Beukers: Arithmeticalproperties of Picard-Fuchs equations, Sdminaire de thgorie des hombres, Paris 82-83, Birkh~iuser Boston, 1984, 33-38. F. Beukers: Some congruences for the Ap~ry numbers, J. Number Theory 21 (1985), 141-150.

204

[6] [7] [8] [9]

F. Beukers: Another congruence for the Apdry numbers, J. Number Theory 25 (1987), 201-210. F. Beukers and J. Stienstra: On the Picard-Fuchs equation and the formal Brauer group of certain elliptic K3-surfaces, Math. Annalen 271 (1985), 293-304. L. Carlitz: Advanced problem 4268, A.M.M. 62 (1965) p. 186 and A.M.M. 63 (1956) 348-350. S. Chowla, J. Cowles and M. Cowles: Congruence properties of Ap~ry numbers, J. Number theory 12 (1980), 188-190.

[10] S. Chowla, B. Dwork and R.J. Evans: On the modp 2 determination of~p~al J , J. [11] [12] [13] [14] [15] [16]

Number theory 24 (1986), 188-196. M.J. Coster: Generalisation of a congruence of Gauss, J. Number theory 29 (1988), 300-310. M.J. Coster: Supercongruences, [Thesis] Univ. of Leiden, the Netherlands, 1988. M.J. Coster and L. van Hamme: Supercongruences of Atkin and Swinnerton-Dyer type for Legendre polynomials, to appear in J. of Number Theory in 1990. J. Diamond: The p-adic log gamma function and p-adic Euler constants, Trans. Amer. Math. Soc. 233 (1977), 321-337. C.F. Gauss: Arithmetische Untersuchungen (Disquisitiones arithmeticae), [Book] Chelsea Publishing Company Bronx, New York, reprinted 1965. I. Gessel: Some congruences for Apdry numbers, J. Number theory 14 (1982), 362-368.

[t7] B. Gross and M. Koblitz: Gauss sums and the p-adic F-function, Ann. Math. 109 (1979), 569-581. [18] L. van Hamme: The p-adic gamma function and congruences of Atkin and Swinnerton-Dyer, Groupe d'6tude d'analyse ultramrtrique, 9 e annre 81/82, Fasc. 3 no. J17-6p. [19] L. van Hamme: Proof of a conjecture of Beukers on Ap~ry numbers, Proceedings of the conference of p-adic analysis, Hengelhoef, Belgium (1986), 189-195. [20] M. Hazewinkel: Formal groups and applications, [Book] Academic Press, New York, 1978. [21] Y. Mimura: Congruence properties of Apdry numbers, J. Number theory 16 (1983), 138-146. [22] W.H. Schikhof: Ultrametric calculus, [Book] Cambridge University Press, Cambridge, 1984. [23] H. Weber: Lehrbuch der Algebra, [Book] dritter dand, Friedrich Vieweg und Sohn, Braunschweig, 1908. [24] P.T. Young: Further congruences for the Ap~ry numbers, to appear, 1989.

Witt realization of the p-adic Barsotti-Tate groups; some applications Valentino Cristante Dipartimento di Matematica Pura e Applicata dell'Universitd di Padova via Belzoni , 7 - 35131 Padova , Italia.

O. -

Introduction.

It is now some twenty years since Barsotti's paper "Varieth abeliane su corpi p-adici; parte prima" appeared (cf. [ve]). In it the Barsotti-Tate groups over p-adic rings are studied by introducing an embedding of their affine algebra R in a ring of Witt vectors with components over an algebra R O, of characteristic p, depending functorially on the reduction mod p of R. In my opinion, this embedding is a good tool for handling a great many problems; so I would like first to explain some of the ideas leading to this embedding, and then show how it can be used in situations involving the local deformation of abelian varieties. In order to say something a little bit more precise, let us consider a Barsotti-Tate (B-T) group G over the ring W = W(k) of Witt vectors with components in a perfect field k of characteristic p > 0, let us denote by R the aff'me algebra of G and by R 0 the affine algebra of its special fibre; R 0 is endowed with the usual topology as a profinite ring and we define R 0 as the direct limit R

= ti~m( R o

[p]

[pl

, Ro

~...)

,

where [p] denotes the endomorphism corresponding to the multiplication by p in G 0, with the limit topology. The algebra depending functorially on R O, to which we alluded above, is exactly the completion R 0 of R~. In what follows the R0-algebra structure of R 0 is fixed once and for all by an embedding i: R 0 --," R O, which is a bialgebra homomorphism, so that each bialgebra extension of R 0 gives an analogous

extension of R O. The main point in our construction of the embedding of R in W ( R o ) depends on the following remark: one can functorially associate to R an extension E = E(R) o f R 0 by a Witt bialgebra, and the extension of R 0 corresponding to E splits. As a consequence there exists a bialgebra homomorphism j sitting in the following commutative diagram: R

l

R ~ k =R 0 where

can

J

~W(R 0 )

i

~ RO ,

and p denote the reduction mod p of R and of W(RO), respectively.

206

Now let us start with a fixed R O, and consider the set of the liftings o f R 0 to W: we mean the set of the pairs (R,ff), where R is the affine algebra of a B - T group over W and ff is a bialgebra homomorphism which can be factored in the following way: R

R ®k

,~.~ ~ R0

.

On this set there is a natural notion of equivalence: the liftings (R,~) and (R',o") are equivalent if and only if there exists an isomorphism z of R in R' which makes the following diagram commutative: R

R'

This shows that W ( R o )

17"

~R0

.

is the natural place where all the liftings of R 0 to W live; but there is

something more here, as well. In fact, for each class of equivalent liftings there exists only one representative embedded in W(RO) , so that after our construction, coordinatizing the classes of equivalent liftings becomes quite natural: it amounts to giving the position inside W ( R O) of the corresponding embedded representatives. With these coordinates, the usual structures associated with deformation theory,, e.g. the Kodaira-Spencer mapping and the Gauss-Manin connection on the de Rham cohomology of the universal lifting, can be described in a very natural and simple way. In fact, in the second part of the present paper we will describe the situation which arises when R 0 comes from an ordinary abelian variety; our results are similar, to, and in fact inspired by, those obtained by N. Katz in [KAI, so that in the end we can conclude that our coordinates are essentially the same of those of Serre and Tate. The first part is devoted to giving the definitions and the results about the canonical embedding necessary for intelligibility of the second part: it is in fact an abstract of a work in progress in collaboration with M. Candilera, where the canonical embedding is used in order to discuss the reduction mod p of the theta functions. In the meantime, Candilera's thesis (cf. [CA]), where part of our results were first published with proofs, may be useful. Most of the main ideas used here, are - in a more or less explicit way - contained in the works of Barsotti, and it is impossible for me to think about these arguments without a deep feeling of gratitude for his heritage.

1.

Witt extensions of B-T groups; their trivialization and related embeddings. The notations in the introduction maintain their meaning; moreover we will use the symbol B W ( S )

to denote the ring of infinite (Witt) bivectors with components in any k-bialgebra S. General results about

207

bivectors can be found in [MA] or in [FO].The coproduct, inversion and coidentity of S, will be denoted by JP, ~, and e, respectively. Now we will describe the Witt extension of the affine algebra R 0 of a B-T group G O over k. A Witt

factor system o f R 0 is an element y ~ W(Ro®Ro) with the following properties: (i) (ii)

7 = sc(7); (e ® id)y= (id ® e)y=0;

(iii)

(~) ® id)y+ ( y ® 1) = (id® ~ ) 7 + (1 ® 7),

where sc and id are the twist of W(Ro®Ro) and the identity of W(Ro), respectively. Using ~, we will construct a bialgebra EO(~) in the following way:

e o ( ~ = Ro[[U o, u 1 . . . . ]l, where (U/)ie hr is a family of indeterminates over R 0 and the bialgebra structure is given by extending the coproduct, coidentity and inversion of R O, in the following way: JPU., cU, ~:U., are the components of I

I

l

place i of the following Witt vectors:

~)U= U ® I + 1 ® U + ~, ~U= 0, ~U= -U, where U denotes the vector (Uo, U 1. . . . ). Observe that when R 0 = k and y = 0, then Eo(Y) becomes the usual Witt formal k-bialgebra W; the bialgebra EO(~) just described is an extension of R 0 by W. As the embedding i: R 0 ~

R 0 is fixed, we can speak of bivectors of BW(Ro) whose components

of negative index are in RO; among these the subset of the canonical bivectors, {X: ~X = X ® 1 + 1 ®X}, has a W-module structure: it will be denoted by M(Ro) and called the canonical or the Dieudonnd module of

R 0 . The Frobenius and the Verschiebung maps operating on R O, R O, BW(Ro), M(Ro), etc. will be denoted by F and V. Usually Rot, R~, R 0 denote the 6tale, multiplicative and radical components ofR O.

THEOREM 1.1. Let ~'be a Witt factor system of Ro; let us denote by E O(~) the bialgebra i,(E o(~),

i.e. the extension coming from E O(7) by means of the embedding i:R 0 ~

R o.Then E,O(Y) = R 0 ® W;

more precisely, there exist two elements t ~ W ( R o) and S ~ BW(Ro), such that y= A,~ = AS. Here, and in what follows, A denotes the operator ~) - i d o l - l ® i d both on W ( R O) and on BW(Ro).The vector t is unique, its components ;ti are in v-(i+ l)go; in particular if R 0 -- R'~,~reach extension of R 0 by W is trivial. On the contrary, Sis unique if and only if R 0 = k. REMARK 1.2. If the Witt vectors are identified with the Witt bivectors having components of

negative indices equal to O, then 17 = ,~. - S is a canonical bivector of R 0 whose components of negative indices are in R O, i.e. it is in M(Ro). The bialgebra E O(~ is essentially determined by the bivector 71: in fact if rl" is an element of BW(R O) whose components with negative indices coincide with the corresponding components of 17, then the factor system ATl'gives an extension equivalent to EO(~). If W r denotes the product of r copies of W, we can consider extensions of R 0 by W r. Everything works as in the previous case: if S = (5i)1_
208

BW(Ro), we will consider the submodule N ( ~ of M(Ro) generated by the canonical bivectors 77i whose components with negative indices coincide with the corresponding ones of S.. We will identify two 1

trivialized extension if they give the same submodule N( 6). Now we will try to explain the relation between the Witt extensions o f R 0 and its liftings to W. Let us consider a lifting (R,cr) o f R 0 to W. When the parameters X = (Xi)l_
R = Rett[X 1 . . . . ,Xg]], where R et, which is the lifting of R~0 to W will be identified with W(Rot ) and CrlRel(ao,a I .... ) = a O. Let us denote by F and g)the power series which give the coproduct and the inversion, respectively:

r~=

Fj(XI®I .....

Xg®I , I®X 1 . . . . .

l®Xg), vXj= gj(X 1 . . . . .

Xg).

Observe that the coefficients of F are in W(Rot® Rot), and those of gj are in W(Ro). By means of ) these series we will construct a k-bialgebra: let (Xij) be a family of indeterminates over Reot, where 1-~ i -
and O_~j _
F i ( X l ® I , . . . , Xg®I , I® "~1 . . . . , l®Y,g), and vX..~]= g](~'l' " " ' Xg )" At this point it is quite easy to verify that the subring ROt[[XIo . . . . .

Xgo]] is canonically

isomorphic to R O, so that S can be canonically identified with RO[[(Xij)] ] , where 1_
THEOREM 1.3. Let R 0 be a B-T group of dimension g over k and (R,tr) a lifting. Then the bialgebra

S corresponding to the lifting (R, tr) just constructed is an extension of R 0 by W g. Let S and S" the extensions corresponding to the liftings (R,cr) and (R',cr'), then they can be trivialized in such a way that the W-module N(b') and N(~) coincide (i.e. they are equ&alent in the sense of 1.2) if and only if (R,cr) and (R',c¢) are equivalent in the sense of the introduction. Here we will give an idea of how the module N(6) arises, the same argument will explain the existence of the embedding of R in W(Ro). Let us denote by j': R "---+ W(S) the map which extends the identity of R et and sends X. in ,~,. The map j ' is a bialgebra embedding such that pj" = or. It can be extended to a sub-bialgebra o f R K (the general fibre) which contains the W-module//(R) of the integrals of the fu-st kind. The j'-images of the integrals of//(R) are canonical bivectors whose components with nonpositive indices are in Ro.Thus that to each h~//(R) there corresponds an element jh in M(Ro): the set of such images is just N(6). Now, we have an embedding j: II(R ) -"* BW(Ro), and it can be proved that if

(h i) is a basis of//(R), there exist g power series, G., such that t R' = R et [[Gl(Jh" I . . . . . j h ) . . . . . Gg(jh I . . . . .

jhg)]],

with the structure induced by means of the components, is a sub-bialgebra of W(Ro). If, as in the introduction, p denotes the usual reduction rood p of W(Ro), then (R',p) is the unique lifting of R 0 equivalent to (R,cr) embedded in W(Ro). It will be called the embedded li~ng equivalent to (R,cr).

209

The sub-module N(b') of M(R O) in the previous discussion is the usual filtration appearing in several points of the literature (for instance cf, []}IV],[VP] or DIM]), so that the following result is well known:

THEOREM 1.4. Let R 0 be the affine algebra of a B-T group of dimension g over k; then a W-

submodule N of M(R O) comes from a lifting (R,cr) if and only if rk(N) = g and N + pM(R O) = VM(Ro). REMARK 1.5.Let N be a W-submodule of M(R O) coming from a lifting of Ro to W. Then there

exists a submodule N' such that M(R o) = N~N'. Moreover, for each f ~ Hom(N, pN') the submodule Nf= {x + f x : x EN) comes from a lifting, and all the modules coming from Iiftings of Ro to W are of this form. The pair (N, N ) is a frame for the set of the classes of equivalent liftings of R 0 andf is the

coordinate of the lifting corresponding to Nf. REMARK 1 6. If R

loc

z;

.

.

.

.

.

toc

.

= R ~ e the local component of R ;s multtphcatLve, then M(R ) sansfies o . 0 . . . . O" " " . . toc et the condmons of 1.4., and M(Ro) has the canomcal direct decomposmon M(Ro) = M(R 0 )~M(R 0 ). Hence •

0

in this case we can speak of canonical frame and canonical coordinate. The lifting which has coordinate 0 is called the canonical li~ng. It is the direct product of the canonical liftings""

of R;nand of R 0

We assume R 0 as in 1.6.; let g and c be the dimension and codimension of R O, respectively. Assume 1 _~r -~g and I -~s -~c, denote by (ts) a family ofcg indeterminates over W, fix a basis (r/i) of et

M(RI; c) and a basis (Zi) of M(Ro), then the g elements C

j=l

of M(Ro)®W[[(trs)] ] give a basis for the module of the integrals of the f'trst kind of a B-T group over

W[[(t.rs)]]: this group is the universal lifting o f R 0 to W[[(trs)]], its coordinate f is the homomorphism

M(RIoC)®w[[(ts)]] "--+ M(Rot)®W[[(tr )]], given by C

r/i --+ 2 t i j ~ . j=l Observe that for each point P ~ Hom(W[[(trs)]],W) the map given by C

j=l •

loc

et

is an element m Hom(M(R 0 )" pM(Ro)) and that each such element arises by specializingf by means of a suitable P. DEFINITION 1.7.Let R = Ret[[x 1. . . . . Xg]] be the affine algebra of a B-T group over W; an

210

integral of the second kind of R is a power series h =~avXV with coefficients in B W ( R et o), such that : a(o ..... O) • BW(ReOt)'

v V!av• Ret(=W(Reot)), and

~ h - h®l - l®h • R@R.

The W-module of the integrals of the second kind will be denoted by 12(R). The quotient 12(R)/R is the usual HIR(R). Let (R,o) be an embedded lifiting ofR O. Then the embedding j can be extended to the module/2(R); the images of the integrals are in BW(R~; more precisely, as a consequence of the relation ^

~h - h®l - l®h • R®R, ^

we have ~jh -jh®l - l®jh • W(Ro®Ro), so that we can conclude thatjh has the same components with negative indices as a canonical bivector r/h' The map h ----+ r/h induces the classical isomorphism between

HIR(R) and M(R o). The most important result for applications to deformation theory is the foUowing

THEOREM 1.9. Let (R,cr) be an embedded lifiting of R o; to each integral of the second kind jh, by A

means of the factor system ~jh - jh®l - l®jh e R®R, there corresponds an embedded extension E of R by the additive bialgebra D. The elements of the module II(R) + are the integrals of the first kind of E. More generally, there exist a natural bijective correspondence between the set of the classes of additive extensions of R, and the set of W-submodutes of M(R O) containing the module II(R): it is the correspondence induced by E ~

II(E); in fact for each class there exists one and only one embedded

representative. In particular, for each lifting (R,cr) there exists a class of extensions corresponding to the module M(Ro). Its embedded representative is an extension of R by D c which is independent of (R,o): it is the usal universal extension, U(R), of R. For a definition of U(R) one can look at [BIV] or [MM].

2.

-

Local deformations. Let A be an abelian variety over k and ~[ the corresponding B-T group; let us denote by R 0 =

Ro(A) the affine algebra of ~ , by M(A) and M*(A) the Dieudonn6 modules M(Ro(A)) and M(Ro(A)), respectively: here R;(A) denotes the dual of R 0 . The components of the elements o f M(Ro(A)) are in

Ro(A) (which is constructed as R 0 (A), but starting from R;(A) ). The duality between R 0 (A) and Ro(A ) gives an identification of M*(A) with Horn (M(A), IV). If L is a line bundle on A, then in [MA] a Dieudonn6 module homomorphism

qJL: M*(A)

' M(A)

is defined.This map, which can also be obtained from a theta function associated to a section of L cf. [CC],

211

[BR] or [CR]), is a Riemann form; i.e. the map L -"+ tOt. induces an injective group homomorphism from the Neron-Severi group, NS(A), into the group of alternating 2-forms

M*(A) xM*(A)

, W.

Now, let us consider the following particular situation: A is replaced by A~4*, where A* denotes the dual of A, and L by the Poincar6 bundle jo on A~4*. If we identify M*(A,~4*) with M*(A)~M*(A*) and analogously M(A,~4*) with M(A)$M(A*), then epp can be represented by a matrix

which works in the following way: 7

%171

6t epA~ ( l..l~

%,7') = ( r for each 77 ~ M*(A ), 1t" ~ M*(A * ), a ~ Hom(M*(A ), M(A )), ~9A ~HOm(M*(A *), M(A ) ), etc.. The results about ~PAused later on are described in the following THEOREM 2.1. (i) ~0A is a Dieudonn6 module isomorphism. (ii) if B is another abelian variety, c~a

homomorphism of A in B and or* its dual, then we have the following commutative diagram." M*(B*) lM( a *) M*(A*)

IM(B) M(a) [ iM(A)

(iii) ~0A and (OA. are related by the following duality relation: ¢PA*=- q~A All this was announced in '62, cf [BRXl, then proved with all details in chap. 7 of [MA]. COROLLARY 2.2. Let us denote by eA the map of M(A) x M(A*) with values in W, defined for each

17~ M(A), rl*~ M(A*) by' ~t(rT, 77*) = <~0-hI, r/*>, A

where < - , - > means the usual duality between M*(A *) and M(A *); then ea(rl, 77*) = - ea,(rl*, 77)

(2.3)

PROOF. The duality relation of Th. 2.1., i.e. tpA* = - ~0], means

(2.4)

= -

for each d ~ M*(A), d*e M*(A*). Since q~Ais isomorphism, (2.4) holds if and only if

2~2

(2.5)

<{0"lfl, fl*> = - <~o-1fl *, rl > A A* for each rle M(A), fl*e M(A*); but (2.5) is exactly (2.3), QED. REMARK 2.6. Let us consider the universal lifting A of ~ and its dual A*, which is canonically

isomorphic to the universal lifting of the dual of ~ ; then there is the following relation between the modules of the integrals of the first kind : (PA(//(A*)'L) =//(A). This can be checked by recalling the commutativity of the following diagram: 0 .....

~ 11(R)

, M(A)

,M(A)!I1 (R) ---+0

r 0

, l1 (R*) "1" ~M*(A*)

~Lie(R*)

~0

where R is the affine algebra of any lifting of ~t to W, r is the restriction of tpAand 6 the map induced by

% REMARK2.7. If MZ(A ) and Met(A) denote the muhiplicative and the dtate component of M(A ), and

if A is ordinary, i.e. M(A ) = MZ(A )@Met(A ), then with respect to the bilinear form CA: M(A) xM(A*) ~

W

we have the following orthogonality relations : (MZ(A)) ± = MZ(A*), (Met(A)) ± = Met(a.). In fact, since (PAis a Dieudonn6 module isomorphism,

ep-l(MZ(A)) = (M*(A*))zr= { f ~ Hom(M(A*),W) IfM]r(A)= 0 } = Horn (Met(A*),W); A the analogous relation for Met(A *) is

~o'l(Met(A)) = Horn (MZ(A*),W). A As we said in part 1, if t denotes the family of indeterminates (t/j), l_
f A " MZ(A)®W[[t]]

' Met(A)®W[[t]]'

which has been called the coordinate of A; in what followsfA will be replaced by the W[[t]]- bilinear form

qA " (MZ(A)®W[[t]]) x (MZ(A*)®W[[t]])

, W[[t]]

213

defined by (2.8)

qA (r/' o*) = eA(fA 77, O*),

here, of course, eA means the extension by W[[t]]-linearity of the map previously defined in (2.2). Now we'll write down (2.8) more explicitly, in order to understand the way in which qA too works like a coordinate. Let us denote by (rli)l
(trli)l~_i-
and qA are related as follows:

for each 7/e Mn(A)®W[[t]].

j qA(Oi , rlj)

PROOF. It suffices to show thatf A (r/i) = (2.2) we have

, for i= 1 . . . . . g. Now by (2.8) and

j

qA%"

0,"

Then, if we can show that (tpA,rli)l~_ i -1 - *

0.

-
immediately. Now by (iii) of (2.I), we have -

•

,

= <-~l,0j )' ~A 71i> = <(q}a) • -1rlj) * ' ~Atrli> = = 8..~j ' QED. The following theorem shows why qA is a good coordinate; its point (i) is just a translation of 2.1 in terms of ~4 "

THEOREM 2.10. Let us denote by A and B two ordinary abelian varieties over k, by A and B the

universal liftings of the corresponding B-T groups, and by qA" qB the coordinates; then:

(i)

qA(n, 77*) = ~ , ( n * , ~1),

for each r1 ~ M~r(A)®W[[t]], rl*~ MX(A*)®W[[tl] (ii)

A homomorphism G eHom(A, B) can be lifted to a e Hom(A, B) if and only if qA (M(G))', 77*) = qB(7, M(G*)rl*),

for each T e M~(B)®W[[t]], rl*~ MZ(A )®W[[t]], where M( G) denotes the extension by W[[t]]-linearity of the element of Hom(M(B), M(A)) induced by G.

214

PROOF. The relation betweenfA andfA, is the following: (2.11)

~ ( 7 + f A 7/,7" +fA,7*)=0. In fact (2.11) means

(2.12)

< rPAI(7 +fAr/), 7* +fA,7 *> =0,

II(A • )± , as remarked m 2.6. Now by (2.12), recalling the ~l-orthogonality of M~r(A) with M~r(A*) and of Met(A) with Met(A*),

i.e. q~-I (ll(A)) =

A

cf. 2.7, we conclude: <~A1 7, fA,Fff > + < (P-la( f A T ] ) ' 7 * > = - < ~0-1A*fA*7*'/~>+ <~°i l f A T ' 7 * > = 0 so that, using the definition 2.8, the last relation can be translated as follows:

qA (7, 7*) = qA.(q*, 7), this is just what we claimed in (i). Now we will prove (ii): G can be lifted to G if and only ifM(G)(II(B)) C_ll(A); therefore for each ~,s MX(B) there must exist 7' c MJr(A) such that (2.13)

M(O)(:'+I~7) = 7 ' + f A 7'. In view of the decomposition M(B) = M(B)~r~ M(B) et and of the analogue for M(A), (2.13) implies

(2.14)

M(G)(fB ~) = fA(M(G)Q,)). Now, by (2.14), and recalling the commutativity of the diagram in (ii) of 2.1, we can write:

qA(M(C)(r).

<

> --<

O* > --

<M*(G*)(~P~(fB~)), 7* > = < q)'B1%7), M(G*)7* > = qB ( ?', M(G*)(r/*)), QED. Our next goal is to compute the differential of qA with respect to the parameters. By the results described in the first part, cf. in particular 1.9., the universal extension of A, U(A ), can be canonically embedded in biv(R( ~)®W[[t]] ). And the well known constance of U(A) implies that, for each homomorphism P of W[[t]] in W, the universal extension U ( ~ p ) of the lifting ,~p corresponding to P, doesn't depend on P; so that, once U(A) is identified with its embedded image, there is the decomposition

215

U ( A ) = U®W[[t]],

(2.15)

where U denotes U(~ie) for each P e Hom(W[[t]],W); of course, the same holds for the integrals of the first kind:

11(U(A))= M(A)®W[[t]].

(2.16)

REMARK 2.17 Let us denote by V the connection on II(U(A)) whose module of the horizontal

sections is M (A ). By means of the canonical identification ~ :HDIR(A )

(2.18)

, I I(U(A)),

17: H 1 R ( A )

~H 1 ( A ) ® g2........... becomes the Gauss-Manin connection on H 1 ( A ) ; the DR w[[t]] ~tttll/w DR corresponding module of the horizontal sections, llt-1 (M(A)), will be denoted by M.

-//~R1 (,~e) for each

In fact, when all things are embedded, it is easy to check that M -

P ~ Hom(W[[t]],W). Now, let us consider the decomposition of H 1 (A) induced by the identification (2.18):

DR HDIR(A) = II(A ) ~ (Met®w[[t]]),

(2.19) w l i e r e M et =

Met(A).

In this situation, the natural isomorphism t: Met®Wilt]]

~ Lie(A*) can be described explicitly :

PROPOSITION 2.20. The map t is determined by eA in the following way: for each rl~Met®w[[t]]

and

h* ~I1(A*), we have: [t(r/)](h*) : ea(11, O*),

where 11" is given by 17" + fA.rl* = h*. This result can be deduced from theorem 8.1 of [VP].

REMARK 2.21. The map

Kod: Ii(A) ~ defined by h ~

Lie(A*) ® f~wmulw w[[ t]] . . . . . . . .

ffPr2(17(h))), where pr 2 is the projection relative to the decomposition (2.19), is the usual

Kodaira-Spencer map. For our next application we prefer a bilinear variation of Kod :

[ -,-]:

xi(A)

, awLt,]v w

defined as follow : [h*, h] = (<-, - > ® id)((id ×Kod)(h*, h)), where id denotes the identity map and <-, - > denotes the usual pairing between//(A*) and Lie(A*). The following is the main result:

216 THEOREM 2.22. F o r each (71,71") ~ (M~(A)®W[[t]]) x (MX(A*)®W[[t]]), w e have:

(2.23)

,~( o, rT*) =

[~* +~,~*, 71+fA rl].

PROOF. First, by 2.2t., recalling 2.9., we can compute Kod :

Kod('¢'l+fA 'r"])= '[Pr2(~( T/+~_,, qA (//, 'r/j)* 8 i))] =~.,~.(~/) ~, dqA(T/, /Tj)j

Y

Then we observe that rl) <

*-1-~

*

r/j J,l, rlj, z(~i) >

:

*

eA(8 i, 11j) = 8 zj..;

in fact,

eA(8 i, rl ) --eA(~At,Ti, rlj> = < Oi' Oj> ' and ( t 7/. ) is . the basis . dual to ( ~J ). l

As a consequence,

f'mally (2.23) comes from (224) by linearity,QED.

REFERENCES [MA] I. Barsotti, Metodi analiticiper le varieM abeliane in caratteristica positiva, seven chapters divided into five publications in Ann. Scuola Norm. Sup. Pisa, from 18 (1964) to 20 (1966). [VP] Varietd abeliane su corpi p-adici, Symp Math. 1 (1968), 109-173. [BIV]Bivettori, Symp. Math. 24 (1981), 23-63. [BRX] Analytical methods for abelian varieties in positive characteristic, Colloque sur la th6orie des groupes alg6briques, Bruxelles, 1962, 78-85. [EO] J-M Fonlaine, Modules galoisiens, modulesfittrEs et anneaux de Barsotti-Tate, Asterisque 65 III (1979), 3-80. [KA] N. Katz, Serre-Tate local moduli, in "Surfaces Alg6briques", Lecture Notes in Math., vol. 868, Springer-Verlag, 1981, 138-202. [BR] L. Breen, Function tlffta et th~or~me du cube, Lecture Notes in Math., vol. 980, Springer-Verlag, 1983. [CC] M. Candiler~ and V. Cristante, Biextension associated to divisors on abelian varieties and theta functions, Ann. Scuola Norm. Sup. Pisa 10 (1983), 437-491. [CA] M. Candilera, Riduzione difunzioni theta, Thesis, University of Padova, 1986-87. [CR] V. Cristante, Thetafunctions and Barsotti-Tate groups, Ann. Scuola Norm. Sup.Pisa 7 (1980), 181-215. [MM] B. Mazur and W. Messing, Universal extensions and one dimensional crystalline cohomoiogy, Lecture Notes in Math., vol. 370, Springer-Verlag, 1974.

POLYEDRES

DE

DE SOMMES

NEWTON'ET EXPONENTIELLES

J. Denef Department of M a t h e m a t i c s University of Leuven Celestijnenlaan 200 B B-3030 Leuven Belgium

1.

Introduction

et 6nonc~

POIDS

et

F. Loeser Universit~ Paris 6 et Ecole Polytechnique Adresse : Centre de M a t h 6 m a t i q u e s Ecole Polytechnique F-91128 Palaiseau Cedex France

des r6sultats

1.1 On note k le corps fini Fq £ q 616ments, et t u n nombre premier ne divisant pas q. Si K est un corps, on note/x" une cl6ture alg6brique de K . On fixe un caract~re a d d i t i f non trivial ¢ : k -* C x et on note £ ~ le ( ~ / f a i s c e a u sur A t associ6 £ ~b et au rev~tement d'Artin-Schreier tq - t = x. Soit X un sch6ma de type flni sur k, si f est un morphisme f : X ~ A~, on eonsi&re la somme exponentielle S(f) = ~ ¢(f(x)). rEx(t) D'apr~s la formule des traces de Grothendieck, on a

S(f) = E ( - 1 ) i

i Tr(F,

H~(X ® ~:, f * £ , ~ ) ) ,

F ddsignant le morphisme de Frobenius (g~om~trique) de k. Dans cette note nous expliquons comment on peut d~tenniner expticitement le module des valeurs propres de F quand X est un tore et f est non d~g~n~r~ p o u r son poly~dre de Newton £ l'infini. 1.2 Si A est un anneau commutatif, on note T ~ = S p e c A [ x l , . . . , x , , x ~ l , . . . , x ~ l] le tore de dimension n sur A. E t a n t donn~ f : T ~ --~ A ~ un A-morphisme, on dcrit f comme un polyngme de Laurent f = ~ cix'. iEz ~

Le poly~dre de Newton A ~ ( f ) de f £ l'infini est l'enveloppe convexe dans Qn de { i E Z " ; c i 7 ~ 0 } U { 0 } . Pour t o u t e f a c e r d e A~c(f) o n ~ c r i t f r = E cix i. O n d i t q u e f

iEr

est non dfig~nfir~ pour Ao~(f) si, p o u r toute face r de A ~ ( f ) ne contenant pas l'origine, le sous-sch~ma de T ~ d~fini p a r

Of~. Oxl

--

.

*

*

Of~Ox,, -

-

- - 0

est vide, On note Vol ( A o ~ ( f ) ) le volume de A ~ ( f ) . 1.3 -

Notre premier r~sultat est le suivant.

218

T h 4 o r ~ m e 1 . - - Soit f suppose que A o o ( f ) -- n.

: T'~ ~ A~ un morphisme non ddgdn&d pour Ao~(f) . On

Alors a)

b)

H ci ( T n~ , f * £ : ~ ) = 0

pour i~n,

dim Hn~W c~ ~' ~ , J':*~E~ )~- - n ! Vol (Ao~(f) ).

Si de plus I'origine appartient ~ l'intdrieur de Ao~(f), aJors c) H ~ ( T ~ , f * / : ¢ ) est p u t de poids r~ (i.e. toutes les valeurs propres de F sont de module v~n ). Ce th~or~me a ~t~ ddmontrd par Adolph.son et Sperber ([A-S 1], [A-S 2]) pour presque tout p et ils ont conjectur~ que le r6sultat ~tait vrai pour tout p. Contrairement £ leur approche, notre preuve est purement g-adique el: utilise des compa.ctifications toro'/dales. 1.4 - D'apr~s les r~sultats fondamentaux de Deligne ([De]), les valeurs propres de F sur H~"(T~, f * £ ~ ) sont de module v ~ ~ avec w e N, w < n. Dans [D--L] nous utilisons la cohomologie d'intersection pour d6terminer le nombre e~ de valeurs propres de module x / ~ (compt~es avec multiplicit~s). Autrement dit, on d~termine le polyn6me E(W~,f) = ~

e,~T ~' .

w~0

Ce problSme a 6td pos6 p a r Adolphson et Sperber ([A-S 3]) qui ont 6galement trait6 certains cas particuliers. Pour ~noncer nos formules, il nous faut introduire encore quelques notations. 1.5 - Pour tout c6ne convexe polyddral a , ayant l'origine p o u r sommet, on d$finit poly ( a ) comme le polytope convexe obtenu en c o u p a m a avec un h y p e r p l a n dans Q'~ ne passant pas par l'origine et intersectant toutes les faces de dimension 1 de a. Ce polytope est bien d~fini £ Squivalence combinatoire pr~s. Si A est un p o l y t o p e convexe darts Q~ et r u n e face de ,5,, on note coneA(r) le c6ne convexe poly6drai dans Q~ engendrd par A - r = {x - y ; x E A , y E r}. On d6finit cone°A(r) comme le c6ne convexe polyddral de sommet l'origine obtenu en coupant cone z~(r) p a r un sous-espace affine passant par l'origine compl6mentaire du sous-espace engendr6 p a r r - r . Ce c6ne est bien ddfini 5~ Squivalence affme pr~s. Avec les notations prdcddentes on peut maintenant ddfinir p a r rdcurrence sur la dimension de a et de A des polyn6mes a ( a ) e t / ~ ( A ) en la variable T p a r (1.5.1)

a(a) =

tronc
(1.5.2)

((1 - T 2 ) ~ ( p o l y ( a ) ) )

si

d i m c~ > 0

a-1

/3(A) = ( T 2 - 1) dim ~ +

Z

(T2 - 1)dim ~a(c°ne°~(r))

r face de A

(~.~.3) (Ici tronc(

~({0}) = L ) dSsigne le polyn6me obtenu par troncation en ne gardant que les monSmes

de degr$ < k.) Les polyn6mes a ( a ) et ~ ( A ) ne dSpendent que du t y p e combinatoire de a et A, et ont dt6 introduits p a r Stanley (IS]). On v~rifie que a ( a ) -- 1 si a est un c6ne simplicial. On note a ( a ) ( 1 ) la valeur de a ( a ) en T = 1.

219

1.6 - Soit f : T~ ~ A ~ un k-morphisme e t r une face de dimension d de Aoo(f). Si 0 E r on peut 6crire f~ = f ~ ( z ~ . . . . . x ~ ), avec ]'~ un polynSme de Laurent e n d variables et e l , . . . , ed une base du r6seau obtenu en intersectant Z" et l'espace affine engendr6 par r. Remarquons que JY~est d6fini £ un a u t o m o r p h i s m e de T d pros, et que s i f est non d6g6n6r6 pour Aoo(f), f~ est non d6g6n6r6 pour A ~ ( f ~ ) . Le r6sultat suivant permet de calculer explicitement, de faqon r6cursive, les e~. T h ~ o r ~ r n e 2 . - - Soit f : T ~ ---* A~ un morphisme non d6g~n~r~ pour A = Aoo(f). On suppose que dim A = n. Alors a) e . = n ! Vol(A) + E ( - 1 ) " - ~

~(d~m r ) ! V o l ( r ) a ( c o n e ~ ( r ) ) ( 1 )

b) E ( T ~ , f ) = e,,T n - E ( - 1 ) n-dim r E ( wdim ~, Y~)a(c°ne°(~')) • 0~r

(On somme sur les faces propres r de A contenant l'origine ; Vol( r) est le volume de r dans le sous-espace engendr6 par r, normalis6 pour prendre la va/eur 1 sur un domaine fondamental du rbseau induit par Zn.) En f a t Adotphson et Sperber ([A-S 3]) ont conjectur~ une formule explicite pour les ew. Notre r6sultat entraine que cette conjecture est fausse, d~j~, pour w -- n = 5 et un poly~dre A ~ ( f ) dont toutes les faces sont des simplexes ([D-L D. 2. Nous allons donner dans ee paragraphe quelques indications sur les principales 6tapes et les principaux ingr6dients de la d6monstration des th~or~mes 1 et 2. Le lecteur trouvera t o u s l e s d~taits dans [D-L]. 2.1

-

Schgmas torlques

(2.1.1) Soit A un anneau commutatif. A tout c6ne convexe poly6dral a darts q " on associe le A-sch6ma torique a/fine X A ( a ) := Spec A[a MZ"]. De plus, £ tout 6ventail (fini) ~ dans q n on associe un A-sch6ma torique X A ( E ) obtenu en recollant les sch6mas X A ( d ) , a E P', d6signant le c6ne dual de a. Cette construction est donn~e dans ([Da, § 5]) lorsque A est un corps, mais elle se g6n6ralise de fa~on ~vidente. Le sch6ma X A ( ~ ) est lisse sur A si et seulement si l'~ventail E est r6gulier, il est propre sur A si et seulement si le support de ~ est 6gaJ £ Q " (cf. [Da D. A tout c6ne a de dimension r de ~ on associe un A-tore X ~ ( Z ) C XA((7) C XA(V~) de dimension n - r. Ces tores torment une partition de X A ( Z ) . En prenant a = {0}, on voit que W~ est un ouvert dense de X A ( ~ ) . On note X ~ ( Z ) l'adh~rence de X ~ ( E ) dans XA(V@ Si a est un c6ne poly6dra] convexe et r u n e face de a, on pose X~(.) : = X A ( ~ -- ~') C XA(O). (2.1.2) D 6 f l n i t i o n . - Soit Y un schema sur A et y E Y. On d/t que Y est torofdal sur A en y si y a un voisinage 4tMe isomorphe sur A & un voisinage ~tale d'un point dans un A-sch6ma torique atone X A(a). Le lemme suivant est bien connu quand A est un corps. (2.1.3) L e m m e . - Soit Z un diviseur de Cartier effectif de X A(E) et z E Z. Soit a l'unique c6ne de E avec z E X ~ ( E ) , Supposons que 1'intersection sch6matique Z M X ~ (E) soit lisse sur A en z et qu'elle ne soit pas 6gale ~ X ~ ( E ) en z. Alors Z es~ toroi'dal sur A en z. Si de plus l'6ventaJ1 ~ est rdgulier, alors Z e s t lisse sur A en z. (2.1.4) Soit A un p o t y t o p e convexe de dimension n dans Q'*. Pour b E Q " on d~finit Fz~(b) comme l'ensemble des x E A tels que le produit scalaire b.x soit minimal. C'est une face de A. Pour a C Qn on pose F a ( a ) = Ne~F(b ). On d6fmit la relation d'6quivalence

220 suivante sur Q~ : b ~-, b' si et seulement si Fn(b) = F a ( b ' ) . Les adh6rences des classes d'6quivalence d6finissent un Eventa~l dans Q'~, l'6ventail associ6 £ A. On le note E ( A ) . L'application a -* F A ( a ) 6tablit une bijection entre les c6nes de E ( A ) et les faces de A. On note X A ( A ) le a - s c h 6 m a torique X A ( E ( A ) ) . Pour a • E ( A ) on 6erit X ~ ( A ) au lieu de X ~ ( E ( A ) ) . Si ~" est une face de A on 6crira X ~ ( A ) pour t'unique X ~ ( A ) avec F A ( a ) = r. 2.2 - C o m p a c t i f i c a t i o n s toro'idales Soit f : T~ ---* A~ un morphisme non-d6g6n6r6 pour A = A ~ ( f ) . On suppose que dim A = n. Si E est un 6ventail plus fin que E ( A ) on peut construire une compactification g~z de f de la fa~on suivante. On pose A : kIT]. Le tore T ~ : T~ x A~ est un ouvert dense de la vari6t6 torique X A ( E ) : Xt,(E) x A 1. On pose G = f - T consid6r6 comme potyn6me de Laurent sur A. On note Y~: l'adh6rence dans X A ( E ) de l'hypersurface G = 0 dans T ~ , et g~: te morphisme propre g~ : Y~ --+ A~ induit p a r la projection X A ( E ) --+ S p e c A = A~. C o m m e le graphe de f e s t te lieu de G = 0 darts T ~ , on volt que T ~ est un ouvert de Y~: et que la restriction de gzc ~ T~ coincide avee f . Pour cr • E on pose } ' 6 = Yn n

Xff(E) et 176

= Ys n

R~(E).

L'~nonc~ suivant est crucial. L e m m e - c l ~ [ D - L ] . - - Supposons que pour tou~ ~ • E tel q u e 0 • FA (a) on air a • E ( A ) . Alors Yz est toroMat sur k[T] en dehors d'un ensemble tini de points. En particulier, g~ e s t localement acyctique en dehors d'un ensemble fmi de points. 2.3-

Ramification mod6r6e On ddmontre le r$sultat suivant dans [D-L].

Th6or~me. Soit f : T~ ~ Alk un morphisme non ddg~ndrd pour Aoo(f). Ri f,.Qt est mod~r~ment rarni£~ ~ l~nfmi, quel que soit i.

Alors

Pour d6montrer ce rEsultat, on utilise en particulier que si R e s t un anneau de valuation discr6te de c a r a c t ~ r i s t ~ u e nulle ayant k pour corps rEsiduel, on peut relever f e n un polynSme de Laurent f £ coefficients dans R avec A o o ( f ) = A ~ ( f ) qui est non d~g~n~r~

pour n~(}'). 2.4-

Cohomologie d'intersection

(2.4.1) Soit X un schema s~par~ de type fini sur k = Fq purement de dimension n. Soit 5r un (~t-faisceau constructible localement constant sur un ouvert de Zariski dense U de X . Alors, d'apr~s ([B-B-D], 1.4.14), il existe tm unique objet K ° de la cat~gorie d&iv~e Db( X, Qt) v6rifiant K*Iu = .T[n] tel que K ° et son dual de Verdier D K ° satisfassent (2.4.1.1) H i K ° = 0 pour i < - n . (2.4.1.2) dim Sup H i K ° < - i pour i > - n . Cet objet unique est le complexe d'intersection sttr X associ~ & .g, on le note I*x(.T). On note I ° x : = IOx(Qt). La cohomotogie d'intersection de X est l'hypercohomologie

H'(X ® ~, I'x[-~]). (2.4.2) Le rdsultat suivant permet de calculer la cohomologie d'intersection des vaxi~t~s toriques et fournit une interpretation cohomologique des polyn6mes a ( a ) et fl(A).

221

Th~or~me.-Soit g u n cdne poly6dral dans Qn, ayant l'origine pour sommet, et soit A un polytope convexe dans Q". On suppose que a e t A sont de dimension n. On pose k = Fq. Pour tout entier i, on a (2.4.R1) dim H i ((I~.~(~)[-n])o) = coefficient de T i dmls O~(ff ) (2.4.2.2) dim H i ( X k ( A ) , I ~ , ( a l [ - - n ] ) = coeNcient de T i dens ~ ( A ) (15ndu'ce 0 d~signent la fibre g l'o•gine).

De plus on a

° ) o est pur de poids n. (2.4.2.3) ( I x~(~) Ce r6sultat est d~montr~ dens [D-L]. L'~nonc~ (2.4.2.2) pour k = C est ~nonc~ sans preuve par Stanley dans IS] off il est attribu~ £ I.N. Bernstein, A.G. Khovanskii et R.D. MacPherson. Apr~s avoir r4dig4 une version pr~liminaire de [D-L] nous avons requ un preprint de K.H. Fieseler ([F]) dans lequel (2.4.2.1) et (2.4.2.2) sont d~montr~s pour k = C. La preuve de [F] utilise ta th6orie de Morse ~quivariante et le th6or~me de d6composition. Pour finir, notons que (2.4.2.1) et (2.4.2.2) pour k = C sont une consequence formeUe du cas off k est fini. 2.5 Pour d~montrer le th4or~me 1, on utilise la proposition suivante, jointe h (2.2) et (2.3), ainsi qu'un th~or~me de D. Bernstein, A.G. Kushnirenko et A.G. Khovanskii ([B-K-Kh]). P r o p o s i t i o n [ D - L ] . - - Soit Y un schdma de pure dimension n sur k et g : Y ---* A~ tm k-morphisme propre. On suppose que g est 1ocaJement acyclique en dehors d'tm ensemble fini de points et que Ri g . Q t est moddrdment rarnifid g l'infmi pour tout i. Alors (2.5.1) H~(Y ® k, g*£,,) = 0 pour i > n, (2.5.2) les applications natureI1es H~(Y ® [', g * £ , ) --* H i ( Y ® k , g*L¢) sont des isomorphismes pour tout entier i. Si de plus Y est lisse sur k, alors (2.5.3)

H i ( Y ® it, g*Z~o) = 0 pour i ~ n,

(2.5,4)

H g ( Y ® k , g*£¢) est p u t de poids n.

2.6 - Pour d~montrer le th~or~me 2, on utilise (2.4) et la proposition suivante, que l'on applique h Y = Y~(a) et g = gn(A). (On peut d~duire de (2.2) et (2.3) que les hypoth&ses de la proposition sont alors v~rifi~es.) P r o p o s i t i o n [ D - L ] . - - Soit Y un schema purement de dimension n sur k et soit g : Y -~ A~ un k-morphisme propre. On suppose qu'en dehors d'un ensemble ~ f i de points g est 1ocalement acyclique relativement ~ I ' y et que tons tes Rig, I ° y sont moddrdment ramifids g 1'infmi. Alors (2.6.1) H ; ( Y ® k, I ' v [ - n ] ® g*£.,) = 0 pour i # n, (2.6.2) H~'(Y ® k, I ' v [ - n ] ® g*/:¢) est pur de poids n.

222

R6f~rences

[A-S 11

A. Adolphson, S. Sperber : ExponentiM sums and Newton polyhedra. Bull. Amer. Math, Soc. 16 (1987), 282-286.

[A-S 2]

A. Adolphson, S. Sperber : ExponentiaI sums and Newton polyhedra : cohomology and estimates. Annals of Math. 130 (1989), 367-406.

[A-S 3] [B-B-D]

A. Adolphson, S. Sperber : Exponential sums on G ~ . Inventiones Math. (A paraitre.) A.A. Beilinson, J. Bernstein, P. Deligne : Faisceaux pervers. Socidt6 Math6matique de France, Ast6risque 100 (1983).

[B-K-Kh]

D. Bernstein, A.G. Kushnirenko, A.G. Khovanskii : Newton polyhedra. Usp. Mat. Nauk., 31 n°3 (1976), 201-202.

[Dal [Del [D-L]

V.I. Danilov : The geometry of toric varieties. Russ. Math. Surveys 33 (1978), 97-154.

[F] IS]

P. Deligne : La conjecture de Well II. Publ. Math. IHES 52 (1980), 137-252. J. Denef, F. Loeser : Weights of exponential sums, intersection cohomdog/" and Newton p @ h e d r a . (A paraitre.) K.-H. Fieseter : RationaJ intersection cohomology of projective toric varieties. (Preprint.) R. Stanley : Generalized H-vectors, intersection cohomology of toric varieties, and related resuits. Advanced Studies in Pure Math. 11 (1987), Commutative Algebra and Combinatorics, 187-213.

DE RHAM COHOMOLOGY AND THE GAUSS-MANIN CONNECTION FOR

DRINFELD M O D U L E S Ernst-Ulrich Gekeler Institut des Hautes Etudes Scientifiques 35, route de Chartres 91440 Bures-sur-Yvette, FRANCE

Introduction. The purpose of the present article is to give a survey on new developments in

the theory of Drinfeld modules, in particular on those points of the theory that are related to non-archimedean analysis. Since 1974, when Drinfeld introduced the notion of "elliptic module" [2], now commonly called "Drinfeld module", this theory showed increasing importance in the arithmetic of function fields. Drinfeld modules are "motives" over global function fields K that may be considered as analogues of "abelian varieties without polarization" over number fields. Their schemes of torsion points and the associated modular schemes have properties close to those of abelian varieties. By means of rank one Drinfeld modules, one can construct all the abelian class fields of K . For higher ranks, there exist (proved or conjectured) reciprocity laws relating Galois representations on the 2-adic cohomologies of modular schemes with automorphic representations of adele-valued groups over K

([2],[t],[3]). But there is an important

difference : Whereas abelian varieties are related to the representation theory of Sp(2n) , Drinfeld modules correspond to GL(n). For abelian varieties A , the interplay of the different cohomology theories (Betti, 2-adic, de Rham) is crucial for the arithmetic of A . Thus it is natural to took for Drinfeld module substitutes of these theories. Concerning Betti and ,e-adic cohomologies, the constructions are evident and known since Drinfeld's original paper. From a seminar held at the Institut for Advanced Study in Princeton 1987/88, a definition HDR for "de Rham cohomology for Drinfeld modules" evolved, due to the efforts of several people (e.g.G. Anderson, P. Deligne, J. Yu, the author). Strictly speaking, HDR is not a cohomology theory but rather a functor corresponding to 1 HDR , but with essentially all the properties one would expect from a first de Rham

cohomology of an abelian variety (except there is no canonical polarization). Among them, let us mention :

224 (a) HDR commutes with base extension ; (b) over the K-analogue C of the complex numbers, there is a natural de Rham isomorphism

of HDR with HBcai; (c) over special fields L (including L = C ), there are GAGA-type isomorphisms ; (d) one disposes of a formalism of "vanishing cycles" that relates HDR of a degenerate Drinfeld module with that of its stable reduction ; (e) there is a Kodaira-Spencer isomorphism that expresses HDR through the tangent sheaf of certain modular schemes. We will survey these properties (and prove some of them), focusing on the analytical aspects, and then apply them to the study of modular forms. The plan of the paper is as follows : In sections 1 and 2, we fix the notation, give general definitions (section 1) and the well-known analytical constructions of Drinfeld modules (section 2). In section 3, we introduce the functors HDR and discuss the base extension problem. Due to the base change property stated in Thm. 3.5, a Drinfeld module • of rank r over a scheme S will define a locally free ®s-sheaf HDR(~) provided with a canonical decomposition into a one-dimensionalpart HI(O ) and an (r-1)-dimensional part H 2 ( ~ ) (3.12). In section 4, we consider Drinfeld modules defined over fields complete w.r.t, a non-archimedean absolute value. Over such fields, there exists an obvious analytical version AADR r4*'an of the de Rham modules. Theorems 4.3, 4.6, and 4.9 describe the relationship between HDR and T4*'an "'DR

HDR

'

and HBctti , and the formalism of vanishing cycles for degenerating Drinfeld modules,

respectively. In section 5, we give a new construction of the universal additive extension of a Drinfeld module, based on the decomposition of HDR. Section 6 introduces the Gauss-Martin connection. Drinfeld modules • may be differentiated with respect to tangent vectors of their base schemes. There results a Kodaira-Spencer map (6.6) from the tangent bundle to Horn (H ( ~ ) , H2(t~)), which for modular Drinfeld modules is an isomorphism (Thm. 6.11).

This fact corresponds to the isomorphy of the sheaf of modular forms of weight two with the sheaf of differentials on elliptic modular schemes. In the last section, restricting to the rank two case, we apply the machinery developed to the study of modular forms. Drinfeld modules of rank two are very close to elliptic curves, and both the analytic and the algebraic part of elliptic modular forms theory may be translated to the Drinfeld case ([11] [12] [5]). Modular forms for an arithmetic subgroup F of GL(2,K) are rigid-analytic functions defined on an "upper half-

225

plane"

f~ C C , transforming nicely under elements of

F , and satisfying "cuspidal

conditions". For an elliptic curve E in Weierstrass normal form, the differentials to = dX/Y and rl = X dX/Y constitute a canonical basis of H1R(E), which is essential for the discussion given in [13], A1. In our context, "co" ,~xists a priori, but "rl" has to be constructed, using the Gauss-Martin connection (Prop. 7.7). "Ilaese forms transform under F like modular forms of a certain weight and type. This leads to a description of the ring M(I") of modular forms for the group F and the "Serre derivation" ~ : M(F) ~ M(F) in terms of HDR of the "generic" rank 2 Drinfeld module over f~. As a by-product, we obtain a nice formula for the analogue of the classical A'/A as a conditionally conve~rgent lattice sum (7.10). As mentioned above, the point of v~iew is analytical, so we avoid all the complications that result e.g. from choosing the proper "~trithmetic" uniformizers at the cusps ([5], p. 38-39), We do not study the nature of the t-expansion coefficients. Further, we limit our efforts to considering modular forms for maximal arithmetic subgroups F of GL(2,K). Thus our approach is merely a fast step in investigating M(F), in that it introduces the technical tools for an arithmetical study. The following terminology is used without further referenceu For a ring R and r ~ R , R * , R/r is the multiplicative group, the factor ring R/rR, respectively. L is the algebraic closure of the field L , and #(S) the cardinality of the set S . "Locally free sheaves" and "vector bundles" over a scheme are used synonymously. In general, we do not distinguish between polynomials (or convergent power series) and the mappings they induce. This article was written during a stay at the "Institut des Hautes Etudes Scientifiques" in Bures-sur-Yvette. The author takes the opportunity to thank the IHES and its staff for their hospitality. Thanks are also due to De~tsche Forschungsgemeinschaft for its support through a Heisenberg grant.

1. T h e basic set-up. Throughout the paper, we use the following notation : F q = finite field with q elements, q = pf with a prime number p ; K = function field in one variable over Fq (we suppose Fq algebraically closed in K ) ; A = subring of elements of K regular ',away from ,,o = place of K , fixed once for all, of degree ~ over F q ; 1?1= normalized absolute value on K ~ssociated with 00 (i.e., lal = #(A/a) for 0 ~ a ~ A) ; deg : K* ~ ~ Z the corresponding degree function deg x = logqlxl (we also put deg 0 = -**) ; K** = completion of K w.r.t. I?1 ;

226 A

C = K = completion of

~,,

w.r.t, the canonical extension of

I?1

to

K**

= smallest field extension of K., that is complete and algebraically closed. 1.1. EXAMPLE : If K is the rational function field IFq(T) and "oo" the usual infinite valuation, we have A = ]Fq[T], "deg" is the usual degree of polynomials, and K.~ the field of formal Laurent series IFq((T'I)). The finite places of K (i.e., those different from

oo

) correspond bijectively to the (non-

zero) prime ideals of A . We shall not distinguish between the "valuation" and the "ideal" point of view, thus using "place", "valuation", and "prime ideal" as synonyms. (1.2) Recall that for any field L of characteristic p , the endomorphism ring EndL(G a) of the additive group scheme GalL is the set of additive polynomials, i.e., polynomials of the form i 5". a i X p , where the multiplication is defined by insertion of polynomials. Denoting by "tp the polynomial X p , EndL(Ga) is the twisted polynomial ring L{'Cp} subject to the commutation rule "Cp a = a p %p for constants

a e L . We let x = 'Cq = ~fp (q = pf) be the operator

corresponding to Xq and L{z} the subalgebra generated by z . According to x a = a q z , we always write "polynomials" in "r with left coefficients, and let deg x the degree in x. (1.3) Next, assume that L comes equipped with a structure Y : A -o L as A-algebra. Let S = Spec L . A DrinfeldA-module of rank r over L (or over S ) is given by a group scheme GIL, isomorphic with GalL, and a structure of scheme in A-modules • : A --~ EndL(G) a ~ Oa on G such that the following hold : (i) The two structures of scheme in A-modules on Lie(G) induced by •

and by Y agree ;

(ii) For each non-zero element a of A , the group scheme Ker(O a) is finite of degree lalr .

Choose an isomorphism c~ : G -~ G a , and let • (c0 : A -~ EndL(Ga) be given by a ~, 0 (a) = o~. 0 a , a -1 . Writing for a ~ 0 a

227

a

i
l

(ca) 1-qi (¢0 we have a. =c a. (c e L*), and our conditions (i), (ii) boil down to l

l

(i') a (a) = 7(a) ; i

(ii') N = r. deg a . Thus e.g. in the situation of (1.1), a Drinfeld module of rank r is given by O(.(a) =

a! a) z i , I

where a!a)l ~ L , a(0a) = ~ T ) , and a (c0r ~ 0 .

Let now S --~ Spec A be an arbitrary A-scheme and ~t : A ---) ®s(S) the corresponding ring homomorphism. A DrinfeldmoduIe of rank r over S is given by a line bundle 91S and a structure of scheme in A-modules : A ---) Ends(G) a~

a

on the additive group scheme G underlying

9

such that for any local isomorphism

ct : 9 ---) ® S ' 03 and (ii') hold, where the "leading coefficients" a (a) are units. A r- deg a

morphism u : (G,(I)) ---) (G',~') of Drinfeld modules is a morphism u : G ---) G' of group schemes that commutes with the A-actions on G and G ' . Non-vanishing morphisms have finite kernels, they are called isogenies. Being isogeneous (connected by an isogeny) is an equivalence relation on Drinfeld modules ; it implies equality of ranks. The scheme of adivision points (a e A) , or more generally, the scheme of ~-division points of (G,~) (0; C A an ideal) is a(I) = Ker ~ a or a(I) = n Ker ~ a ' respectively. If (G,~) is defined ae a over the algebraically closed A-field L and ct is prime to Ker ~/, a(I)(L) is isomorphic with (~'l/A)r as an A-module. This follows easily from (i') and (ii'). (1.4) From now on, we abbreviate "Drinfeld module over S" by "DmodlS". An "r-Dmod" will be a Drinfeld module o f rank r . Given a Dmod (G,(~) over S and a base extension S' --) S , there is an induced Dmod (G',~') over S ' . If there is no danger of confusion, we will write (G,~) or even •

for (G',~').

228

(1.5) Given any locally free sheaf 9tS with underlying group scheme G I S , we will write 9 = L i e ( G ) . If 9 is a line bundle, we will not distinguish between (a) isomorphisms of 9 with ®S ; (b) isomorphisms of G with GalS ; (c) nowhere vanishing sections of 9 ^ = dual bundle of 9 • Considering Dmod's over a field L , we usually choose without reference an isomorphism ct : G --~ G a , or, what is the same, assume that G = G a .

2. A n a l y t i c a l c o n s t r u c t i o n s . The contents of this section is entirely due to Drinfeld [2]. The constructions given should be compared with the Weierstrass parameu'ization of elliptic curves over E (2A), or with Tate's parametrization of elliptic curves over local fields (2.10). An A-lattice in C is a finitely generated (thus projective) discrete A-submodule A o f C . The discreteness of A means that its intersection with each bounded subset of C is finite. For an A-lattice A , put (2.1)

cA(z) = z

II (1 - z/L). O~k~ A

It is easy to verify that (2.2) (i) the product converges for any z in C , uniformly on bounded sets, thereby defining an entire, surjective, A-periodic map e A from C to C ; (ii) e h is Fq-linear, thus deffmed by a convergent power series cA(z) = ~ oti z q~ ; ~0 (iii) the zeroes o f e h are the points of A C C , and are simple. In what follows, r is the projective rank of the A-lattice A . Fix some non-zero a e A , and let ~

a

~ C{x} be defined by the commutative diagram (whose rows are exact by the above) : eA O ~ A ~ C ~ C ~ O

(2,3)

Sa 0 ~A-~

.l.a

a

eA C ~ C~O

By diagram chasing, we see that (2.4) (i) q~A = a + . . . + a N x"N , where N = r . deg a , a N ~ 0 , and a

229

(ii) a ~ • A is a ring homomorphism • A : A ~ C{x}, a

i.e., ~

defines a structure of Drinfeld module of rank r on GalC. The construction may be

inverted : 2.5. T H E O R E M [21 : The association A ~* dPA defines a bijection of the set of lattices of rank r in C with the set of structures of Drinfefd module of rank r on GalC. 2.6. R E M A R K : If A , A' are rank r lattices and c e C* is such that c A C A ' , there exists a unique u e C{x} , u = c + o(x) that for a e A satisfies u * • A = • A' . u , i.e., a a

morphism

u : ~

a

~ OA' . Thus if we define morphisms o f lattices by

Hom(A,A') =

{c e C I c A c A'} , we obtain in (2.5) an equivalence of categories. Also, we could easily give a coordinate-free version of the above, i.e., without assuming •

to be defined on GalC,

(2.7) Next, we try to imitate the construction for local fields other than C . Let y : A --~ L be an A-field with a rank one valuation v , valuation ring B and residue class field L ( v ) . Reduction m o d v and everything derived from it will be denoted by a bar ".9". W e assume that y takes values in B , and, for simplicity, that L is complete and algebraically closed (e.g. L = c o m p l e t e d algebraic closure o f K((t)) , or o f K p

with some p e Spec A ). Let

• : A --~ L{x} be a Drinfeld module of rank r over L . Replacing •

by the isomorphic

module O' = c * • * c -1 with c e L* if necessary, we may assume that @ takes values in B{x} , and that • : A ---}L(v){x} is different from T : A -~ L ( v ) . In that case, ~} is a Dmod over L(v) o f some rank

0 < r 1 < r , whose isomorphism class depends only on that o f O .

We call O the stable reduction of O . If r 1 = r , we say that @ has good reduction. (2.8) Now assume that the Dmod •

over L , of rank r 1 , has good reduction. A lattice in •

is a discrete, finitely generated projective A-submodule of ( L , O ) . As in (2.1), we form the lattice function e A for the lattice A in O of rank r 2 . It has properties similar to those described in (2.2). Defining ~ a e L{x} by the commutative diagram

(2.9)

eA 0 --) A --) L --) L --) 0

eA 0 ---> A -~ L --> L --~ O, we get

230

(2.10) (i) g h : a ~ g r ~a defines a Dmod of rank r = r 1 + r 2 over L ; (ii) @

has stable reduction of rank r I .

Furthermore, 2.11. THEOREM [2] : The association (~,A) ~ @ defines a bijection of the set of isomorphism classes of pairs (~,A), where • is an rl-Dmod over L with good reduction and A is a lattice of rank r 2 in dO, with the set of isomorphism classes of r-Dmod's over L with stable reduction rank r 1 . We call (~,A) the Tate data of gt = @ . 2.12. REMARK : Although we will not need this fact, let us mention that the above bijection also comes from an equivalence of categories, as follows from [4], §3.

3. De R h a m cohomoiogy. In this section, we associate an L-vector space HDR(~) of dimension r with any r-Dmod •

over the A-field L . In the analogy of Drinfeld modules

with elliptic curves, it plays the role of the first de Rham cohomology module. The construction is functorial and gives rise to a locally free sheaf HDR(~) over S , whenever •

is defined

over the A-scheme S. Perhaps, some of the definitions to follow seem to fall out of the blue. They are motivated in [9], section 2.3 (see also [14]). We first assume that S = Spec B is an affine A-scheme and "~" = (G,~) a DmodlS. "Ga" denotes ishe additive group scheme over S with the tautological A-action induced by the structural map ]t : A --~ B . In what follows, "Horn" and "®" without subscript are relative to

Fq. (3.1) We put M(q)) = Hom((G,~),G a) = A-bimodute of 1Fq-linear morphisms of S-group schemes from G to G a ; N(~) = {m ~ M(O) I Lie(m) = 0}. Note that the left and right actions of lFq on M(~) agree. We therefore may regard M(q~) and N(~) as A ® A-modules.

231

3.2. E X A M P L E : Suppose that G = GalS = GalB. Then M(q)) = HomB,Fq_lin (Ga,Ga) = B{t}. The left action of a ~ A is left multiplication by ~ a ) , its right action is right multiplication by O a ' and N(~) is the sub-A-bimodule B{¢} x.

(3.3) An IFq-linear map 11 : a ~ rl a from A to N(~) that satisfies the derivation rule

"qab = ~a) rl b + rl a . Ob

will be called a "derivation". Each m ~ M(O) defines through lq(am) = ~ a ) m - m * • a a derivation .q(m), which is called inner (strictly inner if m ~ N(~) ). We put D ( O ) , Di(O), Dsi(~)

for the A-bimodule of derivations, inner derivations, strictly inner derivations,

respectively. Note that M(q)) ..... Dsi(O) have natural B-module structures compatible with their left A-actions, thus may be considered as B ® A-modules.

3.4. DEFINITION : The de R.ham module of • is the B-module

HDR(CD) = D(cD)/Dsi(CD).

Obviously, our modules

M(O), N(O), D(~), Di(~), Dsi(O), HDR(O) are contravariant

functors in O , i.e., for u : • --4 ~ ' a morphism of Dmod's over S , there results a map u* : M(O') --~ M(O) , etc. If the need arises, we label M(O) = M(O,B) . . . . . HDR(F) = HDR(O,B ) . All these modules are also covariant functors in B (or contravariant functors in S ), where for a base change

S' = Spec B' -~ Spec B = S , M(O,B')

is short for

M((G × S',~'), B') (see (1.4), similar notation for N ..... HDR ). We have canonical base

change morphisms

232

B' ® M(O,B) --> M(O,B') B B' ® HDR(C~,B) ~ HDR(~,B' ) . B The following results are proved in [9], section 4. 3.5. THEOREM : The functors M(~,?), N(~,?), D(~,?), Di(~,?), Dsi(~,?),

HDR((I),?) o n

B-algebras commute with arbitrary base changes, i.e., the above base change morphisms are isomorphisms. 3.6. PROPOSITION : Let the r-Dmod ~ be defined over the A-field B . Then HDR(~,B)

is an r-dimensional B-vector space. Now allow ~ to be defined over a not necessarily affine A-scheme S . It follows from (3.5) that, up to unique isomorphism, there exist unique quasi-coherent S-sheaves M (~) . . . . . HDR((I)) whose sections on open affine subschemes S' = Spec B' c S are given by M (O) (S') = M(~,B') . . . . . HDR(O) (S') = HDR(~,B'). Evaluating on residue class fields of S yields 3.7, COROLLARY : Let ¢b be an r-Dmod over the A-scheme S . The sheaf HDR((I)) is

locally free of rank r on S . Its formation commutes with arbitrary (not necessarily affine) base changes. We conclude the section in describing HDR(O) as a direct summand of D ( ~ ) . Suppose for the moment that •

is def'med on GalB, where B is an A-algebra. Recall that in this case

M ( ~ ) = B{z} and N(O) = B{x} x with the B ® A-structure determined by • : A --->B{x} . As usual, we write

m = 5". m i ~i ~

M((I))

with left coefficients

m i , so

deg x m = max{i I m i ~ 0} is well-defined. Let a be an arbitrary non-constant element of A , and r the rank of O . (3.8) A derivation ~q ~ D(~) is reduced (strictly reduced), if deg x rla < r. deg a

(resp. deg x 11a < r. deg a).

For 11 reduced, we let def I'1 = r . deg a - deg x rl a be the defect of rl. Let Dsr(~) C Dr(~ ) C D(~) be the B-submodules of strictly reduced, reduced derivations, respectively.

233

3.9. PROPOSITION ([9], sect. 5) : (i) The notions o f reducedness and o f defect do not depend on the choice of a . (ii) For each 11 ~ D(O), there exists a unique n ~ N(O) such that ri' =ri - ri(n) is reduced.

Therefore, we have a canonical decomposition D(O) = Dr(O) • Dsi(O), and Dr(O) maps bijectively to HDR(O). The derivation ri(1) given by rlO) = a - • a is reduced but not strictly reduced. Since the leading coefficient of ri(a1) is a unit, subtracting a suitable multiple of riO) from rl e Dr(O) defines a decomposition Dr(O) = B ri0) ~ Dsr(O) "We let (3.10)

HDR(O) = Hi(O) • H2(O)

be the corresponding decomposition of HDR. (3.11) Finally, we allow • = (G,O) to be defined over an arbitrary A-scheme S , where G belongs to the line bundle 9 = Lie(G). Locally choosing isomorphisms 0~ : 9 "-> ®S ' we may apply the above considerations to define (strict) reducedness of local sections of the sheaf D (O). There result subsheaves (in fact, locally direct summands) Dsr(O) C Dr(O ) C D (O), -- . whose sections are the (strictly) reduced sections of D (O). Clearly, Dr(O) ~ HDR(O). An isomorphism ct : 9 --->®S defines a section ~q(c0 of Dr(O). Viewing o~ as a nowhere vanishing section of the dual bundle 9 ^ = Lie(G)^ , ¢c ~ ~q(ct) defines an injection of Lie(G)^ into Dr(O). (Note that 11(cct)= c ri(ct), c a unit). The local considerations show that (3.12)

D (O) = DI(O) @ Dsr(O) @ Dsi(O)

with DI(O) = image of Lie(G)^ = subsheaf generated by the sections ri(c0, and

DI(O) @ Dsr(O) = Dr(O) --> HDR(O) = HI(O) ~ H2(O).

4. Analytical properties of HDR. Throughout the section, we assume Drinfeld modules • : A --~ L{x} to be defined on the additive group G a over the A-field L , where either

234

(a) L = C , or (b) L is as in the end of section 2, and y : A --->L takes values in the ring B of integers in L. In any case, L will be an algebraically closed A-field complete with respect to a nonarchimedean absolute value. (4.1) Let L { {x} } be the L-aigebra of non-commutative power series in x with coefficients in x (subject to the usual rule '~a = a q x for constants a e L ). Further, let Lent{ {x} } be the subalgebra of those power series that define entire functions on L (i.e., power series qi f = Y. a i xi such that for every z e L , the series f(z) = Y~a i z converges). For example, the functions e n lie in Lent{ {z} } . First, we define analytical versions of the de Rham modules HDR(O). Put (4.2)

Man(O) = L{ {x} ]

(with the A-bimodule structure described by (3.2)) ;

Nan(O) = L{ {x} } x ; Dan(O)= {derivations q : a ~ q

a from A to Nan(O)}

(i.e., TI is ]Fq-linear and satisfies /'lab = '~a) lqb + 1]a * • b ) ; D~n(O)

(resp. D~(O))

= {derivations of the form .q(m), m ~ Man(o)

(resp. Nan(o)) ]

(recall that rl~m) = ~ a ) m - m . • a) ;

H;~n(o) = Dan(o)/Ds~(O) .

The relationship between "algebraic" and "analytic" de Rham modules is given by the following GAGA-type result. 4.3. T H E O R E M : Let •

be a Drinfeld module over L , and suppose that (a) or (b) hold. •

* ,an

The canonical map c~ : HDR(O) --~ HDR (O) is an isomorphism.

For the proofs in the two cases, which are quite different, see [9], sections 6 and 7.

235

Let us now give, in case a), a description of HDR((I)) related to "path integration on (I)" By Thm. 2.5, (I) corresponds to some A-lattice

A . Comparing with the Weierstrass

parametrization of elliptic curves, we regard HBetti(~) : = A as a Betti homology group of • IC. Hence it is natural to expect some kind of cycle integral pairing HDR((P) x A ---) C , or, equivalently, a de Rham map DR : HDR((I),C) ---) HomA(A,C) = : HBetti(~,C)

with reasonable properties. Let e = eA~ Cent{ {z} } be the function associated with A. The next lemma is taken from [8], section 2. 4.4. LEMMA : (i) Given rl ~ D(~) and a ~ A non-constant, there exists a unique solution Fl1 ~ Cent{ {'c} } of the functional equation (*)

Fa(az) - a F~(z) = rla(e(z)).

(ii) Fll is independent of the choice of a .

As we read off from (*), the restriction Xrl of Frl to A is A-linear, so we have a Cmorphism r 1 ~ xrl from D((I)) to HomA(A,C). It factors through HDR((I),C) since for 1"1= 13(n) strictly inner, Frl(z) = -n(e(z)) vanishes on A. The induced map DR : HDR(~) ---) HomA(A,C) is called the de Rham map. If r I ~ D((I)) and v ~ A , we formally write f ' q

for DR(rl) (v).

The pairing is functorial. If u : (I) ---) (I)' is a morphism, rl e D((I)'), and v ~ HBetti(~),

as is immediate from definitions. 4.6. T H E O R E M ([8], Thm. 5.14) : For any Drinfetd module

•

over

C , DR is an

isomorphism.

Actually, part (a) of Thm. 4.3 follows easily from (the proof of) the present theorem. Also,

236

we may describe the effect of DR on the decomposition (3.10) : 4.7. COROLLARY ([8], Thm. 6.10) : (i) The class of rl o ) is mapped under DR to -id,

where id : A ~ C is the canonical inclusion. (ii) An A-character X : A --~ C corresponds under DR to an element of F~2(~) if and only if the limit S(1,X) = lim Y. ;¢(k)/k

(0 ;~ k ~ A , I~.1< s)

S--.>oo

vanishes. Functions F~I of the type considered above are similar to quasi-periodic functions for complex lances. (4.8) Now consider case b), i.e., the situation of (2.7)-(2.12). We relate the de Rham modules of ~ and of ~ , where ~g has Tate data (~,A) (see (2.11)). Some function F : L ---) L is

quasi-periodic

for the lattice A in •

if

(i) F is Fq-linear and entire (given by a power series in Lent{ {x} } ) and (ii) for each a ~ A , F satisfies a functional equation F(O a (z)) - 7(a) F(z) = via(e(z)) with some via e L{x} x. Here, e = e A : L ~ L is the lattice function of A . One easily verifies that for F given, the map a ~ vla is a derivation "q ~ D ( ~ ) , and that F restricted to A is A-linear. We let QP(A) be the L-vector space of quasi-pericxtic functions for A . The next results are proved in [9]. 4.9. THEOREM ([9], Thm. 7.7) : The sequence of L-vector spaces s t 0 --~ N(~) --) QP(A) --~ HomA(AJ~) --) 0

is exact, where s(n) = n * e and t(F) = FlA. Therefore, there exists a canonical map i : HomA(A,L) --) HDR(~) , defined by the commutative diagram

237

(4.10)

0 --~ N(~)

--~ QP(A) --~ HomA(A,C ) -~ 0

~11,I, 0 ~

$

,l,i

Dsi(~i/) --> D(~I/) ~

HDR(~t/) ---> 0 .

Next, we define j : HDR(~/) --->HDR(~ ) by composing the map [rl] ~ [1"1* e] : HDR(W) *,an

HDR (~) with the inverse of c~ (see Thin. 4.3). Here, 71 * e ~ Dan(~) is the derivation ( r l . e)a : = rl a * e . The wanted relationship between HDR(~) and the Tate data (~,A) of is given by 4.11. T H E O R E M ([9], Thm. 7.12) : The sequence of vanishing cycles i

*

i

*

0 --> HomA(A,L) --->HDR(~) -~ HDR(~) ~ 0

is exact. 4.12. R E M A R K : For ease of presentation, we assumed

~

to be defined on G a l L .

However, it is purely formal to write all the results of this section in a coordinate-free language. The sequence of vanishing cycles is functorial in ~ , since isogenies u~ : ~ --->~Y induce isogenies u~ : • --->~ ' of the stable reduction parts ([4],§3). Also, the assumption on L to be algebraically closed is not essential. Statements correponding to (2.11) and to (4.11) hold true for Drinfeld modules ~ defined over arbitrary A-fields L complete with respect to a nonarchimedean absolute value, but are more complicated to formulate.

5. The universal additive extension. Here, we give a more geometrical description of H D R ( ~ ) , proposed by Deligne. Let (G,~) be defined over the affine A-scheme S = Spec B . For ~ ~ D(~) and a ~ A , consider the matrix

The derivation rule translates to ~T!ab= ~rla * ~ b . With r l , we associate the extension of schemes in A-modules

238

[11]

0 --o G a - o G a @ G - ~ ( G , O ) --~ 0 ,

ac'su 0 eo t via' ema c s

a

:(:' 0 l'' e

sequence of Lie algebras of [q] splits canonically. It is easy to see that [q] (provided with its Lie splitting) is trivial if and only if q belongs to Di(~) (resp. Dsi(O ) ). Associating [1"11 with its canonical Lie-splitting to rl e D(O) yields a commutative ,

(5.2)

~

HI(O)

diagram ~

~--

Lie(G) ^

Di(~)/Dsi(O)

$

$

,

Ss =

HDR(O)

=

D(O)/Dsi(O )

3, H~(O)

--o

Ext#((G,O),Ga)

$ ~

Dsr(O)

--o

D(O)~i(~ )

$ -->

Ext((G,~),G a) ,

where all the horizontal arrows are isomorphisms. Here, Ext (resp. Ext # ) is the B-module of extensions of schemes in A-modules over S (provided with a Lie splitting), and s is described as follows : A local section ct of Lie(G) ^ is a map from Lie(G) to Lie(Ga), thus a Lie splitting for the trivial extension of (G,O) by G a , whose class is s(~). (5.3) Next, let VIS be the additive group scheme underlying the dual of the locally free Bmodule Dsr(O) --* Ext((G,q~),G a) . We construct an extension (#) of schemes in A-modules of (G,O) by V , the latter being equipped with the tautological A-action. According to (5.1), we define (#) by

(#)

0 ~ V --* V • G ~ (G,O) --* 0 ,

where a e A acts on V @ G through the matrix

o~ = a

Here,~a)

(: 1 qa.

Oa

is the scalar action of a on V , O a the Drinfeld operator on G , and

239

via# e Hom(G,V) is def'med by VI~(x) (1"1)= Via (x)

(x e G , 1"1~ Dsr ( ~ ) ) .

In fact (5.4) (i) rl a (x) (11) is B-linear in Vi, hence 1"1 (x) ~ V ;

(ii) rl~ (x) is Fq-linearin x;

(iii) 1"1# satisfies the derivation rule Viab # = y(a) Vi~ + q ~ . ~b

'

so ~# = ~# * • # ab

a

b"

5.5. PROPOSITION : (#) is the universal additive extension of (G,~) (see [14]). This means : Let X be the additive group scheme underlying a line bundle % over S , provided with the tautological A-action. For any extension of schemes in A-modules (*)

0 --) X --~ Y --~ (G,~) --~ 0 ,

there exists a unique morphism f : Lie(V) ~ % such that (*) is induced by (#) through f , and "(*)~' f ' defines an isomorphism Ext((G,*),X) ~ Horns(Lie(V),%). PROOF : Since Dsr(*) is canonically isomorphic with Ext((G,~),Ga), we have

Horns(Lie(V),% ) = HOms(Dsr(~)A,%) ---) Dsr(~) ~B % --~

Ext((G,~),Ga) ® % ---) Ext((G,~),X) , B

which is the inverse of the stated isomorphism. (5.6) Finally, we drop the assumption that S be affine. Everything remains valid except that we have to replace locally free B-modules by locally free S-sheaves. The universal additive extension of (G,*)IS is an extension by the additive group V of the sheaf Ext ((G,~),Ga)^, which is locally free of rank r - 1 (r = rank of • ). We have canonical isomorphisms of sheaves ,

~

=

HDR(~) = D (~)/Dsi(~) ~ Ext#((G,tb),Ga ) ---) Lie(V ~ G) ^ ,

24O

where (V ~ G,t~ ~) is the maximal additive extension. Applying Lie(?) ^ to (#) yields the canonically split short exact sequence 0 --* HI(O) ~ HDR(q~) --~ H2(q~) --* 0 of (3.12).

6. T h e Gauss-Manin connection. We show how tangent vectors of the base scheme S of a Drinfeld module (G,qb) act on HDR(~) . Assume first that S = Spec B is affine. If D e DerA(B) is an A-derivation of a and f = 5 " . f i z i e a{x} , w e p u t D ( f ) = Y . D ( f i ) x i . 6.1. L E M M A : L e t

f, g e B{x}.Then D ( f o g ) = D ( f ) o g + f 0 D ( g ) .

qi qi i PROOF : D ( f . g) = D(k~ i+j=k f i y ~ gJ ~ ) = Y~k i+j=k y~ (D(fi) gj + fi D(gj q )) ~k qk = ~k i+j~__kD(fi) gj Next, we define an operator

xk + f0 ~ D(gk) ~ = D ( f ) . g + f0 D(g). V D on N(O) . Locally, chose an isomorphism

=

cz : Lie(G) ---> ®S" Then n ~ N(~) may be written n = f * ct with f e B{x} x. Define (6.2)

VD(n ) = D(f) . o~.

6 . 3 . LEMMA : (i) V D(n) is independent of the choice of ct (in particular, is globally

defined). (ii)For b ~ B , VD(bn ) = D0a)n + b VD(n) holds. PROOF:(i)

Let t~'=b "1~ with some unit b e

B*,and f=~fi~i.Wehave

n=fot:t=

f . bt~' = ~ b qi fi ~i. o~'. Since f* = 0 , D(b qi fi ) = b qi D(fi), thus calculating V D(n) with respect to ct' gives the same result ~ b q' D(f i) ~.i o ~, = ~ D(fi ) ~i. ~ as with respect to or. (ii) Product rule for D ! For "q ~ D(O) and a ~ A , p u t (6.4)

(V D(TI))a = V D('qa) .

6.5. LEMMA : (i) VD(TI) e D(O) (i.e., it is IFq-linear in a and satisfies (3.3)).

241

(ii) If r I = 1"1(n) /s strictly inner, VD(rl (n)) = rl(VD (n)) . (iii) For b ~ B , we have VD(bI1) = D(b)rl + b VD(rl).

PROOF : (i) Let a,b ~ A , and write locally rla = f . c~. VD(l'l)ab = VD(~qab) = VD(~a) ~b + lla * ° b ) = 7(a) VD(TIb) + VD(f * ~ * • b * (I "1 * (I) = ~ a ) VD(rl) b + D(f • o t . Ob * (z'l) * ~ = ~ a ) VD(rl) b + D ( f ) . c t . • b (by (6.1), since the "constant" coefficient of f vanishes) = ~ a ) VD(II) b + VD(rl) a . Oh" (ii) and (iii) follow the same lines. By abuse of notation, we also call V D : HDR(O) --> HDR(O) the mapping induced on the de Rham module. The Gauss-Manin connection f o r • is the collection of operators {V D I D ~ DerA(B)}. Note that the deviation from B-linearity of V D is annihilated in the composite mapping

XD :

• , VD , projection__.(O).H 2 HI(O) ~ HDR(O) --.> HDR(O ) -->

Therefore, the Kodaira-Spencer map (6.6)

KS : DerA(B) ~ H o m B ( H I ( O ) , H2(O))

D~r~ D is well-defined. Let now a ~ A have degree d > 0 , and write locally O (c0 a =

E a(.c0 zi i_
l

(recall that r = rank of O , so a ~ ) is a unit).

6.7. L E M M A : The quotient E ~ ) : = D(a~))/a~ ) does not depend on a .

PROOF : Let b ~ A have degree e > 0 , b (c0 and c (cz) the leading coefficients of • (a) re rd+re b '

242

~),

respectively. Then

c (a) = a (et) (b(a))q rd ' rd+re rd re thus its logarithmic derivative w.r.t. D equals that of a (a) . Now def'me D(~) a e Ends(G ) by (~) (a) D(~) a = o~-1 o (VD(q(c0) a- E D rl a ) .

(6.8)

=

6.9. LEMMA : (i) This definition does not depend on the choice of cc : Lie(G) ~ ® S "

(ii) For a,b e A , the derivation rule holds : D(~)a b = y(a) D(~) b + D(~a) * ~ b " PROOF

: Straightforward, using rl (ba) = brl (a) (b e B*) and (6.5).

Hence, applying the procedure in the last section, D defines an extension of (G,~) by the

(

tautological A-module G . An element

./

"~a) I ~ ) . Mapping 0 q~, ]

D

a of A acts on G • G

through the matrix

to the corresponding extension class in

Ext((G,q~),G) --~

Ext((G,~),Ga) ® Lie(G) yields a B-morphism D e r A ~ ) ~ Ext((G,~),Ga) ® Lie(G), which, tracing back our canonical isomorphisms, is nothing else than the Kodaira-Spencer map (6.6). Everything said so far extends immediately to non-aff'me base schemes S (compare (3.11) and (5.6)), replacing DerA(B ) by the relative tangent sheaf ~ S I A ' and Hom B , Ext by their sheaf versions Horn S , E x t . (6.10) We now investigate the Kodaira-Spencer map in the case of modular Drinfeld modules. Let a be a non-trivial ideal of A with support supp(a) C Spec A . We consider M = Mr(a) x (Spec A - supp(cr)), where Mr(a) is the modular scheme for r-Dmod's with a level a structure (see e.g. [2], [5], [1]). In this context, a level a structure on an r-Dmod is an isomorphism of schemes in A-modules of ( a ' l / A ) r with the scheme ct~ of a-division points. It is known ([2], 5.4 Cot.) that MIA is smooth of relative dimension r - 1 , hence the tangent sheaf ~ = ~r'MiA is locally free of rank r - 1. Let (G,~) be the universal r-Dmod

243

over

M.

6.11. T H E O R E M : In the above situation, the Kodaira-Spencer map KS : ~ MIA --~ H°mM(HI(do) ' H2(do))

is an isomorphism. Before giving the proof, let us state some lemmata. Let T : A --o L be an A-field, L[e] the ring of dual numbers over L (i.e., e 2 = 0 ), and do : A --->L{x} an r - D m o d over L . 6.12. L E M M A : The set of extensions o f do to an

r - D m o d over

LIE] corresponds

bijectively to Dr(do). P R O O F : Let

a ~ ~a

be an extension

~ : A ---> L[e] {x} of do to L[E] . Then

~a =

doa + erla ' where rla ~ L{z} , and (i) rla e L{z}x ; (ii) degxrla < r . deg a ; (iii) "qab = ~ a ) rl b + 11a * dob " Conditions (i) and (ii) result from (i'), (ii') of (1.3), and (iii) from doab = ~ a * d)b and 82 = 0 . Therefore, rl ~ Dr(do) , and conversely, any 1"1¢ Dr(O) defines an extension ~ as above. 6.13. L E M M A : Suppose that

Ker(7) does not divide

~ . Let

~ be an extension of the

r-Dmod do on L to L [ e ] . Each structure of level tl on do extends uniquely to ~ . P R O O F : It suffices to verify this for ~ = (a) principal. Let ~ a = Oa + e-'rla " F o r u + ev Lie] , ~a(U + ev) = doa(U) + e ( ~ a ) v + rl a , u ) , so u ~ u + ~v with v = -?(a) "1 rl a . u defines a canonical isomorphism of Ker(doa) with K e r ( ~ a ) , which gives the assertion. 6.14. LEMMA

: Let

do~, doe~ be two extensions o f

do to

L[E] associated

rl,q) e Dr(do). They are isomorphic if and only if modulo Di(do), lq and

with

(p are conjugate

under the automorphism group Aut(do) of dO. P R O O F : Note f'~st that the invertible elements of L{'~} are those of the form u + e v , where u ~ L* and v ~ L { x } . N o w (u + ev) * don = do~. (u + Ev) a

a

is equivalent with the system of equations in L{x} : (i) u O a = 0 a * U

(ii) url a + v . doa = T(a)v + (Pa * u .

244

Equation (i) says that u is an automorphism of ~ . Writing (ii) in the form U~a -

(Pa * u = " / ( a ) v

and counting ':-degrees, this forces

- v , • a (= @v))

deg x v < 0 , i.e., v ~ L , and rl - u" 1 (p, u modulo

Di(~ ) n Dr(q) ) . PROOF OF THE THEOREM : It suffices to show the bijectivity of KS in maximal points x ~ M . Let L be the residue field at x, x = Spec L[e], and OIL the corresponding Drinfeld , --module. We may suppose that ~ lives on GalL, thus HI((I)) ~ L . For the fiber ~I'(x) of t~"MIA in x , we have _=_ ~ O'(x)---) {morphisms of x to M centered at x} ---) {isomorphism classes of extensions (~ of (I), provided with its level structure, to x } by the modular property of M . But (I) with its level structure is rigid, i.e., has no non-trivial automorphisms, so by the preceding lemmata, this set is in canonical bijection with ___-. . Dr(O)/Dr((l)) n Di((I)) ---) Dsr((I)) ---) H2((I)) . The composite is easily verified to agree with KS(x). 6.15. REMARKS : (i) Theorem 6.11 is most naturally formulated in terms of the modular

stack

M r for rank r Drinfeld modules. Here, no need arises to introduce rigidifying level

structures. Essentially the same deformation argument works to give a Kodaira-Spencer isomorphism over M r . (ii) Let R be an arbitrary ring extension of A . Since both ~r'MlA and the H i (~,?) (i = 1,2) commute

with

base .

changes,

there

result

Kodaira-Spencer

isomorphisms

*

0"MxRIR ~ HomMxR(HI(O ) , H2((I))) between the extended data. In the next section, we will use this to investigate modular forms.

7. Applications to m o d u l a r forms. In this final section, we only consider Dmod's of rank 2. We first briefly review the description of modular schemes and modular forms which, in this special case, is quite similar to that of elliptic modular schemes and forms (see [13], A1). The standard reference on the subject is [5], labelled D M C , in particular chapters V,VI.

245

(7.1) Let Y be an A-submodule of K 2 , projective, of rank 2, and F = GL(Y) the subgroup {'#Y7 = Y} of GL(2,K) . Without restriction (see discussion pp. 71 of DMC), we may assume that Y = ~(1,0) + b(0,1) with ideals a , b of A . In this case, F is the group

((ab)la'd~A'ad-bc~F~/) F= cd

be a

-1

b,c~

tXb

-1

/

-

Each z in the "Drinfeld upper half-plane" f2 = C - K , defines by (1,0) ~ z , (0,1) ~ 1 an injection i z of K 2 into C . We let Yz = iz(Y) ' which is a 2-lattice in C . Recall that f2 is a rigid analytic space over C , on which GL(2,K.,) acts through fractional linear (ab) transformations : c d (z) = (az + b)/(cz + d). The corresponding action of F on f~ has finite stabilizers, so the quotient I ~

of ~ by F exists as an analytic space. Furthermore,

z ~ @(z) = 2-Dmod corresponding to Yz induces a bijection of ~

with

Mr(C) = set of isomorphism classes of 2-Dmod's • of type Y (i.e., HBetti(@) A-isomorphic with Y ). M F is a component of the base extension M 2 x C of the coarse modular scheme for 2D m o d ' s , and the above bijection is an isomorphism of smooth one-dimensional analytic spaces over C . 7.2. DEFINITION : A (holomorphic) modular form of weight k and type m ( k a nonnegative integer, m a class in Z/(q-1)) for F is a C-valued function f on fl that satisfies (i) f transforms under 7 = ( acb d) ~ F according to f(yz) = (det ~/)-m (cz + d) k f(z) ; (ii) f is holomorphic on l) ; (iii) f is holomorphic at the cusps of F . Condition (iii) is explained in detail in DMC, pp. 44. For example, the holomorphy of f at the cusp "oo" of F means that for large values of inf{Iz - xl I x ~ K**} (= small values of

246 It(z)l ), f(z) has a convergent power series expansion in 1

t(z) = ec (z), where e c is the function associated with the one-lattice c = ~ ' l b the uniformizer

in C . Thus t is similar to

q(z) = exp(2~iz) in the classical case. We put M~(F)

for the finite-

dimensional C-vector space of modular forms of weight k and type m for F , and M ( ~ = ~ M~n(F). k,m It is easy to write down some examples (see e.g. [10], DMC, [6]) : 7.3. EXAMPLES : (i) For a ~ A , let ~ )

=

Y~ 2i(a,z) xi . Then (z ~ 2i(a,z)) e M0(F), i_<2dega

where k = q l . 1 . We define the (nowhere vanishing) modular form Aa(Z) as 2 2 deg a(a'z) " (ii) The Eisenstein series (fzrst studied in [11]) E(k)(z) =

E ~-k 0 ~ . e Yz

def'mes an element of M k0 (non-zero if k -- 0(q - 1) ). (ab} (iii) Let H be the subgroup of elements 0 1 of F , and put (Zk,m(y,z) = (det 7) "m (cz + d) k

Pk,m(Z) =

Y. 0~klm(?,z) tm(yz) 7e H',F

(k,m e N, k =- 2m (q - 1), m < k/(q + 1)) defines an element of M m rood q-1 -

k

(iv) If A = Fq[T] as in (1.1), the C-algebra M(F) is a polynomial ring C[g,h] in g = E (q'l) and h = Pq+l,1

([6] 5.13. We would like to know a similar presentation of M(F) for base

rings A other than Fq[T] . Unfortunately, all we know about the C-algebra M(I-) is its Hilbert function. For the subalgebra M°(F) = ~k M0(F), it is given in DMC, p. 86, and it may be calculated for M(r) following the same lines).

247

(7.4) In what follows, ~ is the "generic" Drinfeld module over f~, defined by the lattice Yz ' where z varies. By (7.3) (i), we can also regard •

as being defined over the ring R of

holomorphic functions on f~, which is the point of view of [13], A1.. We want to describe a canonical basis of HDR(C',R) = HI(t~,R) • H2(O,R). The class [co] of co = rl (1) ~ Dr(C,,R ) spans H I ( ~ , R ) . We use the Gauss-Manin connection to construct a basis vector [1]] of H2(C,,R). Henceforth, we identify D r and HDR, i.e., we omit the brackets "[ ]". d Let 0 be the differential operator ~ - , a e A non-constant, and put

(7.5)

E(z) = A'a(Z)/Aa(Z) = 0 (Aa)/Aa .

It follows from (6.7) that E is independent of the choice of a. 7.6. REMARK : There exists a canonical "discriminant function" A, which is a modular form of weight q25-1 and type 0 for F , and such that E = 0 (A)/A. That A has remarkable properties : It has a product expansion like the classical A = (2ni)12q ~ (1 - qn)24, and up to constants, all the Aa are powers of A. Thus E is similar to the "false Eisenstein series of weight 2" in the theory of elliptic modular forms. From A, it inherits a simple t-expansion ([6], 8.2). See also (7.10) and (7.35) ! 7.7. PROPOSITION : The map (V 0- E) restricted to H](C',R) is an R-isomorphism with H2(C',R).

PROOF: Since V0(c0 a) =

]~ 2 'i(a,z) ~i = A'a(Z) ,c2 deg a + terms of lower degree in x, i_<2dega

the derivation (V 0- E)(co) is strictly reduced, i.e., V 0- E agrees on H 1 with the KodairaA

Spencer map, and in particular, is R-linear. Let R z be the completed localization of R at z , which agrees with the completed localization of M ~ a ) (C) = F(tl)xO in z, provided that Ct is a non-trivial ideal of A . Here, F(0;) is the principal congruence subgroup Ker(F = GL(Y) -4 GL(Y/c~Y)), which contains no non-trivial elliptic elements, so f~ -4 F(0;)x.Q is 6tale around A

z (DMC, p. 50). Now the stated bijectivity over R follows from that over Rz(Z e f~), which in turn results from Thin. 6. i1. We let 1"1= (V 0- E)(o~) a Dsr(~,R) -4 H2(~,R) be the corresponding basis vector.

248

Denoting by r z : D(O,R) ~ D(~(z),c) the restriction mappings, we have, for each point z e f2, a basis {co(z) = rz(co),rl(z) = rz(rl)} of HDR(O(z),c). Next, we give an ahemate description of ~70 . Using Thm. 4.6, we extend the cycle integral pairing to a perfect R-pairing I

(7.8)

HDR(O,R) × Y ® R ~ R A cp

where the integral evaluated on z is f i

v

~' f v 9 ,

(9)" Equivalently, we have the bijective de Rham

,tv

map over R _=

.

DR : HDR(O,R) --~ HomA(Y,R) given by 9 ~ (v ~ fvcp). It is straightforward to show that under D R , V 0 corresponds to d * 0 = ~l~' i.e., that V 0 is the unique C-endomorphism of HDR(~,R) that verifies

(7.9)

fvV°(cP)= d f v c P

( v ~ Y , ¢ 0 ~ D(O,R)).

7.10. COROLLARY : The function E(z) on ~2 satisfies E ( z ) = - s--9,,~ lira Z ~Yl

(0 ¢ y ~ Y , liz(y)l < s).

With our particular choice Y = ¢(1,0) + b(0,1), this may be written

E ( z ) = - s---~,,~ lim Z az a+ b

((0,0) ~ (a,b) ~ a × b , laz + bl < s).

PROOF : Let )~(z) e HomA(Yz,C ) be the image of rl (z) under D R . From the above, if Y = (Yl,Y2), z(z)(iz(Y)) = ( d _ E(z)) (yl z + y2) = Yl - E(z) (yl z + y2). Due to (4.7), Z (z) has the average property :

249

z(z)(iz(Y)) 0 = s-->o*lirn~, ~ _

= lim (Y. ~

Yl

(summing over 0 # y ~ Y , liz(y)l < s)

- #{y ~ 0 , liz(y)l < s} E(z)).

But the number # is congruent to -1 mod q if s is large enough, thus the result. Let for the moment • be an arbitrary 2-Dmod over C with lattice A = HBetti(~) and de Rham map DR : HDR(~,C ) ---> HomA(A,C). Each pair (~.,IX) of K-linearly independent elements of A ® K defines through A (7.11)

X,~t = Z'0-) ~"(~t) - Z'(~) Z"O-)

(2' = DR(q)'), Z" = DR(q)")) a non-degenerate (Thm. 4.6 l) alternating pairing on HDR(~,C). The dependence on (X,l.t) is described by (7.12)

< >~V,~ = (det "/) < >Z4t

(Y ~ GL(A ~A K) acting from the right).

7.13. R E M A R K : These pairings are a substitute for the cup-product (or the Legendre determinant) in the case of complex elliptic curves. Let e.g. Y = A 2 and x = (1,0), y = (0,1). If DR('q (z)) = Z (z) E HomA(Yz,C), then )~(Z)(z) and z(z)(1) are the quasi-periods of O(z). The case of A = F q IT] is studied in detail in [7]. Up to a constant, the function z " z(z)(1) agrees with E(z) ! Back now to our general situation. We let x = (Xl,X2) , y = (yl,Y2)

be K-linearly

independent elements of Y , and deffme the perfect alternating R-pairing < >x,y on HDR(~,R) by (7.14)

<

'

q)'q)

,,>

x,y (z) =

z(q)) , rz(q) ) iz(X),iz(y) .

Then <¢o,'q>x,y = <¢o,(V 0- E) ¢O>x,y <m, V0m>x,y evaluated at z gives =

(7.15)

d . d <~,rl>x,y (z) = iz(X) ~ - iz(y) - lz(y) ~ - iz(X) = x2Y 1 - xlY 2 = const. ~ 0 .

{ab) Let y = c d be an element of GL(Y) and u = (cz + d) "1 . Multiplication by u yields an isomorphism Yz -'~ Y'~z) • By (2.6), u defines an isomorphism, also denoted by u : ~(z) ~_~

250

• ('¢z) , i.e., for each a ~ A , u * ¢~> = O(Yz> * u . Let u * : HDR(O('Cz)) ---) HDR(O(z)) be the a induced arrow on HDR. Then from (3. I I ) , (7.16)

u*(to(~tz)) = u. to(z) = (cz + d) "I to(z).

We will show at once the transformation formula for rl : (7.17)

u*(rl (Yz)) = (det y) -I (cz + d) rl (z) .

If we def'me a C-linear operation q) )-) (p[y] of F on Dr(O,R) by

(7.18)

rz((p[~t]) u*(r~z ((P)), =

these formulae say that to (resp. "q) transform like modular forms of weight -1 , type 0 (resp. weight 1, type 1). PROOF OF (7.17) : u*(rl (~)) and rl (z) are proportional since H~(~ (z)) is one-dimensional. Let z' = Tz. An elementary calculation yields u iz(X) = iz,(X) T-1 , so iz(x),iz(y) = uiz(X),Uiz(y)

(by definition of < > and (4.5))

= (det y)-I iz,(X),iz,(y) = (det y)-i x,y (z') = (det y)-I x,y

(by (7.12) and (7.15)). Comparing with (7.16), we obtain the

stated proportionality factor. Next, we consider the action of 0 on t-expansions. Recall that t(z) = e'l(z), where e = e¢ de(z) as in (7.2). Since y = 1, (7.19)

d 2d e=~=-t ~.

The next theorem is proved, in the special case A = IFq [T], in ([6], 6.10, 3.10). The proof given there generalizes without complications to arbitrary A. 7.20. THEOREM : Let the t-expansion of the Eisenstein series E (k) be given by E (k) (z) = 5".ait i . I f k = q J - l , a i ¢ 0 implies i=-O or q - l m o d q ( q - 1 ) . In view of (7.19), this implies

251

7.21. COROLLARY : 02 annihilates E oK) (z). 7.22. COROLLARY : 02 annihilates all the coefficient forms ,ei(a,z ) . PROOF : Define the Eisenstein series E (°) (z) as the constant -1. Then for k > 0 , the relation a E (qk-1) =

Y. E (qi'l) i i+j=k 2J(a'z)q

holds (DMC, p. 15), which together with (7.21) implies the assertion. 7.23. COROLLARY : 0 ~ ) = - E 2 . PROOF : Let a ~ A be non-constant, so E = 0(Aa)/Aa and 003) = -(0(Aa)/Aa) 2 = - E 2 since

o2( )

= o.

We can now completely describe the Gauss-Manin connection with respect to the basis {c0,rl} of HDR(~,R ) . By definition of 1"1,

V 0 (m) = Ec0 + il. 7.24. C O R O L L A R Y :

V 0 (1"1)= - E'q.

P R O O F : V 0 ( r 1) = V o ( V 0- E) (m) = V~(m) - 0(E) co- E V 0 ( m ) . The term V2(co) vanishes by (7.22), and due to (7.23), the rest gives - Erl We use these results to construct a "Serre derivation" ~ on M(F). Let (7.25)

S'e(R) = Symm 2 (HDR(@,R))

be the 2-th symmetric power. Since HDR = H 1 (~ H 2 , it decomposes

(7.26)

S'e(R) =

(3 S'e-v'v(R), 0
where SU'V(R) = H]@u(@,R) ® H *2~ v (@,R) with basis m@u ® rl@V The same type of isomorphism holds over any ring R' containing the coefficients of @ and stable under o=d= t 2 d~ , -

252

ff i,j _> 0 , k = i - j , j = -m (q - 1), condition (i) of Definition 7.2 is equivalent with (7.27)

fco~i ® r l ~

is F-invatiant.

Correspondingly, we may express the "cusp condition" (iii) using HDR . As in (7.2), we explain this only for the cusp "oo". Let R t be the subring of R consisting of functions f on invariant under transformations z " z + a (a e c = a "119) and possessing a finite-tailed Laurent expansion with respect to t , i.e., R t = {functions "meromorphic at o~"} . Let further Boo = C[[t]] and Loo = C((t)). Then R t embeds into Loo. As stated in (7.3) (i), the genetic Dmod qb is already defined over R t . In HDR(O,Loo), consider the Boo-lattice W = span of {c0,q} . Its 2-th symmetric power is a lattice in

S2(L~,

direct sum of lattices W u'v =

Boo co~u ® rl ~v (u + v = 2 ) . The holomorphy of f at the cusp oo (condition (iii) of (7.2)) is now equivalent with (7.28)

fco®i ®q®j e W i'j .

Since the module Diffc(R ) of C-differentials of R is free with basis d z , the Gauss-Martin connection defines an arrow (7.29)

V : HDR(O,R) --->HDR(O,R) ® Diffc(R)

~o~ VO (~o) ® d z which is equivariant for the action of F (see (7.18)). The transpose of the R-isomorphism given by (7.7)

Derc(R ) --~ HomR(HI,H2) 0 ~' (V 0- E)IH 1 is =

HomR(H2,H 1) --~ D i f f c ( R ) ,

where

"dz"

corresponds to

(rl ~ co) . Thus, taking (7.26) into account, V

connection, also denoted by V V : S2 =

@ S 2"v'v --) S 2 ® HomR(H2,H~) = ~ S 2-v+l'v'1 0
defines a

253

on S "~ = S y m m ' ~ ( H ; R ( O , R ) ) . Let now f e M~n(F), i,j e I~ satisfy k = i - j , j -= -m ( q - 1). Then f to~i ® 1"1~ is r-invariant and contained in W iJ , as is its image under V . The latter is

(7.30)

V ( f co~i ® 1"1~j) = 0(f) o~~(i+1) ® .q~c.j-1) + if o~i(Et.o + 1"1)® .qtl~j- 1) + jf ~ ( i + l ) ® rI~j-2)(_Erl ) = (0(f) + kfE) o~(i+l) ® .qc~(j-1) + if o~i ® rl~j.

(Note that "®dz" simply amounts to replacing a factor co by rl ). Thus the s i + l j ' l - c o m p o n e n t (7.31)

0k(f) = 0(f) + kfE

satisfies conditions (i), (ii), (iii) "at ~*" of (7.2) with (k,m) replaced by (k + 2, m + 1). The h o l o m o r p h y of 0k(f) at cusps other than %0" follows the same lines. Therefore, 0k(f) e Mkm+21(F). Direct calculation shows : m.

(7.32) If fi e MkiX (i = 1,2), f = flf2 , and k = k 1 + k 2 then

0k(f) = 0k 1 (fl) f2 + fl 0k 2 (f2)" Hence the 7.33. P R O P O S I T I O N : The C-linear map 0 : M(]?) -~ M(F) given on M km by 0 k = 0 + kE is a derivation. It maps Mkm(r) to /~fm+l¢l""k+2 '," ~-' "

There exists a very simple proof avoiding HDR. Writing E = 0(Aa)/A a , and applying 0 to the transformation equation Aa('yz) = (cz + d) q2

deg a_ 1

Aa(Z )

yields the transformation law for E : (7.34)

E('yz) = (det T) "1 (cz + d) 2 E(z) + c(cz + d)/det T

which, together with the law for Off) and some considerations at the cusps, implies that Ok

254 m

l~m+ 1

maps M k to "'k+2 " Nevertheless, it is important to have the "de Rham" interpretation of modular forms, and of 0 (see below). W e finish with some concluding remarks. 7.35. R E M A R K S : (i) From the product expansion of A (DMC, p. 76), it is easy to see that the t-expansion of E is t + 0(t 2) . Since 0 = - t2 ~d- , Ok increases the order of f at ~ by at least 1. The same holds for the other cusps. (ii) The uniformizer t at the cusp ",,o" does not properly correspond to q(z) = exp(2xiz). All the results presented so far remain true if we replace t(z) by c - t(z) and 0 by c • 0

with

some c e C* . Choosing c carefully (i.e., c some analogue of (2xi) "1 ), the normalized Eisenstein series c k E (k) and the c ( q l l ) 2i(a,z)

will have t-expansions with algebraic

coefficients (in fact, with bounded denominators). For the case A = IFq[T] , see [6]. For general A , there are some still unsolved questions of "best choice" of c to get "canonical" texpansions (DMC, p. 38-39). The operator 0 should then act on the A-module of

algebraic

modular forms [11]. The definition given there has to be generalized to include non-trivial "types". (iii) Having chosen the "correct" uniformizer

t , the coefficients will have interesting

arithmetical properties. One may apply the "de Rham" description of modular forms, together with the formalism of vanishing cycles and of the Gauss-Martin connection, to investigate their congruence properties. I hope to come back later to that problem.

255

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N. Katz : P-adic properties of modular schemes and modular forms. Lecture Notes in Mathematics 350. Springer-Verlag. Berlin - Heidelberg - New York 1973.

[14]

B. Mazur - W. Messing : Universal extensions and one dimensional crystalline cohomology. Lecture Notes in Mathematics 370. Springer-Verlag. Berlin - Heidelberg - New York 1974

THE NONARCHIMEDEAN EXTENDED TEICHM~LLER SPACE

Frank Herrlich Fak. u. Inst. Postfach

Let k be an algebraically to a nonarchimedean moduli

scheme~_

f. Math.,

Ruhr-Universit~t

1o2148, D-463o Bochum

closed field which is complete with respect

valuation.

For an integer g ~ 2 consider the

of stable curves of genus g and denote by M

g

analytification open subset of M

of ~

g

× k. Finally

let~

g

curves of genus g over k. In other words: ~g(k)

if and only if the canonical

curve C

(k) the

(k) be the k-analytically

(k) whose points correspond

g

g

to the stable Mumford

a point x • M

g

~k) is in

stable reduction of the stable

represented by x, is totally degenerate,

i.e. has only

x

rational

irreducible components.

In this note we present a "Teichm~ller means Out F

g

theory" f o r ~

that we construct a k-analytic manifold S of outer automorphisms

g

(k): this

(k) on which the group

of a free group of rank g acts

discontinuously with orbit space analytically points of S

g

g

isomorphic

tOng(k).

The

(k) are stable Mumford curves of genus g together with a

basis of the uniformizing the construction

Schottky group.

includes,

in contrast

The advantage here is that

to classical

complex Teich-

muller theory, not only smooth Mumford curves but also those with nodes. On the other hand,

of course,

the s p a C e ~ g ( k )

is only a part of the

whole moduli space M (k). A similar construction has been carried out g over the complex numbers Mg(~),

in [I]. Here S g ~ )

maps surjectively onto

but the map is not the quotient map for the action of Out Fg .

The result of this paper has been obtained by M. Piwek [5]

in a

257

slightly different

form:

for totally d e g e n e r a t e d

he c o n s t r u c t s stable curves

n e i g h b o u r h o o d of this space analytifies

this formal

§ I. U n i f o r m i z a t i o n

an a l g e b r a i c T e i c h m 0 l l e r over k, then considers

in a suitable

larger scheme,

[3]:

PGL2(k)

the k - a n a l y t i c u n i v e r s a l

of a S c h o t t k y group,

covering

p1(k);

As m e n t i o n e d

~ is the set of o r d i n a r y

i.e. a free h y p e r b o l i c

in the introduction,

The k - a n a l y t i c u n i v e r s a l

covering

each i r r e d u c i b l e

domain in a p r o j e c t i v e F L = ~y ~ F :y(L)

lines,

line

component ~l
given a tree of p r o j e c t i v e

(of some rank ~ g) F

to C.

lines X and a d i s c o n t i n u o u s

F of rank g on X such t]~at the r e g i o n of

d i s c o n t i n u i t y ~ of r is connected, arithmetic

genus of C. More

of F L. In particular,

on ~, and ~/F is isomorphic

action of a free group

together w i t h the

and the subgroup

= L} acts as a S c h o t t k y group

acts d i s c o n t i n u o u s l y

stable reduction.

L of ~ is an u n b o u n d e d Stein

with L the set of o r d i n a r y points

Conversely,

is k-analyti-

of such a curve C is an open dense

action of a free group F of rank g = a r i t h m e t i c

P1(k),

F of

by a stable M u m f o r d curve we

c o n n e c t e d part ~ of a tree of p r o j e c t i v e

on

subgroup

to C.

m e a n a stable curve over k w i t h totally d e g e n e r a t e

precisely:

is well known,

of a, M u m f o r d curve C

of rank g = genus of C, and the orbit :space ~/F

cally isomorphic

and finally

of stable M u m f o r d curves

is an u n b o u n d e d S t e i n domain g in points

a formal

scheme.

The u n i f o r m i z a t i o n of M u m f o r d curw~s by S c h o t t k y groups see e.g.

space

genus g w h i c h becomes

the orbit space ~/F is a curve of

stable

if all i r r e d u c i b l e c o m p o n e n t s

258

are c o n t r a c t e d

that are isomorphic

i n t e r s e c t i o n points with other

These c o n s i d e r a t i o n s correspond

~pl(k) and have only one or two

i r r e d u c i b l e components.

show that stable M u m f o r d curves

in a unique way to "stable"

free group on a tree of p r o j e c t i v e projective

to

discontinuous

lines.

over k

actions of a

The notions

of "tree of

lines" and " s t a b i l i t y " will be made precise

in the next

sections.

§ 2. Trees of p r o j e c t i v e

lines

In this s e c t i o n k may be an a r b i t r a r y stand for "tree of p r o j e c t i v e

field.

The l e t t e r s T P L

lines".

In the f o l l o w i n g we give some d e f i n i t i o n s a n d p r o p e r t i e s and sketch their classification. in the f o r t h c o m i n g

shall

More details

and proofs

of TPLs

can be found

article [4].

A finite TPL over k is, as in [2],

§ I, a p r o j e c t i v e

k - v a r i e t y C,

such that (i) all i r r e d u c i b l e (ii) i r r e d u c i b l e

components

components

of C are isomorphic

intersect,

if at all,

to

1 ~k

in k - r a t i o n a l

o r d i n a r y double points (iii) the i n t e r s e c t i o n graph of C is a tree A morphism

f: C ~ C' of finite TPLs

(i) for each i r r e d u c i b l e

component

map or an i s o m o r p h i s m onto (ii) for each i r r e d u c i b l e component

over k is called a c o n t r a c t i o n

L of C w i t h f(L)

L of C, flL is either a c o n s t a n t

f(L)

component = L'

if

L' of C' there is at most one

259

Obviously Hence

the c o m p o s i t i o n of c o n t r a c t i o n s

the finite TPLs

TPLf(k)

is again a contraction.

over k w i t h c o n t r a c t i o n s

of the k - v a r i e t i e s .

A projective

form a s u b c a t e g o r y

system in this c a t e g o r y has

a limit in the c a t e g o r y of locally r i n g e d spaces over k, and any locally ringed space w h i c h

is the p r o j e c t i v e

limit of finite TPLs over

k will be called a TPL over k. It turns out that TPLs over k can be ¢,haracterizedas

Proposition

follows:

I: A c o n n e c t e d

locally ringed

space

(C,60) over k is a TPL

over k if and only if (i) each i r r e d u c i b l e

component

of C is either

or is a single point x w i t h ~

isomorphic

to

p1 k'

~ k x

(ii) the i r r e d u c i b l e

components

of C that are isomorphic

to

1 P k are

a dense subset of C (iii) for any closed point x ~ C, the local ring ~9

is i s o m o r p h i c

to

X

k or to

,0 or to o v ,(0,0)' where V = V(tlt2)

(iv) for any x e C, C - {x} has at most (v) if LI, L2, L 3 are i r r e d u c i b l e L 1 n L 2 and L 1 n L 3 is (vi) For any two i r r e d u c i b l e unique m i n i m a l

two c o n n e c t e d components

components

nonempty,

c A k2

of C such that

then L 2 n L 3 = ~.

components

L I , L 2 of C there is a

closed c o n n e c t e d subset S(LI,L2)

containing

LI

and L 2 . A c o n t r a c t i o n of TPLs over k is d e f i n e d TPLs.

in the same way as for finite

In p a r t i c u l a r we have the f o l l o w i n g observation:

P ropo:sition 2: Let f: C ~ C' be a c o n t r a c t i o n of TPLs over k, and let (Ci)i~i and

(C~)j~j be p r o j e c t i v e

that C = l ~ m C f

systems of finite TPLs over k such

C' = l l m C ~ . Then there exists J

i(j) ~ I and a c o n t r a c t i o n

fj:

Ci(j)

for each j E J an index

C~] such that

260 f

C

~ C'

+

is

+

commutative.

fj

C i (j~ If L is an i r r e d u c i b l e (unique)

contraction

x l , x 2 , x 3 of C there

) C!]

component ~L:

of a TPL C over k, there is a

C ~ L. For any three distinct

is a unique

called the m e d i a n c o m p o n e n t

irreducible

of xl,x2,x3,

and =n(X3)

are three different points

for finite

TPLs,

in p a r t i c u l a r

is obvious

limits of finite TPLs.

of three points

(iv) of Prop.

C a TPL over k. A stable M - m a r k i n g 9: M ÷ C(k)

of L. This p r o p e r t y

that the m e d i a n c o m p o n e n t (see also

L = L(Xl,X2,X3),

such that = L ( X I ) , ~ L ( X 2 )

hence also for p r o j e c t i v e

a one-point-component

component

closed points

Note

can never be

]). Let M be a set and

of C is an injective map

satisfying

(i) for any i r r e d u c i b l e

component

L of C there are ~i,~2,~3

in M

such that L = L ( ~ ( v l ) , $ ( v 2 ) , ~ ( v 3 ) ) . (ii) for each v • M, ~Llv(~(v)) = {~(v)}, where irreducible

component

The second c o n d i t i o n ensures marked.

It is possible,

that for finite TPLs

denotes

the

of C w i t h ¢(v) • L v. in p a r t i c u l a r

however,

that no double points

that end points of C are marked.

this d e f i n i t i o n agrees w i t h the one in [2],

Let F be a group and M a set on w h i c h of a TPL C over k is called F - e q u i v a r i a n t p: F + Aut C

L

are Note § ].

F acts. A stable M - m a r k i n g if there is a h o m o m o r p h i s m

such that

p(y)(~(v))

:

~(yv)

for all v e M, y e F.

We shall apply this concept

in the following

situation:

free group of rank g, and M = ~ is the set of p r i m i t i v e on w h i c h

F acts by conjugation.

If then 9: ~ ÷ C(k)

F is a

elements of F,

is a stable

261

F-equivariant fixed point

marking

of p(y)

of

for each y • F. Since y-1

y is; we see that each

§ 3. Classification

p(y)

has at least

of stable marked

In [2] we classified approach

TPL C ow~r k, it is clear

a

stable

can be extended

that O(y)

is

a

is in ~ if and only if

two fixed points

on C.

TPLs

n-pointed

TPLs by cross

to stable M-marked

TPLs

ratios.

This

for arbitrary

M as

follows: Let q = (C,~) be a stable

M-marked

TPL over a field

~) = (~)l,V2,v3,~4)

e Q(M):=

Here we normalize

~L(X)l,~)2,~)3) by an isomorphism

~L(~,(Vi))

cross

{(v i ''" . ' v4) e M4: ~ i • v j for i * j} let

to 0, °° and J for i = ],2,3~

AS for finite M (see [ 2], ration

k. For any

1.3)

these

L -~ ]Pkz that sends

resp. l~ = Xv(q)

satisfy

the following

relations: =

(i) I )I ,~)2,~)3,v4

l-i

~)2'~)I,v3,v4

(ii) ~vl,x)2,~)3,~)4 = ] - I~)2,~)3,x)4,~)1 (iii) I l = X ~)i ,~)2,~3,~ 4 ~)i ,~)2,v4,v5 ~)l ,x)2,v3,v5 for all distinct ~i'''''v5 in M. If ¢, is F-equivariant in addition

for a group

the .F-invariance ..........

F that acts on M the Xv(q)

satisfy

equations__

Ov) for all

I~{(V l) ,~((V2) ,Y(V 3) ,Y(V 4) = I~ I.,D2,g3,D 4 (Vl,...,v4) e Q(M) and y • F.

The moduli

spaces

the following

B n in [ 2] are in the general

"provarieties"

situation

replaced

by

B : M

For Q := Q (M) let of locally all finite

ringed subsets

~Q

=

spaces)

~ Q he the projective of the proiiective

Q' of Q. Thus

limit

schemes

(in the category ~Q,:__ 7z.

H

~i. for

'6Q '

for any ~ e Q(M) we have a well

262

defined p r o j e c t i o n

Iv

: ~Q ~ .

P Q defined by all equations the p r o j e c t i v e • zg Q'

hence

locally

(i)

(iii).

limit of Zariski-closed,

carries

ringed

Now let B M be the closed

a natural

space

It is clear that B M is itself

i.e. projective

structure

subset of

of provariety

that is limit of a projective

subsets

over

~

system of

of the

(i.e ~-

varieties). In the s i t u a t i o n where equations

(i) - (iv).

Proposition

F acts on M we define B F in the same way using M

In particular,

B~ is a s u b p r o v a r i e t y

3: For any field k, the k-valued

are in I-I c o r r e s p o n d e n c e

with

TPLs over k (resp.

F-equivariantly

The c o n s t r u c t i o n

stable

the isomorphy

of the universal

points of B M (resp. classes

M-marked

natural

projection

B~, ~ BM,

PM

:= M u {m'}.

: B~ ~ B M. Since PM factors

B~)

of stable M-marked

TPLs over k).

family of stable M - m a r k e d

over B M is carried out in the same way as in [2], choose an element m' e M, and let M

of B M.

TPLs

§ 3 for finite M: Now consider through

the

the

for finite M' C M it is easy to see that for q e B M the

fibre p~1(q)is

exactly

with q by Prop. is universal

the stable M - m a r k e d

3. It is also possible

for a suitable

If F acts on M we extend

definition

this action

TPL over k(q)

associated

to show that this family PM of the notion of family of TPLs.

to M by y(m')

= m'

for all

y e F. Then F acts on B M and on B~, and PM is F-equivariant. Thus F over BM, which is the fixed point set of the F-action on BM, F acts on the fibres. stable

This

shows

F-equivariantly

F MF = p M- 1 (B~)

c B~.

that PM

: FFM + BFM is the universal

marked TPLs, where we denote by

family of

263

§ 4. The p-adic,,,,,,,,extended Te,,,,ichmOller_~ace

To come back to the situation of stable M u m f o r d

of § l, i.e.

curves we have

to consider

for 17 and M as at the end of § 2, i.e. and M = ~. To simplify

of B g defined

uniformization

several

F of B M

subspaces

F = F , free group of rank g, g

notation we write

Let U g be the subset

Schottky

B g for B[g. Fg

by the inequalities

iy,y_l,~,y~y_ I # l for all y,~ E F. U g is not open for thie p r o j e c t i v e - l i m i t - t o p o l o g y but there

is a reasonable

topology

if U g and B g are c o n s i d e r e d is open in the k - a n a l y t i c

Also,

over a field k as in the introduction,

Ug

topology.

For q ~ B g, l¥,y-l,~,y~y-l(q) m e d i a n component

on B g for which U g is open.

on B g,

is the m u l t i p l i e r

of ~(y),qS(y-l),~(~).

of p(¥)

on the

That it should not be equal

to

one means

that p(y) may not act as a ]parabolic t r a n s f o r m a t i o n

on any

component

of the TPL C a s s o c i a t e d with q. In particular,

is not

the identity on any component. two fixed points

that are not singular points

F-action on C we have exactly ¢(y)

and ~(y-1)

elements

c F-orbit C

a projective vv

)

•

(gecmetrlc") Obviously

variety, quotient

c on ~, i.e.

for any conjugacy

of ~. Then clearly

Zg

~

as a lim:Lt of p r o j e c t i v e

a finite quotient

Z (n). Thus

varieties

of B g by Z g. and the above remarks

B (n),

B(n)/z (n) is again

and the limit of the B(n)/z (n) is the

U g is zg-invariant,

of F given

= y if y ~ c. Let Z g be the group of

permutations of F g e n e r a t e d by all o

Z g acts on B (n) through

to put the markings

of F, let Oc be the b i j e c t i o n

for y e c, oc(y)

acts on B g. If B g is w r i t t e n

p(y) has exactly

of C. Thus given the

two p o s s i b i l i t i e s

on C. For any F-orbit

class c of p r i m i t i v e by Oc:(y) = y-i

As a consequence,

p(y)

yield

264

Proposition conjugacy

4: The points

classes

"stable" means to

~i

:= ug/z g c o r r e s p o n d

g

of stable n o n p a r a b o l i c

that any irreducible

is m e d i a n

classes"

in U

refers

component

F-actions

component

of fixed points

to c o n j u g a t i o n

let S

g

(k) be the subset

ponding Note

respect

g

TPL is discontinuous

section points

(k) where

implies

of irreducible

components

point

of C.

stable M u m f o r d

curve over k. M o r e o v e r

(in fact Out Fg acts on B g product,

,

g

closed

valuation,

region of discontinuity. that all

inter-

are ordinary points

of C that are reduced

that for q = (C,~) e ~

and E g is a s e m i d i r e c t

nonarchimedean

in particular

It is clear

on ~g(k)

of C that is isomorphic

the F-action on the corres-

components

that all irreducible are end points

(here

of F, and "conjugacy

with c o n n e c t e d

that the last condition

on TPLs

i.e. a l g e b r a i c a l l y

to a n o n t r i v i a l

of U

to

in Aut C ).

For a field k as in the introduction, and complete with

one-one

(k) the orbit Out F

g

of F, and

to a single

space C/F is a

acts in an obvious

and the group generated

as ~ o ~c o ~-I = O~(c)

manner

by Out F g

for ~ e Out F g

O

and a c o n j u g a c y

class

analytic u n i v e r s a l

c in Fg). As for each q =(C,~) e Sg(k),

covering

of the stable M u m f o r d

we conclude

that C/F and C'/F are isomorphic

and

are in the same Out F

(C',~')

g

-orbit.

curve C/F

C is the

(see § I),

if and only if (C,~)

In other words we have set

theoretically (*) If

S

g

(k)/Out F

q = (C , ~) • S g (k)

fixed point

an arbitrary

=~

and y •

C, o n e o f t h e m a t t r a c t i n g attracting

g

(k).

F,

t h a n y has

and t h e

other

exactly

repelling.

of y, the m u l t i p l i e r

a • ~, ~ ~ y ~)

if c(y) denotes

g

the conjugacy

has absolute

two f i x e d If

~(y)

points is

on

the

ty = ~T,T_l,~,y~y_ I (for

value < I. On the other hand,

class of y, we have c¢(y)(ty)

= ty I. This

265

shows

that the inverse

copies

of Sg(k).

family of TPLs

image of Sg(k)

As a c o n s e q u e n c e

over B q to Sq(k),

we can restrict and forming

F-action we obtain

a family of stable

Hence once we have

shown

can conclude canonical

quotient

map Sg(k)

isomorphism.

Teichm@ller

P r o p o s i t i o .n.........5: Sg(k) . dimension

3g - 3.

The proof

is m o d e l l e d

space

for totally

corresponding

Therefore

is an analytic

structure we

of Mg(k) i.e.

we call Sg(k)

submanifold

on M. Piweks proof [5]

degenerated

curves

that the

that

(~) is

the k-analytic

These

subspaces

of U g (k) of

that the Teichm~ller

is a locally n o e t h e r i a n

E = (EI,...,~g)

E is a "geometric" TPL.

over S (k). g

an analytic

is analytic,

for the

space.

he first fixes a basis U gc of U g where

curves

space property

~q(k)

union of

the universal

the quotient

(Mumford)

(k) carries

g

from the coarse moduli

also an analytic extended

that S

in U g is a disjoint

of F and considers

basis

scheme:

subspaces

for the F-action on the

are given by inequalities

of the

form k v , ~ for certain v. Thus U q is open in U q in the same sense as U q itself

is open in B q. He then restricts

multipliers

are O, and proves

But following to see that has

his arguments

the same holds

that U q is an affine n o e t h e r i a n

(in p a r t i c u l a r for U q itself:

to show that the m u l t i p l i e r

expressed one glues, result.

by the m u l t i p l i e r s

to the subset U q where

lemma

in addition

of an arbitrary

of the E

i

1.8)

and the k

as in [ S] ' the spaces U Cg for various

scheme.

it is not hard to [5]

element £i'~J

all

one only

of F can be

'~k'~l

Finally

e and obtains

the

266

References

[I]

Gerritzen,

L. and Herrlich,

J. r. angew. [2]

Gerritzen,

Math. "389 (1988),

L., Herrlicb,

trees of projective [3]

Gerritzen,

Herrlich,

Piwek,

M.:

LNM 817, Springer

F.: Moduli

lines. IS]

lines.

space.

19o-2o8.

F. and van der Put) M.: Indag. math.

L. and van der Put, M.:

curve, [4]

F.: The extended Schottky

Schottky

Stable n-pointed

5o(1988),

groups

131-163.

and Mumford

198o.

for group actions

on trees of projective

To appear.

The formal

Teichm~ller

space.

Preprint

Bochum,

1989.

SUR LES COEFFICIENTS DE DE RHAM-GROTHENDIECK DES VARIETES ALGEBRIQUES Mebkhout Z.,

Narvaez-Macarro L.

M.Z., UFR de MathEmatiques LA 212, UniversitE de Paris 7, 2 Place Jussieu 25175 Paris. N.L. Departamento de Algebra C/Tarfia s/n Universidat de SeviUa, 41012 Sevilla 03spagne). Sommaire § 0. Introduction. § 1. I_~ formalisme des faisceaux d'opErateurs diffErentiels. 1.I. Les operations sur les categories de modules sur les faisceaux d'opErateurs diffErentiels. 1.2. t~opriEtEs de f'mitude. § 2. Le thEor~me des coefficients de de Rham-Grothendieck en caractEristique zero. 2.1. Le thEor~me de finitude. 2.2. Le thEor~me de comparaison. § 3. Le thEorEme des coefficients holonomes complexes d'ordre infini. 3.1. Le faisceau des opErateurs diffErentiels d'ordre infini. 3.2. Le thEor~me de bidualit& 3.3. Le thEor~me de finitude. § 4. Sur les coefficients p-adiques d'ordre infini. REfErences.

§ O. Introduction. Dans son article fondamentaI qui date de rannEe 1966 [G2] Grothendieck a propose de construire une thEorie des coefficients de type de de Rham pour les variEtEs algEbriques dEfinies sur un corps de caractEristique p. On se propose dans cet expose de faire le point sur les rEsultats de la thEorie des coefficients de Rham-Grothendieck et sur les mEthodes de demonstrations. Grothendieck a montrE que la cohomologie de de Rham d'une variEtE algEbrique non singuli~re sur un corps de caractEristique nulle fournissait les bons nombres de Betti [G1], il a dEfini la notion de connexion de Gauss-Manin d'un morphisme lisse sur un schema de base quelconque [G2] et enfin il a soulignE la nEcessitE d'avoir des coefficients gEnEraux qui "joueraient le rEte de C-vectoriels transcendants alggbriquement constructibles, et seraient stables par les operations habituelles (telles que

les Rif.(®x/s)) ~ ([G3], p. 105). I1 s'agit, bien entendu, du cristal unite et de l'image directe cristalline.

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Deligne [D1] a defini sur une variEtE algEbrique non singuliEre sur un corps de caractEristique nulle une catEgorie de coefficients lisses, les fibres h connexion intEgrable ii singularitEs rEguli~res, qui quand le corps de base est C est Equivalente h la catEgorie des syst~mes locaux transcendants d'espaces vectoriels de dimension finie par le foncteur de de Rham transcendant. Darts [D2] Deligne ~ defini une catEgorie de coefficients sur une variEtE alg6brique complexe, les cristaux discontinus, qui est 6quivalente ~ la catEgorie des faisceaux d'espaces vectoriels complexes algEbriquement constructibles. La thEorie des dgx-modules a complEtE les rEsultats de Grothendieck-Deligne. Sur toute varlet6 X non singuli~re sur un corps k de caractEristique nulle on dispose de la catEgorie D~(lgX/k) des complexes holonomes stable par les six operations de Grothendieck f*, f., f., f!, ®, hom et par la dualit& De plus D~(dgx/k) contient comme sous-catEgorie pleine la catEgorie des complexes holonomes rEguliers D~r(dgx/k) qui est aussi stable par les six operations de Grothendieck et qui est Equivalente quand le corps de base est C par le foncteur de de Rham transcendant ~ la categories des coefficients algEbriquement constructibles D~(Cx). Ceci remplit le programme de Grothendieck en caractEristique nulle. Ce rEsultat de la thEorie des dgx-modules est indEpendant du rEsultat de Deligne sur les cristaux discontinus. Le manuscrit de Deligne [D2] n'a malheureusement pas EtE publi6. Du point de vue des demonstrations la stabilitE de la categories D~(dgX/k) par les operations cohomologiques, en particulier le thEor~me de finitude de la cohomologie de de Rham, n'utilise que rEquation fonctionnelle de Bernstein-Sato et est purement algEbrique alors que la stabilitE de la catEgorie D~r(~x/c) par les operations cohomologiques et de leur comportement par le foncteur de de Rham transcendant, en particulier le thEor~me de comparaison, fait intervenir la thEorie des &tuafions diffErentielles complexes et reposaient de fa~on essentielle sur le thEor~me de resolution des singularitEs de Hironaka [HI. Si en caractEristique nulle on avait depuis longtemps une thEorie satisfaisante il ne va pas de rhyme en caractEristique p > 0. La cohomologie cristalline ([G2], [BI]) fournit une bonne thEorie pour les variEtES propres et lisses mais prEsente des pathologies pour les autres variEtEs. Grothendieck avait signalE [Gz] que la cohomologie de Dwork-Monsky-Washnitzer devraient fitre considErEe comme la base d'une thdorie des coefficients de type de Rham en caractEristique p > 0. En effet Berthelot a dEfini une catEgorie de coefficients lisses les F-isocristaux surconvergents sur une variEtd ddfini sur un corps de caract6ristique p > 0 qui donnent naissance/~ la cohomologie rigide qui est de dimension finie pour le crystal unite si l'on dispose de la resolution des singularitEs en caractEristique p > 0 [B2]. Si la th6orie des dgx/c-modules est une th6orie de coefficients au sens de Grothendieck,/~ plus d'un titre, elle introduit deux nouvelles theories cohomologiques qui n'Etaient pas prEvues dans le programme de Grothendieck et permettent d'aller de l'avant. D'une part elle donne naissance ~ la thEorie des faisceaux pervers complexes et de la cohomologie perverse. Cette thEorie garde un sens pour les

269

faisceaux l-adiques pour I diff6rent de la caract6rique p du corps de base (cf. [B-B-D]). D'autre part elle donne naissance A la th6orie des coefficients complexes holonomes d'ordre infini. La cat6gorie des complexes holonomes d'orcLre infini qui est aussi 6quivalente h la cat6gorie des complexes alg6briquement constructibles rend compte des ph6nom6nes qui sont de nature transcendante tout en gardant un sens dans le cas p-adique. En particulier cette cat6gorie est stable par immersion ouverte. Ce r6sultat est le demier pas dans la th6orie des dgx/c-modules et peut ~tre consid6r6 comme le r6sultat le plus profond. Grace ~t la th6orie des faisceaux pervers et ~ la th6orie de la ramification sur le corps des complexes on a pu montrer que le th6or~me de la r6solution des singularit6s n'6tait pas ~ au fond du probl~rne ~ dans la th6orie complexe [Me7]. Le thdor~me de r6solution des singularit6s 6tait consid6d6 jusqu'~t alors comrr~ le th6or6me de base dans l'6tude de la cohomologie des vari6t6s alg6briques et on a toujours pas de d6monstration en caract6ristique p > 0. Ceci encourage, au moins spychologiquement, ~ chercher ddmontrer directement si les r6sultats de la th6orie complexe ont des analogues dans le cas p-adique. C'est ce que nous avons essay6 de falre depuis quelques temps.Cependant il nous semble qu'on se heurte ~ un probl~me de fond : alors que darts te cas complexe le prolongement analytique, c'est ~tdire la monodromie, est au coeur du probl~me en derni6re analyse, c'est prdcisemment son analogue qui pose probl~me dans le cas p-adique. Darts ce but nous avons r6dig6 cet expos6 h l'intention du lecteurchercheur en analyse p-adique pour contribuer au d6bat qui ne sera clos que le jour o~ on aura une th6orie des 29x-modules en caract6dstique p > 0 qui a toute la souplesse de la th6orie en caract6ristique z6ro. Voici le contenu de cet expos6. Dans le § 1 nous avons rassembl6 quelques propri6t6s alg6briques des faisceaux op6rateurs difffrentiels. En partant de la d6finition de ([E.G.A.IV], §16) du faisceau •tgiffx/s(®x) des op6rateurs diff6rentiels sur un S-sch6ma X nous avons rappel6 les op6rations cohomologiques que l'on peut faire sur les catdgories de modules sur le mod61e de la caract6ristique nulle. La diff6rence avec la situation en caract6ristique nulle est que d9/ffx/s(@x) est un faisceau de @x-alg6bres qui ne sont pas de type fini m~me dans le cas d'un S-sch6ma lisse. Pour rem6dier it cet inconv6niant Chase [Ch] et Smith [Sm] ont utilis6 dans le cas d'un corps de base k alg6briquement clos de caract6ristique p > 0 ce qu'ils appellent la p-filtration du falsceau 29/ffx/x(@X) pour une vari6t6 non singuli6re X. C'est une filtration par des sous-faisceaux d'anneaux qui sont na~th6riens. Its ont montr6 que la dimension homologique de dg/ffx/k(@X) est 6gale ~ dim(X) et on peut d6duire de leur r6sultats que le faisceau d~tffx/k(Gx) est cohdrent, t. Haastert [Ha] a 6tudi6 les op6rations cohomologiques sur les dgx/k-modules par passage ~t la limite h partir de la p-filtration. De m~me dans le cas d'un Zpsch6ma de base S, Berthelot [B3] a d6fini la filtration de ,19 iffxls(@x) par les sous-faisceaux d'op6rateurs diff6rentiels d'6chelon m. En fait la reduction modulo p de la filtration par les 6chelons est la p-filtration d6cal6e d'une unit6. Dans le cas d'ordre fini les extensions d'un 6chelon au suivant n'6tant pas plates on ne peut pas en d6duire que le faisceau dO~ffx/s(@x) est coh6rent. Mais d'apr6s Berthelot [B3] ceci est vrai pour la limite inductive des compl6t6s p-adiques des faisceaux d'op6rateurs d'6chelon fixe tensoris6e par Q. I1 y a tout Iieu de croire que tout se passe bien du point de vue de la dimension homologique. Tout semble indiquer que ron a de point de vue alg6brique une th6orie coh6rente de

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29x/s-modules. Dans le § 2 nous rappelons la th6orie des 29x/k-modules en caract6ristique nulle sans th6or6me de r6solution des singularit6s, ce qui se fait surtout grace & une bonne g6n6ralisation en dimension supErieure du nombre de Fuchs attach6 ~t un point singulier d'une 6quation diff6rentielle. Dans le § 3 nous rappelons la th6orie des 29 x-modules complexes ~ partir du § 2. Enfin clans le § 4 nous d6crivons la th6ofie des 29xt/w®zQ-modules et ce que nous savons de la th6orie des ,19 ~/w®zQ-modules. En particulier nous montrons en dimension un que la cohomologie locale analytique de Dwork-MonskyWashnitzer d'un point ~t valeur clans un 29 txt/w®zQ-module d6finie par une 6quation diff6rentielle est 29 txt/w®zQ-coh6rente si et seulement si cette 6quation admet un indice darts respace des fonctions analytiques dans la boule de rayon un qui est le tube de ce point. On peut donc appliquer les th6or~mes de Robba [Rbl], [Rb2]. Tout se passe comme dans le cas complexe et fait penser que les propri6t6s de finitudes des 29 ~/w®zQ-modules sont li6es ~t leurs propri6tds de finitude de leurs solutions ~t valeur dans les espaces de fonctions analytiques dans des tubes convenables. Dans ce demier § on se place dans le cadre des sch6mas formels falbles de Meridith [Mr] pour 8tre un peu original par rapport ~t Berthelot [B3] qui se place dans le cadre des sch6mas formels. Mais il est clair que les deux points de vue sont nEcessaires. Le premier auteur a commenc6 ~t 6tudier la th6orie des 29x-modules et en particulier la th6orie des 29x-modules en juin 1972. I1 nous a sembl6 d6s 1975 que la th6orie des 29x-modules est le cadre naturelle de la th6orie cristalline. En mai 1983 et durant l'ann6e 1984-85 nous avons eu des discusions avec Grothendieck sur la th6orie g6n6rale qu'il appelle "des coefficients de de Rham". Grothendieck pense d'ailleurs qu'il dolt exister une th6orie des coefficients de de Rham sur Z. En 1985 [N-M 2] nous avons cherch6 ~ montrer le th6or~me de finitude de la cohomologie de Dwork-Monsky Washnitzer sur le modble de la caractfristique nulle &ralde de la th6orie de polyn6me de Bemstein-Sato. A parfir de ce moment lh ayant constat6 que cela est insuffisant nous avons cherch6 h developper la th6orie des ,19 ~t/w®zQ-modules qui devait ~tre ranalogue p-adique de la th6orie complexes des 29x-modules et en particulier montrer que l'image directe par une immersion ouverte du fibr6 trivial est un 29 ~ / w ® z Q module de pr6sentation f'mie. Nous aimerions remercier P. Berthelot, F. Baldassarri et G. Christol des discussions que 1' on a eues ces demi~res ann6es qui nous a permi de s'introduire h la thdorie p-adique. Notations. Si el est un faisceau d'anneaux nous noterons par el-mod la cat6gories des el-modules

d gauche et mod-el la cat6gorie des el-modules d droite. Si A est une cat6gorie ab61ienne nous noterons par D(A) la cat6gorie d6riv6e de la catdgorie A et Db(A) sa sous-cat6gorie des complexes cohomologie bom6e. On note par f* le foncteur image inverse pour un morphisme d'espaces annel6s f et par dim(f) sa dimension relative quand elle a un sens.

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Nous conseillons au lecteur pour life cet article d'avoir ~ la main le cours [Mes] pour tout ce qui concerne ia th6orie algdbro-g6om6trique des 29x-modules. §1. Le f o r m a l i s m e des f a i s c e a u x d ' o p ~ r a t e u r s

diff~rentieis.

1.1. Les op6rations sur les cat6gories de modules sur les faisceaux d'op6rateurs diffdrentiels. 1.1.1. Soit h : X--->S un morphisme d'espaces annel6s et m un entier positif ou nul. Le faisceau • m) 29 tff(x~s(@ x) ([E.G.A. IV], § 16) des op6rateurs diff6rentiels d'ordre m est d6fini par r6currence sur m

comme le sous-faisceau du faisceau des endomorphismes h ' l ® s - l i n S a i r e s de @x en posant 29 tff(°)s(®X) := ®x et un endomorphisme P e s t un op6rateur diff6rentiel d'ordre m si et seulement pour toute section locale a de ®x l'endomorphisme [P,a] := Pa - aP est un op6rateur diff6rentiel d'ordre m 1. On pose

d~iffx/s(~x) := Umd~l/ff~m)((~X) XtS 1.1.2. Si h : X--->S est un morphisme de schdmas notons par X (m) le m-Sme voisinage infinitesimal de X pour le morphisme diagonal Ah : X --> X XsX, par h m le morphisme canonique x(m)---)X XsX et par p~m), p(2m) les deux morphismes compos6s

p~m): x(m)_.~X XSx -")X p(2m) : x(m)--)X XsX "-)X.

Notons par P (xn)s)(®x)le faisceau des parties principales d'ordre m du S-sch6ma X. Par d6finition on a

X/S((~x)(m):=(p~m)), (p(m))*((S)X)' Le faisceau des parties principales d'ordre m est muni d'une structure de ®x-algSbre h gauche et d'une structure de ®x-alg6bre h droite induites par les deux projections p~m), p (2m).On a alors l'isomorphisme ([E.G.A.IV], 16.8.4) . m) homox(~/~(@x),Gx ) ~ 29/ff(~/s((DX)

et le faisceau 29/ffx/s(®x) des op6rateurs diff6rentiels est alors un faisceau •

m)

sous-faisceaux 29/ff(X~S)(®x).

d'anneaux filtr6 par ses

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1.1.3. Supposons que le morphisme h est lisse et soient x I . . . . . x n des sections du faisceau ®x au dessus d'un ouvert U telles que dx 1..... dx n forrnent une ®u-base du faisceau f2x/s. Alors les opErateurs diffErentiels

A q ""--A - q~l

q~ / ql !)"" (2 % ....A qnn := (~x~ x,/%!) forment u n e ® o-base du faisceau

.19/ff(x~s) (®x) pour q = (ql ..... %) tels que [ql -< m ([E.G.A. IV], 16.11.2).

1.1.4. Supposons que le morphisme h est lisse et que te schema S est localement n~ethErien. Notons par o h le complexe dualisant relatif pour le morphisme h. Le thEorEme de dualitE [R.D] pour le morphisme

fini p~m)s'Ecrit : (m

(m T

-

(Pl))*(Pl b'((gX) =

h°m®x(PXlS(®X)'@x)" (m)

Mais on ales isomorphismes de dualit6

(p~m))!((gX) -= (hm)!((pl)!(OX)) -= RhOm®XxsX(®X(m), (Pl)!(CgX)), (pl)!(@X) ~ o)pl[dim(pl)] -= (p2)*(oh)[dim(h)]. D'ofi l'isomorphisme Rhomc9 XxsX(@x(m), (p2)*(C0h))[dim(h)] = ~ i f•f x (m) /s(OX). Soit Extdim(h)® XxsX((gx(m), (P2)*(O)h)) ~ ,19/ff(X~)(OX). Prenons la limite avec m on trouve lim_,Ex¢iim(h)o XxsX(@X(m), (p2)*(O~)) - dD/ffx/S(@x). In

Soit Hxdim(h)((p2)*(Oh)) -= ,I9iffx/S(~X) qui est la definition de Mikio Sato des opErateurs diffErentiels (IS], [S.K.K]), incarnation en thEorie des hyperfoncdons du thEor~me des noyaux de Schwartz. 1.1.5. Notons 29X/S le faisceau des opErateurs diffErentiels dg/ffx/s(@x). Si le morphisme h est lisse la

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donnEe d'une structure de gDx/s-modules d gauche sur un un ®x-module quasi-cohErent ~ 6quivalente ~t la donn6e d'une *-stratification relativement ~t S sur ~ isomorphisme pour tout m

est

([G2], [BI]) c'est ~t dire d'un

satisfaisant i~la condition de cocycle sur les images *-inverses sur les m-rme voisinages infinitEsimaux de la diagonale du produit triple XxsXxsX et compatibles quand m varie. Le faisceau structural ®x est donc un £gx/s-module d gauche.

De m~me si le morphisme h est lisse la donnEe d'une structure de £gx/s-module d droite sur un un ®xmodule quasi-cohErent ~ est 6quivalente ~ la donnre d'une !-stratification relativement ~t S sur ' ~ c'est ~tdire d'un isomorphisme pour tout m p~m~,'~ ~ p(2m~T~, satisfaisant ~tla condition de cocycle sur les images !-inverses sur les m-~me voisinages infinitEsimaux de la diagonale du produit triple XxsXxsX et compatibles quand m varie. Cette description des 29x/smodules tt droite est due ~ Grothendieck et ~ Berthelot. I1 rEsulte du throrEme de dualitE pour les morphismes p ~m)p (2m)que le faisceau dualisant relatif o~nest un dDx/s-module d droite. De m~me par dualit6 il rrsulte que le foncteur 'IlL ---)o~®cgxTlL est une Equivalence de categories entre la catEgorie des dDx/s-modules d gauche quasi-coh6rents et la catrgorie des dOx/s-modules d droite quasi-cohrrents. Le foncteur % ---->home~x(O~a,'lt,) 6tant un quasi-inverse. Dans [Ha] Haastert drcrit, sur un corps de base de caractrristique p, les dgx/s-modules d gauche en terme de limite projective et les £gx/s-modules d droite en terme de limite inductive et en dEduit que ces deux categories sont Equivalentes.

1.1.6. Les dgx/s-modules ~tgauche peuvent se drcrire comme les cristaux de modules, objets du topos critaUin ([G2], [B1] ). De m~me, d'aprrs Grothendieck et Berthe!ot les dgx/s-modules ~tdroite peuvent se drcrire en terme de co-cristaux de modules, objets du topos co-cristallin. Le topos co -cristallin garde un sens quand le S-schEma X est singulier et fournit un substitut intrinsrque ~t la catEgode des dgx/s-modules ~ droite. En caractEristique nulle on dispose de tout ce qu'il faut pour drvelopper une thEorie des coefficients de de Rham-Grothendieck, les co-cristaux justement, pour un morphisme f de varirtrs algrbriques.

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1.1.7. Soient f : X-->Y un morphisme de S-sch6mas lisses et Af : X.--0.XxsY le morphisme graphe de f. Nous supposons la base S localement n~etherienne. Notons par ql et q2 les projections de XxsY sur X et Y respectivement. Posons Jgx_ow s := qpHxdim(Y/S)(@ x ®S0~y/s) d~lS/y,_X := qI,Hxdim(Y/S)(O}X]S@SOy). La structure de ,19x/s-module d gauche de ®x induit une structure de ,iDx;~s-module d gauche sur le faisceau ggx~y/s et la structure de dgy/s-module ct droite sur COy/s induit une structure de f-ldgy/smodule ~ droite sur le faisceau ,19x_oY/S qui est donc un ( ~ x / s , f-l,tgy/s)-bimodule. De m~me la structure de dgx/s-module d droite de O~x/s induit une structure de dDx/s-module d droite sur le faisceau dgs/y,_ X et la structure de dOy/s-module cl gauche sur (gy induit une structure de f-lJgy/smodule d gauche sur le faisceau dDs/y~X qui est donc un (f-129y/s-29x/s)-bimodule. En vertu de 1.1.4 le faisceau d'anneaux dOx_ox/s pour le morphisme identique de X est isomorphe au faisceau d'anneaux dgx/s. Et l'involution naturelle de XxsX induit un isomorphisme de faisceau d'anneaux de 29x__,x/s sur dOs/x~X. 1.1.8. Toujours pour un morphisme f : X ---)Y de S-sch6mas lisses consid6rons le diagramme naturel : X --~XxsY

$

+

Y --)YxsY. On ales isomorphismes, avec un abus de notations 6vident,

Lf*dgy/s~f*dgy/s~L f*R Fy((gy®s¢-oy/s))[dim(Y/S)]~R Fx(Gx®sO)y/s)) [dimC//S)]~ dDx.oy/s. I1 en r6sulte que le faisceau f*dgy/s est un (dOx/s,g-1 d9 y/s)-bimodule. Si 51L est un @y-module muni d'une *-stratification relativement ~tS son image *-inverse par f est muni d'une *-stratification relativement ~t S de fa~on naturelle. C'est donc un dgx/s-module ~t gauche. L'isomorphisrne naturel de (9 x-modules f*~[], ~ f*,lgy/s®f-l~y/sf-lcJ]~ ~ JDx~Y/S@f-ldBy/sf'l~], est en fait un isomorphisme de Z)x/s-modules h gauche. Ce foncteur d'image *-inverse est exact droite et se d6rive ~ gauche pour donner naissance ~ un foncteur exact de cat6gories triangul6es Db(dgy/s-mod) --~ Db(29x/s-mod)

275 L

--* Lf*~llk ~ dgx~y/s @ ~1~ [

~WS

f-l'lll,.

Le faisceau "P~Y/S6rant ®y localement libre tout dgy/s-module plat reste plat en tant que ®y-module. Le foncteur Lf* de la cat6gorie Db(,19y/s-reod) est donc la restriction du foncteur Lf* de la cat6gorie Db(®y-mod) ce qui justifit les notations. 1.1.9. Pour un morphisme f : X --* Y de S-sch6mas lisses le foncteur f entre les catdgories Db(Cgymod) et Db(®x-mod) est d6fini en factorisant f par une immersion ferm6e i : X --~XxsY suivie d'un morphisme lisse p : XxsY-~ Y. Pour un morphisme tisse p le foncteur p! est d6fini comme p*(-)® cop[dim(p)]. Pour une immersion ferm6e i le foncteur i ! est d6fini comme l'adjoint ~t droite du foncteur i,. On pose alors ~ := i ! p!. Si ~

est un (gy-module muni d'une !-stratification relativement ~ S les

objets du coreplexe ~qt, sont reunis d'une !-stratification relativement ~ S. C'est donc un complexe de 29x/s-modules ~t droite que nous allons d6crire. En fait on a l'isomorphisme naturel de complexes de ,19x/s-modules ~ droite pour tout complexe dgy/s-reodules h droite 'J~ : L f!~'I~ = f'l~Nfl,@f-l~y/sdgSZV,_x[dim(X/S) - dim(Y/S)]. On a ainsi d6fmi un foncteur exact de cat6gories triangulges : Db(reod-dgv/s) ~ Db(mod-dgx/s)

Le foncteur naturel de la cat6gorie des modules ~t gauche dans la cat6gorie des modules ~ droite transforme le foncteur *-image inverse en !- image inverse et r6ciproquement. 1.1.10. Pour un morphisree f : X ~ Y de S-sch6mas lisses on d6finit les foncteurs *-images directes (cristaltines) if, Db(,29x/s-mod) -o Db(dDy/s-mod) par 'IlL ---~ff ~

L :~ Rf,29s/y,__x®,Vx/sTfl,, et

ff Db(mod-,19x/s) --~ Db(mod-dgy/s) L 'Ill, ~ if, ')11, := Rf, gll, ®29X/s,~X__~y/s.

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I1 rtsulte de la formule de projection que le foncteur naturel de la catdgorie des modules ~t gauche dans la cattgorie des modules ~ droite et compatible aux foncteurs *-images directes. Pour un morphisme lisse de variE~s lisses le foncteur ~ coincide avec la d6finiton de la connextion de Gauss-Manin de Grothendieck quand la base est de caractEristique nulle. Mais bien entendu ce n'est pas le cas en caractEristique non nulle. Darts ([G2], 3.5, p.330 ) Grothendieck donne un example off la connexion de Gauss-Martin d'un morphisme propre et lisse n'a pas une structure de 29-module. En fait la connexion de Gauss-Manin correspond ~ l'image directe des modules sur le faisceau des optrateurs difftrentiels d'tchelon nul ou de p-filtration un cf. § 1.2 ci-dessous. 1.1.1 1. Si ~

est un complexe de la catEgorie Db(29x/s-mod) on dtfinit son complexe dual 'IlL* de la

catEgorie Db(29x/s-mod) en posant nO,* := hornO x(Ob~/s, Rhom ex/s(~L,29x/s))[dim(X/S)]. De mtme si ~ est un complexe de la cattgorie Db(mod-29X/S) on dtfinit son complexe dual n&* de la cattgorie Db(mod-29x/s) en posant T~* := O~s ®o x Rh°m'~x/s(T& ,d9 x/s)) [dim(X/S)]. 1.1.12. Si f : X --+ Y est un morphisme de S-schtmas lisses on d6finit les !-images directes (cristallines)

if(Db(29x/s-mod) --+ Db(29y/s-mod) par ---~f~ ~1, := (~1~*)* et ffl Db(mod-29x/s ) -+ Db(mod-29Y/S )

~ ~ ~ := (~ ~*)*. Donc pour un morphisme f : X -+ Y de S-schtmas lisses et pour les cat6gories Db(29x/s-mod) et Db(mod-29X/s) on dispose des op6rations f*, ~, f., fl et de la dualit6 (-)*. 1.2. Propri6tts de finitude. En gtntral l'anneau F(X;29X/S) pour un S-schtma affine lisse n'est pas nmth6rien. Mais si tousles hombres premiers sont inversibles sur S sauf un nombre p e t six 1..... x n sont des coordonntes locales

277 m

la F(X;@x)-alg6bre F ( X ; H x / s ) est engendr6 par Aip pour m parcourant tousles entiers. En effet si qi est un entier dont le ddveloppement p-adique s'dcrit qi = a0 + alP 1+ ...+ azPt alors le nombre (pl!)al...(pt!)adqi! est une unitd p-adique u et ron a donc ,~

l

Si on note par F(X;dgX/S,rn) pour tout entier rn la F(X;®x)-alg6bre est engendr6 par APm' pour m' < m on obtient pour m variable la p-filtration de F ( X ; H x / s ) quand S est le spectre d'un corps algdbriquement clos k ([Ch], [Sm], [Ha]) et la filtration de Berthelot [B3] par les dchelons quand S est un Zp-SChdma. Nous allons voir que ces filtrations ne d6pendent pas des coordonn6es et permettent d'dtudier l'anneau F(X;,19 x/s). 1.2.1. Cas off S est le spectre d'un corps k algdbriquement clos de caract6~Sstique p > 0. Pour dtudier le faisceau Hx/k on introduit, ~ cotd de ta filtration naturelle par rordre, la p-filtration Hx&.m (m ~ N) ([Ch], [Sm], [Ha]). Pour tout "niveau " rn on note par F m := X ~ X mle m-dme it6r6 du morphisme de Frobdnius. La sous-algdbre Hx/k. m :=hom®xm+l(@x,@ X) de hOmk(®X,® X) est en fait une sous-algdbre d'opdrateurs diffdrentiels et ron a rdgalit6 ([Ch],[Sm]) k-Ad~X/k,m =d~X/k.

On peut consider les catdgories Db(Hx/s,m-mod),Db(Hx/s-mod),Db(mod-Hx/s, m) et Db(mod-Hx/s). Pour tout m e t tout morphisme X ~ Y on ales faisceaux Hx..~y/$,m et d9 S/Y~_X,m de fa~on dvidente. Pour tout m on les op6rations f~, f~, f*,m, f!,m, (-)* pour les catdgories Db(Hx/s,m-mod),Db(mod Hx/s,m) qui donnent par passage ~tla limite quand m tend vers rinfini les opdrations analogues pour les catdgories Db(Hx/s-mod), Db(mod-Hx/s) ~tcondition de se restreindre aux coefficients quasi-coh6rents en tant que ® x -modules [Ha]. Proposition ( 1 . 2 . 2 ) . - - S i X est k-schdma affine non singuIier la F(X;@x)-atgdbre pk F(X;HxN,m) est engendrd par les op~rateurs A i , k = 0,...,m, i = 1 ..... n, oa les A sont associ~s d un systdme de coordonn~es locales x = (x 1..... Xn) au dessus de X. Preuve. I_es op6rateurs A~k. k = 0,...,m sont dans F(X;Hx/k,m) puisque Ieur ordre est < pm+l ([Sm], pm+l

Th., 2.7). Rdciproquement soit P u n opdrateur de F(X;Hx~,m). On a[P, x i

] = 0 pour i = 1..... n.

Soit d son degr6 nous allons montrer que d

pro+l, dcrivons P = P0 +'''PdAnd pm+l

ob les Pi commutent avec Xn; par hypothdse [P, x n

i

pm+l

I = 0. Donc ZPi[z~r~ x n

d

] = 0. Mais [An,

278

xPm+i]=A~-Pm+l+Xpm+l d pm+l d n An, Donc 0 = PaXn Am + termes de plus bas degr6 en An. Ceci entraine que Pa = 0. Donc si Pest un op6rateur de p-filtration m+l la plus grande puissance de Ai; i = 1...n, qui apparait darts sa d6composition est strictement plus petite que pm+I En vertu du d6veloppement padique d'un entier s i d < pro+l, A~ appartient ~ l'anneau engendr6 par A~k, k = 0 ..... m. La p-filtration est croissante, pour m > m' l'extension ,IgXlkan, ---) ~X/k.m est plate ~t droite et ~ gauche ([Ch], [Sm]) et pour tout m l'anneau F(U;dgX&,m) est n~eth6rien a droite et ~t gauche pour tout ouvert affine U de X. On en deduit en particulier que le faisceau ~gxlk,m d'anneaux est coh6rent pour tout m. L'anneau F(U;Jgx/k) n'est pas n~eth6rien. Cependant de ces remarquables r6sultats on obtient le th6or6me :

Thdordme (1.2.3).-- Pour toute vari~td X non singuIidre sur un corps k alggbriquement clos de carast#ristique p > 0 le faisceau JOXlkest un faisceau d'anneaux coh6rent. Preuve. I1 faut montrer que le noyau de tout morphisme d~x&-lindaire : (dgx/k)q ~ ,19X/k(~ droite ou gauche) est un dgx/k-module de type fini, Mais le faisceau dgx/k 6tant quasi-coh6rent en tant que ®xmodule il suffit de montrer que le noyau du morphisme F(U;(dgX/k)q) --~ F(U;dgx/k) pour tout ouvert affine U est de type fini. Mais pour U affine F(U;dOx/k) est r6union des anneaux F(U;,19X/k,m) pour m e N. I1 en r6sulte que les op6rateurs diff6rentiels dormant le morphisme pr6c6dent sont dans l'anneau F(U;,19 X/k,m) pour m assez grand. Mais l'extention F(U; dgX/k,m) --* F(U; dgx/k) 6tant plate ([Ch],[Sm]) te noyau du morphisme F(U;(dgX/k)q ) --4,F(U;,IS~x/k) provient par extention des scalaires du noyau du m o r p h i s m e F(U;(d~X/k,m)q)----) F(U;dgx/k,m) pour m assez grand. Mais l'anneau F(U;,Igx/k,m) 6tant

n~eth6rien ce noyau est de type fini. On peut donc comme en caract6ristique nulle consid6rer les cat6gories ab61iennes des dgxtk-modules b b coh6rents et tes cat6gories triangul6es Dc(dgx/k-mod ), Dc(mod-,ggxlk) des complexes de 29x/k-modules

~tcohomologie bom6e et coh6rente. Remarquons cependant que le faisceau structural @x et le faisceau dualisant C0xlk ne sont pas ,Igx&-coh6rents. On a alors le th6or~me de Chase-Smith ([Ch],[Sm]): Th#ordme (1.2.3).-- Pour tout ouvert affine U d'une vari4t# X non singulidre sur un corps k alggbrique ctos de caractdristique p > 0 la dimension homologique de l'anneau F(U;,IDx&) est dgale d la dimension de X. On peut alors d6finir la catdgorie des dgx/k-modules holonomes: Ddfinition (1.2.4).-- On dit alors qu'un dgXlk-module coherent ~ faisceaux Extiex/k(]'lL,dgX/k) sont nul pour i ~ dim(X).

est holonome si les

279

I1 n'est pas clair que la categories des dOx/k-modules holonomes soit abElienne bien que cela semble probable. On note par D~(dOXak-mOd), D~(mod-dOx/k) les categories des complexes de dOx/k-modules cohomologie bornEe et holonome. I1 n'est pas clair que les categories D~(dOx/k-mod), D~(mod-dOX/k) soient triangulEes. De mEme pour tout m puisque le faisceau dOX/k,m est coherent de dimension homologique 6gale ~t dim(X) on a l e s categories D~(dOx/k,m-mod), D~(mod-dOx/k,m). On obtient des categories de coefficients dont on a pas encore exorciser Ies propriEtds de finitude si cela est raisonnable. 1.2.5. Cas d'un Zp-schEma de base S . Soit X un S-schEma lisse. Sur te faisceau dOx/s on a la filtration de Berthelot [B3] par les faisceaux des opErateurs diffErentiels "d'Echelon " m pour m e N qui donne la p-filtration par reduction modulo p e n vertu de (1.2.2). Si on fixe un Echelon m on remplace le faisceau des parties, d'ordre 1, I£ (xlls(@X) par le faisceau des parties principales, d'ordre 1 et d'Echelon m, P (~sm(@X). Pour sa definition precise nous renvoyons le lecteur h l'expos6 de Berthelot darts ce m~rne volume. On pose alors dOiff~s,m((~X) := homtDx(~ OX))S,m(~)X),Ox ). C'est le faisceau des opErateurs diffErentiels d'ordre 1 et d'Echelon m, voir aussi (1.23). C'est un faisceau de ®x-algEbres localement libre de type fini. Pour m > m' on un morphisme naturel

qui fait des dOtff(~san (@x) (pour 1 fixE) un systEme inductif essentiellement constant ~t dOtf-~s(®X). On pose alors dO/ffx]s,m ((~X) := uldOiff~s,m (~X)" Le faisceau

doiffxlS.m (@X) est

un faisceau de ®x-algEbres et le faisceau ® x est doiffxts.m (®x)-

module ~ gauche. La limite inductive des do/ffX/S,m(@x) (me N) est alors canoniquement isomorphe

dOiffx/s(®x).

En coordonnEes locales rimage de dOiffx/s,m(@x) dans dOiffx/s(®x) est engendrEe en

tant que ®x-algEbre par les opErateurs APk, k = 0 ..... m et les morphismes de transition ne sont autre que les inclusions canoniques. Le faisceau est dOiffx/s,m(®x) coherent pour tout m. La reduction modulo p de la filtration par les Echelons de dO/fix/s(@ x) n'est autre que la p-filtration.

280 Remarque (1.2.6).--- L'extension 29 iffx/s,m,(®X)-->29/ffx/s m(@X) pour m > m' n'est pas plate bien que sa reduction modulo p est plate!. Proposition (1.2.7).-- On peut aussi remarquer que l'anneau des opErateurs d'Echelon m est engendr6 par les opErateurs d'ordre au plus d'ordre prn aussi bien en caracterisfique pure p qu'en in6gale caractErisfique p. Ceci donne une desciption intrins6que simple de ta filtration par les Echelons. Proposition (1.2.8).--Si S est le spectre d'un anneau de valuation discrdte W complet d'indgale caractdristique p pour tout ouvert affine U de X et pour tout m le complEt6 p-adique de l'anneau r(u;291ffx/w~(@x) ) est de dimension homologique ~gale d dim(X/W) + 1. A

Pmuve. Notons 29iffx/W,m((~-)X)le complEtE p-adique de 29iffx/w,rn(@x). La reduction modulo p de ranneau 1-'(U;29tf-fx/w.m(@x))est l'anneau des opErateurs diffdrentiels de p-filtration m+l sur un corps de caractiristique p qui est de dimension homologique Egale ~ dim(X/W) ([Ch],[Sm]). Un vertu de ([Rt], thm. 9.33) on a l'in6galitE dh(r(u;29iffx/w,m(@x))) > dim(X/W)+l. D'autre part le graduE associ6 ~ la filtration p-adique de F(U;29/ffxAv~(@x)) est de dimension homologique dim(X/W)+l car isomorphe ~ l'algEbre des polyn6mes ~ une variable sur un anneau de dimension homologique dim(X/W). Donc dh(F(U;~tffx/w,m(®X))) < dim(X/W)+l en vertu de ([N-V], VII-1 1, p. 315). De m6me la dimension homologique de F(U 29iffx/w m(@x)) est moins Egale ~t dim(X/W)+l. 11 est probable qu'elle est Egale ~ dim(X/W)+l.Si on pose avec Berthelot [B3] 29x/w . - ~

29/ffx/w,m(@x) •

m

On trouve que la dimension homologique de F ( U ; 2 9 ~ @ x ) ) est au pIus dim(X/W)+2 car limite inductive d'anneaux de dimension homologique dim(X/X)+l en vertu de [Bet]. D'autre part sa dimension homologique est au moins dim(XjW)+l car sa reduction modulo p e s t de dimension homologique dim(X/W)+l. I1 est probable que dh(F(U;29~w(®x)) = dim(X/W)+1. § 2. Le th6or~me des coefficients de de Rham-Grothendieck en caract~ristique z~ro.

On suppose dans ce § que le schema de base est un corps k de caract6ristique zero que nous supposerons, pour simplifier, alg6briquement clos. Le faisceau des opErateurs 29x/k:= 29/ffx/k(@x) diffErentiels sur un variEtE algEbrique X non singuli~re sur k est alors un faisceaux de @x-atgEbres de type fini tel que le faisceau graduE gr(29x/k) associE ~t la filtration 29 (xm/~):= 29 if~)~ par l'ordre des opErateurs diffErentiels est un faisceau de ®x-alg6bres commutatives de type fini. La variEtE au dessus de X associEe ~t la ®x-algEbre gr(29x/k) est le fibre cotangent T*X de X. L'anneau des sections

globales au-dessus de tout ouvert affine de coordonnEes de X du faisceau 29x/k est n~ethErien. On en dEduit que le faisceau quasi-cohErent 29x/k est en fait un faisceau coherent d'anneaux.

281

2.1. I ~ th6or%me de finitude. 2.1,1. Soit U un ouvert affine de X. On a alors le th6orSme ([Ro], [Bj], [Ch]) : Thdordme (2.1.2).-- La dimension homologique de l'anneau F(U;Igx/k) est ~gale clla dimension de X.

Tout 29X/k-mOdule coh6mnt ~

admet une fillration 'In,(m) (m e N) par des @x-modules coh6mnts qui

est bonne c'est ~t dire dont le gradu6 associ6 g r ( ' ~ ) est un gr(dgX/k)-modtfie cohdrent. La vari6t6 r6duite Ch('Jll,) du fibr6 T*X associ6 h gr('Ill) ne d6pend pas de la bonne filtration choisie. Par ddfinition c'est la varidt6 caractdrisfique de ']m,. En faite la multiplicit6 de la vari6t6 caract6ristique Ch(~],) en chacun de ses points ne d6pend pas de la bonne filtration choisie. En particulier le cycle CCh(]A) associ6 ~t gr(~[IL) ne d6pend que de ~ . C'est par ck!finition le cycle caract6ristique de 7rl,. Voir par exemple ([Mz] ou ([Mes, 1.2.2, 2•4))• On a alors rin6galit6 dite de Bernstein cf. [Be] : Thdor~me ( 2 . 1 . 3 ) . - - L a dimension de la varidtd caractdristique CCh(71L ) d ' u n £gX/k-mOdule cohdrent non nul 7P~est au moins ~gale d dim(X).

Pour une d6monstration g6omdtrique du th6or~me (2•1•3) voir ([M 2] ou ([Mes], I•2•3))• En fait la th6orie des bons anneaux filtr6s voir par exemple ([M 2] ou ([Me], 1.4•1•3, 4,2.14)) montre que le th6or~me atg6brique (2.1.2) est 6quivalent au th4or~me g6om6trique (2.1.3). Si on pose pour un dgx/k-module coherent non nul Til grade(Tfl,) := inf{i; EXt~xlk(~t, i ,d9 X/k) ~ 0}. On a g r a d e ( ' ~ ) = codirnT.x(Ch('ITk)) cf. ([Mes], 1.4.2.14). Cela am6ne h poser la d6finition : Dgfinition ( 2 . 1 . 4 ) . - - On dit qu'un ,6)Xlk-mOdule cohdrent 71% est holonome s'iI est nul ou si la dimension de sa varidtg caractdristique est dgaIe d dim(X)• De fagon dquivalente s'il est nul ou si les •

i

fa~sceaux Ext ~xf(?l~,~gx/k) sont nuls pour i ¢ dim(X).

On note par Mh(dgx/k- ) la cat6gorie des dgx/k-modules ~t gauche holonome qui est alors une souscat6gorie pleine de la catfgorie dgx~-modules coh6rents. Darts une suite exacte courte de 29x~-modules coh6rents le terme m6dian est holonome si et seulement si les termes extremes sont holonomes. En particulier la cat6gorie des dgx~-modules holonomes est ab61ienne. On note D~(,tgx~-mod) la cat6gofie des complexes ,Igx/k-modules ~ cohomologie bom6e et holonome. C'est alors une sous-cat6gorie pleine et triangul6e de la cat6gorie D~(,19x&-mod).

282

Si ~

est un ,IDx/k-module ( ~t gauche) holonome le dgx~-module (~ droite) Extn~ _(Tfl,,dgX/k) est x/K encore holonome. Autrement dit le foncteur de dualit6 'JTk --+ 'rfl,* est une anti-6quivalence de catfgorie de la catEgorie des dDx&-mc~ules (~t gauche ou ~ droite) holonomes dans eIle m6me. 2.1.5. Le point clef des propri6t6s de finitude en caract6ristique nulle pour les dDx/t-modules holonomes est l'6quation fonctionnelle dite de Bemstein-Sato. Motiv6 par le probl6me qui ~ priori n'a rien i~ avoir avec la cohomologie des vari6t6s alg6briques, pos6 par I. Gelfand au congr6s d'Amsterdam (1954), du prolongement analytique de la distribution s --~ ps off P est un polyn6me rdel I.N.Bernstein a d~montr6 le th6or~me suivant. Soit P u n polyn6me ~ n variable ~t coeficients dans k. Notons Dkn/k l'algEbre de Weyl c'est "a dire F(kn;agkn/k) et Dkn/k[S]P s le Dkn/k[S] := Dkn&®kk[s] module engengr6 par le symbol p s II faut voir Dkn/k[s]P s comme sous-module de A[s,P-1]P s off A := k[xt,...Xn] et off l'action de Dkn/k[S] sur A[s,P-1]P s est celle qu'on pense. Thdordme (2.1.6) [ B e ] . - - Soit P un polyname non nuI de A il existe alors un plyname non nul fl une variable B(s) d coefficient clans k et un op~rateur diff~rentiel de Q(s) de Dkn/k[S] tels que l'on air l'gquation fonctionnelle B(s)P s = Q(s)PW. Corollaire (2.1.7).-- Si P e s t un polyn6me le Dkn/k-module A[P -1] est de type fini. En effet pour tout entier m on a un morphisme de spdcialisation de A[s,P-1]P s dans A[m,P-1]P m : (I:~m : A[s,p-x]P s -+ A[m,P-I]p TM. De l'6quation fonctionnelle on d6duit que le Dkn/k-module A[P 1] est engendr6 par p-m pour m assez grand. On peut localiser l'6quation fonctionnelle. Soient U un ouvert affine de X off l'on dispose de coordonnEes locales, A := F(U,®x), Du/k := F(U,dgx/k) et M l'espace des section globales sur U d'un ~gx/k-module holonome. Pour tout 616ment non nul u de M e t toute fonction r6guliEre P sur U on considEre le Du/k[s]-module Dua~[s]P~u. On alors le rEsultat suivant : Thdordme (2.1.8).-- Soit P une fonction non nul de A it existe alors un polyngme B(s) non nul d u n e variable d coefficient dans k et un op~rateur diff~rentiet de Q(s) de DuN[s] tels que t'on ait l'~quation fonctionnelle B(s)PSu = Q(s)PPSu. Le th6or6me (2.1,8) se r6duit h (2.1.6) si U = k n, M = A et u = 1. La d6monstration de (2.1.8) est aujourd'hui 616mentaire. A partir de (2.1.2) et de la th6orie des bons anneaux filtrds elle repose sur le fait que A est une k-alg6bre de typefini et sur le fait qu'en g6om6trie alg6brique tous les faisceaux coh6rents en dehors d'une sous-vari6t6 se prolonge en faisceaux coh6rents cf, ([Mes], 1.4.2).

283

Corotlaire

(2.1.9),-- Si ~fi, u est un 29 u/k-module holonome sur U compldmentaire d'une

hypersurface Z de X l'image directe de effk u par l'inclusion canonique de U dans X est un 29X/k" module holonome.

En effet la question est locale sur X. On peut supposer que X est affine et que Z dEfini par une Equauon P. Si i dEsigne l'inclusion cananique de U dans X, i,'lll, U est un 29Xfk-mOdules quasi-cohErent. Pour montrer qu'il est 29X/k-cohErent il suffit de montrer que le Dx/k-module de ses sections globales est de type fini. Mals 'Ill, U admet un plongement holonome cf. ([Mes], 1.4.1.8) ~ X. L'Equation fonctionnelle (2.1.8) appliquEe aux sections globales d'un tel prolongement et ~ ses gEnErateurs montre que F(X,i, Tfl,u) est un Dx/k-mOdule de type fini. Une fois acquis le fair que i,'}'t],U est 29X/k-cohErent un argument de spEcialisation montre qu'il est holonome cf. ([Me5], 1.8.2, p. 101). On peut dEduire facilement de (2.1.9) que les categories sont stable par images inverses cf. ([Mes], I. 8.7). Si f : X ---->Y est morphisme de vari&ds algEbriques sur k avec les notations du § 1 : Corollaire (2.1.10).-- Si ~

est un coefficient de D~(29y/k-mod) a/ors f*']~ et f!Tfl, sont des

coefficients de D~(29x/k-mod).

Si ~ est un complexe de 29X/k-modules posons DR(TfL) := Rhomzgx&(@x,~). En caract&isque z3ro on on l'isomorphisme (notations du § 1) : L DR('I1L)[dim(X] ~ '1~ ®29X/k29speek<.__X On dEduit de (2.1.8) le thEor~me de fmitude : Th~ordme ( 2 . 1 . 1 1 ) . - - S i ~ est un 29X/k-module holonome Ia cohomologie de RF(X;DR(Tfl,)) sont des espaces vectoriels sur k de dimension f'mie.

Preuve. La suite spectrale de Cech-de Rham perrnet de supposer que X est affine. En Prenant un plongement de X dans un espace numErique, on est r~duit ~t supposer que X = k n. En vertu de (2.1.8) l'image de 'Ill, par l'inclusion de k n dans l'espace projectif pn de dimension n e s t un 29 pn-module coherent. On est rEduit ~t montrer que le complexe RF(Pn;DR('II~)) est h cohomologie de dimension finie sur k pour un 29X/k-module coh6rent 'JTI,. Mais tout 29pn-module coherent est quotient d'un module de la forme 29~®Oen:y pour un @pn-module coherent ~ . Mais RF(Pn; DR(29pn@O vnff')) est isomorphisme h RF(P~; o~x~®opn~r)[-n]. On est rEduit au thEor~me de finitude de Ia cohomologie d'un

284

faisceau algEbrique coherent sur un espace projectif. Mais puisque le dOX/k-module @x est holonome en caractErisque zero on u'ouve que la cohomologie de de Rham d'une variEtE alg6brique non singuli~re sur un corps de caractEristique nulle est de dimension finie. Plus gEnEralement si f : X ~ Y est un morphisme de variEtfs algEbriques non singuliEres sur k on a, avec les notations du § 1, le thEor~me : Th~or~rne ( 2 , 1 . 1 2 ) . - - Si ~

est un coefficient de D~(dOx/k-rnod) alors f~'IIt, et ffl 711, sont des

coefficients de D~(dOy/k-mod).

Preuve. On factorise f e n une immersion fermEe suivie d'une projection. Le cas d'une immersion fermEe ne pose pas problbme. Si f est la projection XxY ~ Y la suite spectrale Cech-de Rham relatif nous rEduit ~t supposer que X est affine. Un plongement dans un espace numErique rEduit ~ supposer que X est l'espace affine k n. Le corollaire (2.1.9) rEduit ~ supposer que X est l'espace projectif pn. Mais tout dor,n×y-module coherent est quotient d'un module de la f o r m e d~pn×y®®pnxyI~' pour un @gt, y-module coherent ~ . La doy/k-cohErence rEsulte alors de la formule de projection et du thEor~me d'images directes par un morphisme projectif des faisceaux algEbriques cohErents. Une fois acquis la coherence le thEorEme de dualitE relative pour un morphime projectif qui dit que le foncteur image directe commute ~ la duatit6 ([Mes], I. 5.3.13) montre que les faisceaux Ext sur ,19Y/k et contre dOY/k des faisceaux de cohomologie des images directes sont concentrEs en degr6 dim(Y) ([Mes], I. 5.4.1, p. 76). Ils sont donc holonomes. En particulier le thEorEme (2.1.12) 6tablit l'holonomie des modules de Gauss-Martin [G2] qui sont les faisceaux de cohomologie du complexe ~ @x et cela pour un morphisme f arbitraire de variEtEs algEbriques non singuli~res sur un corps de caractEristique nulle. L

Si 3'I1,1 et ~ 2 sont des complexes de doX/k-modules (& gauche ) Ie complexe ~ 1 ®Ox 'Ill' 2 est un complexe de dox/k-modules (~t gauche) et il en rEsulte facilement de (2.2.4) que le complexe est ~t cohomologie holonome si ~ 1 et 'm, 2 le sont. Autrement dit pour un morphisme f de variEtds algEbriques non singuliEres sur k les categories L

D~(doX/k-mod) sont stables par les cinq operations f*, f., if, f~ et ®Ox parmi les six operations de Grothendieck. Les categories D~(dox/k-mod ) n'Etant pas stable par l'opEration interne Rhom®x('gl, I, ¢1~,2) qu'il faut remplacer par l'adjoint ~ gauche du produit tensoriel qui existe ([Mes], I1.9.1, p. 185). On dispose d'une thEorie des cycles 6vanescents, la thEorie de la V-filtration, pour les dox/k-modules qui est ranalogue de la thEorie des cycles 6vanescents de SGA 7, cf. par exemple [S-M].

285

2.1.13. Si on part d'un morphisme f de variEtEs analytiques complexes ou p-adiques rigides on a encore L les categories D~(£gx/k-mod) qui sont stables par les operations f*, f!, ~ et ®®x " Pour rimage directe il faut supposer le morphisme propre et imposer une condition d'existence de filtrations globales. Mais les demonstrations sont algEbriques [M-N2]. 2.2. Le thEorEme de comparaison. 2.2.1. Soit X une variEtE algEbrique complexe non singuli~re et X a~ la variEtE transcendante associEe ~t X. On a un morphisme GAGA : (*) RF(X; DR(®x)) --~ RF(Xan; DR(®xan)). Thdordme (2.2.2) (Grothendieck).-- Le morphisme (*) est un isomorphisme.

Comme te complexe DR(@xan) est une resolution du faisceau constant Cxan en vertu du lemme de Poincar6 la cohomologie de de Rham algEbrique d'une variEtE algEbrique complexe non singuliEre est isomorphe ~t sa cohomologie de Betti. Si X est propre le thEor~me (2.2.2) est une cons&tuence du thEorEme GAGA de Serre. Mais si X est affine sa demonstration n6cessite la thEorie des Equations diffErentielles complexes [i points singuliers r~guliers. Plus gEnEralement soit ~ un fibre sur X muni d'une connexion intEgrable. On a encore un morphisme GAGA : (**) RF(X; DR(~)) --->RF(Xan; DR(~aa)). Thdor~me (2.2.3) (Deligne).-- Si le fibrd image inverse de ~ sur toute courbe non singulidre audessus de X n'a que des singularitds r#gulidres d l'infini le morphisme (**) est un isomorphisme.

La demonstration du thEor~me de comparaison de Grothendieck-Deligne ([G1], [D1]) reposait de fa9on essentielle sur le thEorEme de resolution des singularitEs de Hironaka [H]. En ce sens elle restait largement inaccessible vu la difficultE de la demonstration du thEorEme de Hironaka. Nous aUons expliquer que le thEor~me de Hironaka n'est pas indispensable ici. 2.2.4. Dans les thEorEmes de comparaison prEcEdents, comme beaucoup de thEorEmes de gEomEtrie algEbrique, il s'agit de dEmontrer que les fibres h connexion en question ont une ramification moddrde rinfinie. En dimension un le thEorEme de Fuchs affirme qu'un point singulier d'une Equation diffErentielle est rEgulier si et seulement si le nombre de Fuchs de cette Equation attache ~tce point est nul. Le calcul de ce nombre n'offre pas de difficultE. En dimension supErieur il faut faire la m~me chose. Etant donne une variEtE algEbrique non singuliEre X sur un corps k de caractEristique nulle et un fibre ~ connexion ~ u sur le complEmentaire U d'une hypersurface Z dventuellement singuli~re de X nous allons attacher au triplet (X, Z, ~u) une stratication u Z i de Z et pour chaque strate Z i un entier mi

286

positif ou nul qui mesure parfaitement la ramification de ~ u le long de Z. Si X est une courbe ces nombres m i ne sont au autre que les nombres de Fuchs du fibr6 ~ u aux points singuliers Z. Puis il faut donner un crit6re facile ~ v6rifier assurant que le cycle positif ]~miZi est nul. 2.2.5. La d6finiton de ce cycle a pour origine la d6monstration de Grothendieck [G1] de son th6or~me de comparaison. En effet c'est dans cette d6monstration qu'on trouve la manisfestation la plus tangible d'un objet d'une cat~gorie d#riv#e discr6te du type D~(C x) qui est ~t la base de la notion defaisceau

pervers. Rappelons que si X est varidt6 algEbrique complexe un faisceau transcendant 9 r d'espaces vectoriels complexe de dimension finie sur Xan est dit alg6briquement constructible s'il existe une stratification alg6brique telle la restiction de ~ ~ chaque strate est une syst6me local. On note alors D~(C x) la cat6gorie des coefficients complexes constructibles c'est h dire des complexes ~ cohomologie bom6e et constructible. On dit qu'un coefficient constructible ~ a la propriEt6 de support si ses falsceaux de cohomologie hi(~ r) sont concentr6s en degr6s [0, dim(X)] et si la dimension du support du faisceau h i ( ~ ) est au plus 6gale ~t dim(X) - i pour tout i dans [0, dim(X)]. On dit qu'un coefficient constructible a la propri6t6 de co-support si le complexe dual R h o m c x a n ( ~ , Cxan) ( X 6tant non singulier) a la propri6t6 de support. On dit qu'un coefficient constructible est unfaisceau pervers s'il a la propriEt6 de support et de co-support. On note Perv(Cx) la cat6gorie des faisceaux pervers. C'est alors une souscatEgorie pleine de D~(Cx) et ab#lienne cf. [B-B-D]. Un coefficient constructible ~ sur une sousvari6t6 Z de X est un faisceau pervers sur Z si ~ [-codimx(Z)] vu comme coefficient constructible sur X est un faisceau pervers sur X. On note Perv(Cz) la catEgorie des faisceaux pervers sur Z. 2.2.6. Soit ~

un ,19x/c-module holonome sur une vari6t6 alg6brique complexe non singuli6re. Alors

son complexe de de Rham transcendant DR('III, an) := R h o m ~xan/c(@Xan,'lll, an) est un coefficient constructible ayant la propri6t6 de support [K1], voir aussi [N-M1]. D'autre part il r6sulte du th6or6me de dualit6 locale qu'il la propri6t6 de co-support [Me3]. C'est donc un faisceau pervers sur X. 2.2.7. Si Z e s t une sous-vari6t6 ferm6e de X notons par i : Z---) X +- U :j les inclusions canoniques et par ian : zan--~ X an e - U an : jan les inclusions transcendantes correspondantes. Rappelons que le foncteur i an! de D~(C X) dans D~(Cz)

287

est adjoint ~t gauche du foncteur ian, de D~(C Z) dans D~(Cx). Ici ian!(9¢) est simplement i-lRFzan(~'). Soit un 29xtc-modute holonome 'IlL posons I R z ( ~ ) := ian!(DR((j,j-l~lL)an))[1]. En vertu de (2.1.9) j , j - i T ~ est un $gx/c-module holonome et donc IRz('I]L) est un coefficient constructible sur Z. On a ainsi d6fini un foncteur cohomologique de Mh(29x/c-) darts D~(Cz). I1 transforme suite exacte de ,Ox/c-modules en triangle distingu6 de D~(Cz). Pour rassurer le lecteur le complexe IRz(~qL) n'est rien d'autre que le c6ne du morphisme naturel DR((j,j -1TIL)an) ---)Rjan,j-lanDR((']'lL)an et sous cette forme qu'il appara3t dans la d6monstrafion de Grothendieck [G1]. On a alors le th6or~me suivant [Me6] : Thdordme

( 2 . 2 . 8 ) . - - Si Z e s t

une hypersurface et 'IlL est un ,19 x / c . m o d u l e

holonome

le

coefficient IRz('IIL) est un faisceau pervers sur Z.

Comme un triangle disfingu6 form6 par des faisceaux est en faite une suite exacte de la cat6gorie des faisceaux pervers il en r6sulte que le foncteur IR z : Mh(dgx/c-) ---)Perv(Cz) est un foncteur exact de cat6gories ab61iennes. Ddfinition(2.2.9).-- Si Z e s t une hypersurface et ~

est un dgx/c-module holonome on appelle

faisceau d'irr6gularit6 de TfL le long de Z le faisceau IRz(TfL). Par exemple si X est une courbe et Z un point le faisceau IRz(Tfl,) est un espace vectoriel complexe de dimension finie. Sa dimension est 6gale au hombre de Fuchs en Z de Tfl,. Cela r6sulte du th6or6me de comparaison de Malgrange [M1]. I1 faut maitenant avoir un crit6re pour que le faisceau IRz(']]L) soit nul. On dit qu'un module holonome ~t support une sous-vari6t6 lisse Y est lisse si sa vari6t6 caract6ristique contient au plus que le fibr6 conormal T~,X. Soient Y une sous-vari6t6 d'une vari6t6 alg6brique complexe non singuli~re X, Z une hypersurface de X dont la trace sur Y contient le lieu singulier de Y e t de codimension un dans Y e t ~ un ,IDx/c-module holonome ~t support contenu dans Y e t lisse en dehors de Z. On a alors le th6or~me [MET] :

288

Thdordme (2.2.10).-- Sous les conditions pr~cgdentes le faisceau IRz('II), ) est nul si et seulement si Ia codimension clans Z ~ Y de son support est au moins 6gale ~t un. Par exemple si on on prend Y = X et ~

le fibr6 ®x ~ v i a l muni de la connexion naturelle on trouve que

le faisceau I R z ( ® x ) est nul pour toute hypersurface Z puisque de fa~on naturelle son support est contenu dans le lieu singulier de Z qui est bien de codimension un darts Z. A partir de 1~ on touve par un agument combinatoire que le faiscceau IRz(H~(®x) ) est nul pour toute sous-vari6t6 Y de X, pour toute hypersurface Z et tout p. Si on applique cela ~ X = l'espace projectif complexe pro, y radhErance d'une vari6t6 algEbrique affme non singuli~re plong6e dans l'espace numErique C ra et Z le diviseur ~tl'infini on trouve que le faisceau IRz(H~(®x) ) est nul. Mais l'hypercohomologie de ce faisceau est pr6cis6ment robstruction ~ l'isomorphisme de la cohomologie de de Rham de la vari6t6 affine en question avec sa cohomologie de Betti [GI], Autrement dit le th6or6me (2.2.10) implique le th6or~me de comparaison de Grothendieck [G1]. Plus gEnEralement soit ~ un fibre ~ connexion intEgrable sur une varlet6 algEbrique complexe non singuli6re muni d'un plongement dans l'espace num6rique C m. Notons Y l'adhErance de cette vari6t6 dans l'espace projectif pm et ~ l'image directe au sens des ,19-modules de ~ par l'immersion dans pm est alors un 29 pro-module holonome ~ support Y e t lisse en dehors du diviseur ~t l'infini Z. Supposons que l'image inverse de ~ sur toute courbe non singuli6re n'a que des singularitEs r6guli6res ~t l'infini et Y est normale (pour simplifier car ceci n'est pas une restiction). Alors le faisceau IRz(H~(®x)) est ~ codimension Y ~ Z dans au moins un comme on le volt en faisant passer une courbe gEn6rale en un point assez g6n6ral de Y. Le th6or6me (2.2.10) implique que le faisceau IRz(~ ) est nul. Mais I'hypercohomologie de ce faisceau est l'obstruction ~t l'isomorphisme entre cohomologie de de Rham de ~ et cohomologie de de Rham du fibr6 transcendant associ6. Autrement le thEor~me (2.2.10) implique le thEor6me de comparaison de Deligne [D1]. 2.2.1 1. Plus g6n6ralement soit ~

un 29x-module holonome sur une vari6t6 alg6brique complexe non

singuli6re X. On a alors le th6or~me suivant [MET] :

ThAordme (2.2.12).-- L'image inverse de ~t, sur route courbe non singulidre au dessus de X n'a que des singularitds rdgulidres ( y compris d l'infini ) si et seulement si son faisceau d'irr~gularitd le long de tout diviseur ( y compris ceux de l'infini ) de tout ouvert affine est nul. Le th6or~me (2.2.12) arn6ne ~ poser la d6finition :

D~finition ( 2 . 2 . 1 3 ) . - - Un 29 x-module holonome ~ conditions dquivalentes (2.2.12).

est dit rdgulier s'it satisfait au deux

289

Le th6or~me (2.2.10) donne un crit6re qui ne ddpendpas de la r6solution des singularit~s pour v6rifier si un 19x-module est r6gulier ou pas. On note par Mhr(29x- ) la cat6gorie des 29x-modules holonomes r6guliers. Dans une suite exacte de 29 x-modules holonomes le terme m6dian est r6gulier si et seulement si les termes extremes sont r6guliers. Cela r6sulte du fait que le foncteur IR z entre cat6gories ab61iennes est exact pour toute hypersurface Z. La cat6gorie Mhr(29x- ) est donc ab61ienne, stable par extension et sous-quotient. On note par D~r(29X/C-mod ) la sous-cat6gorie de D~(29x/c-mod) des complexes ~ cohomologie r6guliSre. La cat6gorie D~r(29x/c-mod) est alors triangul6e. On d6duit du thfor~me (2.2.10) que les cat6gories D~r(29X/c-mod) pour X variables sont stabiles par les op6radons cohomologiques de Grothendieck. Thdordme (2.2.14).-- Si f : X --~ Y est un morphisme de varidtds algdbriques complexe non singuIidres les foncteurs f*, f!, envoient Ia catdgorie D~r(29y/c-mod) dans la catdgorie Dhr(29X/Cb

rood) et pour tout comptexe ]]L de la catdgorie D~r(29 y/c-mod) on les isomorphismes entre coefficients constructibles :

fmtDR(,tlL an) = DR((f*"ffL)an) fandDR('IlL an) ~ DR((~"JIL)an). Preuve. [Me7]. Thdordme (2.2,15).-- Si f : X --> Y est un morphisme de varidtds algdbriques complexe non singulidres les foncteurs if, f~, envoient la catdgorie D~r(29x/¢-mod) dans la catdgorie D~r(29y/c-

mod) et pour tout complexe ~

de la catdgorie D~r(29x/c-mod) on a l e s isomorphismes entre

coefficients constructibles

DR((ff ~'IL)an) _=Rfan,DR(TfL an) DR((~ 'IlLan)) =_Rf~n!DR('IILan). Preuve. [MET]. On d6duit [Me7] des th6or6rnes (2.2.14) et (2.2.15) le th6or~me des coefficients de de RhamGrothendieck ~ savoir que tes cat6gories D~r(29 x/c-mod) sont stables par les six op6rations de Grothendieck et que le foncteur de de Rham transcendant qui est pleinement fid~le respecte les six opfrations analogues dans les cat6gories D~(Cx) de coefficients alg6briquement constructibles. En

290 particulier on obtient une demonstration du thEor~me de la rggularit~ de la connexion de Gauss-Martin en caractEristique nuUe indEpendante du thEorEme de resolution des singularitEs ([Kal], [Kaz])et ce pour un morphisme arbitraire de variEtEs algEbriques. Ceci remplit parfaitement le programme de Grothendieck en caractdristique nulle ([G2], p. 312) " It would be convenient to have a more general duality formalisme of t)23e f!, f , as developed in [R.D], for the De Rham cohomology ". On constate, une fois de plus, que pour avoir une bonne thEorie cohomologique des variEtEs algEbriques il faut d'abord dEgager une bonne thEorie des coefficients qui sont stables par les operations cohomologiques. 2.2.16. La catEgorie D~r(29X/C-mOd ) est purement algEbrique, en effet la condition pour un dgx/cmodule holonome d'avoir que des singularitEs rEguliEres sur toute courbe au dessus de X est algEbrique. Donc elle garde un sens sur un corps k de base de caractEristique nulle. On dEduit alors que les catEgries D~r(,tgx/k-mod ) sont stables par les six operations de Grothendieck. Mais les demonstrations se ramEnent par le principe de Leschetz au cas transcendant. Cependant le faisceau transcendant IRz(~l~ ) qui est la base des demonstrations a un substitut purement algEbrique. Ceci amEne ~tpenser qu'on peut dEmontrer certainement le rEsultat prEcEdent de fa~on purement algEbrique. Le faisceau IRz(71I,), comme tout faisceau pervers, admet un cycle caractEristique CCh(IRz(Tfl,)) qui est un cycle lagrangien positif du fibre cotangent T*X cf. [L-M]. Par definition le faisceau IRz(]~I,) s'incEre dans une suite exacte de faisceaux pervers sur X : 0 ---4IRz(~flL)[-1] --~ DR((j,j-I'IIL)an) --~ Rjan,j-I~nDR('tlL~a) --~ O. Le cycle C C h ( I R z ( ' ) ~ )[-1]) appara]t comme la difference C C h ( D R ( ( j . j - I ' I I ~ ) a n ) ) _ CCh(Rj~.j-lanDR((7tl,)an). Mais chaque membre de cette difference est purement algEbrique. En effet le cycle CCh0)R((j,j-1711,)an)) est le cycle caractErisfique du dgx/c-module holonome j.j-l'~/1, et le cycle CCh(Rjan.j-lanDR('Jl"k an)) est l'image directe au sens des cycles du cycle caractEristique du dgu/c-module holonome j-1]'l~ [Me6]. Nous allons dEcrire le cycle CCh(IRz('Irk)[-1 ]) darts le cas important oh ~'k est un fibre vectoriel de rang r au-dessus de U. A c e moment 1~ le cycle CCh(Rjan.j-lanDR(]'ll,an)) est Egal ~ rCChf,j,j-l@x) et se dEcrit ~ partir de Fhypersurface Z [L-M]. On a donc CCh(IRz(~)

= C C h ( j , j - l ~ ) - rCCh(j,j-lOx).

Sous cette forme le cycle CCh(IRz(]'D,) garde un sens sur un corps de base de camctEristique nulle. Ce cycle &ant positif on obtient une statification de Z = •Z i et pour chaque strate un entier m i positifou nul et le cycle ZmiZi positif promis en 2.2.4 attach6 au triplet (X, Z, ]TI,) qui gEnEralise le nombre de

Fuchs. Le thEor~me (2.2.10) assure alors que le cycle ZmiZ i est nul 0 ...... si les nombres attaches aux strates de codimension nulle sont nuls. Le calcul de ces derniers est gEnEral accessible. En somme on

291

est darts la m~me situation en dimension supErieure que celle du thEor~me de Fuchs en dimension un. § 3. Le th~or~me des coefficients holonomes complexes d ' o r d e infini. Une fonction g de (Qp[x][1/x])? est la somme d'une sdrie ~am/X m+t (am E ~ a , m ~ N) et d'une fonction de ( ~ [ x ] ) t avec la condition de Dwork-Monsky-Washnitzer limm ~ . ] am[ E"m = 0 pour un reel positif e assez petit convenable ([Dw], [M-W]). Mais 1/xm+l = (-1)mAm(1/x) et si on pose P(A) := ~am(-l)mAm on trouve que Y~arn/Xm+l = P(A)(I/x). Autrement dit ( ~ [ x ] [ 1 / x ] ) t est un module de type fmi et m~me de presentation f'mie sur un anneau d'opErateurs diff6rentiels d'ordre infini. La situation est pareille ~ la situation transcendante comptexe qu'on a longuement 6tudiEe depuis maintenant presque vingt ans. En effet si g est une fonction holomorphe sur le plan complexe qui a au plus une singularit6 essenfieUe ~t l'origine c'est alors une somme d'une s6rie Y.aJx m+l (am e C , m N) et d'une foncfion enti~re avec la condition pour tout e positif il existe une constante M~ telle que [ am[ < MeE~rn.On a par le m~me c~dcul a~lgEbrique que ranneau des fonctions holomorphes sur le plan complexe ayant au plus une singul;u'it6 essentielle en z6ro est un module de prEsenfion finie sur ranneau des op6rateurs diff~rentiels d'ordre inf'mi. Ceci sugg6re que si X est une vari6t6 affine non singuli~re sur Fp et Z une hypersurface de X le faisceau jr,® U t [Mr], o~ U := X-Z, est l'an:flogue du faisceau jan,®oaa de la situation transcendante complexe assocife ~ une situation alg6brique complexe. On dispose sur C d'un th6or~me finitude pour ces faisce~Lux et plus g6nEralement d'tme th~orie des coefficients holonomes complexe d'ordre infmi. 3.1. Le faisceau des opfrateurs difI:'6rentiels complexes d'ordre inrmi. Pour une vari6t6 analytique complexe X on note 29x le faisceau 29iffxlc(®x). 3.1.1. Soit (X, ®x) une vari6t6 analytique complexe. Le faisceau des fonctions holomorphes est un faiscean de FrEehet-NuclEaire : pour tout ouvert U de X respace vectoriel complexe F(U;® U) est un espace vectoriel topologique de Fr6chet-NuclEaire et les morphismes de restictions sont continus. e~

D~finition (3.1.2).-- Le pr~faisceau ,des op~rateurs diff~rentiels d'ordre infini 29 x est ddfinie comme sous ~pr~faisceau dufaisceau homcx(®X,® x) des Cx-endomorphismes continus.

oo

Lemme (3.1.3).-- Le prdfaisceau 29 x est un faisceau. En effet un endomorphisme localement continu est continu. Proposition ( 3 . 1 A ) . - - Si U est un ouvert de X oa sont d~finies des coordonn~es locales x :=

292 (x 1..... x n) un endomorphisme au-dessus de U est opdrateur diffdrentiel d'ordre infini si et seulement c'est une somme infinie Z aa(x)A = oa aa(x) (0re N ~) est une suite de fonctions holornorphes au dessus de U telle que lira4 t4 ~ J aa(x~ 1• o4 uniform6ment sur tout compact de U. Preuve. I1 r6sulte des in6galitds de Cauchy qu'une somme infini Z a=(x)A~t qui la propri6t6 de (3.1.4) est un op6rateur diffdrentiel d'ordre infini. R6ciproquement soit P un op6rateur d'ordre infini. On d6fmit la suite de fonctions holomorphes a~(x) en posant

ac~(x) := ~l~
Y.v.~o~(-1)k( ~)x w lCp(xk) = Z~< a (-1)k(~)(x-y) w ke((x-y)k). De cela si pour tout e assez petit et tout compact K de U on trouve que

1a=(x)l K -< MaO ~4 oi~ M Eest une constante qui ne ddpend que K et de e. Ceci est la condition requise pour que la somme infini Zaac~(x)Aa soit un op6rateur diffdrenfiel d'ordre infini. Par construction P e t Z~ac~(x)Act coincident sur les polyn6mes. Par localisation et densit6 P = Zc~a~(x)AC~.

En particulier on trouve que le faisceau 29x est le sous faisceau de 29 xdes opdrateurs diff6rentiels d'ordre localement fini.

Par construction le faisceau d9 xeSt un faisceau d'anneaux et te faisceau @x est un ,/9 x-module gauche. De m6me partant du faisceau des n-formes diff6rentielles coX qui est aussi un faisceau de FrechetNucldaire muni de sa structure de 29x-module &droite et considOrant le faisceau des endomorphismes Cx-lindaires continus on trouve que ce faisceau est canoniquement isomorphe ~t 29xet que la structure de ,19x- module &droite de Ox se prolonge en une structure de 29x-module d droite. On peut aussi remarquer que ron a Ext n ~(Ox,d~ ~ = ~X X--

"

293

3.1.5. Soient f : X---~Y un morphisme de vari6t6s analytiques complexes lisses e t ~,f : X--)XxY le morphisme graphe de f. Notons par q~ et q2 les projections de XxY sur X et Y respecfivement. Posons

29 XoY := ql*Hxdim(Y)(q2mY) 19 y(-x := ql*Hxdlm(Y)(q~O)X)"

La structure de 19x-module d gauche de ® x induit une structure de 19 x-module d gauche sur le faisceau 19 x-~Y et la structure de 19 ~-module d droite sur my induit une structure de f-129 ~-module d

droite sur le faisceau 29 x--,Y qui est donc un (,ID~¢ f-t J9 ~-bimodule. De mSme la structure de ,19xmodule d droite de o)x induit une structure de 19x-module d droite sur le faisceau 19 y(--x et la structure de 19 ~-module d gauche sur ® y induit une structure de f-1 19 ~,modute d gauche sur le faisceau ,IDy~x qui est donc un (f-119 ~ 2) ~-bimodule. En particulier si fest le morphisme identique on trouve que le faisceau pl,Hxdim(X)(p~a)X) est un (,D~ pl,p2q19 ~-bimodule et donc un ( ~

dO ~-bimodule.

Proposition ( 3 . 1 . 6 ) . - - Le (19 ~ ,19 ~x-bimodule pl,Hxdim(X)(p~C0X) est canoniquement isomorphe au faisceau d'anneaux ,IDx" Preuve. Si K(x,y)dy est une classe de cohomologie de pl,HXaim(X)(p~0lX) et g(x) une foncfion holomorphe la formule de Cauchy montre que 2nfSi'~I x-yl=e K(x,y)g(y)dy, pour un cycle I x-yl =e convenable est une fonction holomo12~he de la forme P(g)(x) = Z ac~(x)Aag(x) qui ne d6pend pas du cycle I x-yl

=e.

On a alors un homorphisme de pl,Hxdim(X)(p~0)X ) d a n s 19 x dont on vdrifie par un

calcul directe que c'est un isomorphisme de (dg~e ,19 x~-bimodules.

3.1.7. On ne connait aucune propri6t6 de finitude du faisceau ,19~¢ En particulier on ignore s'il est coh6rent ou pas. Mais on a l e r6sultat suivant ([S.K.K], 3.4.2):

Thdordme (3.1.8).-- L'extension 19 x---) 19 xeSt fid6lement plate d droite et d gauche. La ddmonstration de [S.K.K.] est micro-diff6rentielle. Comme nous l'avons dit dans l'introduction de [Mes] nous connaissons une d6monstration diff6rentielle. 3.2. Le th6orSme de bidualit6.

294

Soient X une vari6t6 analytique complexe et ¢J~ un fibr6 vectoriel sur X de rang fini & connexion intdgrable. Alors son syst6me local des sections horizontales h o m ~ x ( ® x , ' l & ) le d6terminent compldtement par risomorphisme mill, _=_hom Vx(®X,~%)®Cx® x. I1 est plus commode de considdrer son syst~me des solutions holomorphes hom~x(Tfl, ,®x) et l'isomorphisme pr~cgdent devient un devient un isomorphisme de d~x-modules ~t gauche : ¢j~ = homcx(hom~x(CIll,,@x),@x). Mais si ~

est un ,IDx-module holonome son faisceau (pervers) de de Rham DR(']1I,) :=

llhom~x(@X,'}~ ) ou son faisceau des solutions holomorphes S('Ill, ) := Rhom~x('ll~ ,@x) ne le determinent pas en g~n6ral ~t cause de la ramification non moddrde. Cependant ils d6terminent le ~ x module J B x ® ex'l& obtenu par extension des scalaires. Si ~ ~ est un complexe born6 de ,t9 xmodules (h gauche) on a un morphisme canonique

Bd("fll, ~) : " J ~ _~ Rhomcx(Rhom~x']'lL~,®x),(Dx)

de complexes de ,~x-modules. Ce morphisme s'explicite en prenant une r6solution ,IDx-injective ~ de ®x qui reste par platitude @x-injective et le morphisme Bd(Tfl~~) est simplement le morphisme naturel de bidualit6 : Bd(~], ~) : ~m,~ .-~ homcx(hom~ xffJ'fl,~,~),~ ). La terminologie peut prater h confusion avec les th6or~mes de bidualit6 (interne) pour les coefficients coh6rents ou discrets constructibles. On a alors le th6orSme ([Me3], thm. 2.1) : Thgordme

( 3 . 2 . 1 ) . - - Si ~ ~est un complexe parfait de ,19 ;(modules tel que le complexe

R h o m ~ ~ l l , ~ , @ x ) est constructible le morphisme Bd(~], ~) est un isomorphisme. Rappelons qu'on appelle parfait un complexe de modules admettant localement une r6solution finie par des modules libres de type fini [S.G.A. 6]. Soit Z une hypersurface de X etj rinclusion de U:= X-Z dans X.

295

Corollaire (3.2.2).-- Le morphisme 29 x ® ~)x®x(*Z) --) j.j4® x est un isomorphisrne. Par dualit6 ([Me3], Thm. 1.1) le thtortme de Grothendieck [G1] est 6quivalent ~t l'isomorphisme S(®x(*Z)) ~ j,.j-ICx et (3.2.2) est constquence de (3.2.1) parce que Rhomcx@J-lCx,@ X) = Rj.j-I@x =__j.j- 16) x- Le corollaire (3.2.2) montre en particulier que j , j l @x est un 29 x-module de prtsentation finie ce qui est l'analogue transcendant complexe de (2.1.9) pour le fibr6 trivial. Cependant sa dtmonstration, bien que maintenant indtpendante de la rtsolution des singularitts, ntcessite la connaissance des solutions S(®x(*Z) ). Nous ne connaissons pas une dtmonstration algtbrique directe qui puisse nous ~tre utile darts le cas p-adique sauf dans le cas des singularitts trts simples (croisements normaux, point quadratique ordinaire) cf. 4.4. La dtmonstration de (3.2.1) utitise d'une faqon essentiel les proprittts de constructibilit6 des solutions holomorphes. La cat6gorie des complexes 71~ ~parfaits tels que R h o m s g ~ ' l l k ~ , ® x ) est constructible est triangulte.Une sous-cattgorie, ~t priori plus petite, de complexes qui ont les propritt6s de (3.2.1) est formte des complexes localement de la forme 29 x ® ~ x 'll~ pour un complexe ~ de D~(29x-mod). I1 n'est pas du tout clair que cette dernitre soit triangul~e. Pour 61ucider cette difficult6 il faut un th6or~me de finitude g6ntral 3.3. Le thtor~me de finitude. 3.3.1. Soient ,'3~ un coefficient constructible sur une varidt6 analytique complexe. Alors Ie complexe R h o m c x ( ~ , ® x) est muni d'une structure de 29 x-complexe ~t gauche induite par la structure de 29 xmodule de ®x- On alors le thtor~me [Me4] :

Th~ordme (3.3.2).-- Le complexe R h o m c x ( ~ ,®x) est globalement de la forme 29x®t~x711' pour un complexe 'l~ de Ia cat~gorie D~r(29x-rood). La signification de D~r(29x-mod) est clair. C'est la cattgorie des complexes bornts de 29x-modules cohomologie holonome rtguli~re. Un 29x-module holonome Tfl, est rtgulier si son faisceau IRz(]'fL) est nul pour toute hypersurface Z. Soitj : U ---)X le compltmentaire d'une hypersurface Z de X et 'II~u un fibr6 ~tconnexion inttgrable sur oo

U. La structure de 29u-module ~tgauche de 'IlLUse prolonge en une structure de 29u -module ~ gauche

296

Un cas particulier de (3.3.2) et le Corollaire ( 3 . 3 . 3 ) . - - Le 29 x-module j , ' I ~ u est de la forme 29 X ® ~ x ~ r(*Z) pour un 29xmodule holonome T~ r tel que Ie faisceau IRz('l~r) est nul.

En particulier le 2919x-module j,TfL u est de pr6sentation finie. La d6monstration est hautement transcendante. Bien que ind6pendante aujourd'hui de la r6solution des singularit6s elle n6cessite le th6or~me d'existence de Riemann : trouver un 29x-module holonome T~ r tel que IRz(T& r) est nul et qui a m6me monodromie que ~ u- Le prolongement analytique est au fond de la question dans le th6or6me de firtitude (3.3.2) et semble indiquer que pour la d6monstration d'un r6sultat analogue darts le cas p-adique le Frob6nius est au fond du probl6me. Le th6or6me (3.3.2) utilise toutes les resources de la th6ories des 29 x-modules et en est l'6tape ultime. Mais il a de nombreuses cons6quences et en particulier clarifie la structure de la cat6gorie des complexes parfaits de 29x-modules dont les solutions sont constructibles. D6finition (3.3,4).-- On appelle 2919x-complexe holonome un 2919x-complexe 7& ~ tel qu'il existe

localement sur X un 29x-comptexe holonome Tfl, et un isomorphisme

29 x ® Z x ~ -= ~®" Notons Mh(29 x-mod) la cat6gorie des 29 x-modules holonomes et D~(29 x-mod) ia sous-cat6gorie de la cat6gorie Db(29 x-mod) des complexes holonomes. Les cat6gories Mh(29 ~-mod) et D~(29 X-rood) sont d6finies en terme purement alg6briques ~tpartir du faisceau des op6rateurs diff6rentiels d'ordre infini. Cependant il n'est pas du tout clair que la cat6gorie Mh(29 x-mod) est ab6tienne et stable par extention ni •

b

e~

que la cat6gone Dh(29X-mOd) soit triangul6e. Le th6or~me de finitude (3.3.2) va dire que tel est bien le cas.

Notons T l e foncteur d'extention des scalaires D~r(29X-mOd) "-~ D~(29x-mod); 'l~ --~ T(TI],):=29~® exT&; F l e foncteur fournit par le thEor~me de constructibilit6 D~(29 x-rood) -~ D~(Cx); ~1l,~ ---)F ( ~ ) := Rhom,~ ~T&-,@ X)

297

et G le foncteur fournit par le thEor~me (3.3.2) D~(Cx) -~ D~(29x-mod); IT ~ G(IT):= Rhomcx(IT,@x). Thdor~me (3.3.5)[Me4].-- Les foncteurs F et G sont des Equivalences de categories triangulEes quasi-inverses l'un de l'autre et le foncteur T e s t une 6quivalence de categories.

Comme l'extension 29x---~ 29 xeSt fid~lement plate le foncteur T induit une Equivalence de categories

Mhr(29x-mod) ~ Mh(29x-mod); ¢J~ ~ T ( ~ ) := 29x ®.VxCJ~l,.

En particulier la catEgorie Mh(29 x-mod) est abglienne et stable par extension et la catEgorie D~(29 Xmod) est triangulde. De plus l'inclusion de la catEgorie des 29 X-complexe holonomes darts la catEgorie des ,19x-complexes parfaits dont les solutions sont constructibles est une Equivalence de categories. Ceci rEsulte des thEor~me (3.2.1) et (3.3.2). Enfin l'inclusion de la catEgorie des 29 x-complexes holonomes dans la catEgorie des complexes de 29 X-modules ~ cohomologie bomEe et holonomes est une Equivalence de categories par les mSmes thEor~mes ce qui justifie la notation D~(29 x-mod) pour la categories des complexes holonomes. On a ainsi clarifi6 la structure de la catEgorie des 29 x-Complexes holonomes comme consequence ultime de la thEorie des 29x-modules. 3.3.6. Si X est variEtE algEbrique complexe on dEfinit le faisceau 29 xcomme l'image directe du faisceau des opErateurs diffErentiels d'ordre infini de la variEt6 transcendente associEe par le morphisme canonique X an --~ X. On dEfinit un ,19x-complexe holonome comme un 29 x-complexe localement pour la toplogie de Zariski isomorphe ~ un 29 x-complexe obtenu par extent.ion des scalaires ~t partir Cun 29x-complexe holonome. Alors la catEgorie D~(29 x-rood) des complexes holonomes est Equivalente ~tla catEgorie des coefficients algEbriquement constructibles D~(Cx). L'espoir est qu'un analogue de la categone Dh(29x-mod) garde un sens en caractEristique p > 0 et fournit une catEgorie de coefficients p•

•

b

ela

adiques. Aussi nous allons rappeler ([Mes], II. 9.5.7) le formalisme des six operations de Grothendieck pour les categories D~(29 x-mod). Si f : X --+ Y est un morphime de variEtEs algEbriques complexes non singuli~res on pose

298

L

L

~ :--- Rf.,~W~X®~x~)'rL. pour un complexe % de la catdgode Db(,19~,-mod) et pour un complexe ~ de la catdgorie Db(2) xmod). On obtient ainsi les foncteurs image inverse et image directe entre catdgories de complexes de modules d'ordre infini. Si '111,~ est un complexe de la catEgorie D~(d9 x-mod) il est localement de la forme ,/9 x ®,~x ~1' pour un complexe ~ de la catEgorie D~(29x-mod). Le complexe ZIx ®agXTll,* est un objet de la catEgorie D~(d9 x-mod) donne localement. En vertu du thEor~me (3.3.2) et du thdor~me de dualitd locale c'est en fait un objet globale de la catEgorie D~(~x-mod ) qui est le dual 'Ill,~*. On a donc le foncteur de dualitE de la catEgorie D~(29x-mod) qui est un anti-Equivalence de categories. On pose alors pour un complexe ~1, de la categoric D~(dg~-mod) et pour un complexe ~ de la catdgorie D~(d9 x-rood) f!% := (f*(~*))*

f f ~ := (~(~1,*))*. On alors les foncteurs image inverse et image directe extraordinaires entre catdgofies des complexes de d9 x-modules. Le produit tensoriel interne ® de la catdgorie D~(29 x-mod) est aussi ddfini mais c'est le compl~td du produit tensoriel sur ®x ( cf. [Mes], II, § 9, p. 191). On a alors le thEorEme: Thdordme (3.3.7).-- Soit f : X--+Y un morphisme de varidtds alg~briques complexe non singulidres alors les catdgories D~(dgx-mod) sont stables par les operations cohomologiques ~ ,f~,f*,f!, ® et par

la dualitd. Le foncteur contravariant Rhom,~ ~ ~ % ® x) et le foncteur covariant Rhom,~ ~ @x, ~ ~) qui s'~change par dualit~ sont des dquivalences de categories triangulEes entre la catdgorie des coefficients holonomes d'ordre infini D~(d9 x-mod) et la catdgorie des coefficients constructibles complexes D~(Cx). Le foncteur contravariant Rhom,~ ~'l'g %@x) admet comme quasi-inverse le foncteur Rhomcx(~',®x) et le foncteur covariant R h o m b i ® x, ~ ~) admet comme quasi-inverse le foncteur Rhomcx(~V,@x). De plus le foncteur Rhom,~ ~711, %®x) transforme f* en f-1 mais

299

tran,rforme ft. en f!. Duatement le foncteur Rhom~ ~ ® x, ~**) transforme f* en f! et transforme ft. en f , .

La d6monstrafion de (3.3.7) repose de faqon essentielle sur le th6or~me (3.3.2) et donc sur le prolongement analytique complexe, Dans le cas p-adique il s'agit de trouver des substituts au prolongement analytique!.

§ 4. Sur les coefficients p-adiques d ' o r d r e infini. Soit W un anneau de valuation discrete complet de corps r6siduel k de caract6risfique p > 0 et de corps de fractions K de caract6ristique nulle. Soit X * := (X,® *x) un sch6ma forrnel faible sur W [Mr]. Rappelons que l'espace topologique X sous-jacent est une vari6t6 alg6brique sur k munie de la topologie de Zariski et le faisceau structural @txeSt un faisceau de W-alg6bres. 4.1.On peut consid6mr comme en 1.1 le faisceau des op6rateurs diff6rentiels :

Proposition (4.1.1).-- La cat~gorie des W-schdmas formels faibles admet des produits fibrds finis. Preuve voir [N-M2]. A partir du produit XtxwXt on peut construire comme dans ([E.G.A], § 16) les (b t et l'on a risomorphisme parties principales (s6par6es), d'ordre 1, ~o~v(@ ~) homcgx(~O~v(® ix), ®tx) ~ 29iff(1) ~JX/~(@t~ X ~" D'oh ron d6duit comme dans ([E.G.A.IV]. § 16) que dgiffx/w(® tx) est un faisceau d'anneaux filtr6 de ® tx-alg6bres. Supposons que la vari6t6 X est non singuliSre et que @xt est W-plat. Notons alors " ao _ x~/w (1) et 29xt/w pour d9/ff(~v(O ix)et Z/ffx/w(O ix). Alors pour tout 1 les ® ix-modules ~ ~ () O , ( 1 )t

--(1)d~/X*/W

sont localement libre de type fini. Si (x 1..... Xn) sont des coordonn6es au dessus d'un ouvert aft'me U de X c'est dire telles que dx 1..... dx n forment une base du module des diff6rentielles s6par6es alors le F(U,® tx)-module F(U,29 O{/w) est engendr6 par les op6rateurs ~tpuissances divis6es Aq pour longueur de q < 1 cf.[N-h~2]. Comme on est clans le cas d'un Zp-module de base W, F(U,d9 ~/w) est filtr6 par les

300

opErateurs diffErentiels d'ordre < 1 et d'Echelon m pour m variable cf. § 1. En vertu du developpement p-adique d'un entier ranneau des opErateurs diffErentiels d'Echelon m est engendrE par les opErateurs d'ordre au plus pm On note ,~ Xt/Wm le faisceau des opErateurs diffErentiels d'ordre au plus I e t ~ (1)

d'Echelon au plus m e t '~xt/w,m := u l " ° xt/Wm le faisceau des opErateurs diffErentiels d'Echelon m. Pour m variable les faisceaux forment un syst~me inductif et l'on a risomorphisme

lim ,(gxt/w.ra - 29xt/w. m (1)

Pour tout m le gradu6 gr(,~)Xt/W,m ) associ6 ~t la filtration u l , ~ xt/Wm est un faisceau de (9 tx-algEbres n~ethEriennes commutatives. On a dEduit comme en caractEristique nulle que Ie faisceau d'anneaux dgxt/wcn est coherent et naethErien. P r o p o s i t i o n (4.1.2).-- Pour tout m e t tout ouvert affine U de X 1"anneau F(U;dgxt/w.m) est de dimension homologique dgale au moins d dim(X)+l.

Preuve. C'est la m6me preuve que (1.2.7). Comme rextension 29xt/W,m. ~ dDXt/W,mn'est pas plate pour m' < m on ne peut pas dEduire que le faisceau ,19xt/w est coherent, Mais tensoris6 avec Q ce faisceau ~ toutes les propriEtEs du faisceau des opErateurs diffErentiels sur une variEtE analytique complexe. Le faisceau d'anneaux ,19xt/w®zQ est coherent et de dimension homologique dim(X). On dEfinit naturellement la catEgorie des dgxt/w®zQmodules

hotonomes

comme

la

cat6gorie

des

modules

cohErents

~

tel que

Ext~oxt~W®zQ('II~,,~gxt/w@zQ) = 0 pour i ~ dim(X). La catEgorie Mh(dgxt/w®zQ-) des modules holonomes est abElienne et stable par extension et les cat6gories D~(,lgxt/w@zQ-mod) sont stables par les six operations de Grothendieck et par la dualitE exactement comme dans le cas analytique complexe. Ceci rEsulte du r6sultat purement alg6brique suivant [N-M2]. Si A est une K-algEbre rEguliEre n~ethErienne 6quicodimensionnelle sur un corps K de caractEristique nulle telle qu'il existe x a..... x n 6tEment de A et 01..... On des K-dErivations de A tels que 0i(xj) = 8ij ofa dim(A) = n posons DA/K := A[O 1..... On]. Si les corps rEsiduels des points fermEs de A sont Kalg~briques l'anneau DA/K est de dimension homologique 6gale ~ dim(A) [Bj]. Sous ces conditions un

i DA/K-module M de type fmi tel que ExtDnm(M, DA/K) = 0 pour i ~ dim(A) est dit holonome. Thdordme (4.1.3).-- Sous tes conditions pr~cgdentes soient P u n dldment non nul de A, M u n DA/K-module holonome et u un ~ldrnent non nul de M. AIors il existe une ~quation fonctionnelle non

301

nulle

B(s)Psu = Q(s)PPSu et le D ~ - m o d u l e M[P q] est encore holonome. Le prob16me dans (4.1.3) est que l'algdbre A n'est pas algdbriquement de type fini. Appliquant (4.1.3) ~t une algtbre A t de DMW (Dwork-Monsky-Washnitzer) on trouve que At[p-1]®z Q est un DAt/Kmodule de type fini. Seulement ralg6bre At[P 1] n'est pas faiblement compldte et est donc distincte de (At[p-I])t qui est l'algdbre de coordonndes (dag) du compldmentaire de l'hypersurface ddfinie par la reduction modulo p de P. Le cas complexe cf.§ 3 et les exemples dvidents suggbrent que ce compldt6 est un module de prdsentation fini sur le compldt6 faible en un certain sens de l'anneau DA/w une fois tensorisd avec Q bien stir. Mais c'est l~t un probltme qui semble rel6ver de la thdorie des dquadons diffdrentielles p-adiques et non un probl~me purement algdbrique. Par le m~me thdortme on a une bonne thdorie des 29x/w®zQ-modules pour un W-schdma formel toplogiquement de type fini X. 4.2. Soit un schdma faiblement formel (X,@ tx) sur W, un anneau de valuation discrbte d'indgale caractdristique p, d'iddal maximal m, de corps des fractions K et de corps rtsiduel k. D~finition (4.2.1).-- On appelle faisceau des opdrateurs diff~rentiels d'ordre infini sur le schema faiblement formel (X,®tx) et on note 29 ~ / w le sous-faisceau de

homw(e*x, ® ix)des

endomorphismes P tel pour tout r>l la r~duction modulo m r de Pest un opdrateur difftrentiel sur le schdma (X, (9 ~/mr(gtx) d'ordre infdrieur ou dgale d ~.(r+l) pour une constante r~elle ~ > 0 inddpendante de r.

Darts la notation 29 ~t/w l'indice t sur 29 rappelle qu'on a compldtd faiblement en quelque sorte et l'indice t sur X le distingue du faisceau dOtxrw considdr6 par Berthelot [B3] et construit de m~me ~tpartir d'un W-schtma formel. Le faisceau 29 ~/w est l'analogue p-~Ldique du faisceau des opdrateurs d'ordre infini dans le cas complexe. I1 est tout ~t fait naturel d'examiner dans quelle mesure les rtsultats complexes que ron a ddcrit darts le § 3 ont des analogues dans le cas p-adique. C'est la piste qui nous semble la plus strieuse pour dtmontrer le th6or~me de finitude de la cohomologie de Dwork-Monsky-Washnitzer d'une varitt6 affine non singuli~re sur un corps fini.

302

Cependant il y a des diff6rences importantes. L'extension 29xt/w ~ 29 t / w mSme tensoris6 par ®zQ n'est pas fid61ement plate. Ceci am6ne ~ penser que l'anneau d'une bonne th6orie des coefficients darts le cas p-adique est le faisceau 29 t/w®zQ. Par construction le faisceau (9 test un 29 t/w-module ~ gauche. De m~me en coordonn~es locales le faisceau o)tx/west un 29 t/w-module ~ droite. Mais cette structure ne d6pend pas des coordorm6es parce que l'on a l'isomorphisme mix/w® ®xt/w29 t / w = COx/wtet la structure de 29x,/w-module ~ droite de mtx/w ne d6pend pas des coordonn6es. Soient f : X--~Y un morphisme de sch6mas formels sur W non singuliers et Af : X--+XxwY le morphisme graphe de f. Notons par ql et c12 les projections de XxwY sur X et Y respectivement. Posons

29t_~yt/w:= q,.Hx~
Proposition (4.2.2).-- Le (29 t / w , 29 t / w )'bim°dule Pl*Hxaim(X)(p2t'°xt ) est canoniquement isomorphe au f aisceau d"anneaux 29 ~ /wDarts un travail ult6rieur nous montrerons que le faisceau 29~/w est filtr6 par les compl6t6s faibles des faisceaux 29xt/w~a comme dans la situation de Berthelot. La m6thode de Berthelot [B3] montre alors que le faisceau d'anneaux ,{9 ~/w®zQ est coh6rent. De plus nous montrerons que la dimension

303

homologique de ,[9~/w®zQ est dgale ~tdim(X), Ceci permet de ddfinirles ,/9~Av®zQ-modules holo.omes comme les Z en degrd dim(X).

®zO-modules coh ents telsque leur

Z

®st co.cen

4.3.Pour f : X--+Y un morphisme de schdmas formels sttr W non singuliers, un complexe de ,I9 ~/w" t modules ~tgauche 'It~t et '}l,t tm complexe de a9 yt/w-modules ~tgauche posons : L

f*~l,t:= ,gg~_+yt/w®FI ~tytAvf'l%t L X?/w xq/W On obtient les foncteurs images inverse et directes pour les ~ ~/w-modules ~t gauche. En particulier si Y ®st un point la cohomologie de Dwork-Monsky-Washnitzer d'un ,19~/w-module h gauche~ll, test le complexe de W-modules L

RF(X;d~. ~-xt/w ® ~ $ / w ~j~'*) qui tensorisd avec Q est isomorphe ~t RF(X; D R ( ~ ¢))®zQ[dim(X)] := RF(X; Rhome *xt/w(®Xt,?ll,t))®zQ[dim(X)].

Le point clef sera de ddgager la cat6gorie des ag~t/w-modules stables par les foncteurs images directe et inverse. Comme en caractdristique nuile ceci se rarndne ~tdtablir la stabilit6 par immersion ouverte. Pour rinstant on ne peut clue se contenter de quelques exemple dvidents et d'une conjecture. 4.4.Supposons que le sch6ma formel faible est l'espace affine Spff(W[X] t) de dimension n. Alors ranneau des sections globales Dntdu faisceau Jg~ttw®zQ sont les sdries Z~,~aa,l~XaAI3tel que la sdrie Y.c~,f~aa,l~xt~yI~ soit dans ralgdbre W[x, y]t. Une question naturel qui se pose est de savoir si ranneau Din®zQ est naethdrien. L'anneau Dnt confient l'anneau W[A]t des opdrateurs ~tcoefficients constants et rextenfion W[A] t --+ Din est fid6lement plate. Donc sl W[A]'®ZQ n'dtait pas naethdrien cela entraine t que Dn®zQ n'est pas n~ethdrien. Seulement : Proposition (4.4.1).-- L'anneau W[A]t®zQ est naeth~rien.

304

Preuve. Ceci est une consEquense du falt que l'anneau W [ k ] t ® z Q admet un algorithme de division de Weierstrass-Hironaka. Fixons un ordre total sur N n on a alors ta notion d'exposant pour tout opErateur ~tcoefficients constants dEfini comme l'exposant de son polyn6me initiale pour la valuation m-adique.

Proposition (4.4.2).-- Soient P1 ..... Pr des opdrateurs de W [ A ] t ® z Q alors pour tout opdrateur A il existe des uniques opdrateurs Q1 .... Qr, et R de W [ A ] t ® z Q dont tes exposants satisfassent aux proprigt~s usueIles de la division tels que A = QIP 1 +...+ QrPr + R . La preuve se falt en considErant l'extension W de W tel que s i t est une uniformisante de W e t ~ une uniformisante de W alors t = ~p-1. Un opErateur p(A) est dans W[A]t®zQ si et seulement si la sErie P(r~ex) appartient ~t W[x]t®zQ ofa e est l'indice de ramification absolu de W. En fair le morphime de W [ A ] t ® z Q dans W [ x ] t ® z Q qui tt P(A) associe P(Ttex) est injectif et fait de la K-algEbre DMW une extension libre engendr6 par 1..... r&--2 de la K-alg6bre D M W ?~puissances divisdes. Dans la g.-algEbre K[x] t on a un algorithme de division tel que les quotients et le reste proviennent de W[A]t®zQ si le dividant et les diviseurs proviennent de W[A]t®zQ.

On pourait songer ~tfabriquer un algofithme de division dans ranneau Dnt ® z Q • C'est possible qu'on se restreint ~ l'Echelon zero c'est dire a l'algEbre de Weyl D M W mais non clans le cas gEnEral. En effet considErons ~ une variable le diviseur P = 0 - px. Si on choisi 0 comme exposant de P pour faire le division et on si divise l'opErateur ]~m2"m02m qui est darts W[A] t on trouve que le coefficient constant dans le reste, qui une sErie en x, ne converge pas en caractEristique 2. Si on choisit x cornme exposant et si on divise la sErie Zm(-1)m2mxZm on Irouve que le coefficient constant du reste, qui est une sErie en 2, ne converge pas en caractEristique 2. On peut fabriquer des contres exemples en caractEristique diffErente de 2. Voir en 4.6 rEpilogue pour ce point.

4.5. Une question clef est de montrer que le Dtn®zQ-module (W[xl[p'I])t®zQ est de prEsention fini pour tout polynEme P de W[x].

4.5.1. Si P e s t le monSme xl., ,x r (r _
305 ( W [ x ] [ p ' l ] ) t ® z Q . C'es~ une s6rie Za,~aa,~xttP-~ (cce Nn,13e N, aa, l)~ K ) a v e c la condition DMW v(aa, l~) > ~,([3+ct) asymptotiquement pour une constante positive ~ non nulle. On peut supposer que les aa, l~sont dans W. Pour 1~>2, P'[~= (b~ +...+ b~)13"lP-t/41~'l(13-1)l(I]-r/2)t.En caract6ristique diff6rente de 2 rop6rateur Za,l~act,t~x'X(b~ +...+ ~)I~-1/4t~-1(~-1)!(~-r/2)1 formel est en fait un op6rateur diff6rentiel d'ordre infini. En effet un calcul directe montre qu'il est de la forme Zct,~aa,~xaAV(3t e N r) avec la condition DMW pour la s6rie Zct,,taa,~xC~y"~. En caract6ristique 2 on a x~+...+ x r2 = (x I +...+ xr) z mod(2) et risomorphisme (W[x][P-~])* = (W[x][(xl +...+ xr)'l])*. Darts tousles cas ceci montre que le D~®zQ-module (W[x][Pi])t®zQ est de type fini pour la forme quadratique. Cependant comme dans le cas complexe [Me2] cette m6thode n'a gu~re de chance de montrer les propridt6s de finitude esp6r6es darts le cas g6n6ral. 4.5.3. Nous allons formuler une conjecture dans le cas p-adique calqufe du cas complexe. Soit X un sch6ma affine non singulier formel faible sur W d'alg6bre A t et ~

un 29xt/w-module tel que ~t, ®zQ

soit holonome qu'on peut prendre alg6brique. Si on choisit une section cr : K/Z ---)K de la projection K --) ~ / Z alors 'll~®zQ admet par rapport ~ttoute fonction P de A t u n polyn6me de Berstein-Sato [N-M2] qui permet de d6finir une V-filtration canonique (cf. par exemple IS-M]) ind6x6e par rimage de o dont le gradu6 est localement ind6x6 par un ncanbre fmi d'616ments de K que nous appelerons les exposants de ]q], le long de V(P)®zQ. Faisons rhypoth6se que les exposants de ' ~ le long de V(P)®zQ sont des nombres alg6briques en particulier ne sont pas des nombres de Liouville, ceci ne d6pend pas du choix de c. Alors nous pensons que jr.jr-l(29 i~/w® ~xt/w~j~)®zQ est 29 ~/w®zQ.coh6rent et holonome. o~ j d6signe rinclusion du compl6mentaire de rhypersurface d6finie par P par reduction modulo p. Nous avons v6rifi6 cette conjecture en dimension un grace aux r6suttats de Robba sur les th6or~mes d'indice ([Rbl],[Rb2]) pour un module qui n'a que des singularit6s r6guli~res comp16tement soluble ou est de rang un. De fa~on plus pr6cise :

Proposition (4.5.4),-- Avec les notations prdc~dentes supposons que Z e s t un point d'une courbe affine X alors le cohomologie locale de Z d valeur dans 29 ~/W®~xt/w'll~ ®zQ est ,19 ~/w®z Qcoh~rente si et seulement si le 29xt/w-module ~ ®zQ admet un indice dans l'espace des fonctions analytiques dans le tube de Z. 4.6. Epilogue. Berthelot vient de montrer que ranneau Dnt ®zQ cf. 4.4.2 n'est pas n~eth6rien.

306 REFERENCES

[B-B-D] [Be] [Ber]

[BI] [B2]

[B 3] [B -O] [Bj]

[Ch] [D 1]

[D2] [Dw] [G 11 [G2] [G 3] [Ha] [HI

[K I]

In9

BEILINSON, A., BERNSTEIN, I.N., DELIGNE, P., Faisceaux pervers, Astdrisque 100 (1983). BERNSTEIN, I.N., The analytic continuation of generalized functions with respect to parameter, Funct. Analysis Appl, 6 (1972), 26-40. BERSTEIN, I., On the dimension of modules and algebras (IX), direct limits, Nagoya MathJ. 13 (1958), 83-84. BERTHELOT, P., Cohomologie cristalline des sch6ma de caract6ristique p > 0, Lecture Notes in Math 407, Springer Verlag (1974). - - , G6om6trie rigide et cohomologie des vari6t6s algdbriques de caract6ristique p > 0, Journ6es d'analyse p-adique (1982), in Introduction aux cohomologies p-adiques, Bull. Soc. Math. France, M6moire 23, p. 7-32 (1986). - - , Cohomologie rigide et thdorie des/9-modules, dans ce volume. BERTHELOT, P., OGUS, A., Notes on Crystalline Cohomology, Princeton University Press 21, (1978). BJORK, J.E., The global homological dimension of some algebras of differentiets op6rators, Inv. Math. 17 (1972), p. 67-78. CHASE, S. U., On the homological dimension of algebras of differential operators, Comm. Algebra $ (1974), 351-363. DELIGNE, P, Equations diff6rentielles ~ point singuliers r6guliers, Lecture Notes in Math 163, Springer Verlag (1970). - - , Cristaux discontinus, Notes manuscrites non publi6es qui datent du d6but des ann6es 70 (19 pages). DWORK, B., On the Zeta function of hypersurface I, Publ. Math. I.H.E.S, 12, (1962). GROTHENDIECK, A., On the De Rham cohomology of algebraic varieties, Publ. Math LH.E.S., 29, (1966), p. 93-103. - - , Crystals and the De Rham cohomology of schemes, in Dix expos6s sur la cohomologie des sch6mas, North Holland Company (1968), p. 306-358. - - , Groupes de Barsotti-Tate et cristaux de Dieudonn6, Les Presses de l'Universit6 de Montr6al 45, (1974). HAASTERT B., On the direct and inverse image of ,O-modules in prime caracteristic, Manuscripta Math. 62, (1988), p. 341-354. HIRONAKA, H., The resolution of the singularities of algebraic varieties over a field of caracteristic zero, Ann. of Math. 79 (1964), 109-326. KASHIWARA, M., On the maximally overdetermined systems of differential equations, Publ. RIMS, 10 (1975), 563-579. - - , B-function and holonomic systems, Inv. Math. 38 (1976), 33-53.

307

[Ka 1] [Ka 2] [L-M] [M:]

[Me 1]

[Me21 [Me3] [Me41 [Me5] [Me 6] [Me71

[Mr] [M-W] [N-M 1] [N-M 2]

[N-O] [O] [S-M] [S]

KATZ, N., Nilpotent connexions and the monodromy theorem, application of a result of Turittin, Publ. Math. I.H.E.S. 39, (1970), p. 176- 232. - - , The regularity theorem in algebraic geometry, Actes Congr6s intern. Math.,t970, tome I, Gauthier Villars (1971) p. 437-443. LE.D.T., MEBKHOUT Z., Vari6t6s caract6ristiques et vari6t6s polaires, C.R.A.S. Acad. Paris, t. 296 (1983), p. 129-132. MALGRANGE B., Sur tes points singuliers r6guliers des 6quations diff6rentielles, Ens. Math. 20 (1974), p.147-176. - - , Op6rateurs diff6rentiels et pseudo-diff6rentiels, Institut Fourier 1976. MEBKHOUT, Z., Valeur principale et residu des formes ?: singularit6 essentielles, In Lecture Notes in Math 482, Springer Verlag (1975), p. 190215. --,, Cohomologie locale d'une hypersurface, in Lecture Notes in Math 670, Springer Verlag, (1977), p. 89-119. - - , Thdor~mes de bidualit6 locale pour les 29 x-modules holonomes, Ark. Mat. 20 (1982), p. 111-122. ---, Une 6quivalence de cat6gories et une autre 6quivalence de cat6gories, Comp. Math 51 (1984), p. 51-88. ---, Le formalisme des six op6rations de Grothendieck pour les 29x-modules coh6rents, Travaux en cours Hermann 35 (1989). - - , Le th6or~me de positivit6 de l'irr6gutarit6 pour les ,(9 x-modules, in Grothendieck Festschrift, Birkhatiser (1990). - - , Le th6or~me de comparaison entre cohomologie de de Rham d'une vari6t6 alg6brique complexe et le th6or~me d'existence de Riemann, PubL Math. I.H.E.S, 69 (1989), p. 47-89. MEREDITH, D., Weak formel schemes, Nagoya Math. L 45 (1971), 1-38. MONSKY, P., WASHNITZER, G., Formal cohomology I, Ann. of Math. 88 (1968), 51-62. NARVAEZ L., MEBKHOUT Z., D6monstration g6om6trique du th6or6me de constructibilit6, in Travaux en cours Hermann 35 (1989), p. 248-253. - - , La th~:,orie du polyn6me de Bernstein-Sato pour les alg6bres de Tate et Dwork-Monsky-Washnitzer, Ann. de l'~cole Norm. Sup. ( 1 9 9 0 ) ( h paraitre). NASTASESCU, C., VAN OYSTAEYEN, F., Graded rings Theory, North Holland Comp. (1982). OGUS, A. F-isocrystals and de Rham cohomologie II: Convergent isocristals, Duke Math J. 51 (1984), 765-850. SABBAH, C., MEBKHOUT, Z., 29x-modules et cycles 6vanescents, in Travaux en cours Hermann (I989) 35, p. 200-238. SATO, M., Hyperfunctions and partial diferential equation, Proc. Intern. Conf. on Func. Ana. and Related Topics, 1969, Univ. Tokyo Press, Tokyo,

308

[S-K-K]

[Sin]

[Rb 1] [Rb21 [Ro] [Rt]

(1970), p. 91-94. SATO, M., KAWAI, T., KASHIWARA, M., Micro-functions arts pseudodifferentieI equations, Lecture Notes in Math 287, (1973), p. 265-529. SMITH, S.P., The global homological dimension of ring of differential operators on a non singular variety over a field of positive characteristic, J. of Alg. 107, (1987), 98-105. ROBBA P., Indice d'un oprrateur difffrentiel p-adique IV, cas des syst~mes, Ann. Inst. Fourier t. 35, (1985), p. 13-55. - - , Livre (~tparaitre). ROOS J.E., Determination de la dimension homologique globale des algrbres de Weyl, CYLA.S,, Paris 274, (1972), p. 1556-1558. ROTMAN, J.J., An Introduction to Homological Algebra, Academic Press, New York (1979). Sigles

[E.G.A.IV] E16ments de Gromrtrie Algrbrique, GROTHENDIECK, A., DIEUDONNE, A., Etude locale des schrmas, quatri~me partie, Publ. Math. I.H.E.S, 32 (1967). [S.G.A.6] S6minaire de G 6 o m r t r i e A l g r b r i q u e du B o i s - M a r i e (1966-67), GROTHENDIECK, A.. BERTHELOT, P., ILLUSIE, L., Throrie des intersections et th6or6me de Riemann-Roch, Lectures Notes in Math. 225 Springer V erlag. ( 1971). [S.G.A.7] S6minaire de G 6 o m r t r i e A l g r b r i q u e du B o i s - M a r i e (1967-69), GROTHENDIECK, A., DELIGNE, P., KATZ, N., Groupes de monodromie en gromrtrie algrbrique, Lectures Notes in Math. 2 8 0 , 3 4 0 Springer Verlag (1972-73). [R-D] Residue and Duality, S6minaire HARTSHORNE, Lectures Notes in Math. 20, Springer Verlag (1966).

O n a F u n c t i o n a l E q u a t i o n of I g u s a ' s Local Z e t a F u n c t i o n . * Diane Meuser Boston University Let K be a finite algebraic extension of the field Qp of p-adic numbers, RK the ring of integers of K, PK the unique maximal ideal of RK, K the residue field and q = card -K. Let I IK be the absolute value on K so that the absolute value of an element in PK -- p 2 is q-~. If f(x) is a polynomial in Rg[xl,... ,Xn] the Igusa local zeta function associated to f is defined for s C C, Re(s) > 0 by

ZK(S) =

f ]

JR 7~

If(x)lKldxlK

It is a rational function of q-8, and in fact is in Z(q-l,q-8). The denominator can be written as

E (1--q-m'q-N's) iEI

where (Ni, m l ) are data associated to exceptional divisors El, i E I in a resolution of the singularities of f . The numerator is quite complicated in known examples, and no general result is known. Many of the known examples of ZK(s) are for polynomials associated to irreducible regular K-split prehomogeneous vector spaces. These polynomials are invariant polynomials for a group acting transitively on the complement of the hypersurface defined by f = 0. They have been classified into 29 cases, given in [4]. with some cases containing infinitely many polynomials. Igusa observed that there was a functional equation for ZK(S) in the 20 out of 29 classified cases of the above polynomials for which ZK(s) has been determined. For these cases one has that if L is any finite algebraic extension of K, qL the number of elements in the residue field, and ZL(s) the local zeta function, then ZL(s) is obtained from ZK(S) by the replacement q ~-~ qL. This uniquely defines Z(u, v) E Q(u, v) such that Z % -1

-s

)

His result is that for these cases

(1)

Z(u

v-b

= v d z(u, v)

where d is the degree of f [2]. In order to ask whether a functional equation of this type is true more generally it is necessary to impose a restriction that forces the function Z E Q(u,v) satisfying ZK(S) = Z ( q -1, q-S) to be well-defined. Hence Igusa introduced the following definition.

ZK(S) for f(z) E RK[Xl,...,xn] is universal if there exists a function z(u, v) e Q(u, v) such that ZL(S) = Z(q[ 1, q-£~)

Definition.

* Supported by National Science Foundation Grant Nos. DMS-8702667 and DMS8610730.

310

for all finite extensions L of K. He then posed two conjectures under the assumption that f(x) has "good reduction" modulo PK. By "good reduction" it is meant that if f(x) has coefficients in a number field, then the conjecture should hold for all except possibly a finite number of completions. C o n j e c t u r e I: For homogeneous polynomials, ZK(S) universal implies Z(u, v) has the functional equation (1). C o n j e c t u r e II: If f(x) is one of the remaining nine types of invariant polynomials associated to a prehomogeneous vector space then ZK(S) is universal. T h e o r e m : Conjecture I is true. The general proof will appear in [1]. The proof shows that the functional equation of the Igusa local zeta function is a direct consequence of the functional equation of the Weil zeta function for the smooth, proper varieties given by the intersections of the exceptional varieties in the embedded resolution of the singularities of the projective variety defined by f. More specifically let {Ei}iCT denote the exceptional divisors of such a resolution, and let EI = Aiez El. One can use the theory of resolutions to obtain the following expression for ZL(S) for any finite algebraic extension L of K (2)

zL(

) =

1

1 - qL 1

q;-
bL,

ICT

II(

1-

1 -

1

)

iEI

where bL,I is the number of L rational points of Ez where E1 denotes the reduction of Ex modulo PK. Since Ez is smooth and proper the functional equation of the Weil zeta function states that the map a ~-~ q'~-l-tZl/o~ is a bijection of the eigenvalues of Probenius of E~. Since the definition of universal zeta function has been introduced, the natural question is to ask under what conditions does a given polynomial possess a universal zeta function? In particular the functional equation for the remaining 9 types of polynomials associated to prehomogeneous vector spaces as mentioned above remains a conjecture since one does not know that their zeta functions axe universal. However since the first 20 were shown to have universal zeta functions by Igusa's explicit calculations, it is reasonable to conjecture the others do as well. The following shows that the property of f(x) E RK[Xl,..., xn] satisfying the condition that its Igusa local zeta function is universal implies strong properties for the Well zeta function for the projective variety X over I£ defined by 7 = 0. T h e o r e m 2: Suppose f(z) e t~K[Xl,..., Xn] has the property that it8 Igusa local zeta function is universal. Let Zred(t) denote the reduced Well zeta function obtained from the Well zeta function for X by deleting any common factors in the numerator and denominator. Then the reciprocal zeroes and poles of Zre d are all of the form qi, 0 <_ i
311

degree at most n -- I such that Ne = P(q~). T h e idea is to observe one has

/R~.2 If(x)l'ld~l : Z(q-% q-~9 by universality, and on the other hand oo

/R~2 lf(x)l'ldxl = Vol{x

VoZ{x I tf(x)t

I If(x)l = 1} + ~

=

q-~}- q-~

{=1

Letting Re(s) --+ ~ one gets Vol{x I If(x)l = 1} = Z(q-~,o) but this volume is 1 - q-neN¢. The polynomial expression for N~ is then obtained by considering the power series expansion of Z(u, 0) 6 Q[[u]]. rt--1 Write Ne =- ~ i = 0 aiq e', ai 6 Q. Choosing m 6 Z + such that mai 6 Z Vi, n--1 let Nle = ~i=o rnaiq e'' By considering the Weit zeta function for the affme variety defined by f = 0 we have 3 a i , f l j 6 C such that Ne = ~ a ~ ~ f l ~ , so we let N 2 = Erna~ - Ernfl~. Since N~ = N [ , we have

N•t¢ -

exp E e=l

:

e

N ~: U

exp

'

e=l

which in turn gives

I]~1,,,
-

qi) ml''l

1-ii,a,>o( 1 _ qi)m~,

y[1(1

jgjt) m

-

Ili(1 - ~t)m

Thus after possible cancellation we have each ai,/gj is an integral power of q. Letting N~ denote the number of points on the projective variety X and using Ne : 2~r~(q - 1) + 1 gives the same conclusion for the Well zeta function of X. Let f ( x ) be of degree d in variables. In the case where the variety defined by f ( x ) = 0 is projectively nonsingular the Well zeta function Z(t) is reduced and satisfies

z(t)

N(t)(-1) ("-1) :

(1 --

t)(1 - qt)... (1 -- q("-2)t)

where b

w(t) : ll(1-~,t)

,

i=1

b is the betti number of Hn-~(X, QI) and Ioq] = q(n-2)/2 for all i. Thus n odd implies b = O. But the middle dimensional betti number is given by b = (d - 1) '~ + ( - 1 ) n ( d - 1) So if n is odd we must have d = 2. Conversely these polynomials all have the property that their Igusa local zeta functions are universal since they are special cases of the invariant polynomials associated to prehomogeneous vector spaces.

312

If n is even all the middle dimensional eigenvalues of Frobenius al must equal q(n-2)/2 A further characterization of these varieties can be obtained in terms of properties of tile middle dimensional 1-adic coholnology group. In particular there is a map from algebraic cycles of codimension (n - 2)/2 to H n - 2 ( X , Q~). A cohomology class is said to be algebraic if it is the image of an algebraic cycle. Frobenius acts trivially on the subspace spanned by algebraic cycles so if H n - 2 ( X , QI) is spanned by aJgebraic cycles the Igusa local zeta function is universal. Conversely, Tate's conjecture [6] is that the order of the pole of the Well zeta function at q-('~-2)/2 is the rank of the subspace spanned by the algebraic cycles. Thus one has T h e o r e m 2: A projectiveIy nonsingular variety X defined by 7(x) = 0 where f ( x ) has an even number of variables has a universal Igusa local zeta function if and only if H~-2(X, QI) is spanned by algebraic cycles, with the only if part assuming Tate's conjecture. Any rational variety having a cellular decomposition will satisfy the property that its cohomology is spanned by Mgebraic cycles. Some other examples are given by Fermat hypersurfaces. The Fermat hypersurface of degree d in n variables where n is even and d > 4 has H n - 2 ( X , Ql) spanned by algebraic cycles if and only if there exists an integer v such that pV = - 1 (rood m) where p = char(I(), as was shown by Shloda [5]. These Fermat hypersurfaces are unirationat, ie. there exists a rational map of finite degree from a projective space to the variety. For the case of singular varieties the situation is much less clear since the Weil zeta functions are not as explicit. However this is the primary case of interest since in particular the invariant polynomials associated to prehomogeneous vector spaces define singular varieties. By explicit consideration of the expression of the Igusa local zeta function given in (2) one can easily see P r o p o s i t i o n 1: If {Ei)ieT denotes the collection of exceptional divisors in a resolution of f , then if every subvariety EI = ~i~I Ei for I C_ T has the property that its eigenvaIues of Frobenius are integral powers of q, then the Igu~a local zeta is universal. However, whether or not the converse is true remains open. Even for the prehomogeneous cases whose Igusa local zeta functions have been determined it is not known whether their resolutions satisfy this simple property, since in general their resolutions have not been obtained. Igusa's method of determining the zeta functions for these polynomials does not use a resolution. I have shown that in two non-trivial cases the conditions of Proposition 1 are satisfied. P r o p o s i t i o n 2: If f(x) is the invariant cubic in 27 variables for the group E6 or the invariant quartic in 56 variables for the group ET, then the eigenvalues of Frobenius of every subvariety in the resolution are integral powers of q. The zeta functions of these are known to be universal by methods of Igusa. On the other hand the numerical data of the exceptional divisors was obtained by Kempf in [3]. For instance for the cubic in 27 variables there are three exceptional divisors E1 , E2, Ea with numerical data (1, 1), (2, 10), (3, 27) and the I with ]II > 1 satisfying Ez ~ 0 are {1,2}, {1, 3}. By consideration of the formula for the Igusa local zeta function given in (2) it is clear that the only possibly non-universal contribution comes from the bL,I. By expanding this formula and Z(u, v) as formal power series in v with coefficients in Q(u) one can compare coefficients to show each bn,i is given by a function in Q(u). Then the arguments of Proposition 1 show that the eigenvalues of Frobenius are integral powers

313

of q. A similar but more complicated consideration of the seven exceptional divisors for the polynomial associated to E7 shows that their intersections also have this property. Kempf has indicated to me that he had not been aware of the result of the above property nor was he aware of any geometrical property of the exceptional divisors for the above cases that implied this result. Of course the construction of embedded resolutions for the remaining prehomogeneous cases would enable one to settle Igusa's conjecture, but perhaps there are geometrical properties of these varieties that force theirlocal zeta functions to be universal. References 1. J. Denef and D. Meuser, A functional equation of Igusa's local zeta function, To appear in Amer. d. Math.. 2. J. Igusa, On functional equations of local zeta functions, preprint. 3. G. Kempf, The singularities of some invariant hypersurfaces, Proc. Conference on Algebraic Geometry, Berlin (1985) Teubner-Texte, 92 (1986), 210-216. 4. T. Kimura and M. Sato, A classification of irreducible prehomogeneous vector spaces and their relative invariants, Nagoya Math J., 65, (1977) 1-.155. 5. T. Shioda and T. Katsura, On Fermat Varieties, T6huicu Mat. dourn., 31 (1979), 97-115. 6. J. Tare, Algebraic cycles and poles of zeta functions, Aritl.;metical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963), 93-110, Harper and Row (1965).

On Vanishing of Cohomologies of Rigid Analytic Spaces Yasuo Morita MathematicM Institute Tohoku University Aoba, Sendai 980 Japan

1

Introduction

In our previous paper [ Mo ] , we have studied relative cohomologies of the rigid analytic space k n , and, by using a vanishing theorem of relative cohomologies, we have constructed a p - a d i c theory of hyperfunctions on compact subsets of k n . In this paper, we shall review the results of [ Mo ] , and study relevant vanishing theorems of cohomologies of rigid analytic spazes. In particular, we shall give two open problems concerning vanishing of cohomologies of rigid analytic spaces.

2

Vanishing of the Derived Sheaves

"His((.gx)(i # n)

Let Qp be the p - a d i c number field, and let k be a complete nonarchimedean algebraically closed field containing Qp. For example, the field Cp , which is the completion of the algebraic closure of Qp , satisfies this condition. We denote by I I the standard p - a d i c valuation of k with I P I= p-1 . Let X = k n be the n-dimensional affine space with the standard rigid analytic structure ( cf. e.g. [ B G R ] ) . Let Ox be the structure sheaf on this rigid analytic space X , let U be an admissible open subset of X , and let H'(U, Ox) be the i - t h cohomology group. Let K be a compact subset of X , and let HKnu(U, i Ox) be the i - t h relative cohomology group with supports in K fq U , and let 7-l'K(Ox) be the i - t h derived sheaf of Ox with supports in I( (cf. [ Mo ] for the definition). Then we have proved in [ Mo ] that the derived sheaf 7-t~((Ox) vanishes for any nonnegative integer i ~ n . Further, by using the degeneracy of the spectral sequence i j HKnu(U, 7iK(OX)) ~

h HKnu(U, Ox) ,

we have obtained the following isomorphism :

HKnu(U ,,,+m O x ) = H"(U, 7"l'~¢(0x) for any integer m >__O.

315

Hence the presheaf BK defined by

K(I¢ n U) = H nu(U, Ox) becomes a (flabby) sheaf on K . Furthermore, we have proved that the n - t h relative cohomology group H~.(X, (.gx) gives the dual space of the space of locally analytic functions

on K (of. [ Mo ]) One of the most essential part of [ Mo ] is in the following temma. Let U be an affinoid subset of X = k" containing the product of 1-dimensional affinolds D 1 , . . . , D , and let S b e a p r o d u c t of open balls B, = {z E k I I z-a, I<_r [ } (i = 1 , . . . , n ) such that B; C Di and ri E I kx I . (In [ Mo ] , these conditions on U, Di, B, are not correctly stated.) Then we have proved the following : LEMMA. Let the notation and assumptions be as above. Then : (i) The relative cohomology H~.(U, Ox) vanishes for any nonnegative integer i different from n . (ii) The n - t h relative cohomology H~(U, Ox) can be explicitly calculated. REMARK. B. Chi~ellotto also obtained this result in a more generalized form. He studied the Serre duality in the rigid analytic case (cf. his report in this Proceedings) . Though we have stated in [ Mo ] only for compact subsets in the affine space X = k ~, it is easy to see that, for any compact subset K in the projective space X = P~(k) , we can prove that the derived sheaf 7-lK(Ox ) vanishes for any nonnegative integer i # n . In particular, if we denote by L a locally compact subfield of k containing Qp , then we can prove that the derived sheaf 7-['p~(L)(Ox) vanishes for any nonnegative integer i # n. Further, if K is a compact subset of Pn(k) , then we can prove that the cohomology H~¢nu(U, Ox) vanishes for any integer i ~ n + 1 and for any admissible open subset U of W(k) . In the classical case, it is known that for any n-dimensional real analytic manifold M and for any complexification X of M , the derived sheaf 7"l's(Ox) vanishes for any closed subset S of M and for any nonnegative integer i # n (cf. e.g. [ KKK ] , Theorem 2.2.1). This is the most essential fact to construct the theory of hyperfunctions on M, and we have used an analogous fact to construct a p - a d i c analogue of the theory of hyperfunctions (cf. [ Mo ]) . So we have the following problem : Let L be a locally compact subfield of k containing Qp , let X be an n-dimensional rigid anMytic space defined over L , and let M be the subset of X consisting of all L - v a l u e d points of X . Let S be a closed subset of M such that X - S is admissible open in X . Then : PROBLEM V D S . i different from n ?

Does the derived sheaf

~'s(Ox)

vanish for any nonnegative integer

We have already noted that the answer is yes for the projective space X = Pn(k) .

316

3

Relation with Vanishing of Cohomologies

Let (X, O x , T ) be a rigid Let U be an admissible open is admissible open in U . Let supports in S . Then we have

H'snu(U, 7-lJs(.T)) ~

analytic space, and let .T be a sheaf of Ox-module on T . subset of X , and let S be a subset of U such that U - S H's(U,.T) be the i - t h relative cohomology group of ¢- with a spectral sequence

H~nu(U, Y') .

Hence we have an isomorphism n+m n H~.u (u, ~') _~ H m (U, -U.(~-))

for any integer m _> 0 if the derived sheaf ~ ( ~ - ) vanishes for any nonnegative integer i ¢ n (cf. [ G ] , TMor~me 4.4.1 ) Since ~}(.T) is the sheafication of H'snu(U,.T) , the derived sheaf 7~}(.T) vanishes for any nonnegative integer i ¢ n if and only if (1) H'snu( U, .T) vanishes for any integer i _> n + 1 and for any admissible open subset U , and (2) H)nu(U,.T) vanishes for any integer i _< n - 1 and for any sufficiently small adntissible open subset U . Now we have the following exact sequence (cf. [ Mo ]) : o

, ~°~(u,.r)

~ H°(u,f)

~

H°(U-

s,:r)

....

H ~i ( U , J : )

, s'(u,7)

--. ~'(u - s,J:) ....

(exact).

Hence, His(U,.T) vanishes for any nonnegative integer i ¢ n if and only if the following three conditions are satisfied :

Hi(U, Y:) ~- H'(U - S ~ )

for any integer i ¢ n - t, n ,

ffn-l(u, .~) ----+ H n - I ( u - S, .~) H"(U,.T)

, H'~(U - S,.T)

is i n j e c t i v e

,

is surjective.

Hence "H~(~') vanishes for any nonnegative integer i ~ n if and only if

(1')

H"(U,.T) ~ H"(U - S,.T) is surjective, and H ' ( U , y ) ~_ H i ( U - S , . T ) for any integer i_> n + l

holds for any admissible open subset U of X , and

H"-I(U,.T) (2')

----) H'~-I(U - S,.T)

H i ( U , ~ ") ~_ H i ( U - S , . T )

is injective, and for any integer i < n - 2

holds for any sufficiently small admissible open subset U of X . Therefore, if the derived sheaf 7[is(Ox) vanishes for any nonnegative integer i # n , then we have

317

H'(U;Ox) ---

H~(U- S, Ox)

for any integer i >_ n + 1 and for any admissible open subset U of X . In the classical case, it is known that for any coherent O x - m o d u l e 7" on an n-dimensional complex manifold X , the cohomology group H'(X, Ox) vanishes for any integer i > n + 1 (cf. [ Mal ], TMor~m 3 ). Further, it is known that the cohomology H"(X, ~') also vanishes if X is not compact and if 7" is a locally free coherent Ox-module (cf. [ M a l l , Probl~mes 1 ) . Malgrange used the Dolbeault complex to prove these facts, so it is not easy to translate the proof into the p - a d i c case. We note that the cohomologies Hi(X, .7) (i > nq- 1) may not vanish if 7" is not coherent. On the other hand, by a result of A. Grothendieck, the cohomology Hi(X, ~) vanishes for any integer i > n + 1 and for any sheaf of abelian groups 7" on X if X is a noetherian topological space of dimension n . In our case, M. van der Put obtained the following (cf. [ P ] , p.174, L e m m a ) : THEOREM V C

Hi(X, .T') vanishes

, If X is an n-dimensional affinoid space, then the cohomology group for any integer i >_ n + 1 and for any sheM" Y of abelian groups on X.

Van der Put expressed the cohomology H i ( X , ) t') in terms of the Cech cohomology group ~Ii(X, .T') , and expressed it by the Cech cohomology kri()(, Y) of the reduced affine variety X . Then he proved the vanishing of the last cohomology. It seems that his result holds not only for affinoid spaces but also for rigid analytic spaces satisfying a countable condition (cf. [ P ], p.172, Proposition 1.4.4), but the author could not check it because only a sketch of the proof is given in [ P ] . REMARK. Though the cohomological dimension of a paracompact topological space can be calculated locally, its proof seems to have a difficulty to translate into the rigid analytic case (cf. [ G ] , Th4or~me 4A4.1 ) . In the classical case, we can prove that the cohomology H'(X, .T) vanishes for any integer i _> 1 if .T is a soft sheaf. But soft sheaves seem to be not so nice in the rigid analytic case. Related with this question, we are also interested in the following problem : PROBLEM VA.C. Does the cohomology group Hi(X,.T') vanish for any integer i _> n and for any locally free coherent O x - m o d u l e 5c if X is an admissible open subset of a quasi Stein space ?

318

REFERENCES

[A]

M. Artin, Grothendieck topologies, Lecture Notes of Harvard University, 1962.

[ BGR ] S. Bosch, U. G/inter and R. Remmert, Non-Archimedean Analysis, Springer-

[G] [ KKK]

[Mall [Mo] [SM] [P]

Verlag, 1984. R. Godement, Topologie atgdbrique et tMorie des faisceaux, Hermann, 1958. M. Kashiwara, T. Kawai and T. Kimura, Foundations of algebraic analysis, Princeton Univ. Press, 1986. B. Malgrange, Fasceaux sur vari4t4s analytiques r4elles, Bull. Soc. math. France, 83(1955), 231-237. Y. Morita, A p-adic theory of hyperfunctions, It. In : Algebraic Analysis, Vol. I, 457-472, Academic Press Inc., 1989. W. H. Schikhof and Y. Morita, Duality of projective limit spaces and inductive limit spaces over a nonspherically complete nonarchimedean field, Tohoku Math. a., 38(1986), 387-397. M. van der Put, Cohomology on affinoid spaces, Composito Mathematica, 45 (1982), 165-198.

A p-adic Analogue of the Chowla-Selberg Formula Arthur Ogus * University of California, Berkeley

Let X/C be an elliptic curve, with complex multiplication by a quadratic imaginary field E. Then X/C automatically and uniquely descends to the algebraic closure Q of Q in C and has good reduction at every prime 9 of Q. Let Kp denote the algebraic closure of Qp in the completion of Q at 9, and let W, denote the set of automorphisms of Q which preserve 9 and which act as some integral power of the Frobenius automorphism on the residue field kp at 9. These automorphisms extend to automorphisms of K~, by continuity. As explained in [3], there is a canonical semilinear crystalline action of this Well group on H1R(X/'Q) ® Kp, compatible with the action of E. The principle goal of this paper is to give an "explicit formula" for a matrix representat'ion of this action, with respect to a basis for H~R(X/-Q ). As we shall see, our formula, 3.15 below, bears a striking resemblance to the classical Chowla-Selberg formula [15] for the classical period matrix of X/Q, with the classical gamma function replaced by Morita's p-adic gamma function. As a matter of fact, our proof is inspired by and patterned after Gross's proof of the classical formula in op. cir., in which the formula for elliptic curves is deduced from a similar formula for the periods of Fermat curves. In our case, we use the formalism of absolute Hodge cycles to reduce to the computation of the Frobenius matrix of Fermat curves. When p does not divide the degree, this computation goes back a long way, but in fact no convenient reference seemed to exist in the literature until Coleman's article [6], which was written in response to my queries. More recently, Coleman has been able to carry out his computations for all odd primes. In fact, in the case of bad reduction he was only able to calculate part of the Frobenius matrix, but, remarkably, this partial calculation is sufficient for the determination of the calculation of the periods of the "semiversal ulterior cyclotomic motive" constructed here (c.f. Theorem 3.1). Consequently, using the results of this paper, Coleman was finally able to bootstrap the calculations and obtain an explicit formula for the entire Frobenius matrix of the Fermat curve. We refer to his paper for the explicit formulas and the detailed story of the relationship between his results and ours [5]. I should point out that our formula is a rather special consequence of the general philosophy that Hodge cycles should be compatible with the action of Frobenius on crystalline cohomology, as conjectured in [17]. In fact, recent work of Fontaine-Messing [14] and Faltings [12] has made it possible to strengthen this conjecture, and, as Don Blasius has pointed out [4], to prove it for abelian varieties. The strengthened conjecture, which includes a p-adic analogue of Shimura's monomial period relations, has been considered by other authors as well [7]. I only emphasize the Chowla-Selberg formula itself to try *Partially supported by NSF Grant No. DMS-8502783

320

to render crystalline cohomology more "explicit" and "concrete," at least in the setting of the theory of complex multiplication. Our paper is organized as follows. The first sections review the terminology and formalism of motives of CM-type and their classical and crystalline periods. In particular we introduce the concept of a "marked motive," which is a motive endowed with a choice of basis for its cohomology. We construct an abelian group out of the category of marked motives and state the main theorem, which asserts that the periods of a marked motive depend only on the corresponding marked CM-type. This more delicate formulation is essential for Coleman's bootstrap in the determination of the Frobenius matrix for Fermat curves in the case of bad reduction. Section 2 reviews the classical and crystalline cohomology of Fermat curves, being of necessity rather careful about specific choice of bases for cohomology and about rational factors. Section 3 includes the construction of the semiversat ulterior motive M alluded to above, which can probably be viewed as a geometric incarnation of Anderson's constructions in [1]. Finally, section 4 contains a proof of the general result on absolute Hodge cycles on which our formula relies. I would like to thank Don Blasius for sharing a preliminary version of his proof of the "De Rham conjecture" for absolute Hodge cycles upon which our results are based, as well as Robert Coleman for the dedication with which he pursued the precise forms of the formulas I needed. Thanks also go to Hendrik Lenstra and Ken Ribet for useful conversations and hints about Galois cohomology and CM-types.

1

Complex Multiplication

We begin by recalling the basic facts about complex multiplication and motives of CMtype. Let E/Q be a CM-algebra--that is, a product of CM-fields--and for any Q-algebra A, let S(A) =: Mot(E, A) for any. Then E ® K ~ K s(K) for any algebraically closed field K of characteristic zero. Let X/-Q be an abelian variety with complex multiplication by E, so that (by definition) E operates on X/-Q and the corresponding action of E on De R h a m cohomology makes H1R(X/-Q) a free E ® Q-module of rank one. For each s E S ( Q ) , the fiber Y~)R(s ) of H~)R(X/-Q) at s has dimension one over Q , and there is a canonical isomorphism: HbR(X/-Q) ~ II{HbR(s) : s e S}. We shall find it extremely convenient to consider motives of higher weight constructed from abelian varieties of CM-type by linear algebra operations. We refer to [10] for the formal definition, contenting ourselves here with the examples and properties we need. Associated with an E-motive X/-Q of rank r are its various cohomological realizations:

• HB(X/Q)--a free E-module of rank r. • HDR(X/Q)--a free Q ® E-module of rank r. • H~(X/Q)--a free Qp ® E-module of rank r. These realizations are functorial and come equipped with other functorial data as well. For example, the De R h a m cohomology comes with the Hodge filtration, and there are canonical isomorphisms between the De Rham and Betti cohomologies (after tensoring with C). We shall be more explicit about these and other data later. Here are some examples of constructions of E-motives:

321

A/'Q is an abelian variety width complex multiplication by E, then the degree one part of its cohomology is an E-motive of rank one.

• If

• If a: E --* E r is a homomorphism of CM-algebras and X / Q is an E-motive of rank r, then a*X =: X ®E E ' is an Er-motive of rank r. • E ® H i ( P 1) is an E-motive of rank one and weight i if i = 0 or 2. We denote this motive by E if i = 0 and by E ( - 1 ) if i = 2. • If X and Y are E-motives, we can define X ®E Y and HomE(X, Y) so that

H(X ®E Y) ~- H(X) ® H(Y)

and

H(gom(Z, Y)) ~- H o m ( g ( x ) , H(Y)) Here the "H" stands for any cohomological realization, and the tensor product and Horn are taken over the appropriate coefficient ring. There are several ways to define morphisms, and in particular isomorphisms, of Emotives. For the sake of definiteness, let us choose here to use the strictest definition, originally proposed by Grothendieck--morphisms of motives are induced by algebraic correspondences. The point is that any such morphism is necessarily compatible with the all the structures on and compatibilities between the various cohomological realizations. The ttodge filtration on the De Rham cohomology of an E-motive X/-Q is especially important. If X/-Q has rank one, this filtration can be described purely combinatorially. We identify S(Q) with the set of maximal ideals of Q ® E, and the fiber HDR(S) of HDR(X/-Q) at s can be identified as g / ) R ( s ) ~ {z e HDR(X/-Q):ex ----s(e)x for all e e E} Then for each s E S(Q), we define

r(X)(s) =: sup{i: HDn(S) e Fi HDn(X/-Q) } Since each stalk is one dimensional, it follows immediately that

F'HDR(X/-Q) = E {HDR(S) : v(X)(s) >_i} $

Attached to each motive is a weight, i.e. the weight n of the underlying IIodge structure, which can be expressed in terms of r and the natural action of complex conjugation on S(Q). Namely, one finds easily that r(s) + r(~) = n for every s e S(Q). If E is a field, n is just an integer, and in general it is a function on S ( Q ) which is constant on the orbits of GaI(-Q/Q) acting on S(Q). In general, a function r: S(Q) ---+Z such that r ( s ) + v(l) = n is called a "CM-type of weight n." Evidently the sum of a CM-type of weight n and a CM-type of weight m is a CM-type of weight n + m, and the set of all CM-types forms a subgroup of the group of all functions S(Q) ~ Z. We denote this subgroup by CM(E). Let Mot(E/-Q) denote the set of isomorphisms classes of E-motives of rank one. Using the isomorphisms:

X ®E Y "- Y ®E X

and

( x ®E Y) ®E Z "*--X ®E (Y ®E Z),

322

we find that Mot(E/-Q) becomes an abelian group. Since the Hodge filtration of X ® ~ Y is the tensor product filtration, we have r ( X ®E Y) = r ( X ) + r(Y). Similarly, one finds that if a: E --* E' is a homomorphism with corresponding map a*: S ' ( Q ) ---* S(Q), then r ( a * X ) = v ( X ) o a*. We summarize: P r o p o s i t i o n 1.1 Formation of the CM-type induces a surjective homomorphism of abelian groups:

r: M o t ( E / Q ) ---* CM(E) Ira: E --* E' is homomorphism, tet a* denote the corresponding maps S'(-Q) ---* S(-Q),

and

C M ( E ) --* CM(E'),

Mot(El-Q) ---* Mot(E'/-Q)

Then there is a commutative diagram: Mot(El-Q)

~....... CM(E)

1o

1o - -

Mot(E'/Q)

"r

,

CM(E')

Proof: The only part of this proposition that is not yet clear is the surjectivity of r. This is classical; we recall the outline of the argument. Notice that if r E C M ( E ) takes its values in {0, 1}, then the weight w o f t is either 1 or 2. In the latter case, r is constant and equal to v ( E ( - 1 ) ) . If the weight is one, we can regard r as defining a partition of S(Q) into two subsets ~ and ~ . Then there is a well-known complex analytic construction of a complex torus on which the ring of integers O of E operates: O is a lattice in C ~:, and the quotient is the desired torus. One finds that the action of E on cohomology is such that one recovers r as the CM-type. Then one proves that the complex torus is in fact an abelian variety, and that this variety, together with the action of O, descends uniquely to Q. To prove the proposition, one now is reduced to checking that the group C M ( E ) is generated by elements r which take their values in {0, 1}, which is easy. II It is sometimes possible and useful to make more precise statements, with respect to a specific basis. To this end, we define a "bimarking" of an E-motive X to be a pair ((, ~7), where ~ is an E ® Q basis of HDR(X/-Q) and T/is a Q-basis for HB(X). Similarly, we define a "De Rham marking" of X to be an E ® Q---basis ~ of HDR(X/-Q). For example, HDR(E) is just E ® Q, which has a canonical basis 1, and HB(E) is just E, which also has a canonical basis 1. Similarly, H o R ( E ( - 1 ) ) is E ® Q , with basis 1, and H B ( E ( - 1 ) is (2rci)-lE, with basis (2rri) -1. Then we have the obvious notions of the categories of bimarked and De Rham marked motives, and the tensor product operation induces a group structure on the corresponding sets of isomorphism classes, which we denote by B M o t ( E ) and DMot(E), respectively. Notice that an element e of T/~(Q) induces an isomorphism of bimarked motives

(x, ~, 7) ~ (x, e~, e~), so that these two define the same element of BMot(E). If A is any Q-algebra, we let T E ( A ) denote the group of units of E ® A; we have a canonical isomorphism T E ( K ) ~- K *s(K) if K is an algebraically closed field. There are

323

then evident exact sequences:

0

-*

Ts(Q)

0

-*

Ts(Q)/TE(Q)

0

---*

TE(Q)

---* BMot(E)

-*

DMo~(E)

~

0

DMot(E)

--*

Mot(E)

-*

0

---, BMot(E)

---*

Mot(E)

~

0

-*

Let us next review the formalism of periods of E-motives. The theory of integration defines a canonical isomorphism: a s : HDR(X/Q) ®-QC ~ H B ( X / Q ) ®Q c If we choose a basis ~ for HoR(X/-Q) over E ® Q and a basis rl for HB(X/'Q) over E, then the matrix for a s with respect to these bases is a unit of E ® C, which we denote by 7(X,~,rl). Its image in T E ( C ) / T E ( Q ) depends only on X and (, and its image in T E ( C ) / T E ( Q ) depends only on X; we denote these by 7(X,~) and 7(X), respectively. One checks easily that in fact 7 defines homornorphisms: m

BMot(E, Q)

--,

TE(C)

DMot(E,Q)

--*

TE(C)/TE(Q)

Mot(E, Q)

-,

TE(C)/TE(Q),

all of which we denote simply by 7. For example, 7(E, 1, 1) = 1 and

7 ( E ( - 1 ) , 1, (27ri) -1) = 27ri

For the crystalline version, we let p be a prime of Q and let ¢ be a member of the Weil group W~. Since we are working over Qp, we can identify W~, with the "crystalline Weil group" constructed in [3]. Since abelian varieties of CM-type have potentially good reduction everywhere, the crystalline Weil group acts on the completion of their De Pdaam cohomology, and hence also on the completion of the cohomology of any motive construtted from abelian varieties of CM-type. If again ~ is an E ® Q-basis for HDR(X/Q), we let 7 ~ ( X , ~ , ¢ ) E T E ( K p ) denote the matrix for the action of ¢ with respect to ~. (This also makes sense for any E ® K~-basis.) The group Wp also acts on T E ( K ~ ) through its action on K~, and if A E TE(K~,) and ~ E HDn(X/Kp) is a basis, we have the following formulas:

¢(~) = ¢(~)¢(~) ~p(x,~,¢'¢) = - r A x , L¢')¢'(~(x,L¢)) ~Ax, A~,¢) = ¢(~)~-'vAx,~,¢).

(1)

The second of these says that 7p(X, ~) defines a crossed homomorphism

W . ( K . ) --* T E ( K . ) L e m m a 1.2 The crossed homomorphism 7~(X,~): Wp(K~) ---* T s ( K p ) i.e., it vanishes on a subgroup of finite index of the inertia group Ip. K C_ K~ is a finite extension of Qp such that X is defined and has good K and such that ~ E HDR(X/K), then 7p(X,() vanishes on GaI(Kp/K)

is continuous, Specifically, if reduction over n I~.

324

Proof: This follows immediately from the definitions in [3]. Namely, if W(I~) is the Witt ring of the residue field k of K and Xk is the special fiber of X / K , there is a canonical isomorphism: HDR(X/K) ® Kp ~- Hc~,(Xp, W(p)) ®w(p) Kp Then if ¢ e Wp(Kp) has degree n, ¢ is defined to act on H¢ri,(Xp, Wp) ® Kp as g c r i , ( F ~ ) ® ¢. In particular, if n = 0 and ¢ • Gal(gt~/g), then ¢(~) = ~ and - t A x , (, ¢) = 1. I We find that calculation of periods induces a a homomorphism of abelian groups: 7p: DMot(E/-Q) --~ ZI(Wp, TE(Kt~)) The formulas 1 above imply that the image of 7t~(X, ¢) modulo coboundaries from T E ( Q ) is independent 6f the choice of basis ¢. We will therefore find it convenient to make the following definition. D e f i n i t i o n 1.3 Let G be a group and M and N are G-modules with N C M. Then Hi(G, M; N) is tile quotient oft/he module ZI(G, M) of crossed homomorpfiisrns G ---* M by the coboundaries BI(G, N) coming from N. Perhaps it is worth remarking that there are natural exact sequences

0 ~ WIN G --* ZI(G, M) --~ Hi(G, M; N) ~ 0 0 ---* M / ( N + M G) ---* H I ( G , M ; N ) ~ HI(G,M) ---*0 0 ---* HI(G,N) ---* H I ( G , M ; N ) ---*Z I ( G , M / N ) Thus, 7t~(X) should be viewed as an element of g l ( w p , wE(gp); T E ( Q ) ) . For typographical reasons, we shall abbreviate this by HI(Wt,,TE(Kp;-Q)). L e m m a 1.4 The groups HI(Wt~, T E ( K p ; Q ) ) and HI(Wp, WE(Q)) are torsion free. Proof: The group ZI(TE(Kp)/TE(-Q) is evidently torsion free, so we see from the exact sequences above that it suffices to show that HI(W~,, TE(Q)) is torsion free. To this end, let T E ( Q ) n denote the kernel of the surjective map multiplication by n on T E ( Q ) . Let Dp C_ GaI(-Q/Q) be the decomposition group at p. We have a commutative diagram:

HI(Dp,TE(-~)n)

---. HI(Dp, TE(-~) )

I.... HI(Wt~,TE(-~)n)

n,

i

I-.

TM

--.

HI(W~,TE(~))

HI(Dp, T m ( ~ ) )

n

HI(Wp,Tm(~))

Since T E ( Q ) n is a finite set, any cocycle z E ZI(Wp,TE(-Q)n) automatically factors through a finite quotient, and hence prolongs uniquely to a cocycle in Z 1(Dt~, TE(Q)n). Thus, the map resn in the diagram above is surjective (and even an isomorphism). On the other hand, Hilbert's theorem 90 implies that HI(D~, T E ( Q ) ) vanishes. A diagram chase now shows that multiplication by n on HI(Wp, T E ( Q ) ) is injective. II We can summarize the situation so far as follows:

325

Corollary 1.5 Let Kp =: Q

n

Qp. Then there/s a commutative diagram with exact

rOWS:

0

.-.* TE(-Q)/TE(Kp)

--,

ZI(W~,TE(Kp))

T

-*

HI(Wp,TE(Kp;-Q))

0

T

T

0-..

TE(Q)/TE(Q)

--*

DMot(E)

-*

Mot(E)

o

TE(Q)/TE(Q)

--,

TE(C)/TE(Q)

--

TE(C)/TE(Q)

-~

--

--+ 0

--,

O.

R e m a r k 1.6 If E / Q is Galois (as we may as well assume), then GaI(E/Q) operates in a rather trivial way on the right on the category of bimarked E-motives, via the base change operation 1.1. GaI(E/Q) also acts on CM(E), T E ( C ) and on T~(K~,) on the right, and it is clear that the maps r, 7, and 7p are compatible with these actions. The main result which underlies our proofs of Chowla-Selberg states that both the period homomorphisms of Corollary 1.5 depend only on the CM-type. For the classical period 7, this result is essentially equivalent to Shimura's monomial period relations and is due to Shimura and Deligne [11]. T h e o r e m 1,7 The homomorphisms 7 and 7~ above factor through r. That is, there is

a commutative diagram: TE(C)/TE(Q)

T Mot(E)

r

CM(E)

~

I~ HI(W,~,TE(Kp;-Q))

|

To express a convenient analog for marked motives, we let B C M ( E ) and D C M ( E ) denote the fiber products in the diagram below:

DCM(E)

---*

I TE(C)/TE(Q)

CM(E)

BCM(E)

I,

--,

I

--~ TE(C)/TE(-Q)

TE(C)

CM(E)

I, --~

(2)

TE(C)/TE(-Q)

It is clear that we have natural maps:

rDM:DMot(E,-Q) --~ D C M ( E )

and

rBM:BMot(E,-Q) --, B C M ( E ) .

We can now formulate the following slightly stronger version of Theorem 1.7. T h e o r e m 1.8 The homomorphism 7p factors through r D M , and the induced map 7p

fits into an exact ladder:

326

0

---*

TE(Q)/TE(Q)

---*

DCM(E)

--*

0

.--* T E ( - Q ) / T E ( K . )

--*

ZI(W~,,TE(K.))

--

CM(E)

HI(W.,T.(K.;-Q))

-=* 0

---* 0

We shall explain the proofs of 1.7 and 1.8 in section 4. Let us close this section by mentioning the following natural question. P r o b l e m 1.9 Are the maps 7 : C M ( E ) -+ T E ( C ) / T E ( Q )

and

7)~:CM(E) --* H I ( W p , T E ( K t ~ ; WE(Q))

injective?

2

Periods

of

Fermat

Curves

In this section we calculate the periods described in the previous section for the motives constructed from the dacobians of Fermat curves. These calculations are due to other authors; it is simply a matter of licking the formulas into shape. We have to be a little careful to keep track of the rational factors neglected in [11] and to be specific about the choice of bases. It seems best to recall in detail the situation, following the notation of [11] as closely as possible. Let X m denote the Fermat curve of degree m, with homogeneous equation x~n + x ] n + x~n = 0. Let #m denote the group of roots of unity in Q and let Am denote the group /a~ modulo the diagonal, acting on the right on X m by -1

(Z0 : Xl : ~:2)(~0 : ~1 : ~2) = (~0~O 1 : z l ~ i -t : X212 ) Then Am acts on the left on coordinates (and cohomology), and in particular if ( E A, (xi = (~-lxi The group A~ of Q-valued characters of Am can be identified in an obvious way with the subgroup of ( m - I Z / Z ) 3 whose coordinates sum to zero. I f x =: (Xo,X1,X2) is such a character, we let H x denote the x-eigensubspace of H D R ( X m , Q). Recall that H x has dimension one if no Xi = 0 and is zero otherwise; in the former case we say that X is a "Fermat character." To describe the Hodge level of H×, let (a) denote the representative of a in Q 1"3(0, 1]. (Note that the endpoint convention is that of [11] rather than [6]; it makes (a) + ( - a ) equal to 1 if a is a nonzero element of Q / Z and to 2 if a is zero.) For any Fermat character X, we let (X) denote ~ i ( X i ) . L e m m a 2.1 ([11](7.6)) I f x is a Fermat character, the Hodge level of H x is (X) - 1. That is, GrJFgx is zero unless j = (X) - 1, and r ( X ) ( x ) = <X) - 1. The group algebra Q[Am] acts on the Jacobian of X m and its various realizations; in fact, even Q[A] acts on HDR(Xm/-Q). Each Q-valued character X: Am --+ Q defines a homomorphism Sx: Q[Am] ~ Q, and there is a corresponding idempotent e x of Q[Am]: e× =: [Am1-1Y':.~X(~)A. We will allow ourselves to identify X and s×, and we let Fm

327

denote the group of all Fermat characters. The Galois group G a l ( ' Q / Q ) operates on the set of characters A~ and on the group algebra; this action factors through the cyclotomic character GaI('Q/Q) --* Z / m Z * , and g% = %×. Thus, the Z/mZ*-orbits of A~ induce Galois equivariant idempotents of Q[Am] i.e. idempotents of Q[A,~], and we can decompose the Jacobian of Xm (and its corresponding motive) according to these idempotents. The sum ~ e× Q [Am] over all the Fermat characters is Gal(Q/Q)-invariant, and hence descends to a Q-algebra ~m; the characters of ~m are precisely the Fermat characters. Thus, H B ( X ) is a free
=

dyl Yl

Let a0 =: - a l - a2. Then if no al is divisible by m, wa is a differential of the second kind on Xm, and hence defines a cohomology class ~a in H ~ R ( X m / Q ). One sees immediately that ~a E HX, where Xi is the image of - a J m rnod Z. Following Coleman's observations in [6], we shall find it slightly more convenient to consider

=: -';a0a l(x0) -Ixl

(3)

and its corresponding class ~x E HDR. Let A denote the standard one-simplex in R ~, let e,; = e ri/rn, and let ~r : A ---, Ym

by

(.

.~ i / r n

(tl,t2) ~-* ~cm~l

.~ i / r n ~

,cm~2

)

The group algebra Q[Am] operates on the vector space of Q-chains of Ym(C), and we 2 in the i th place and zero elsewhere. Then let 8i be the element of Am which is em rf =: (I - 6 1 ) ( 1 - 6 2 ) 0 " is a cycle of Ym(C), and hence defines an element r/of H B ( X m ) , dual to the homology basis rf. If rn = dn, there is an obvious map frn,,: Xm -" Xn, defined by its action on points: fm,,~(z0 : xl : z2) : (x0d: z d : x~), and on coordinates by ]~n,n(Y~) : Yds" This map is compatible with the actions of Am and An, via the homomorphism Am --~ An sending an element to its dth power. One finds easily that f~,,~(wx) = w x and fm,,*(~') = 71'. We can conclude: C o r o l l a r y 2.2 If n divides m, the natural map Crn --~ ~n induces an isomorphism of bimarked ¢.-motives

• To describe the classical periods, we shall find it convenient to define, for any z E C,

r . ( z ) =: (2~ri)-tr(z). If z 6 Q / Z , we let F,(z) =: F,((z)). P r o p o s i t i o n 2.3 The cohomology class r1 forms a era-basis of H B ( X m ) , and the class =: ~--~x~x forms a chin-basis of H o n ( X m / Q ) . The classical periods of X m are given by

= r , ( o ) / - I r.(x,) '

i

328

Proof: This formula is essentially classical; we follow the treatment of [11]. It can probably also be found, at least implicitly in [1]. T h e following formula is contained in the proof of L e m m a 7.12 of [11]; it appears on the b o t t o m of page 89. Recall t h a t at and a2 are positive integers; we let a0 = : - a l - a2 and assume that no ai is divisible by m. Then: F(1--~)faw

a

--"'+'2---2rtal'rta2' = ,,2¢,~ .... ~m)-tm

(4)

/

Recall the formula: r(s)r(1

- s) = ~ ( s i . ~ )

-~ = ( 2 ~ 0 ~ - ' " ( I

- ~-~"')-'

0)

Hence, r(

) =

2~ir(1

-

-~)-le;,a'(1 -e~2~')-1

(6)

Now suppose t h a t X is a Fermat character and ai = m ( - x i ) for i = 1,2. T h e n also (Xi) = 1 - a i / m for i = 1,2, and hence

r(~) = r.(x,>-'(;,"'(1

-

~;~,)-i

(7)

Substituting this into (4), we find

r(1-~) f ~a=a~m-2r.(xl)-~r.(x2)-l(1-c;,2"')-1(1-~2~)-~

(8)

Continuing to follow [11], we note that if A E Am, A*Wa = X(A)Wa. Hence the group algebra element 6i operates by 6~wa = eVn2a'wa . Let ~ =: (1 - 6t)(1 - 62); then we can write:

= =

f(1 - ~1)'(1 - ~)'~a )/(1 - ~72°,)(1 - ~;2o~)~a

=

a~rn-2r(1-

~)-~r.(x~)-~r.(x~)

-~

(9)

m

Note that 1 - ~o = (x0) + 3 - ( x ) m

T h e Hodge level ( X ) - 1 of H x is either zero or one, and using the fact that F ( s + l ) = s t ( s ) one checks t h a t in either case,

F(I - ~) = 2~i-a°m(X°)2-{×)F,(Xo)

(I0)

Substituting 10 into 9 gives 2

f wa = -a2m-lao-l(Xo)(x)-2(2ri) -1 H r'(Xi)-I t"

i=0

(I1)

329

Recalling that r , ( 0 ) = (2~ri) -1 and the definition (3) o f w x, we find that

~

= r , (o) l ] r . (x,)-l,

as claimed.

|

For the crystalline analogue, we define, for a E Q / Z with denominator prime to p,

rv(a) =: rp((a)), where Fv is Morita's p-adic F-function. Proposition

2.4 I r e E W~i8 has degree one and p does not divide m,

¢ ~ = (_p)Ix)-I I ]

rA-vx,)-%~

i

Proof: This follows from Coleman's formulas in [5]; we simply need a dictionary comparing his notation to ours. His (r,s) is ( - ; ( 1 , - X 2 ) , so that r + s = Xo, and his e(r,s) is 2 - (X). Furthermore, let us note that al

a~

al

a2

af

a~-rn

d(~ Y2 ) = [(~ + ~2)y~ Y2 + a2U~ Y2

]dyl/y~,

and hence

a~(o,,o~-.~) = ,,o~a + d(u;' u~) Using also the fact that dyl/yl = -y~(y2/yl)d(yl/Y2), we find that wx is cohomologous to

-m(Xo)2-(x)~(~,,.~_.~) = m(Xo)~-(×)y~'y~u~lyld(y~ly2) This is just mr]r,~ in Coleman's notation, and as ( ( - r , - s ) = 2 - ( - X ) = (X) - 1, we find t h a t his Corollary (1.9) of [5] is precisely our formula. • Let ~ -- ~ x ~ X , which forms a ~m-basis for HDR(X/Q) as a era-algebra. Recall that CM(q~m) is a subset of the set of functions F,~ ~ Q and that T v , , ( C ) can be viewed as the set of functions Fm --~ C* (and analogously for Kp). It is clear t h a t the cocycle 7~(X,~, 7/) in ZI(Wp, Tcm(K~,)) is determined by its values on elements of degree one. T h e n in the language of the previous section, we can state: Theorem

2.5 The CM-type and periods of the marked ~m-motive (Xm,~, ~) are given

as follows:

¢(x~)(x) = ~ ( x , ) - 1 i

~(x~,~,~) = r~(o)y[r~(x,) -1 i

and, if deg ¢ = 1 and p does not divide m,

~.(x~,~., ¢)(x) = (-v) I¢-'~)-11-I r~(-x,) -1 i

We shall discuss the case in which p divides m in this next section. T h e symmetric group $3 also acts on Xm, and it is clear that ~r*H× = Ha× for o" E $3 and X a Fermat character. However, we shall need the following more delicate remark:

330

Lemma

2.6 I r a E Sa and X is a F e r m i ! character, i'~× = sgn(a)~a×

Proof: It suffices to check this for the two transpositions r =: (1, 2) and i = : (0,2), as these generate $3. Let z =: yl/y2, and ai =: ( - X i ) , so that cox = m ( x o ) 2 - ( × ) Y ~ ' Y ~ d z / z " Since r ' z = z -1, we have r * ( d z / z ) = - d z / z and

~'~x = -re(x0) 2-('~t v~' v?" d~ / z = - ~ , ~ We have i * y l = Yl/Y2 and o'* y2 = y~1, so i* z = Yl and i * ( d z / z ) = d y l / y l = - y ~ d z / z . Let X' =: a)C and a i = : ( - X : ) . Then

i'co~ = - . , ( x o ) ~- I~ v?~ v~?- °~- ~ dz / ~

(12 )

We consider separately the two possibilities (X) = 1, 2, i.e. al + a 2 > m and al + a 2 < m, respectively. In the second case, !

and we have if*co X

=

-my~'y~-a'-a~dzlz

=

-mv~,v;~d~/~

=-

--COX~

I

!

If (X) = 1, we have a~-'m(-x0 ) rn(x0)

= =

2m-al-a2 ! al -4-a2-- rrt-- r n - - a 2 !

Substituting these expressions into equation 12 and using also ,

,

,

.i

.~

,

,

=

(a!'~- m)V~V2

=

' ' -4- d(y ~i y 4 - . , - ( a ' 1 + a'2 - m)y~'y;~

)

=

-,~(x~)v~ ~v~; + d(v°; v,~--,).

d(v°'v % - ' ) = ( ~ + ~ - m)vl v~ + ( 4 - ,~)y~o, V~°~-m, we find that

i'cox

i

a~-rrt

Thus, we see that i*~× = -~o× in this case also.

II

Each element of $3 induces an automorphism i of (I),,~, and hence we can pull back any ~,,,-motive by i , obtaining an action of $3 on the group of bimarked (I)m-motives. C o r o n a r y 2.7 If i E 83, i * ( = s g n ( i ) ~ and i * q = sgn(~l)T1. Consequently there is an isomorphism of bimarked ~m-motives cr* (X, ~, 71) ~ (X, ~, r/). Proof: It is clear from L e m m a 2.6 that i*(~) = s g n ( i ) ~ . Since f o . , , ( = f , , i * ~ sgn(cr) f , , ~, we must also have i,.r/' = s g n ( i ) q ' and hence i * ~ = s g n ( i ) q .

= •

331

3

T h e A b e l i a n Case

In this section we give the "explicit" formula for the periods described in section 1 when E is abelian and p is unramified in E. It suffices to treat the ease of a cyclotomic field Em =: q ( # m ) C Q. Actually we shall find it more convenient to work with the algebra Cm =: Q [ T ] / ( T 'n - 1) ~ I'Idl,n Ed. If m = en, the map T ~ T e induces a map Cm ~ Cn. This map can be identified with the obvious projection l~dl,,~ Ea --~ rIaln Ea. For any Q-algebra R, Mor(Cm, R) is canonically isomorphic to the set /~m(R) of mth roots of unity of R. In particular, if we fix an orientation on C, there is a natural bijection:

m - I Z / Z "*/~m : a ~-* e 2'ri~ and if K is a field containing Q, we have a natural isomorphism

T

(K) "~ K ""~ "~ K m - I Z / Z

LFrom now on we make these identifications without further comment. Notice that U,~ acts naturally on/~m ~ m - l Z / Z . Furthermore, if n divides m, we have natural maps r:Um---*Un

and

i:#n--*#m

and i(~r(u)a) = ui(a) i f u E Um and a e /J~,. Thus we can let U =: limUm act on Q / Z = li__.mrn-lZ/Z ~/Jco, compatibly with our actions at finite levels. Furthermore, we let C stand for the inverse system {Crn : m E Z+}, regarded as a pro-object, so that M o r ( C , R ) =: ~l_~mMor(C,~,R) = #co(R). If K is a field containing Q, we have an identification T c ( K ) "~=K " Q / Z "~=K *~'~ The character group of the torus T c over Q is the free abelian group on Q / Z , which we denote by B Q / Z . Similarly, the character group of T c ~ is the free abelian group on m - I Z / Z , which we denote by Bm Our first goal is to factor the bimarked motive (Xm,~,r/) in a way which reflects the factorization 2.5 of its periods into a product of values of the F-function. This step can be viewed as a geometric incarnation of Anderson's factorization in [1]. Strictly speaking, the factors Mm will not be motives, but "ulterior motives," i.e. elements of the abelian group BMot(Cm) ® Q. If n divides m, the map Crn --* C,~ will induce an isomorphism Mm ®c= Ca "~ M,~. We then say, by abuse of language, that we have constructed an "ulterior motive M on the pro-algebra C." For example, the system {(Xm,~,T/) : m E Z +} forms an ulterior motive on the pro-algebra ¢ =: "li.___m"¢,,~. Because the inverse system {Cm : m E Z +} can be identified with the product I-Ira Era, to give such a motive M on C is the same as giving a system of motives Am =: M ® C Em on each E,n (with no compatibilities). For each i E {0, 1,2}, let j~:/~rn --~ Am be the map inserting ~ into the i th place, and also use jl for the corresponding map C ---*Cm "-'* Q[Am] "-* era. Recall that Xm(--1) is the Tate twist of the Fermat curve of degree m; it has as a bimarking the pair (~, ~l/2~ri). T h e o r e m 3.1 There is a bimarked ulterior C-motive (M,~,r/) of rank one with the following properties:

332

1. The bimarked CM-type rBM( X (--1),~, ~/2~i) o / X ( - 1 ) is isomorphic to the product [Lj~rsM(M,£,,7) in BCM(O) . 2. The CM-type rM 6 Q Q / Z of M is the function ( ) . 3. The complex period 7 6 T o ( C ) ® Q of(M,~, 7) is the function F~( ) - 1 Proof: Let E by the cyclotomic field Q((m). If a:C ---* E is any homomorphism, the ulterior CM-type ( ) on C induces by pullback an ulterior CM-type a ' ( ) on E. As T c ( E ) is the free abelian group on the set of all such a, we obtain in this way a homomorphism rE: T c ( E ) ---* C M ( E ) ® Q : aa ~-* a ' ( ) We construct a homomorphism FE( ): T c ( E ) ---* T c ( C ) ® Q in the same way. Specifically, if we view T c ( E ) as the free abelian group on Q / Z , and if a =: ~ a~ is an element of T o ( E ) , then we have: tea(,.,)

= =

If X: (I, --* E is a Fermat character, let X x ( - 1 ) =: x ' ( X ( - 1 ) , ~ , ,7/2ri) 6 BMot(E), and let a× =: ~ i Xi. It follows from Proposition 2.7 that X× is independent of the ordering of (X0,X1,X~), i.e. depends only on a x. We shall prove Theorem 3.1 simultaneously with the following result. T h e o r e m 3.2 Let E be the cyclotomic field U,~Q(~m). There is a homomorphism

F: B Q / z ® Q --* BMot(E) ® Q with the following properties: 1. Ira 6 q / z z Mor(C,E), F(cra) = a'M. 2. The CM-type r 6 Qu o f F ( a ) is the function rE(or). 3. The complex period 3' 6 T E ( C ) ® Q o f F ( a ) is the function F~(a) -1. 4. F is Gal(E/Q) equivariant and maps B~n ® Q to BMot( Ern) ® Q. 5. For any Fermat character X, rBM(X

(- 1)) =

Proofs: If we have defined a Galois equivariant homomorphism F satisfying 3.2.2-3.2.5, then in particular for each m 6 Z + we have an ulterior Era-motive F(al/m). As explained above, the set of all these defines an ulterior C-motive M such that 1/m*M = F(al/,n). Then for any u 6 U,~, u / m * M = u*(1/m'M) = u'F(al/r,,) = F(au/,n), by the Galois equivariance, and hence 3.2.1 holds. Then 3.1.2 and 3.1.3 follow from the corresponding statements 3.2.2 and 3.2.3; 3.1.1 follows immediately also, e.g. from Theorem 2.5. To prove 3.2, it will suffice to define the map F on any basis for B Q / z ® Q. The natural basis Q / Z is not as convenient as the one provided by the following lemma.

333

L e m m a 3.3 For each b E Q / Z , let fb =: 2at + a-2b. b E Q / Z , / s a basis/'or the vector space B Q / z ® Q.

Then the set of all fb, /'or all

Proof: It suffices to show that the set of all fb's for b E m - X Z / Z spans B,,. Let B " be the space thus spanned. It suffices to show that for each for each a E m-XZ/Z, a~ belongs to B ' . It is clear that - 2 a ~ ~- a - 2 , mod B ' , and by induction (-2)Jct~ ~ a(_:), a for all j. Since m - l Z / Z is a finite set, there exist two distinct integers j and k such that ( - 2 ) i a = ( - 2 ) ~ a , whence (-2)1c~a -~ ( - 2 ) k c ~ mod B " and a . E B ' . It Returning to the proof of 3.2, we note first that Eo = El = E2 = Q. We define

F(fo) =: ( 3 Q ( - 1 ) , 1,(27ri) -3)

and

F(fl/~) =: (2Q(-1),2i,(27ri)-2) •

(13)

Because fo = 3So, F(c~o) is ( Q ( - 1 ) , 1 , ( 2 r i ) - 1 ) , and we can check 3.2.1 and 3.2.2 for (~o instead of )co.

tQ(~0)

=

(0>

=

1 r(Q(-1))

=

r Q ( ~ 0 ) -1

=

( r , ( 0 ) ) -1

= =

2:¢i 7(E(-1),1,(2~ri) -1)

Checking fl/2 = 2ai/2 + a0, we note that

tQ(fl/2) F Q ( f l / ~ ) -1

= = = = =

2(1/2) + 1 = 2 2~(Q(-1)) ( 2 ~ r i ) a r ( 1 / 2 ) - 2 r ( 1 ) -1 (2ri)2(2i) 7 ( 2 Q ( - 1 ) , 2 i , ( 2 7 r i ) -2)

Now i f a ~ {0, 1/2} and we choose an ordering of the set of elements { a , a , - 2 a } we find a Fermat character X. Notice that the isomorphism class of the bimaxked E-motive X× =: x*X,~ is in fact independent of the ordering, by Corollary 2.7. It is also independent of the choice of m, by 2.2 We define F(fa) =: X × ( - 1 ) . Then Theorem 2.5 shows that the CM-type and periods of F(f~) are as predicted by the statement of the theorem. The Gal(E/Q)-equivariance of F follows from Remark 1.6. Statement 3.2.5 is just a restatement of the first two, using the definition of B C M and the formulas 2.5. II R e m a r k 3.4 Although we deduced Theorem 3.1 from Theorem 3.2, let us note that it is also true that Theorem 3.2 is an easy consequence of Theorem 3.1. Indeed, we can use 3.2.1 above to define the homomorphism F. Then properties 3.2.2 and 3.2.3 follow from their counterparts 3.1.2 and 3.1.3, and 3.2.5 follows from theorem 2.5. The Galois equivariance follows from Remark 1.6.

334

R e m a r k 3.5 The Hodge conjecture for all products of Fermat varieties would imply that the factorization 3.1.1 above takes place in BMot(C,~) ® Q, not just in BCM(Crn)® Q. In fact, it seems likely that one could in fact verify the algebraicity of the specific Hodge cycles needed for this statement directly. This would free Coleman's calculation of the Fermat period matrix in the bad reduction case from its dependence on Theorem 1.8, if not entirely from the formalism developed here. Unfortunately we have not had time to complete this task, and we will simply refer to the paper [2] in which a similar calculation was carried out based on Shioda's "inductive structure" of Fermat varieties [18]. Let us also note that our construction of F can be refined in an obvious way to produce a homomorphism from the subgroup BF of BQ/Z generated by the set of all a x for all Fermat characters X to BCM(E). One simply defines F(a×) to be the bimarked E-motive of Xx, and verifies that F prolongs to a homomorphism by inspection from Theorem 2.5. Conjecturely, one would like to say that F factors through BMot(E). We are now ready to look for our p-adic analogue of the Chowla-Selberg formula. Attached to the marked ulterior C-motive (M,~) is the cocycle 7e(M,~) E Z'(W~, Tc( K~)) ® Q ~ Zl(We, T c ( K e ) / l ~ ) Our aim is to give an explicit formula for this cocycle in terms of the Morita p-adic gamma-function. We shall need the following proposition. P r o p o s i t i o n 3.6 Suppose G E Zl(W~,Tc(Kp)/#oo), and let

B

=:

I-IjTa i

Suppose that for every ¢ E Wp , 7p(X(-1),~,¢)(a,a,-2a)

= B(¢)(a,a,-2a),

a(¢)(0) = v eg¢

,

if

a(¢)(1/2)

a E Q/Z\{0,1/2}

and

= v d°g¢/2

Then *[p(M,() = G and 7~,(Xm(-1),() = B. Proof: Fix ¢ E W e of degree one, and define homomorphisms 3`¢ , G 0 : BQ/Z ---*K~,/l~oo 7¢(a) =: H [ 3 ` p ( M , ( , e ) ( a ) ] '~"

and

by :

G ¢ ( a ) =: H [ G ( ¢ ) a ] c'' a

For any Fermat character X, let

and

Be(X) =: B(¢)(X) = I ] a t ( x , ) i

We have to prove that 8¢ = Be and that 3'¢ = G¢. By the constructions in Theorems 3.2 and 3.1, we also have the bimarked motive Era-motive F(a×) = IL x~M.

335 Suppose first that a E Q / Z \ { 0 , 1 / 2 } . Then X =: (a, a , - 2 a ) is a Fermat character, and by hypothesis, Be(X) = ~¢(X). On the other hand, F was constructed so that F(fa) = x ' ( X m ( - 1 ) , ~ ) . It follows that 7¢(fa)

= = = =

7p(r(fa), ¢)(1)

"/~,(X"(X (- I), ~, ".,b)(I)

"r,,(x(-i), ¢)(x) ~(x)

=

B(¢)(x)

=

Go(f~).

Furthermore, G¢(Y0) = C¢(~0) 3 = p3

and

7¢(f0) = [Tp(M,¢)(0)] a = 7 p ( Q ( - 1 ) a, 1) = p3

Similarly,

G¢(II/~) = p2 and r~(Yt/2) = 7 A Q ( - 1 ) L

2i) = - p 2 ¢ ( i ) i

(Recall that we are working modulo roots of unity, so -psi(i)i = 1.) As the set of all f , ' s forms a basis for BQ/Z, it follows that 7¢ = G¢. Now 3.1.1 tells us that the marked CM-type of ( X , , ( - 1 ) , ~ ) is l a rDMj~(M), and hence by Theorem 1.8 we can conclude that 8¢ = I l i J~7¢ = Be. II R e m a r k 3.7 Coleman has asked me to point out that Proposition 3.6 can be refined in the following way. Let S C Q / Z be a subset stable under GaI(-Q/Q), corresponding to quotient C ---* Cs of the pro-algebra C. If S is stable by multiplication by 2, then Lemma 3.3 still applies, and we find that the Bs ® Q has as a basis the set of elements fa : a E S. Let @s be the quotient of @ whose characters consist of the Fermat characters X such that each Xi belongs to S. Finally, let Ms and X s be the bimarked ulterior motives on Cs and @s respectively obtained from M and X by base change. Then Proposition 3.6 applies to Ms and X s , with the obvious modification. Proposition 3.6 shows that a determination of the action of W r, on the part of the cohomology of X,n corresponding to the Fermat characters of the form (a, a,-2a) is enough to determine the entire action. Remarkably, this is precisely the portion of the action that Coleman is able to compute in the case in which p divides m, and so he is able (with the aid of an additional congruence to pin down the roots of unity) to obtain an exact formula for the entire matrix. We refer to his paper [5] for the details and formulas. In anticipation of the aesthetics of our eventual Theorem 3.13, we make the following definition (in the general case). D e f i n i t i o n 3.8 I r e • W~ and a • Q / Z , F~,(¢)(a) =: 7~,(M,~,¢)(a) • Kr,/pao. We shall justify this definition by giving an explicit formula for Fp in terms of the Morita p-adic F-function in the unramified case. Thus, we let C' denote the limit of the Cm's over the multiplicative set of m's which are prime to p, and let M ' be M ®c C' Then T c , ( K p ) ~ K~ Q/Z', where Q / Z ' is the subgroup of Q / Z consisting of the primeto-p torsion. Note that Wo operates on Q/Z1 by the rule ¢(a) = pdeg~a. Since we are working modulo torsion, we can neglect roots of unity--in particular, signs. For any rational number r, we have a well-defined element pr of Kp/poo.

336

r, e zx(w~, Tc,(Kp)/~oo) is the unique co-

T h e o r e m 3.0 With the above notation, cycle Fp

such that rA¢)(a) =

p(¢-'")r~(~)

whenever deg ¢ = 1

Explicitly, ifd =: deg ¢, d-1

rA¢)(~> = 1-Ipl¢ .... olr;(¢,~> i=0

Proof: Since the elements pr 6 K~/poo and the values of the p-adic gamma-function are fixed by W~,, it is clear that the formulas above do indeed define a unique cocycle Fp with the prescribed values on elements of degree one. To prove that the formula is correct, it suffices to apply Proposition 3.6 and Remark 3.7. Indeed, if X is any Fermat character, Theorem 2.5 tells us that for all ¢ of degree one and all a 6 Q / Z ' 7p(X(-1),(,¢)(X)

P(¢-'×)1-Ir.(-x,>-'

=

/

Since Fp(1 - z)Pp(z) = -4-1 and we are working modulo roots of unity, we find

= p(¢-'~l 1-I r~(x,>

7p(X,~ ( - 1 ) , ~, ¢)(X)

i

=

1-Ip(¢-'~')r,(x,> i

i

Finally, we note that

rA¢)(0)

=

v(°lrA0)

--- p and Fp(¢)(1/2)

=

p(ll2Irp(I/2) pi12

Thus Proposition 3.6 applies, with g =: F~. Knowledge of the periods of the ulterior motive C determines the periods of all ulterior E-motives (with E abelian), because of the following fact. P r o p o s i t i o n 3.10 I r E is a cyclotomic field, the map ~E: T c ( E ) ® Q --~ C M ( E ) ® Q

described above is surjective.

337

Proof: This proposition is well-known, but we give the proof in a form which is computationally useful, for example, for our proof of the Chowla-Selberg formulas for elliptic curves. The argument is essentially taken from [16]. We use the action of the group Urn on the set m - I Z / Z . For each divisor d of m, the stabilizer subgroup of the image of 1/d in m - I Z / Z is the subgroup Id C_ Urn of elements congruent to 1 mod d, and the orbit is naturally isomorphic to Ud ~ Um/Id, The orbits are thus in bijeetion with the divisors of m, and hence we obtain a canonical direct sum decomposition &dim where A~ is the free abelian group generated by Ud. If a E B,~, we write ct = ~ d a~ under this decomposition. Thus, a~ = a~/d for v E Ud. To show that our map t E is surjective, we may first complexify it to obtain a map Bm® C ~ CM(Em) ® C, also denoted by t. Regard CM(Em) ® C as a linear subspace of the space of all functions from the abelian group Um --* C. This space has a natural inner product structure, corresponding to the Haar measure on the group U,n in which Um has measure 1, and the set of complex characters of U,n forms an orthonormal basis. Observe first that C M ( E , , ) ® C is precisely the span of the trivial character and the set of odd characters. Indeed, it is clear that each of these characters belongs to C M ( E , 0 ® C. On the other hand, if X is an even character and f E CM(E,n) ® C has weight w, then one finds easily that that < f, X > vanishes if X is nontrivial and is w/2 if X is trivial. Thus, to show that t is surjective, it suffices to show that its image contains the trivial character and every odd character. As we have seen, too is twice the trivial character. Suppose that e is nontrivial and odd, and suppose that e is the smallest divisor of m such that e factors through U,,, ~ U~; let e~ be the corresponding character of U~. C l a i m 3.11 t(tr~) = - L ( 0 , e~)e~, where

e~(u)a~,/~ e Bm® C

c~ =: Z ufiU.

Proof: Suppose a is any member of Bm® C and X is a character of U,~. We recall from [16] that <

x >=

-

L(O,

<

>,

d

where the sum is taken over all the divisors d of m such that X is pulled back from a character Xd of Ud. But ~t lives on the orbit of I/e, and so a~ = 0 if d # e, while a~ = ee. Thus, < t~,X > is zero unless X = e, and < Q,e > = -L(O, ee). As the set of all such X'S form an orthonormal basis, the claim is proved. Since e~ is an odd primitive character, the factor L(0, ee) does not vanish, and thus 3.11 shows that e lies in the image of t. II R e m a r k 3.12 An explicit set of generators for the kernel of

t E was

given in [16].

T h e o r e m 3.13 Let E be the cyclotomic field Q((,,). Suppose that X E Mot(E) ® Q

and a E Bm® q are such that r ( X ) = rE(a). Then

338

I. 7 ( X ) is the class o f F , ( a ) - '

in

2. 7p(X) is the class o f F , ( a )

in

TE(C)/T~(Q)

HI(Wt~,TB(Kp;'Q))

Proof: Consider the bimarked ulterior E-motive F(a) constructed in 3.2, and let X' be F(cQ stripped of its markings. It is clear that r ( X ) = r ( X ' ) , and hence Theorem 1.7 implies that 7(X) = 7(X') and that 7,,(X) = 7p(X'). Thus the formulas 1 and 2 follow from 3.2.3 and 3.8, respectively. | Here is a slight refinement: C o r o l l a r y 3.14 Suppose X, E, and a are as above. H n R ( X / Q ) and 'Ifor HB(X) such that

i. ~ ( x , , , , ~ ) = r / ( ~ , ) - '

2. 7,~(X,Q=F~(a)

in

in

Then there exist bases ( for

TE(C)®Q

ZI(Wp, T £ : ( K p ) ) ® Q

Proof: The first statement of the previous theorem tells us precisely that there exist and 7/such that 3.14.1 holds. It follows that rBM(F(a)) = rBM(X,~,~) in B M ( E ) ® Q , and hence Theorem 1.8 implies that 7~(F(a)) = 7~(X,~). | Let us make this explicit for elliptic curves. Let E be the quadratic imaginary field with discriminant - m and let X/-Q be an elliptic curve with complex multiplication by E. Let r be the CM-type of X, which in this case amounts to just an embedding of E in C. If we use s to obtain an identification t of the set of embeddings of E in C with GaI(E/Q) ~- { i l } , then r = (t + 1)/2. The image of E under s is contained in the cyelotomic field Era, and the corresponding mapping on Galois groups Urn - - #2 can be regarded as a primitive character e of Urn. The pullback of X to Em then has as CM-type (e + 1)/2. According to 3.11 we can write -L(O,e)e = t~,, where a~ =: ~ u e(u)a,lmWrite L for the integer L(0, e), so that r ( X ®E Era) = r E ( l / 2 - a J L ) . T h e o r e m 3.15 With the above notation, there exist bases

and a basis 7}' for the singular homology of X such that

f f,

~,

=

'

(2=i)'/2 i~ r.
,,~ =

¢~,

(2~i1'/~ l-I r . < - ~ / m ) <~)/2L

m c'/~oo

uEU,~

= p~,~/2 l-I r A c ) ( ¢ ~ l m ) - ' ( " ) / 2 ~

, in K~l~oo

U

N

Proof: If we view s as a m a p E --~ Era, s * X is an E,~ motive, and its CM-type is tE(c~0 - a,/L). Then our formulas follow immediately from Theorem 3.13 and the

definitions.

|

339

4

Absolute Hodge Cycles

In this section we explain the proof of Theorems 1.7 and 1.8 from the point of view of the theory of absolute Hodge cycles, following the theme of [11]. T h e o r e m 4.1 Let X =: (X1,X2....Xn) be a family of abelian varieties and projective

spaces over K~, with good reduction and T a tensor construction as explained in [17]. Suppose that ~Da 6 Toa(X) is a Hodge c/ass, i.e. ~Dn 6 F°Ton(X) and crn(~oR) 6 TB(X). Then ~Da is fixed by W,,(K,,). Proof of 1.7 and 1.8: If (X, ~) is a marked E,~-motive of rank one, its cohomology groups can be expressed as a tensor construction involving abelian varieties and projective spaces having good reduction everywhere. If r(X) = O, HoR(X) is purely of type (0,0) and so HDR(X) ® C is trivially generated by Hodge classes. In particular, if rDM(X,~) = 0 in T E ( C ) / T E ( Q ) , an(~) 6 H e ( X ) is a Hodge class, so 4.1 implies that Wp fixes ~ and hence 7~,(X,~) = 1. This proves 1.8. To prove 1.7, we note that Deligne's theory of absolute Hodge cycles [11] tells us that the Hodge classes of HDR(X) ® C lie in fact in HDR(X/-Q). Hence we may multiply ( by an element of T E ( Q ) then and apply the previous argument. II Theorem 4.1 follows from a stronger result, conjectured by Deligne and proved by Blasius. In order to state it, we have to use the additional structure on motives provided by the Fontaine-Messing-Faltings machinery comparing De Rham and 6tale cohomologies [14,12]. Recall that there is a field BDR D_K~ endowed with a canonical filtration F and an action of the Galois group G := Gal(K~,/Qp). Furthermore [12], for each smooth projective scheme X/K~, there is a canonical isomorphism:

a~t: HDR(X/-Q) ® BDR --' H~,(X/Kt,) ® BDR If X / K ~ is obtained by base extension from X/-Q, we also have natural isomorphisms (which we regard as identifications):

H~,(X/-Q, Qp) H~t(X/-Q, Qp) Het(X/C,Qp)

H~,(X/K~,Qp) H~,(X/C, Qp) ~- HB(X)®Qp

~ ~

T h e o r e m 4.2 (Blasius) In the notation of Theorem 4.1, suppose that ~ fi TDR(X) is a Hodge class. Then cr~t(~) = ~ro(~). Proof: For the reader's convenience we include a simplified and (somewhat generalized) version of Blasius's proof. Following Deligne's strategy in [11], we have to establish principles A and B in the above context. Principle A is quite trivial: Suppose that 0i 6 Ti,B for i = 1 . . . n - 1, and let G C_GI(HB(X)) be the subgroup consisting of those elements g such that gr/i = rh for all i; regarded as an algebraic group over Q. Suppose that each r/i is a Hodge class and let ~i denote the corresponding element in Ti,DR(X/C); recall that in fact ~i necessarily lies in T~,DR(X/-Q) by the main result of [11]. Now let (~,, r/n) be the DeRham and Betti components of another Hodge class, and let G' C_ G be group fixing (r/x,... ,r/n-l,rln). Suppose that a~(~i) = rli for all i < n and that G' = G. Then I claim that a~t(~,) = On ahso. To see this, we adapt the argument of [11] and [17].

340

For any Q'--algebra A, let P(A) denote the set of isomorphisms ¢: HDR(X/'Q) ~

A ---*

HB(X) ®Q A such that ¢(() - r/. It is clear that P is representable by a scheme of finite type over Q, and that P(A) is a G(A)-pseudotorsor, i.e. is either empty or a G(A)-torsor. The similarly defined P ' is representable by a closed subscheme of P, and is a G' pseudotorsor. By hypothesis, G' = G, and it follows that P'(A) = P(A) provided that P'(A) is not empty. Since the classical period isomorphism crB belongs to P ' ( C ) , we see that P(C) is not empty, and by Hilbert's Nullstellensatz P(Q) is also not empty, so in fact P' = P. By hypothesis, a~, belongs to P(BDR), and hence also to P'(BDR); this says exactly that cr~,((n) = r/n. We next must prove Principle B. The following is a generalization and simplification of Blasius's proof. P r o p o s i t i o n 4.3 Let f :X --~ S be a smooth proper morphism of Q-schemes of finite type, with S connected, and let 77 be a global section of F°Tfa.'*Q, where T a tensor construction as above. Suppose that for some so E S(Q), a~(~)(so)vR) = 71(so)~t. Then the same is true at every point. Proof: Let X be a smooth compactification of X; one knows by the theory of weights [9] and the degeneration of the Leray spectral sequence [8] that the image of

i;:T(X) --* T(X(s)) can be identified with H°(S, TIa'~Q). In particular, the kernel K(s) of i; is independent of s. Moreover, as i] is strictly compatible with the Hodge filtrations, one can choose an element ~ of F°T(-X) which lifts 7. It suffices to prove that a~,(r/DR) -- r/~, 6 K(s) for every s. As this is true for So by hypothesis, the proposition is proved. II The relationship between Theorems 4.2 and 4.1 uses also the existence of the filtered subring B¢~i, of BDR with its Frobenius endomorphism F. We obtain a semilinear action Pcris of Wv on Boris ® Kv in the obvious way, and there is a natural injection B~i, ®Kp ---* B D R . The following is an expression of Fontaine's conjecture on potentially crystalline representations [13], proved by Faltings in [12]. Messing has pointed out that for abelian varieties, (the only ease we need), this result can in fact be deduced from results in the literature. T h e o r e m 4.4 (Faltings) Suppose X / K p is a smooth projective variety with good reduction. Then there exists a natural isomorphism

Hon(X/K•) ® B~.i~ ® K~ ~- H~t(X/Kp, Qp) ® Beri~ ® K~, This isomorphism is compatible with the isomorphism a~t above, and under it the action of W~ on De Rham cohomology corresponds to idH~ ~ Peris. | It is clear that this result and Theorem 4.2 together imply Theorem 4.1.

References [1] Greg Anderson. Cyclotomy and a covering of the Taniyama group. Mal.herna~icae, 57(153-217), 1985.

Compositio

341

[2] Greg Anderson. Torsion points on Fermat varieties, roots of circular units, and relative singular homology. Duke Mathematics Journal, 54:501-561, 1987. [3] Pierre Berthelot and Arthur Ogus. F-isocrystals and de Rham cohomology I. Inventiones Mathematicae, 72:159-199, 1983. [4] Don Blasius. P-adic ttodge cycle on abelian varieties, in preparation. [5] Robert Coleman. On the Frobenius matrices of Fermat curves. This volume. [6] Robert Coleman. The Gross-Koblitz formula. Advanced Studies in Pure Mathematics, 12:21-52, 1987. [7] Ehud de Shalit. On monomial relations between p-adic periods. Journal f~r die Reine und Angewandte Mathematik (Crelle), 374:193-207, 1987. [8] Pierre Deligne. Th~or~me de Lefschetz et crit~res de d~g~n~scenee de suites spectrales. Publications Mathgmatiques de I'I.H.E.S., 35:107-126, 1968. [9] Pierre Deligne. Th~orie de ttodge II. Publications Mathgmatiques de 17.H.E.S., 40:5-57, 1972. [I0] Pierre Deligne. Valeurs de fonetions L et p~riodes d'int~grales. Symposia in Pure Mathematics, 33(part 2):313-346, 1979.

Proceedings of

[11] Pierre Deligne. Hodge cycles on Abelian varieties. In Hodge Cycles, Motives, and Shimura Varieties, volume 900 of Lecture Notes in Mathematics. Springer Verlag, 1982. [12] Gerd Faltings. Crystalline ¢ohomology and p-adic galois representations. To appear in the American Journal of Mathematics. [13] Jean-Marc Fontaine. Sur certains types de representations p-adiques du groupe de Galois d'un corps local, construction d'un anneau de Barsotti-Tate. Annals of Mathematics, 115:529-577, 1982. [14] Jean-Marc Fontaine and William Messing. P-adic periods and p-adi¢ ~tale cohomology. In Current Trends in Arithmetical Algebraic Geometry, volume 67 of Contemporary Mathematics. American Mathematical Society, 1985. [15] Benedict Gross. On the periods of Abelian integrals and a formula of Chowla and Selberg. Inventiones Mathematicae, 45:193-211, 1978. [16] Neal Koblitz and Arthur Ogus. Algebraicity of some products of values of the 7-function. Proceedings of Symposia in Pure Mathematics, 33:343-346, 1979. (Appendix to "Valeurs de Fonctions L e t P~riodes d'Int~grales" by P. Deligne). [17] Arthur Ogus. ttodge cycles and crystalline cohomology. In Hodge Cycles, Motives, and Shimura Varieties, volume 900 of Lecture Notes in Mathematics. Springer Verlag, 1982. [18] Tetsuji Shioda and Toshiyuki Katsura. On Fermat varieties. Tohoku Mathematical Journal, 31(1):97-115, 1979.

THE COMPLEMENTATION

PROPERTY

OF g ~

IN p-ADIC BANACH SPACES

Wim H. Schikhof Department of Mathematics, K.U. Nijmegen Toernooiveld, 6525 ED Nijmegen, The Netherlands

A B S T R A C T . Let K be a non-archimedean valued field whose valuation is complete and nontrivial. It is shown that go° is complemented in any polar K-Banach space (Theorem 1.2) which opens the way for various descriptions of complemented subspaces of gc¢ (Theorem 2.3).

I N T R O D U C T I O N . (For unexplained terms see below.) Let K be as above. If K is spherically complete and E is a K-Banach space then, like in the 'classical' theory, each closed subspace is weakly closed, each subspace D of E has the weak extension property (WEP) i.e. each f E D' can be extended to an y E E'. It is also well known that these statements are false if K is not spherically complete. In fact, the following questions are posed in [1]. Q.1. Does every weakly closed subspace have the WEP ? Q.2. Is every closed subspace having the WEP weakly closed? It was a close look at these questions (for the (negative) answers, see the Remark following Proposition 1.5 and Example 3.3) that revealed the above-mentioned complementation property of g~.

Again, for spherically complete K this is nothing new as it follows directly from the

spherical completeness of gc~. However, for nonspherically complete K the results came as a surprise. PRELIMINARIES.

Let K be as above. From now on until §5 we suppose that K is NOT

spherically complete. We use the terminology of [2]. P.1.

For an absolutely convex subset A of a K-vector space we set

A* :--- N{),A : A E K, IAI > 1}. A is edged if A = A% P.2.

Let E, F be K-Banach spaces. I:(E, F ) is the Banach space of all continuous linear maps

E --* F with the operator norm, E ' := I : ( E , K ) .

E and F are isomorphic if there exists a

surjeetive linear isometry E ~ F; We write E ,,~ F if E, F are linearly homeomorphic. We shall say that E and F are almost isomorphic if for each e > 0 there exists a linear homeomorphism

T:E--*Fwith

[[T[] < 1 + ¢ , liT-all < l + e .

343

T • £ : ( E , F ) is a quotien~ map if T maps {x • E : Hx[[ < 1} onto {x • F :

llx}[ < 1} . The

closed unit ball {x • E : [[x[[ < 1} is denoted BE. The n o r m closure of a set X C E is X. Its weak closure, i.e. the closure with respect to the weak topology (w-topology) a ( E , E ' ) , is ~'~. Similarly, if Y C E ' then ~--w' is the w~-closure of Y i.e. the closure with respect to the w'-topology

a(E',E). For X C E we set X ° : = { f • E ' : If(x)l _< I for all x • X } For Z C E ' we set X0 := {m • E : If(m)[ _< 1 for all f • Z} X is a polar Beg if (X°)o = X . Let A • •(E, F). If ~

is a neighbourhood of 0 in F then ABE = ABE. (The 'classical' proof

of (one half) of Banach's Open Mapping Theorem yields that ABE is a zero neighbourhood. By absolute convexity ABE is open, hence closed.) P.3. Let E , F be K - B a n a c h spaces. The n a t u r a l map j z : E --* E " satisfies HJEtl < 1. We call

E a polar Banach space (and the norm a polar norm) i f j E is an isometry into E". Following [2], E is topologically pseudoreflexive if JE is a linear homeomorphism into E". E is reflexive if jE is a n isomorphism. Recall ([2], 4.17) that co and ~oo are reflexive and duals of one another via the pairing (x, y) = ~ xny,, (x • co, y • £oo). In the same spirit, co(I) and g°°(I) are reflexive if the cardinality of I is nonmeasurable ([2], 4.21). For A • £ ( E , F )

we define its adjoint A' • £ ( F ' , E ' ) by A ' ( f ) = f o A ( f • F').

We have

HA'I[ < HAIl. The diagram

E

A.~

1~E E"

-Art '~

F

1" F"

commutes. A subspace D of E has the W E P in E if tim adjoint E ' --* D ' of the inclusion map D --+ E is surjective. P.4. The following two statements axe not hard to prove. 1. Let (E,[[ [[) be a K - B a n a e h space. Let ~1 > ~2 > . . . , l i m n ~ o o ~ , , = 0. I f p l , p 2 , . . . are polar norms on E such that l[xll < pn(x) < (1 -k e,~)llxl[ (n e N , x Z E ) then III[ is a polar norm. 2. If E is a polar K - B a n a c h space and E ,-, g ~ then E is almost isomorphic to £oo

(By

reflexivity, E s .-~ co. By choosing t-orthogonal bases of E ~, where t is close to 1 one proves that co and E ~ are almost isomorphic, hence so are E and goo.) P.5. Let A be a complete metrizable absolutely convex compactoid in a Hausdorff locally convex space E over K . Let A E K , ]AI > 1. Then there exist el~e~,.., in AA with lim,,--,oo e , = 0 such that, with B the closed absolutely convex hull of {el, e2 . . . . }, (i) d C B C ),A. (ii) Each x e B has a unique representation x = ~-'].n°°__lA , e , , where A,, • K , I)~,1 < 1 for each n. (iii) If An • K , [A,[ G 1 for each n then )-'~,°°=1A,,e, • B. (Proof. By [4], 6.1 A, as a topological module over {A • K : ]A] < 1}, is isomorphic to a closed compactoid of co. T h e n apply [2], 4.37 ( a ) =~ (~).)

344

P.6. (p-adic Alaoglu Theorem) Let E be a K - B a n a c h space. T h e n BE, is, for the w'-topology, a complete compactoid. (The proof is standard.)

§1 T H E C O M P L E M E N T A T I O N P R O P E R T Y OF goo. Proposition 1.1. Let E be a p,olar K-Banach space, let D be a closed subspace with inclusion map i : D --* E. Then (

~

)~ = BD,. .

P r o o f . Since

.

t

Ili'll < 1 and BD, is wl-cXosed and edged we have S := ( i ' ( B E , ) ) ~ C BD,. Suppose

we had a n f E BD,, f q~ S. As the w'-topology on D ' is of countable type and S is w'-closed and edged, S and {f} can be separated by a w'-continuous linear function D ' --* K ([3], 4.4 and 4.7). But such functions are evaluations ([5], Th. 4.10) so there is an x E D with [f(x)l > i and

Ih(x)l < 1 for all h E S. We have 1 < If(x)l _ Ilfll Ilxll ___ Ilxll so I1=11> 1. On the other hand, by polarity,

II=II= IIi(~)II=

sup{19 o i(z)l : g E B E , }

= sup{li'(g)(z)l : g e BE,}

_< sup{Ih(x)l : h E S} _< 1, which is a contradiction.

T h e o r e m 1.2. Let E be a subspace of some polar K - B a u s c h space X . Let E ~ goo. Then, for each t 6 (0, 1), E has a t-orthogonal complement in X . P r o o f . Let i : E -+ X be the inclusion map. I. We show that the adjoint i' : X ' --+ E I is a quotient map.

By reflexivity the w'-topology

on E ' equals the weak topology. The norm topology on E ' is strongly polar ([3], 4.4) so every norm-closed absolutely convex edged set is weakly closed. ([3], 4.9). Together with Proposition 1.1 this leads to .~1/1I

BE, = i ' ( B x , )

)~ = ( ~ ) ~

= (i'(Bx,)) ~

implying that i ' ( B x , ) is a norm neighbourhood of 0 in E ' . Then (see P.2) i ' ( B x , ) is closed so we find BE, = i ' ( B x , ) ~, which is what we wanted to prove. II. Let t E (0, 1). The space E I is of countable type so it has a v'~-orthogonal base el, e2, .... Now i' : X ' ~ E ' is a quotient map so we can choose zx,z2,... E X I such that i ' ( z , ) = e , , i i z , I] _< t-½[len][ for each n. The formula oo

T(Z n=l

oo

A~e~) = Z

A~z~

(A. E K, lim [[A.e.[[ = O) n~oo

n=l

defines a T E ,C(E',X') for which IITII _< t - I and i ' o T is the identity on E'. Then T ' o i" is the identity on E " and it is easily seen from the diagram Ell

~i"

X"

T-

1,;

---*

X

E

Y E

i

~T'

Ell

3;45

that P := j E z o T' o j x is in L:(X, E), has norui _< t -1 and is a projection onto E. R e m a r k . The space E of Theorem 1.2 is almost isomorphic to ~o~ (see P.4.2). C o r o l l a r y 1.3. Let E be a subapace of some topologically pseudoreflexive K - B a n a c h space X . Let E ,~ £o¢ Then E is complemented in X .

C o r o l l a r y 1.4. (i) Let E , X

be as in 1.2. Then, for each e > O, each f E E r can be extended to an ] E X r with

II]11 < (1 + e)ll$11. (ii) Let E , X

be as in 1.3. Then E has the W E P in X .

Corollary 1.4 yields the following result that may look bizarre at first sight. Proposition

1.5.

Suppose # K

is nonmeasurable.

Let E be a K - B a n a c h space such that E '

separates the points of E. Suppose E r ,'~ g~. TJ~en E ,,~ co.

P r o o f . j E : E -+ E " is injective, so # E < # F " = #co = # K a n d # E is nonmeasurable. Thus we can, in a s t a n d a r d way, construct a quotient map ~r : co(I) -+ E where # I is nonmeasurable. Then r~' : E ' -~ co(I)' is an isometry. By Corollary 5.4 ~r'(E') has the W E P in co(I)' so r " : co(/)" --+ E " is surjective. From the reflexivity of co(I) amLdthe commutative dlagraxn c0(I)

__L+ E

co(I)"

---* I¢H

E"

it follows that j E is surjective. So, E "- E'* ~', co. R e m a r k . It may be worth noticing that the 'dual' statement of Proposition 1.5 is false! In fact, in [2], 4.J a closed subspace D of ~

is constructed for which D ' "-, co b u t D # £°°. This D

furnishes also a negative answer to the question 2 in the Introduction: it is easily seen that each f E D r has a unique extension ] E (£oo), so D has the W E P in £oo, but also D is weakly dense so that D is not weakly closed.

§2 C O M P L E M E N T E D

SUBSPACES

OF' g°%

We first prove two useful lemmas, Let E, F, G be K - B a n a c h spaces, let i E £ ( E , F ) , ~r E £ ( F , G). Suppose Im i = Ker rr, ~r is surjective. A standard application of the O p e n Mapping Theorem shows that for the adjoint sequence G j ---+ ~'' F t -+ i' E ' we have Im r ' = Ker i r. If i' is surjective we can apply the same argument to G' -+ '~' F ' -+ i' E ' yielding the following. L e m m a 2.1. Let D be a closed subspace of a K - B a n a c h space E with inclusion map i : D -+ E and quotient map 7r : E --+ E / D .

Suppose D has the W E P in E.

Then in the commutative

346

diagram D

-~

E

_L~

E/D

D" ~ E" itt we have Im i" = Ker rr", and i" is injective.

---*

(E/D)"

lrH

L e m m a 2.2. (Compare Mso Example 3.3) Let D be a closed subspace of a polar K-Banach space E, and let D have the W E P in E. (i) If D is reflexive then D is weakly closed. (it) I f E is reflexive, D is weakly closed then D is reflexive. P r o o f . Straightforward from Lemma 2.1 and 'diagram chasing'. (Observe that D is weakly closed if and only if JE/O is injective.) Remark.

The following corollary is a partied positive answer to question 2 (see also the remark

following Proposition 1.5). Let E be a K-Banach space with a base whose cardinality is nonmeasurable. Then every closed subspace with the WEP in E is weakly closed. ( P r o o f . Without loss, assume E = co(I) where # I is nonmeasurable. By Gruson's Theorem [2], 5,9 any closed subspaee is isomorphic to Co(J) for some set J. Then # J <_ # I so co(J) is reflexive. Now apply Lemma 2.2.(i).) T h e o r e m 2.3. For a closed subspace D of goo the following are equivalent.

(a) For each t E (0,1), D has a t-orthogonal complement. (b) D is complemented. (c) D is weakly closed. (d) Either D is finite dimensional or D ~ goo. (e) Either g°° / D is finite dimensional or g°° / D ,,., goo. (f) g°°/D is a polar K-Banach space. (g) (g°°/D)' separates the poinU of g°°/D. (h) D is reflexive and has the WEP in go~. (i) BD is weakly closed in goo Proof.

We first prove (a) =~ (b) => (c) => (d) =ez (a). Trivially (a) => (b) =* (c); (d) =ez (a)

is Theorem 1,2, To establish (c) =*. (d), let D be weakly closed, D infinite dimensional. By reflexivity, D = A ° where A is a closed subspace of co. Then A has an infinite dimensional complement B in co ([2], 3.16(v)) so B ~ co a~ld D ~ B' ~ goo. Next, we prove (a) =:> (f) =~ (g) =~ (c) of which only (a) ==~ (f) needs some attention.

Let

0 < tl < t2 < .,. < 1, lim t , = 1. For each n 6 N there is a p r o j e c t i o n P , : goo ..., D with n~oo

I1P, II < t~-1. Then Q , := I - Pn is a projection with norm _< t~ 1 mad kernel D. So there is a bijection An 6 f~(g°° / D , Qng ~ ) making

coo

_~

Q,\

e~/ D ~/A,

Q,~Oo

347

commute. For each n the norm z ~

IIA.zll (z

e g°°/D) is polar. Also we have

Ilzll _< ItA.zll ___tZallzll

(z • g°°/O)

so that (see PA.1) the quotient norm on e ~ / D is polar. We have also (b) =*- (e) ( E / D is linearly homeomorphic to a complemented subspace of g°°) and (e) =~ (g). So at this stage we have proved that ( a ) - (g) are equivalent. The implication (a) =~ (h) is easy, (h) =~ (c) is Lemma 2.2 (i) and (c) =~ (i) is obvious. So we shall complete the proof by showing (i) =~ (d). Let D be infinite dimensional. The 'closed' unit ball of goo is for the w'-topology a(g °°, co) (which equals the weak topology) a metrizable ([3],8.3), complete (P.6), edged compactoid, hence so is BD. Let A • K, ]A[ > 1. By P.5 there exist fl,f2,...inABD with lirnoofn = 0 weakly such that BD C -c-5w{fl,f2,...} C ABD and such that each element of c-OW{fx,f2, ...} has a unique representation f i )~nfn where An • K, [~,1 _< 1 (the summation is with respect to the weak topology of g~). The formula

(~1, ~,...) £ f i ~,f, defines therefore a linear bijection T : goo ~ D. It is easily seen that []x[[ < I[Tx[[ < [)q [Ix][ (x • g°°). Thus, D is linearly homeomorphic to g°°. R e m a r k . The implication (i) :0 (c) is an ultrametric version of the classical Banach-Dieudonn6 Theorem which states that a subspace D of the dual of a complex Banach space is w~-closed as soon as the closed unit ball of D is wt-closed. In the Appendix we shall prove a stronger version (Krein-Smulian Theorem) for Banach spaces over a spherically complete ground field.

§3 H O W A B O U T £¢~(I)? It is natural to ask to what extent the previous results can be generalized to g°°(I). The 0nly positive result we have is in fact Proposition 3.1. The examples 3.2 and 3.3 show that several implications in Theorem 2.3 fail if we replace goo by g°°(I).

Proposition 3.1. Let I be a set whose eardinality is nonmeasurable. For a closed subspace D of g°~(I) the following are equivalent. (a) D is complemented. (b) D ",~ goo(j) for some set J where # J is nonmeasurable. D has the WEP in g°~(I). Proof, (a) =~ (b). Clearly D has the WEP, is weakly closed, so D is reflexive by Lemma 2.2 (ii). (l°~(I) is reflexive). Let P : g°°(I) --* D be a linear continuous surjection. Then P' : D' --* Co(I) is a norm homeomorphism into co(I) so by Gruson's Theorem D' "~ co(J) where # J < # I is nonmeasurable. So D ,-, D" ,,~ goo(j ) .

348

(b) =~ (a). Let i : D "-* g ~ ( I ) be the inclusion map. By (b) the adjoint c0(I) ¢ D' is surjective. Now D ~ g ~ ( J ) , where # J is nonmeasurable so D' ~ co(J) and there is a map T e £(D', co(I)) such that i' o T is the identity on DL Then T' o i" is the identity on D" and jD 1 o T' o Jtcc(z) is a projection of g ~ ( [ ) onto D. E x a m p l e 3.2. Let

# I = # K be nonmeasurabIe. Then there exists an infinite dimensional closed

subspace A1 of g ~ ( I ) that has the W E P in g ~ ( I ) and is of countable type. (Hence, D is weakly closed, reflexive, but not complemented (Lemma 2.2 (i) and Proposition 3.1).) Proof.

We can make, in a standard way, a quotient map co(I) -~ g~.

By reflexivity It" is

surjective, so A1 := ~r'((g~) ') has the W E P in co(I)' and is of countable type. E x a m p l e 3.3. (Negative ar~swer to question 1) Let I, K be as above. Then there exists a weakly closed subspace A2 of ~ ( I )

such that A2 is of countable type, but A2 does not have the WEP in

P r o o L Let D be as in the Remark following Proposition 1.5. Again, make a quotient map ~r : co(I) --* D. It is easily seen that A2 := 7r'(D') is weakly closed, of countable type. If A2 haA the W E P then ~r" would be surjeetive. Then, by reflexivity of c0(I), jD would be surjective conflicting the nonreflexivity of D.

§4 S O M E C O N S E Q U E N C E S

FOR STRONGLY POLAR SPACES.

Recall that a K-Banach space E is strongly polar ([3], 3.5) if sup{If t : f E E ' , [fl <- P} = P for each continuous seminorm p on E. It is proved in [3], 4.2 that E is strongly polar if and only if for each continuous seminorm p, for each subspace D C E, for each f E D' with Ifl -< P, for each ¢ > 0, there exists an extension ] E E' such that I]1 < (1 + e)p. It is still an intruiging open problem whether each strongly polar K-Banach space is of countable type. The previous theory yields the following results.

Proposition 4.1. I ] E is reflexive and strongly polar then each closed subspaee is reflexive. P r o o f . Any closed subspace is weakly closed. Now apply Lemma 2.2(ii). P r o p o s i t i o n 4.2.

Let E be reflexive and strongly polar. Let E' be a subspace of some polar

K-Banach space X . Then E t has the W E P in X and E' is weakly closed in X . P r o o L The first statement follows from Part I of the proof of Theorem 1.2. For the second statement apply Lemma 2.2(i).

Proposition 4.3, Let E be a reflexive strongly polar space. If E' is linearly homeomorphic to a subspace of too then E is of countable type. P r o o f . Assume dim E = co. By Proposition 4.2 and Theorem 2.3 (h) =:~ (d), E' ,~ too. Then E ~ E " ~ co.

349

§5 A P P E N D I X " T H E p - A D I C K R E I N - S M U L I A N T H E O R E M . T h r o u g h o u t §5 K is s p h e r i c a l l y c o m p l e t e . By modifying 'classical' techniques we shall prove: T h e o r e m 5.1. Let E be a K-Bauaeh space, A convez subset C of E' is w'-closed if and only if

for each n e N the set C n {f e E ' : llfll -< n} is w'-closed. P r o o f . We only need to prove the 'if' part. (1) From the assumption on C one easily deduces that C r i B is wl-closed in B for every bounded set B C E'. (2) Let bw' be the topology on E t of uniform convergence on compact subsets of E. Then bw' is locally convex, stronger than w', but coincides with w' on bounded subsets of E'. As

j E ( E ) = ( E ' , w ' ) ' and bw' is admissible we have (Et,bw') ' = j B ( E ) so a convex subset of E' is w'-closed if and only if it is bw'-closed. (See [5] for details) (3) Let a e E ' \ C ; it suffices by (2) to find a bw'-neighbourhood V of a for which V C E ' \ C . We may assume a = 0, see (1). For each r > 0 set BT := {x • E : lixll < r},

B'~ :=

{f E

E ' : Ilfl] < r}. We shall find finite subsets F0,F1,... of E such that Fn C B1/n for each n • {1,2,3,...} and F~o n F ° n ... n F ° n Bt,+l C E ' \ C for each n • {0,1,2,...}. (Then X := U Fn U {0} is compact so U :-- X ° is a bw'-zero neighbourhood, U C E ' \ C . ) As C MB~ n

is w'-closed there is a finite set F0 C E for which Fo° V1B~ C E ' \ C . Suppose we have chosen F0, F1,..., Fn-1 with the required properties, in particular (*)

F0° n F ° n .,. n F ° _ l n B" C E ' \ C

and suppose there is no Fn that meets the requirements. Then, for each finite subset F of

Ba/n we have A F := F ° n.~o nF~ n... nF~_~nB'+~ nC # 0 The sets AF, where F is a finite subset of B1/n, are c-compact in the w'-topology and have the finite intersection property. So there is an f •

n AF.

Then f e C and If[ -< 1 on each

F

finite subset of B1/,,, so []f[[ < n i.e. f • B ' . Then, by (*),

S ~ F° n ~ n...nr°~_~ nB'. c E'\C contradicting f E C. C o r o l l a r y 5.2. A subspace D of E' is w'-ei'osed if and only if BD is w'-closed. Proof.

Suppose BD is w'-closed, Let $ E K, I~1 > 1. For each n E N the set D n {f E

E' : IlYtl < 1~t"} = ~ " B o is w'-closed. :Let r > 0. For large n we have IAin > r so that D M {f E E ' : Ilfll < r} = A"BD N {f E E ' :

Ilfll < r} is w'-closed. Now apply Theorem 5.1.

350

REFERENCES [1] N. de Grande-de Kimpe and C. Perez-Gaxcia: Weakly closed subspaces and the Hahn-Basaach extension property in p-adic analysis. Proc. Kon. Ned. Akad. Wet. 91,253-261 (1988). [2] A.C.M. van Rooij: Non-archimedean functional analysis. Marcel Dekker, New York (1978). [3] W.H. Schikhof." Locally convex spaces over nonspherically complete valued fields. Groupe d'6tude d'analyse ultram6trique 12 no. 24, 1-33 (1984/85). [4] W.H. Schikhof: A connection between p-adic Banach spaces and locally convex compactoids. Report 8736, Department of Mathematics, Catholic University, Nijmegen, 1-16 (1987). [5] J. van Tiel: F,spaces localement K-convexes. Indag. Math. 27, 249-289 (1965).

G r o s s - K o b l i t z f o r m u l a for f u n c t i o n fields Dinesh S. Thakur School of Mathematics Institute for Advanced Study Princeton, NJ 08540, USA

The Gross-Koblitz formula, based on crucial earlier work by Honda, Dwork and Katz, expresses Gauss sums lying above a prime p in terms of values of Morita's p-adic gamma function at appropriate fractions (see [GK],[K]). Now various analogies between the global fields have been quite useful, so I will discuss and sketch a proof of an analogue of the Gross-Koblitz formula in the theory of function fields over finite fields. Let K be a function field of one variable over finite field Fq. Fix any place co of K and let A be the ring of elements of K integral outside co. Basic analogies are, Q~K,

Z~A,

C ~ = : ~

(1)

Gauss sums and gamma functions are both closely related to the cyclotomic theory. Over Q, one has basic cyclotomic extensions Q(/tn)'s and the Kronecker-Weber theorem says that any finite abelian extension of Q is contained in one of these. Over K, usual cyclotomic extensions K(#,,)'s are just constant field extensions and there are many more abelian extensions eg. Kummer and Artin-Schreier extensions. Carlitz [C2] in 1930's and Drinfeld and Hayes [D],[H1],[H2] in 1970's produced other 'Cyclotomic families' K(Aa)'s (a E A) where A~ is the set of a-torsion points of suitable rank one Drinfeld module A -~ EndGa, in analogy with #,, which is the set of n-torsion points of Z ~ EndG,~ (where integer n gives n'th power endomorphism). In Gekeler's talk, we have learnt the basics about Drinfeld modules (see [Ge] for more details). So I will just present a simple example, due essentially to Carlitz [C2], in detail. Let A = Fq[T]. It is easy to see that EndG~ is the (non-commutative) ring of polynomials in Frobenius. Consider the ring homomorphism A --~ EndG~(a ~-~ Ca) given by CT(U) =: T u + u ~, Co(u) =: Ou, (u 6 f/, 8 6 Fq) (2) For a 6 A , let A, =: {u 6 ¢/ : Ca(u) = 0} For example, T-torsion points are just solutions of 'T-th cyclotomic equation' u q + T u = 0. For nonzero a 6 A, K(A=) is an abelian extension of K with Galois group (A/a)*. Let me mention in passing, that for general K, the maximal abelian extension of K is the compositum of all such K(A~)'s over a's and A's (i.e. all possible choices of co's). In our case of A = F~[T], it can also be described [H1] as the compositum of constant field extensions, K(A~)'s and 'K(A~)'s for A = Fq[1/T]'. Gauss sums that we will now consider arise in the mixture of cyclotomic families K(#n)'s and g(A~)'s. ClassicalIy, a Gauss sum is defined to be -

~_, X ( x ) ¢ ( T r x )

352

where 2: is a non-trivial multiplicative character X : F~,, -~ C*, ¢ is a non-trivial additive character ¢ : Fp ---* C*. We view ¢ rather as an isomorphism of Z-modules Z / p ---+#p and replace it by an isomorphism of A-modules ¢ : A / 9 ~ A~ (Here 9 is a monic irreducible polynomiai of positive degree h of A and hence is a prime of A). Notice that ¢ is no longer a character in usual sense. Let k be a finite field of 'characteristic 9' i.e. a finite extension of A l p . To obtain non-trivial Gauss sums, we restrict the class of multiplicative characters to those giving Fq homomorphisms ¢ : k ~ L, where L is a field containing K(Ap). Then Gauss sum [T1],[T2] is defined as

g(¢) = - ~

¢(x-1)¢(Trx)

(3)

xEk*

It's easy to see that one only has to consider k = A l p and that there are h basic Gauss sums, say gi (jmodh) with ¢ = Xi being Fq-homomorphisms A / p --* L, indexed so that X~ = Xj+l(jmodh). One can prove [T1],[T2] analogues of various results on classical Gauss sums. We just state here a weak form (without congruences) of an analogue of Stickelberger's theorem. Let Kh =: K(#qh_l) and L =: K(A~)Kh. Now 9 splits in Kh completly into 9i =: T - X~_j(T)(jmodh) and 9j tota~y ramify to power qh _ I in L, let ~-~ be the unique prime above 9j in the integral closure of A in L. T h e o r e m 1 ('Stickelberger factorization'): With the notation described above, we have, in L, (gJ) =

••

h-j

(4)

Even though the proof is quite different than the classical case, we omit the proof, as the classical version is easy to prove and well-known. Now we turn to the gamma side. Classically, the exponential function e z is nothing but the entire function (normalized) satisfying the functional equation e"* = (e') '~ corresponding to Z ---} EndGm. Similarly, in this game, the exponential e(z) is defined to be the entire function (normalized to be tangent to identity at Lie algebra level i.e. linear term is z) satisfying the functional equation Ca(e(z)) = e(az) corresponding to A --} EndGa of (2). Classically, e" = ~ z"/n!, here e(z) being linear, one can write e(z) = ~ zq"/D,, (normalization corresponds to Do = 1) and hence one can regard D,,'s as factorials of qn by analogy. One gets the recursion relation

D i = [i]D~_I,

[i] = T q ' - T

(5)

by equating the coefficients of z q~ in the functional equation for e(z), for a = T. For n E N, define the factorial of n to be (due to Carlitz)

where n = ~ njq J is the base q expansion. W h y is this a good notion of factorial? For one thing, classically n! = [Ip"p, nn = E~>>.l[n/Np~], where N is the norm and the product is over positive primes. In our case

353

also, as Sinnott noticed (see [Go1]), the same formula holds, if p's are replaced by the monic primes of Fq[T]. Now from (5), it's easy to see that [i] is the product of all monic irreducible polynomials of degree dividing i and D~ is the product of all monic polynomials of degree i. Hence 'removing p-factors', Goss [Go2] made Morita-style p-adic factorial II~ as follows. Define f)i to be the product of all monic polynomials, prime to p, which are of degree i. Goss showed that -/9~ ~ 1 in p-adic topology and hence defined, for n E Z p , n = ~ n s q J , O <_ n j < q,

(7)

II~(n) =: I I (-/gJ) ~j c A~ Now

we can state our main

result:

T h e o r e m 2 ('Analogue of the Gros~-Koblitz f o r m u l a ' ) : •

gj =

l

J

o < j < h

(s)

Proof: (For simplicity, we will not describe which qh _ 1-th root we have chosen and how we choose the embeddings). Clearly, 19~ = D ~ / D a _ h p l, where I is such that /~a is unit at p. Hence, using the base q .expansion qJ/(1 - qh) = ~ q j + i h we get, q1 n A _--=1

=

b ÷mh = Iim(-tF+lDi÷,

,,/p

where w = ord~Dj+,,h. Applying the recursion formula (5), h times, we have

Let T = au be the decomposition in K~ of T, a unit at p (without loss of generality, p ~ T), as the product of its 'Teichmuller representative' a and its one unit part u. As a q'~ = a and u q' ~ 1 as t -~ oo, we have, as m ~ oo, [I + mh] = ((au) q'~h+' - T ) (a q' - T ) , which is just (negative of) one of the pj's above. Using this in the limit above and counting powers of p, using the description of [i] given above, we see that II " qJ

) 1-qh =

"(-p-t)

/p

Comparing with the Stickelberger factorization (4) (and using more precise information of Stickelberger congruences, which we have not stated, to fix units), one immediately deduces (8) Q.E.D. Note that this proof is quite direct and does not need a lot of machinery, unlike the proof in the classical case. Gauss sums described above in detail for Fq[T] can be similarly defined for any A using p-torsion points for suitable Drinfeld modules, but in general they might also have prime divisors not above p. For more peculiarities of the genera/theory, see IT4]. Now, if we drop the insistence on the analogy described above, then 'Gauss sums' i.e. 'elements providing correct Stickelberger elements', were shown to exist in the general

354

situation by Tare and Deligne; Hayes provided an explicit construction. (See [H3],[H4] for the details). Interesting feature of his construction is that 'Gauss sum for a prime Sa' occurs as a torsion point for some rank one Drinfeld-A-module, where the infinite place for A lies above p. Hence an infinity-adic formula for a torsion point is a 'p-adic' formula for Gauss sums. Again, for simplicity, we assume A = Fq[T], though it is not really necessary. Classically, n-th roots of unity are special values e2"~I'~/"of the exponential function. Now 2rriZ can be defined to be the solution set of e* = 1. Since we are dealing with the additive rather than the multiplicative group, and since the kernel of e(z) is of the form grA, for some ~- E [2, we consider # to be an analogue of 2rd. Then a-th torsion points for C are given by e(~rb/a), as is easy to verify using the functional equation

c~(e(~)) = ~(a~). In our non-archimedean situation, meromorphic functions are determined, up to multiplication by a nonzero constant, by their divisors, so that Z

e(z) = ~ II '(~ - ~)

(9)

AE~A

But from the point of view of the divisors, classically, gamma function is a meromorphic function with no zeros and simple poles at zero and negative integers. So we define (using analogy between positive and monic)

r(z) = : -

I I (1 +

)-1, n(z) =: zr(,), (~ e a)

(10)

Z n mordc

With these definitions, we have Theorem 3

('Reflection formula'): 1-I n(e,)_

e(#z)

(11)

06A*

P r o o f : If one compares (9) and (10), apart from the scaling factor ~-, the main difference between the two is that in (9) all the signs are allowed, whereas in (10) we restrict to the monic elements. But the product over 0's takes care of this difference and the formula follows. Q.E.D. Compare this with its classical counterpart 1-I (Oz)! = OEZ*

7rz sin ~rz

which is nothing but r ( z ) r ( 1 - z) = z/sin rrz in disguise. (Note that from the point of view of the divisors, e(z) is an analogue of the classical sine function). On the other hand, one can also interpolate [Go2],[T1],[T3] the factorial function (6) oo-adically to get H~o : Zp --+ Koo, and show [T1],[T3] that essentially

r~(~-) = v~, r~(o) = ~

(12)

355

where Foo(z) =: Ho~(z - 1) and the first equation holds only for odd characteristic. This also generalizes to general A [T1],[T3]. Observations above together with (11) and (12), express 'Gauss sums' in terms of values of 'p-adic gamma functions' at fractions. In this sense, this is an analogue of the Gross-Koblitz formula, the main difference being that in this case the gamma function is constructed by uniform procedure at all places, instead of being interpolated from a fixed gamma function. This gamma function can also be interpolated IT3] and one might ask for Gross-Koblitz phenomenon using the interpolated gamma function. For more information on a two variable gamma function of Goss, functional equations, algebraicity and transcendence results, Chowla-Selberg phenomenon, more analogies and some striking differences, see the references below. References [C1] L. Carlitz, On certain functions connected with polynomials in a Galois field, Duke Math.J. (1935),pp.137-168. [C2] L.Carlitz, A class of polynomials, Trans. A.M.S. 43 (1938), pp.167-182. [D] V. Drinfeld, Elliptic modules (English translation) Math. Sbornik, vo123,(1974),561592. [Ge] E. U. Gekeler, Drinfeld modular curves, Lecture notes in math. 1231. [Gol] D. Goss, von-Staudt for Fq[T], Duke Math.J. 45 (1978),pp.885-910. [Go2] D. Goss, Modular forms for F~[T]. J. reine angew. Math. vol.317, (1980),163-191. [Go3] D. Goss, The F-function in the arithmetic of function fields. Duke Math J. vol. 56 (1988), 163-191. [G-K] B.H.Gross, N. Koblitz, Gauss sums and the p-adic F-function. Ann. Math. 109, 569-581 (1979). [H1] D. Hayes, Explicit class field theory for rational function fields. Trans. Am. Math. Soc. 189, 77-91(1974). [H2] D. Hayes, Explicit class field theory in global function fields. G.C. Rota (ed), Studies in Algebra and Number theory. Academic press 173-217 (1979) [H3] D. Hayes, Stickelberger elements in function fields. Comp. Math. 55, 209-235 (1985). [H4] D. Hayes, The refined ~-adic abelian Stark conjecture in function fields. Inv. Math.

94, 505-527 (1988). [K] N. Katz, Crystalline cohomology, Dieudonne modules and Jacobi sums. In Automorphic forms, representation theory and arithmetic. (TIFR, Bombay, 1979), 165-245. [T1] D. Thakur~ Gamma functions and Gauss sums for function fields and periods of Drinfeld modules. Thesis, Harvard University(1987). IT2] D. Thakur, Gauss sums for F~[T]. Inv. Math. 94,105-112 (1988). IT3] D. Thakur, Gamma functions for function fields and Drinfeld modules, To appear in Annals of Math. [T4] D. Thakur, Gauss sums for function fields, To appear in Journal of Number Theory.

Three generalizations of Mahler's expansion for continuous functions on Zp. Lucien V A N H A M M E

Vrije Universiteit Bmsset, Faculty of Applied Sciences Pleinlaan 2, B-1050 Brussels, Belgium

1. I n t r o d u c t i o n

- Notations.

Let p be a prime number. Zp and Cp denote, respectively, the ring of p-adic integers and the completion of

the algebraic closure of the field of p-adic numbers. The valuation on Cp will be denoted I I. Consider a continuous function f : Zp --4 Cp. Let A be the difference operator defined by (Af) (x) = f (x+l) - f(x). Applying A n-times we get n

(Anf) (x) = Anf(x) = Z ( ~ ) ( - l ) n - k k=o

f(x+k)

The continuity of f implies that lira Anf (x) = 0 uniformly. From this it is easy to get Mahler's expansion n--)oo

(see e.g. [2] ) :

n=o

The purpose of this paper is to prove three results which include Mahler's expansion as a special or limiting case. In the next section we state our results and give a few examples. In sections 3, 4 and 5 complete proofs are given.

2. T h e T h e o r e m s .

T h e ~ e m 1 : If f : Zp --4 Cp is a continuous function and g : N --~ Cp is a bounded sequence then ~(X)

g(n)Anf(0)=

n:=o

~(x)

Ang(0) ,snf(t)lt=x_n

n=o

Anf (t) [ t=x-n is the value of Anf at the point x-n. This is a generalization of Mahler's expansion since for g(n) = 1, n = 0, 1, 2 ..... we get Ang(0) = 0 for n > I and the formula reduces to Mahler's expansion. The following special cases are worth mentioning. Take f(u) = (1 + t) u, I t I < 1. This gives n n=o

n=o

t

n

357

Putting x = - 1 and replacing t by -t we get the well-known expansion g(n) tn = ~ Ang(O) tn n=o (l-t) n+l

(2)

Take g(n) = (-1) n (Y) in (1) and replace t by -t. This gives ( nx) ( n )yt n = ( l _ t ) x Z

(x) (-lnY)

n=o

t n ; ltl
n=o

which can be written as a (known) relation between hypergeometric series 2F1 (-x,-y; 1; t) = (1-t)-x 2F1 (-x, l+y; 1; t_--ftl) Finally' take f(Y) = (Ym +z) then Anf(0) = (rnZ-Zn) , Anf(t)lt-x n - - = (x+z-n~t\m--n, and the theorem becomes m m Z (n)(m~-Zn) g (n)= Z \(x'~(x+z-n'~ n / \ m-n / Ang(0) nmo

n=o

For x = z= m this gives m

£

(nm) 2g(n)= Z ( n )m (

n=o

m+n n ) A m-ng(0)

n=o

Theorem 2 : If f : Zp --->Cp is a continuous function then 0o

x x-~yn

f(x) = f(0) +

(x.o) Anf(t) [ t=-yn n

n=l

For y = 0 we again get Mahler's expansion. The third generalization gives an expression for the remainder in Mahlefs expansion. We need the notion of convolution of two continuous functions. The convolution of two sequences f(n) and g(n) is, by definition, the sequence given by n

(f,g)(n)= 5". f(k) g(n-k) k=o In order to extend this definition to continuous functions defined on Zp, we need the following lemma. l_~mma t : ff the sequences f(n) and g(n) are the restriction to N of two continuous functions from Zp to Cp then lnim_~An(f,g)(0) = 0. Corollary : The sequence n-->(f,g) (n) is the restriction of a continuous function from Zp to Cp, denoted by (f,g). This corollary is well-known (see e.g. [2] p. 106) but the published proofs are somewhat complicated. An easier and more direct proof can be based on (2).

358 Proof of l~mma 1 :

(f,g)(n) Tn =

f(n) T n

g(n) T n

n=o _

1 (l-T) 2

Anf(0)

Ang(0)

k, n=o

n=

Define numbers a n by ~ (f,g)(n) T n = rl=o

e,o

1

Tn

(l_T)2

an

1 (

(l_T)n-l_ T

1 +

I'~FT) £

Tn a n (l_T)n

n=o

n=o

a© (l-T) +

(an + a n - l ) (l_T)n+l n=l

Then An(f,g)(0) = a n + an_1 for n _> 1. Since lim Anf(0) = lim Ang(0) = 0 we see that lim a n = 0 which proves the lemma. 11---)':~ I1~ 11--+oo We now introduce another convolution which is, in a sense, more natural. Definition : (f©g)(x) = (f,g)(x-1) x ~ Zp If U is the function defined by U(x) = 1 for all x e Zp then (fOU)(n) = f(0) + f(1) + ... + f(n-1) and fOU is the "indefinite sum" of f i.e. A (fOU) = f We can now state our third result. Theorem 3 : Mahler's expansion with remainder. If f : Zp -~ Cp is continuous then

f(x) = f ( 0 ) + Af(0) ( ~ ) + . . .

+ Anf(0) ( x ) +

( x ) © (An+lf)(x)

Corollary : If n tends to infinity we get Mahler's expansion. To see this, observe that l[ f o g 11-< II f II. II g II where II f II - sup I f(x) I. x ~ Zp

Hence II ( x )

oAn+lfll
359

3. Proof of Theorem 1 Since lim Anf(x) = 0 uniformly, the series on the l.h,s, and on the r.h.s, are uniformly convergent which ll---->c,o

means that their sums are continuous functions of x. Hence it is sufficient to prove the theorem for x = n c N. Start from the identity (l+vu-v) n = ((v-l) (u-1)+u) n

,)ku°-k u_l)k

k=o k=o The linear map L 1 : Cp [u] -~ Cp : u k --> f(k) maps ( u - l ) k on Akf(0). If we apply first L I to (3) and then the map L 2 : Cp [vl --~ Cp : v k ~ g(k) we get ( ~ ) g(k)Akf(0)= ~ ( ~ ) Akg(0)Akf(t)[ t=n-k k--o k=o which proves the theorem for x = n. Remark : By imposing more severe conditions on the function f one can deduce a formula for f(O) from theorem 1. Write the theorem in the form oo

n=l

o,o

(x-l) n-1

g_~ Anf(0)= g(0). f(x)-f(0)+ ~ / ' x - l ' ~ ~ n x \ n-l] n n=l

Anf(t)l

x-n

tflim a~(x) = 0, uniformly in Zp, one can take the limit for x --~ 0. n.-->oo n This gives

( - 1 )nn - I [ g ( n ) A n f ( 0 ) - Ang(0)zx"f(t)lt =-n]

g(0)f(0) = n=l

If, moreover, f(x) = if-x) then Anf(t)lt~ n = (-1) n Anf(0), f(0) = 0 and we get a formula involving only differences at 0 :

£

(-1)n ---if-- g(n) Anf(0) =

n=l

-

Ang(0) Anf(0) n--

n=l

4. Proof of Theorem 2 x (x+yn) x/x+y n-l'~ Put An(x,y) = x-~yn = n k n-1 ] for n _> 1. Ao(x,y) = 1 We first need a lemma. Lemma 2 : I An(x,Y) I < 1

x,y ~ Zp

Proof : There are three cases to consider. 1e case : I x I > I ny 1. Thenlx+ynl=lxlandtAn(x,y)

l= x@ynt [ ( x + y n ) l

<1.

360

2e case : l x l < l n y l . Then I x+yn I = I nyland x-~yn = n-~ < 1. Hence I A n (x,y) I < 1. 3e case : I x l = l n y [ . ThenlXl=lYl<-land IAn(x'y) l= We can now prove theorem 2. We first consider the case where

x I/'x+n y-l'~l Inl'lk n-1 ./I _<1,

f(x) = ( x )

n > 1.

The theorem then reduces to n

(x)_~

~ (x7~) (xy~) = ~: A~(x,y~(xy~)

k=o

(4)

k=0

This identity follows from the more general identity n

(X+n+nY) = ~ Ak(X'Y)(X+(nnkk)Y) k=o by taking z+ny = 0. Formula (5) can be found in [1] p. 172.

(5)

We now use (4) to prove the theorem. N N

~°,~o) (x)

Ak(X,y) Anf(0) ( n yk)

_-

n=l

n=l

k=l

N = k=l

N Ak(X,y) ~ A n f ( 0 ) ( n yk) n=l

Fromf(x)=~Anf(0)(x) n=o

we

getAkf(x)= ~ A n--k

n f(0)(n_k) x

This means that we can write N Z Anf(0) (nXk) = Akf(x) + rk,N with liNm_~rk,N = 0 n=k Hence

~.f(0)(x)

~

= Z Ak(x'Y) Akf(-Yk) + Z Ak(x'Y) rk,N

n=l

k=l

k=l

Noting that Irk,N I < I r0,N I and using lemma 2 we see that the last sum tends to zero if N---~,. This proves the theorerrL

361

5. Proof of Theorem 3 The proof will follow from two lemmas. Recall that U is the function with U(x) = 1 for all x e Zp. Lemma 3 : (i)

(x) © U = (x) n+l

(ii)

(U©U©U©...©U)(x)=(x)

,n~ N,xe

(n+l) factors Proof : Since all functions involved are continuous it is sufficient to verify the statements for x = k ~ N. k-1 For n = 0 the assertion (i) reduces to (U © U)(k) = k and this is equivalent with ~] i=k. (ii) is trivial for n=0 i=o For n > 0 we use induction on n. The lemma follows from k-1 • { i + l ' ~ { i "~= (kn) O U ( k ) = Z ( ' n ) = ( n k l ) s i n c e k n + l , - \ n + l . / (in) i=o Lemrna 4 : f = f(0). U + U © Af Proof : Since f is continuous it is sufficient to verify this when the variable x = n ~ N. But in that case the lemma reduces to n-I fin) = frO) + Z Af(k) k=o which is obvious. Theorem 3 now follows by using lemma 4 repeatedly. f = f ( 0 ) U + U © Af Af = Af(0) U + U © A2f Hence f = f(0) U + AfrO). U © U + U © U © A2f Continuing in this way we get f = f(0) U + Af(0). U © U + ... + Anf(0). U © U © ... © U + U © U © ... © U © An+lf (n+ I) factors

(n+ 1) factors U

Evaluating this at the point x we obtain (from lemma 3) theorem 3.

REFERENCES

[1]

J. RIORDAN : Combinatorial Identities. Robert Krieger Publishing Company, Huntington, New York 1979.

[2]

W. SCHIKHOF : Ultrametric Calculus. Cambridge University Press, 1984.

P-ADIC SYMMETRIC DOMAINS

Harm Voskuil Department o f Mathematics, University of Groningen P.O. Box 800, 9700 AV Groningen, The Netherlands

§0 INTRODUCTION We will study p-adic symmetric domains. The complex symmetric domains are well-known (See [C] and [H]). We will briefly recall their construction.

Let G be a real non-compact semi-simple and connected Lie group. The group G contains maximal compact subgroups K, they are all conjugated. Now X=G/K is the symmetric space associated to G. An arithmetic subgroup FcG acts properly on G/K. Now X/F need not to be compact, but there exists a space XDX such that X/F is compact (See [BB], [BS], [N] and [AMRT]). The construction above does not work for p-adic Lie groups, since the maximal compact subgroups are

not all conjugated in this case. Our construction is based on the work of

Kurihara (See [Ku]). Let G be a split simply-connected linear algebraic group defined over a non-archimedean local field K. Let F c G be a discrete co-compact subgroup. Now we call X a symmetric space

for the pair (G~F) if X/F is a proper rigid analytic variety. To construct such a space X, we start with a projective variety Y=G/P, here P c G is a maximal parabolic subgroup defined

over K. We then try to construct a space X c Y ,

which is a symmetric domain. In fact the

construction is such that we have:

Y-(HBnY)cXcY Here HB is defined as follows. Let Y be imbedded in a projective space [~. Let T c G be a K-rational torus of maximal rank. Let xl, i = l . . n + l

be the co6rdinates associated to a basis

of P~ such that T acts diagonally on P~(. Now the union of all these hyperplanes xi=O, ~= 1 . . n + l for all maximal K-rational tori T c G is HB. Many examples of discrete co-compact subgroups F of p-adic Lie groups G are known (See [Kan]). Furthermore it is known that the groups F are S-arithmetic if rank(G)>1 (See [Mar]~ [T.1] and

[V]). For more

information about S-arithmetic

groups

we refer

to

[S] and

building B

for

a

the

literature given there. This article is divided into three paragraphs. In

§1 we

briefly

recall

the

construction

of

the

Bruhat-Tits

reductive

linear algebraic group G. Our construction of a symmetric domain X uses the building B. In §2 we give a simplification of the construction of X in the case G=PGL(n,K) acting ;q-1

on Y=P~ . In this case X is very well-known, X=P~-l-{K-rational [Mus] and [Ku]).

hyperptanes} (See [D],

363

G=PSO(f,K), where f is a quadratic form in n variables, acting on the projective space YcP~ -1 defined by f=O. In this case we can not In §3 we study the split orthogonal groups

construct a

symmetric domain for

"symmetric" space for

FoG

discrete and co-compact, but we can construct a

FcH discrete and co-compact. Here HoG is a group isomorphic to

GL(1,K) with l= [ 2 ] ' In

[F] there

is indicated another

approach to

p-adic

symmetric spaces. Last but

not

least I would like to thank Marius van der Put for his help while I was (and am) studying this subject.

§1 THE BRUHAT-TITS BUILDING

Here

we

will

briefly

recall

the

construction

of

the

Bruhat-Tits

building

(or

affine

building) of a reductive linear algebraic group G. For more precise statements and proofs see [BT] and IT.2]. We also will give some extra information about the buildings needed in §2 and §3. Notation: K

a finite extension of Qp or

Fq((t))

V

the additive valuation on K, normalised such that

G

a reductive linear algebraic group defined over K

ToG

v(K*)=Z

a torus defined over K with maximal rank, so

T(K)~ (K*)n with n maximal

rank (G)

the rank of G which is the maximal rank of a K-rational torus

NcG 4)

the normaliser of T

W

a finite WeyI group

4).f,4):f War

an affine root system

ToG

a finite root system

an affine Weyl group

A group G is called

split if rankK(G)=rankL(G) for every finite extension KcL.

We will always assume that G is simple and split.

ToG belongs a finite root system 4)oR n, where n=rank(G). On this Weyl group W. Furthermore we have WeN~T, (#N/T
root system acts a finite

affine root k+c~: (.,.)

system 4):f. The affine root system 4)af consists of the affine-linear functions

R'~R, with c~e4), k ¢ 7 and k-= l m o d 2 if ½c~¢4), defined by: k+c~(x)=(~,x)+k for xeR n, is the

inner product. For the definition of 4):y we refer to

[Mac]. We can identify

364 the affine root k+c~e~al with the halfspace {xeg~n]k+(cx, x)>O}. The boundary {xeRnlk+(a,x)=O} of an affine root is called a wall. The drawing o f all the walls gives a simplicial decomposition of Rn.

J ,/~, type A2

The

(

:L

maximal

simptices

are

triangles). A chamber C is the

the

called

~/,

J

type ,42

chambers (In the

example

intersection of exactly n + l

above,

affine roots

these

are

the

(i.e. halfspaces).

affine roots form a simple basis Ac~/ of the affine root system #a/ (See [Mac]).

These n + l On

~af

affine

root

system

acts

an

Weyl group

affine

W~/, which is generated by the

reflections in the walls. Now we return to our group G. Let T be as before and let TooT be defined by: To: ={t~TIv(tii )=v(tjj)

V i,j}

(Here we assume that T is represented by diagonal matrices). The group N/T o contains an affine Weyl group Wa/. We can fix a set of isomorphisms ~a: U c ~ K ,

c~e#, satisfying certain

conditions (See [BT]). The additive valuation v of K gives us a valuation Vo~c~ on Uc~. We can

use

these

valuations

Vo~c~ to

refine the

set

{Uc~]Cx~#} into

a

set

of

T0-invariant

additive subgroups {Uk+dk+cxe#~(f)}. Here Uk+c~ is defined by:

uk+~: = { u e U ~ l k

+v

o~Au)~0}

Again we have: w(Uk+c~)=U~(k+cO VweWafcN/T o. All K-rational tori ToG of maximal rank are conjugated. Therefore we can associate to each torus ToG as above a space Rn with its simplicial decomposition given by the affine root system. Such a space is called an appartment A. Let A be the appartment belonging to T. The affine building B of G is defined by:

B: = U g . A / ~ gEG

Here g.A

is the

identifying

all

the

appartment affine

belonging to

roots

gTg-1. The equivalence-relation ~

(half-appartments)

belonging

to

the

same

is given by T0-invariant

additive subgroup. The stabiliser of a simplex S e B

is called a parahoric subgroup.

In [BT] it is proved

that the parahoric subgroups of G are compact and that every compact subgroup of G is included in some parahoric subgroup of G. Now the following two statements are clear:

a) b)

FoG(K) discrete <=~F acts with finite stabiIisers of simplices on B. FoG(K) co-compact ~=~B/F is finite. For the next paragraphs we need to know the structure of the Iwahori-groups Pc, the

365

stabiliser of the chamber C.

Proposition: Let C e A c B be a chamber. Let T O be as above and let Aaf be the simple basis of

~al belonging

to C. Now we have: Pc=

For later use we will now give some more detailed information about the buildings of the group SL(n,K) and the split orthogonal groups SO(f,K), where f is a quadratic form. The building o f SL(n,K): (See [BT] §10.2 and [T.2] §i.14) This is a building of type An-1 i.e. the affine root system is of type An-l- Let ToG be a fixed torus and let G=SL(n,K) act on Kn. We choose a basis % i = l . . n of Kn such that T acts diagonally. Now the T0-invariant additive subgroups Uk+%j are: Uk+~ii: ={9eGlg(ed=e=+cej, g(es)=% s ~ i , v(c)_>k} for i ¢ j . On the affine roots we have a relation: c~j=-c~jz For a standard chamber CEA the simple basis Aal of ~al is: cq,1+l, i = t . . n - 1~

l+c%i

The roots in di~I satisfy the following relation: n-1

cq,i+l+ (1 +°%t) = 1

l

The building of SO(f,K), f =

~xix2~+l_i, />1: (See [BT] §10.1) i=1

This is a building of type Aix.4i, /=2, A3, /=3 and D1 for />3. Let TcG=SO(f,K) be a fixed torus and let G act on K2k We choose cobrdinates %

i=1..21 of K~lsuch that T acts

diagonally. The To-invariant additive subgroups are: Uk+~ji: = {geGlg(ei) =ei+cej, g(e~+l_j) = e~m_j - ce2~+i_i, g(es) = es, s ~ i, 2/+ 1 - j , v(c) >_k} for i # j , 2 / + l - j . On the affine roots we have the following relations:

For a standard chamber C e A the simple basis Aay of qSaf is: cq,~+l, i = 1../-1, (xtl+2 , I-{-(x2~,2 The roots in Aai satisfy the following relation:

366

l-2 ~ oq,i+l+Oq_l,l+~l,l+2=l,

cq,2+(l+c~a,2)+2

1>4

i=2

¢xl,2 + ( 1 +o~6,2) +c~,a +eta, 5 = 1

I= 3 /

The b u i l d i n g o f SO(f,K), f = x o +2

~xix~+l_i, / > 1 : (See [BT] §10.1) i=1

This is a building of type Bz. Let TcG be a fixed torus and let G=SO(f,K) act on K a+l. We

choose

a

ei, i = 0 . . 2 /

basis

of

K 2~+I

such

that

T

acts

diagonally.

The

T0-invariant

additive subgroups are:

U~+c,ji: = { g e G ]

g(ei) =ei+cej, g(e~+>j)=e~+>j-ce~+l_i, g(e,)=e,, s # i,21+ l - j , v(c) > k}

for i c j , 2 1 - j and i,j#O and:

Uk+ceO,j: = { g e G I

g(ej)=ej+ 2cxo+dxa+>j, g(%)=%- :ea+l_j, g(es)=e,, s#j,O, c2+d=O, v(d)>_2k}

On the affine roots we have the following relations:

eqj = -o~ji , ccij= c~2z+l_j, ~+1-i, i,j ~ 0 C~o,1=-r~0,~+l_ j (and we define c~j,o=-c~0,j) For a s t a n d a r d chamber CeA the simple basis Aaf of ~a1' is: t+c¢~2,2 , oq,i+l, i= l..l-1, c%,z+I The roots in Aaf satisfy the following relation: ( 1+ e%,a) + 2¢xl,a + C~o,a= 1 ,

/=2

I-1

(1+c¢~2,a)+2 ~

oq,i+l+c%l+l=l~ / > 2

i=l

§2 THE GROUP PSL(n,K) ACTING ON P~ -~ We action

will give on

construction

the

construction

of

a

symmetric

domain

for

y=p~-l. This symmetric domain is also constructed is a simplification

in terms of rigid analytic

of the one given by Kurihara.

geometry

PSL(n,K) starting in [D],

[Mus] and

We describe

(See [BGR] or [FP]). Let us first define

with

its

[Ku]. Our

the construction what we mean

by a symmetric domain. Definition:

Let

G be

local field K. Let F o G

a

reductive

linear

algebraic

group

defined

over

a

non-archimedeaa

be a discrete co-compact subgroup. A symmetric domain X for the pair

( G , f ) is an analytic space X such t h a t X/F is

a proper rigid analytic variety.

367

Notation: Let B be the building of G=PSL(n,K). Let A c B be a fixed appartment and let TAcG

be the K-rational torus belonging to A. We fix a basis {el,...,en} of p ~ - I acts diagonally on FT(-1 with respect to this basis. Let {xl,...,xn)

such that TA

be the associated basis

of 0(1),

To each root c ~ #

belongs a character Xa of TA with which T A acts on the additive subgroup

Ua. To each affine root k+(xeqSay we associate the meromorphic function k . ~

x j ~ where T acts

with character Xa on x-A. X. In the notation of §1 we associate to the affine root k+cqj the subgroup Uk+a,j and the /unction r ~xi.

xj

Proposition 2.1: Let CeA be a chamber, Aay the simple basis defined by C and PccPSL(n,K)

the Iwahori-group stabilising C. Now we have: a)

The

set XC,A: n-1

={peP~-l[ ] ~ ( p ) ]

_<1, i=l..n, VgePc} is an aff!noid subspace of

*

Y=PK . (Here g Xi=X i og-1). b)

The affinoid algebra belonging to Xc, A is:

K< g xi[i= 1..n, g~Pc> =K <~rk~ik+cqieAa~> (So for our standard chamber C of §1 this is K < ~ , Proof:

This

follows directly

from

the

description

x2 Xn-lxn,~x----nxl > .) xs~.., of

Pc and

the

T0-invariant

additive

subgroups given in §1.

HA be the union of the f: p~-IHA.~A (=~n-1) be the map defined by: Definitions:

Let

hyperplanes

defined

by

xi = 0,

i = 1..n.

Let

f(p)=q if and only if v((~)(p))=cqj(q), Vi,j=l..n, i ~ j . Of course here the affine roots are interpreted as affine-linear functions on R"-1 Proposition 2.2: Let f as above. a)

f ( p ) e C .~ peXc, A

b)

XA:=cUXc,A=P~-~--H~

C)

The covering C: ={Xc,nlCeA } of X A is pure. The reduction of X A with respect to the covering C has for every O-simplex S e A exactly

d)

one proper component. Proof: a and b follow directly from the definitions. c)

This is proved by using the function f and statements a and b of the proposition.

d)

This is proved by using a,b,c and the theory of toroidal embeddings (See [KKMS], [0.1]

or [0.2]). Note that one needs, in order to apply the theory of toroidal embeddings,

368 change

the

innerproduct

of

A~-R"-1 such

that

the

lattice

of

vertices of

A

becomes a

sublattice of Lv~-1 with the standard orthogonal inner product. Definitions: Let us define now:

Xc:= Iq Xc,A, A~C

X : = A XA,

He:= (3 HA.

AcB

AcB

Let us fix an appartment A and a basis {Xl,...,Xn} of 0(1) belonging to TA. Now we define a function r: p~-i ,N by: n

if iR x~(p)=O

r(p) = inf

otherwise.

[emma 2.t:

a)

Xc, A n Xc, A, = Xc, A - R-I(flA,\RA) = XC,A, - R-~(RA\ftA, )

b)

Xc=Xc,A-R-I(HB\~IA). (Here R:Xc, A

,XCC,Ais the reduction-map.)

Proof: a) This is proved by using the description of X C , A given in proposition 2.1. b)

This is proved by using statement a repeatedly. Note that infinite

number

of

appaxtments,

we

only

have

a

finite

even though CeA

number

of

for an

different

affinoid

spaces Xc, A for a fixed chamber C. Lemma 2.2: Let r be as above. The function r has the foUcuring two properties. a)

r ( p ) = 0 ~ pert B

b)

r(p)¢O ~ B(gePSL(n,K)) i=t~I ] ~ ( P ) I

=r(p).

Proof: This is clear.

Definition:

We call

an

analytic

space Z

locally proper if Z has two admissible affinoid

coverings {A~lieI } and {A~[i~I } such that:

A~A~, VieI. If I is finite then Z is proper in the sense of Kiehl. If the two coverings axe

invaxiant

under the action of a group F which acts discontinuously on Z and has a finite number of orbits on the Ai, i E l then Z/F is proper. Theorem 2.1: a) X= 13 Xc= n XA=PK~-1 -{K-rational hyperplanes}. CeB

b) c)

AcB

The covering C: = {XcICeB } of X is pure and invariant under the action of PSL(n,K). X is locally proper.

369

r(p)=O then pelt B and pv~Xc VC~B since

Proof: a) We use the function r defined above. If

Xcce~-~-~B. If

r(p)#0

then

]#~:(P)[. one sees b)

peX=P~-I-HB and we can find an element gePSL(n,K) such that There exists a chamber

that in fact

Ceg(A) such that peXc, A. Using lemma 2.1

p~X c.

C,C'eB there exists AcB such that C,C'eA, and proposition 2.2.c. The invariance under the action of PSL(n,K) is clear.

The fact that the covering C is pure follows from the fact that for an appartment

c)

This is proved by using the fact that we have:

Xc={p~p~-ll

Ig~ ( x~ P)I

= 1,

i= 1..n, VgePc}

Xc,~ := {pEPKn - 1 [e -1 -< I . (p)[ <e, i=l..n, VgePc} with e > l is contained in X. Now Xc~Xc,~ and {Xc,~[CeB} is an admissible covering of X. This

Clearly

the

affinoid

space

proves c.

Remark: We can now construct a map A:

X---~B as follows: rl

*

peX we take an appartment g(A)cB such that i / / l l g - ~ i ( p ) l = r ( p ) and we use the function f:P~c-I-HgA---->g(A) to define A(p)=f(p). The proof of theorem 2.].a shows that this is well

For

defined. The map A is the same map as is used in [D] §6 and [Be] §4.3. Now

Xc=A-I(C).

Remarks: 1) It is known that the space X has a reduction consisting of a projective space pn-1 with all K-linear subspaces blown-up for every vertex K

Our

covering C gives this reduction. This can

SeB (See [Mus] and [Ku]).

be calculated by using the

theory of

XA (See proposition 2.2.d). Now XccXc,A is open, so the reduction of Xc is an open subspace of the reduction of XC,A. One can determine the reduction of X by using the fact that the parahoric subgroup PscG is toroidal embeddings. First one determines the reduction of

transitive on the chambers C containing S and also on the appartments A containing S. 2) If

FoG is not co-compact then in some cases one can construct a subcomplex BrcB such

that F acts discontinuously on X r : = ['1

XA and X r / P is a proper rigid analytic variety (See

A cBF

[Mus].). 3)

The following examples of spaces

X/F are known:

a) Mumford-curves (See [Mum.l] and [CP].) b) Mumfords fake projective plane (See [Mum.2] and [I].) (I will give some other examples of surfaces

X/F in my thesis.)

370

§3 THE SPLIT ORTHOGONAL GROUPS t

We take G=PSO([,K),

here f

~xix~+l_ , or [=x~ + ~xix~+l_i.

is a quadratic form f =

i=I

i=l

Now G acts on the projective variety Y defined by f=0 in p~l-1 (resp. P~l). We

exclude the

case f=xlx 4+x2x a.

Then

Y ~-P]~xP~

and

G

is isogenous with

PSL(2,K) × PSL(2,K). In this case Kurihara's original construction does work (See [Mus]).

First we will indicate why the

construction given in §2 does not

work

in this case.

Then we will improve the construction and get a space XA with properties similar to those stated in proposition 2.2. Now the torus TA acts on XA, but the normalizer NA of TA does not. This makes it impossible to construct a symmetric domain for G itself. But we can construct a "symmetric" space for a subgroup ttcG,

which stabilises two disjoint maximM K-rational linear isotropic

subspaces of Y. Clearly H ~-GL(l,K), where I is as above. Now we will show why the construction described in §2 does not work in this situation. I

We only treat the case f =

~ xix21+l_i. The other case is similar. i=1

Let xi, i=1..21 be as above and let TACC be the maximal K-rational torus, which acts diagonally on rD2/-1 "K with respect to these co6rdinates. As in §2 we can now for a chamber C in the appartment AcB belonging to T A define an affinoid subspace XC,ACY. We have:

XC,A: =Sp K< g x ixi [ VgePc, i=l..21>c~Y = Sp K < T r ' ~ t n+c~ijeA~f> nY The differences with the situation in §2 are the following: 1) For each n + ~ q e A ~ f we have exactly two meromorphic functions on p~z-1. They are ~rnxi and ~rnx2t+l-f . Xj X21+l_ i 2) The components of the reduction of C~A Xc'A are not in accordance with the vertices (0-simplices) of A. This is a consequence of the fact that the two meromorphic functions mentioned in 1 need not have the same absolute value for p ecU A Xc, A. There do exist

Pec~A Xc'A such that ~rn~(p) # ~ r n ~ ( p )

3) The components of the reduction of 12 XC,A C~A

. are

not proper.

The second problem can be solved by replacing XC,A by another affinoid subspace XC,A defined by:

371

XC,A: = {peXc,A] [PiP~+l-i] = [PjP~+I-j[ Vi,j = 1..21} The two meromorphic functions mentioned in 1) have the same absolute value for pECyA.~C,A.ffi Still

the

components

of

the

reduction

c~AXC'A

of

are

not

proper.

This

is

because

cUAff[C,A~ and cUAXC,A are rather unnatural subspaces of Y-HA, where HA is the union of the hyperplanes xi=O for i=l..2l. In order to make a space XA with properties as in §2 we need a way to associate to each affine r o o t n+(~ij an unique meromorphic function. This function turns out to be ~rn~., where - *3

the yi are defined by:

3'i --

xi

i = 1..1

xsx2t+l-s

i=l+l..2I

X21+l-i

Here x s is choosen such

that

lxsx~2+l_sI= max Ix~cx~+l_lct. So the function is not uniquely /¢=1 ..I

defined, but its absolute value is. Note that

we have made a choice here. The coSrdinates

xi, i = l . . l define a maximal isotropic subspace x~ . . . . the yi a b o v e replace the xi, i=l..l,

z~=O. We could in the definition of

by some other set of xj's defining a maximal isotropic

subspace. D e f i n i t i o n s : For i= t..l we define for C~A the following affinoid subspace of Y: X C,A. i . =SP K<x~x2~¢~-f x ' 7rnYr [ j=l..l~ n + e t r s e A a f > n Y Xi

2/+1-i

Ys

xr

r= 1..l

xix2t+l-i

r=l+l..2l

Here of course we have: Yr = ~2 l +l-r

Furthermore we define: l

i

I

Xi

We now can define the function fi: X~--.~A( =R ~) by:

fi(p)=q ~, v i

(p) =o~8(q ) r,s=l..2l, r e s

j

Since fi =-f j on XAnX A we can glue the functions fi together into a function f : XA-->A. The function f is well defined on Y-ZA, where Z A is the union of the hyperplanes xi=O for

i= 1..l and the maximal isotropic subspace given by xz+,=xt+~ . . . . .

Proposition

i

i

3.I: a) fi(p)~C ~ p~Xc, A for peXA.

b)

I Xi f ( p ) e C ** pci~_l C,A for p ~ Y - Z A .

c)

The covering C={Xe,nlz=l..l , CeA} of X A is pure.

d)

XA is locally proper.

Therefore X A = Y - Z A. i

.

x2z=O.

372

Proof: a) and b) follow directly from the definitions of fi and f. c) Using the definition of fi one sees that the covering {X~,AICeA } of X~ is pure. So we only have to look at affinoids X i i

C,A

j

and X j

i # j . In this case X ~ nX j

C',A

C,A

C',A

is contained in

XAnX A. Now using fi or fj one sees that the intersection is non-empty if and only if CnC'#¢. Furthermore the intersection is open. d) This can be proved by using the admissabie covering {XclAIZ=l..t , CeA} of X A. Here X~:~:={peY[

~(p).i.21+l_i

<e, l~nYr(P)~s

<e, j=l..l, n+oLrs~Aa/}. Of course we have for e > l :

X~,A g: XC,A i ,~

Remark: Instead of XC,A i we can look at the affinoid subspaces

V~,A: = ~ P K of P~t-I-Z A. ~s ~ XiX2l+l_ i

This

gives

us

a

pure

T-invariant

covering

i . {Vc,AI*--1..I, CeA} of P~*-X-ZA.

We

have

XC,A=Vc,Ar~Y. As in proposition 3.1 one now proves that P~*-~-ZA is locally proper. Remark: For the standard chaznber C of §1 we can give X~, A explicitly as follows:

X~,A=Sp K < ~ ,

x__22 x~__j xzx~-x X3 ~'''~

~rxix2**l-i

Xl ~ XiX21+l_ i

XlX 2

x~x2*+l-i { j=l..l, j # i > n Y ~ XiX21+l_ i

The generators of the affinoid algebra of X~,A satisfy the following two relations: 2 x,

.

2 . . . . . cx,_2

X'-~l "X~2" X 3 X 2 X2 X3 X i X 2 l + 1 - i

b)

.

x,_,x,

7rxix2l +'-i=T(~ / = 3 XlX2

1

~ xTx2z+l-i=O j=l

XiX21+l-i

Relation a corresponds

with the

relation satisfied

by

the affine roots

forming

a simple

basis Aa] of f~af" Relation b corresponds with the equation f= O. This shows that we have: i = Rc,A i × @,A, i XC,A with RiC,A." =Sp K and

S C,A.=Sp i . K<Xix21+l-f I j=l..l>/(f=O). XiX21+I-i

P r o p o s i t i o n 3.2: a) The reduction of XiA with respect to the covering {X~,A[CeA } consists of

one component Ds for every vertex SeA. b) Ds~-Es×]~ z-2 and Es is proper and non-singular. c) The reduction of X A = Y - Z A with respect to the covering {X~,AJi=l..l, CeA} consists of one proper component Fs for every vertex SeA. These components Fs are such that we have a surjective map ~: Fs---,P'~-2 with ~oq(p)~-Es for all peP~ -2.

373

Proof: a) Relation a in the remark above shows that the reduction of XC,A i has exactly one component for every vertex SeC. Glueing these components together for all C e A containing S we get the component Ds of the reduction of X~. b) The function fi: X~-->A maps XC,A ~ into CeA. The image fi(P) of peXc,A=Rc,A×SC,A i ~ depends only on the co6rdinates of the projection of p into RC,A. Now the appartment A is, after having

made

the

lattice

of

its

vertices

a

sublattice

of

Z t,

the

picture

of

the

torus

embedding belonging to {Re,AlCoA }. The theory of torus embeddings tells us that we have for i

every S e A a componen.._ttEs in the reduction of {Rc,AICeA} which is proper and non-singular. The

reduction

S~, m of

SiC,A is

of

course

an

/kz-2~,

The

reduction

X~, A

of

is

X~,A = RC,i A >4 SC, A i = RC,Ai×A*-2~. The reduction of X~ is now easily determined, since we only have to glue along open subspaces U x A ~-2 where UcR~,A. So the reduction of X~ consists of one component Es × A z-2 for every vertex S e A . c) We already know the reduction of the X~ for i=l..l. To determine the reduction of XA we only have to glue these together. The

intersection

XC,ANXc, A is given by

2: j : Y i l + l - j

=1

in

both X C,A i and X~, A. i A for a fixed chamber C glue together into an The reductions So, A of S~,

pZ-2 k

with

!

co6rdinates xlx21+l_i, i = l . . l

satisfying the

~xix~+l_i=O.

relation

For

a

fixed ~ePt~ -2 we

i=l

only have to look at one SC,A with reductions of the Xc,A, ~ CeA

_ - - ' 7, p c S c , A. Now part b of the proposition shows that the

and i fixed glue together into components E s × P , ~-2. Now it is

clear that the reduction of X A consists of exactly one component Fs for every vertex S~A, such that we have a surjective map ~: Fs*P~ -2 with - l ( p ) = E s for all pee~- -2. Remark: Looking more precisely at the appa.rtment A and the associated torus-embedding one sees that the reduction of XA has the properties: 1) F s n F s , ~ ¢

~, {S,S'} is a simplex in A

2) F s n F s , ¢ ~

~ FsC~Fs, is of codimension one in Fs and Fs,

Remark: Now we are going to construct for the subgroup HcG~ stabilising the pair of maximal isotropic subspaces x 1= x 2 . . . . .

x~ = 0 and x~+1= xt+~ . . . . .

x~ = 0 a space X,~c Y such that XH[F is a

proper rigid analytic variety for F c H discrete and co-compact. First we define a subcomplex BHCB on which H acts. Then XH: = n

X A is a locally proper

AcB H

space which has a H-invariant affinoid covering. Definition: Let H be as

above. Let A c B

be a

fixed appartment such that

the maximal

K-rational torus TA belonging to A stabilises the two maximal isotropic subspaces mentioned above, so TACH. Now we define the subcomplex BHCB as follows:

BH:= U g . A c B g~H

Lemma 3.1: B H ~-B × R, where B is the building of SL(I,K).

374

Proof:

It

is

clear

that

H=GL(1,K).

Therefore

all

maximal

K-rationM

tori

TcH

axe

conjugated and correspond with an appartment A c B H. To every maximal torus T c H rank(T)=rank(T)+l

corresponds exactly one maximal torus 7"cSL(I,K).

We have

where Tc is the torus of rank one which

and T is isogenous with T × T c ,

is in the center of H=GL(I~K). Now T c acts on R and T on B. This proves the lemma. Note that we don't care about the simplicial structure. Lemma 3.2: Let A,A'cB H be appartments such that A¢3A' contains a chamber C. I f g e P c such

that g(A)=A' then geH. Proof: Let m and m' denote the two maximal isotropic subspaces of Y stabilised by H. Since

L,TA, CH~ they stabilise m and m'. Furthermore there exist gEP c such that g(A)=A' and TA,=gTAg -1. Now gTAg-l(m)=m implies that

TAg-l(m)=g-a(m).

So the maximal

isotropic subspace g-l(m)

is stabilised by

T A. This

proves that g-l(m)=w(m) for some w~W, where W is the finite Weylgroup NA/T A. This cannot be, unless w=id. So g(m)=m. Of course also g(m')=m' and therefore geH.

Definitions:

Let TACH

be a

torus

and let xi,

i=l..21

be

the

associated coordinates of

P~l-lsuch that T A acts diagonally. Let m be the maximal isotropic subspace Xl=X 2 . . . . . and let m' be the maximal isotropic subspace xz+1= xz+2 . . . . .

xl=O

x22 = O.

Let HoG be the stabiliser of m and m'. Let Z A be the union of the hyperplanes xi=O, i = l . . . t and m'. Now X A = Y - Z A. For every A'CBH there is a geH such that g(A)=A'. Our choice of Z A is such that XA,=g(XA) and ZA,=g(ZA) are uniquely defined. Therefore the following definitions are allowed:

x.:=

n

z.:=

A' cB H

u

z A,

A' cB H

We will now construct a pure H-invaxiant covering of Xn. Lemma 3.3:

a)

p~Xc, A ~

b)

PEXc, A ~

i

(p) <1

VgePc, Vj=l..2l

g*xfg*x2t+l~i(p ) <1 VgePc, V j = l . . l XiX2~+I-i

Proof: a) The definition of the yj is such that

X. i ~j(p) _1 Vj=I..21 for peXc,A. We can now

associate to p~X~, A & point ~p~(l-1 defined by ~i:=yflp), now shows that

j = l . . 2 l . The definition of the yj

peVc.,A: =Sp K . rtX r

AS in lemma 2A.b one can show that V c . a - S p K < g - - ~ x l g e P c , j = l . . 2 t > . Therefore we have:

~

xj(~)

= ~

yj(~)

<1.

375 2t

For gePc fl=

we have: g*xj(~)= ~ajkx~(~)

~(~) xj

with

<1,

since fl=v(ajk)+(~kje4~,+f,

i.e.

~ nifli with niei[>0. ,8i ~ f

So we have:

g x~(~)xj <-k=lmax..21 ajk~j (~))

~j'(p)

=

~ xj(~)

aj kxk(p) ---~.l

< - ~ = 1max ..2z]

<

max

x i(p) - k ~ 1 . . 2 ,

ajkXj:(~)l < 1 xj(~)'

This proves a. i

b) This follows from a since we have for peXc,A:

Ig*xjg*x~+~_j(p)l <_tYjY~+,-flP)I = Ixix~+l-~(p)l Remark: A chamber C~B H is contained in infinitely many appartments A c B H. But there exists a finite number N of appartments A1,...,A N such that every XtC,A is equal to X ic,Aj for some je{1,...,N}.The appartments Aj, je{1,...,N} depend on C of course. Definition: Let C e B H and N be as above. For seI={1,...,l} t' we now define:

x~:

N

=

nx~3

./=

Lemma 3.4: a)

i j i ~ X C,A' j is open. XC,Af3Xc,A'CXC,A

b)

X~cX~.4~is

ope~ for ,Ul j=.I..N.

Proof: a) Let g e P c be such that gA=A'. Now lemma 3.3.a gives us:

p~Xc,AnXc,gA ~

(p) =

(p) = 1, k = 1..1

(*)

From lemma 3.3.b we conclude:

~

pEXc,AnXc,gA ~

g * x ~g*x2!+~-i(p) =1 xix21+l_ i

(**)

The equations (*) and (**) together define an open subspace of XiC,A and XJC,A' which is in

fact

C;,ANXJ,A ,.

b) This follows from the repeated use of a.

376

Definition: Let r: Y-~R be defined by 1

0

if

// lx~(p)l=O

i=1

r(p) =

[geH}

otherwise

Here det(g) is the determinant of g as an element of GL(I,K). Lemma 3.5: l

a)

r ( p ) = 0 ~ 3(geH) +Hllg*xi(p) I =0

b)

r(p)#O ~ 3(geY)

c)

r(p)~O ^ p ~ m ' ** p o X H

1

~~

(p) =r(p)

Proof: a) This follows from the completeness of K. b) The

cohrdinates

of

p e P ~ 1-1 are

defined

over

finite

some

extension

L~K.

The

set

{is I jseL, Isl _>r>0}c~ is discrete. From this b follows c) This follows from a and b.

3.1:a ) X . = C Y B H 8YI X~

Theorem

b) The covering ¢={X~,ICeB m s~l} o/ X H is pure and #-invariant. c) X H is locally proper. Proof: a) We will show that for every p e X H there exists a CEB H such that p~X~ for some s ~ I . Since peXH the set F = { g e H I

1 Idet(g)l

" (p) =r(p)} exists and is non-empty. Now

i=1

g*xig*x2l+l_ i t=inf{imlaxs]

Xk~C2[+l_k (p)] ]geF}

exists

for

a

fixed

k

such

that

xkx21+l_k(p)~a0 and

r e 0 , since p~m'. Let F' be defined by F'= {geFlimalx. jig*x~g*x~+l_i(p)l =t'lx~:x2a+l_k(p)]). Now F' is non-empty and in fact F'=F (we will not prove this). Take an element geF'. There exists a chamber CegA such that peX~,g A for some ie{1../}. In fact p e X ~ for some s e I . This follows from lemma 3.2 and lemma 3.4 and the choice of g. b) The

fact

that

C is H - i n v a r i a n t

is clear. The pureness of

the

covering follows from

proposition 3.1 and the fact that for C,C'eB n there exists an appartment AcBI.I such that

C,C'eA. c) This is proved as in proposition 3.1.d and theorem 2.t.c. Remark: Let Tv denote the torus of rank one in the center of H. Let F 0 c T c be a discrete

co-compact subgroup. Let FlcSL(1,K ) be a discrete co-compact subgroup. Then /~=1`0X/'l is a discrete co-compact subgroup of H~GL(I,K). P r o p o s i t i o n 3.3: Let F c H

1`=1`o×F~,

be a discrete co-compact subgroup without torsion of the form

where 1"0 and 1"1 are as in the remark above.

variety XH /1" is non-projective.

Now the proper rigid analytic

377

Proof:

We

have

a

surjective

map

T:XH---*P]~-l-{K-rational

hyperplanes}:=~2 ~-1,

defined

by T((xt,..,x2,))= (xl,..,xz). The fibre of a point p e $21-1 is isomorphic to / ~ - ~ - { 0 } .

F=FoxF~

Since

A]~-x-{0}/F0.

The

the

morphism

algebraic

T

induces

dimension "of

the

a

~p:XH/F----.d -1/F~

map

Hopf-manifold

/k~-l-{0}/F0

is

with

fibre

strictly

less

than l-1 (See [Mus]). Furthermore I2z-1/F1 is an projective variety of dimension t-1. Therefore the algebraic dimension of XH/F is strictly less than 2/-2. So X H / F can not be a projective variety. Remarks: 1) In the proofs above we never needed the fact that p e X H satisfies the equation

f=O. We can delete this equation in the definition of the Xc~ s CeBH, s e I .

This gives us a

fD21-I 7 ~21-I pure H-invariant covering of ~/~ - ~ w The space rK - ~~ m ~t ~ for F c H discrete and co-compact

is proper. l

2) All we did in §3 is also true for Y defined by f = x ~ + ~xix2l+i_ i in P~/. In this case we i=1

have for A c B , where B is the building of G=PSO(f,K) again a space X A. Now XA: = Y - Z A , where

ZA

is

the

union

Xo=Xt+ 1 . . . . .

of

the

hyperplanes

xi=0,

i = 1..l

and

We can cover X A with the affinoids XC,A, CeA, i=O..I. 2 X ~ "=SpK

NOW

the

space

x2~=O. Note that t o n Y is the maximal isotropic space xt+t . . . . .

C,A"

~ XiX2t+l_ i ~ XiX21+l_ i

m

defined

by

x~=O.

for i=l..l.

Here yj is defined by:

yj =

x j,

j = 1..l

xix2t÷l-i

j=l+l..21

X21+l_j

V/x~x2t+l_j

, j=O

o A by X C,A o =SpK
] n+c~rseA~f, j = l . . l > n Y

and here the yj are

XO

given by: x: Y/=

j = 0..l

2 x°

, j=l+l..2I

X2l+l-j

The

subgroup

xl=x 2.....

H

is

now

the

x~=O and x,+~ . . . . .

stabiliser

of

the

maximal

isotropic

subspaces

defined

by

x~=O.

3) For a discrete co-compact subgroup I'cPSO(f,K)

it is not clear yet how to construct a

space X c Y such that X/I' proper. If we assume that / ' is neat (See [B]) then F n N A c T A , here NA is the normaliser of T A. Now F n N A acts properly on XA. The group F has infinitely many orbits on appartments A c B .

It is not clear wether there exists a F - i n v a r i a n t choice of the

X A such that Af~cBXA/F is proper. If such a choice exists, this would give the iv-adic analogon of the bounded symmetric domain which has SO(n)/SO(n-2)×SO(2),

(n=21(+l))

as its compact dual (See [Ku] §0.4).

This symmetric space is denoted by BDI ( q = 2 ) in [HI p.354 and by IV~ in [N] p.114.

378

REFEP~NCES

[AMRT] A. Ash,

D. Mumford, M. Rapoport and Y. Tai, Smooth compactification of locally symmetric varieties, Math. Sci. Press, 1975. [BB] W.L. Baily and A. Borel, Compactification of arithmetic quotients of bounded symmetric domains, Ann. of Math. 84 (1966), 442-528. [Be] V. Berkovich, Non-archimedean analytic spaces and buildings of semi-simple groups, Preprint I.H.E.S., february 1989. [B] A. Borel, Introduction aux groupes arithm~tiques, Hermann, 1969. [ss] A. Borel and J.P. Serre, Corners and arithmetic groups, Comm. Math. Heir. 48 (1973), 436-491. [BGRI S. Bosch, U. Giintzer and R. Remmert, Non-archimedean anatys~s, Springer Verlag, 1984. [BT] F. Bruhat and 3. Tits, Groupes r~ductifs sur un corps local I: Donn~es radicieUes valu6es, Publ. Math. I.H.E.S. 41 (1972), 5-251. [C] E. Cartan, Sur tes domaines bornes homog@nes de l'espace de n variables complexes, Abh. Math. Sera. Univ. Hamburg 11 (1935), 116-162; Also, Oeuvres Completes part I, 1259-1308. [D] V.G. Drinfeld, Elliptic Modules, Math. USSR-Sb. 23 (1974), 561-592 [F] G. Faltings, F-isocrystals on open varieties: Results and conjectures, Preprint Princeton University, 1989. J. Fresne! et M. van der Put, Geom~trie analytique rigide et applications, Progress in Math. 18, Birkh~user, 1981. [GP] L. Gerritzen and M. van der Put, Schottky groups and Mumford curves, Lect. Notes in Math. 817, Springer Verlag, 1980. S. Helgason, Differential geometry and symmetric spaces, Academic Press, 1962. [HI [I] M.N. Ishida, An elliptic surface covered by Mumford's fake projective plane, TShoku Math. J. 40 (1988), 367-396. [Kan] W.M. Kantor, Reflections on concrete buildings, Geometricae dedicata 25 (1988), 121-145. [KKMS] G. Kempf, F. Knudsen, D. Mumford and B. Saint-Donat, Toroi'dal embeddings I, Lect. Notes in Math. 339, Springer Verlag, 1973. [Ku] A. Kurihara, Construction of p-adic unit balls and the Hirzebruch proportionality, Amer. J. Math. 102 (1980), 565-648. [Mac] I.G. Macdonald, Affine root systems and Dedekind's U function, Inv. Math. 15 (1972), 91-143. [Mar] G.A. Margulis, Aritlm]eticity of irreducible lattices in the semisimple groups of rank greater than one, Inv. Math. 76 (1984), 93-120. [Mum.l] D. Mumford, An analytic construction of degenerating curves over complete local rings, Compositio Math. 24 (1972), 129-174. [Mum.2] D. Mumford, An algebraic surface with K ample, (K2)=9, pa=q=O, Amer. J. Math. 101 (1979), 233-244. [Mus] G.A. Mustafin, Nonarchimedean uniformization, Math. USSR-Sb. 34 (1978), 187-214. [N] Y. Namikawa, Toroidal compactification of Siegel spaces, Lect. Notes in Math. 812, Springer Verlag, 1980. [0.1] T. Oda, Lectures on torus embeddings and applications, Tara Inst. of Fund. Research 58, Springer Verlag, 1978. [0.2] T. Oda, Convex bodies and algebraic geometry, Springer Verlag, 1988. J.P. Serre, Arithmetic groups, in "Homological group theory" edited by C.T.C. Wall, is] London Math. Soc. Leer. Notes Series 36 (1979), 105-136. fT.1] J. Tits, Travaux de Margulis sur les sous-groupes discrets de groupes de Lie, Sere. Bourbaki 1975/76, Lect. Notes in Math. 567, Springer Verlag, 1977, 174-190. ff.2] J. Tits, Reductive groups over local fields, Proc. A.M.S. Syrup. Pure Math. 33 (1979), 29-69. T.N. Venkataramana, On superrigidity and arithmeticity of lattices in semisimple IV] groups over local fields of arbitrary characteristic, Inv. Math. 92 (1988), 255-306.

LIST OF PARTICIPANTS

Alan Adolphson, Department of Mathematics, Oklahoma State University, Stillwater, 74078 Oklahoma, U.S.A. Yves Andr6, Intitut H. Poincar6, 11 rue P. et M. Curie, F-7523I Paris 5, France Jesfis Araujo, Departamento de Matem~ticas, E.T.S.I. Industriales, Castiello de Bernueces, Universidad de Oviedo, 33204 Gijon, Spain Francesco Baldassarri, Dipartimento di Matematica, Universifft di Padova, Via Belzoni 7, 35131 Padova, Italy Edoardo Ballico, Dipartimento di Matematica, Universita di Trento, 38050 Povo (TN), Italy Luca Barbieri Viale, Dipartimento di Matematica, Universit~ di Genova, Via L.B. Alberti 4, 16132 Genova, Italy Jose M. Bayod, Departamento de Teoria de Funciones, Facultad de Ciencias, Avda. de los Castros s/n, 39071 Santander, Spain Vlaclimir Berkovich, Department of Theoretical Mathematics, The Weizmann Institute of Science, P.O.B. 26, 76100 Rehovot, Israel Pierre Berthelot, IRMAR, Universit6 de Rennes 1, Campus de Beaulieu, F-35042 Rennes Cedex, France Daniel Bertrand, Universit6 de Paris VI, UER de Math6matiques, Tour 46 5~me Etage 45-46, F-75252 Paris Cedex 05, France Frits Beukers, Mathematics Institute, State University of Utrecht, Budapestlaan 6, P.O. Box 80.010, 3508TA Utrecht, The Netherlands Siegfried Bosch, Mathematisches Institut, Westf'alische Wilhelms-Universit~t, Roxeler SlraBe 64, 4400 Mtinster, West Germany Abdelbaki BoutabaL G 511, R.U. "Jean Zay", F-92160 Antony, France Stefaan Caenepeel, Vrije Universiteit Brussel, Faculty of Applied Sciences, Pleinlaan 2, B-1050 Brussel, Belgium Maurizio Candilera, Dipartimento di Matematica, Universit~ d![ Padova, Via Belzoni 7, 35131 Padova, Italy Michel Carpentier, Math6matiques, Universit6 Paris VI, 4 Place Jussieu, F-75230 Paris Cedex 05, France Pierrette Cassou Nogues, Math6matiques et Informatique, Univ. Bordeaux 1, 351 cours de la Lib6ration, 33405 Talence Cedex, France Bruno Chiarellotto, Dipartimento di Matematica, Universit~t di Padova, Via Belzoni 7, 35131 Padova, Italy Gilles Christol, Universit6 de Paris VI, UER de Math6matiques, Tour 47 5~me Etage 45-46, 75230 Paris Cedex 05, France Robert Coleman, Department of Mathematics, University of California, Berkeley Ca. 94720, USA

380

Matthijs Coster, C.W.I., Kruislaan 413, 1098 SJ Amsterdam, The Netherlands Valentino Cristante, Dipartimento di Matematica, Universit~ di Padova, Via Belzoni 7, 35131 Padova, Italy Nicole de Grande-de Kimpe, Vrije Univ. Brussel, Faculteit Wetenschappen, Plenlann 2, 10F'/, B-1050 Brussel, Belgium Jan Denef, Department of Mathematics, University of Leuven, Celestijnenlaan 200B, B-3030 Leuven, Belgium Bernard Dwork, Department of Mathematics, Fine Hall, Princeton University, Princeton NJ 08540, USA Alain Escassut, D6partment de Math6matiques Pures, Universit6 Blaise Pascal, Complex des C6zaux BP45, F-63170 Aubi~re, France J. Y. Etesse, IRMAR, Univ. de Rennes 1, Campus de Beaulieu, F-35042 Rennes Cedex, France Jean Fresnel, UER Math6matiques et Informatique, Universit6 Bordeaux 1,351 cours de la Lib6ration, F-33405 Taience Cedex, France Ernst Ulrich Gekeler, Institute for Advanced Study, School of Mathematics, Princeton NJ 08540, U.S.A. Giovanni Gerotto, Dipartimento di Matematica, Universit?t di Padova, Via Belzoni 7, 35100 Padova, Italy Hiroshi Gunji, Department of Mathematics, University of Wisconsin, Madison, WI 53703, U.S.A. Frank Herrlich, Mathematisches Institut, Ruhr Universit~t, Postfach 102148, D-4630 Bochum 1, West Germany Ernst Kani, Department of Mathematics, Queen's University, Kingston K71 3N6, Ontario, Canada Ha Huy Khoai, Institute of Mathematics, P.O. Box 631, Buu dien Bo Ho, 10000 Hanoi, Vietnam Pierre Jarraud, 5 Avenue de la Porte de Villiers, F-75017 Paris, France Bernard Le Stum, 12 rue de Brest, 35000 Rennes, France Quing Liu, UER Math6matiques et Informatique, Universit6 Bordeaux 1,351 cours de la Lib6ration, 33405 Talence Cedex, France Fraqois Loeser, Centre de Math6matiques, Ecote Polyteclmique, F-91128 Paiaisean Cedex, France Werner Lfitkebohmert, Mathematisches Institut, Universit~t Miinster, Einsteinstr. 62, D-4400 Mtinster, West Germany J. Martinez-Maurica, Departamento de Teoria de Funciones, Facultad de Ciencias, Avda. de los Castros, s/n, 3907, Santander, Spain M. Matignon, UER Math6matiques et Informatique, Universit6 Bordeaux 1, 351 cours de la Lib6ration, F-33405 Taience Cedex, France Zoghman Mebkhout, UER de Math6matiques L.A.n. 212, Universit6 Paris VII, 2 Place Jussieu, F25175 Paris Cedex 05, France Diane Meuser, Department of Mathematics, Boston University, Boston, Ma 02215, U.S.A. Yasuo Morita, Department of Mathematics, Faculty of Sciences, Tohoku University, Aoba, Sendai 980, Japan Elhan Motzkin, 196 rue du chateau des rentiers, F-75013 Paris, France

381

Samuel Navarro H., Dpto. de Matem~itica, Universidad Santiago, Casilla 5659 Correo 2, Santiago, Chile Peter Norman, Department of Mathematics, University of Massachusetts, Amherst, 01002 Massachusetts, U.S.A. Arthur Ogus, Department of Mathematics, University of California at Berkeley, Berkeley, Ca. 94720, USA Pier Ivan Pastro, Dipartimento di Matematica, Universit~t di Padova, Via Belzoni 7, 35131 Padova, Italy Meinolf Piwek, Ruhr-Universit~it Bochum, Fakultiit ftir Mathematik, Postfach 102 148, D-4630 Bochum, West Germany Marc Polzin, UER Math6matique et Informatique, Universit6 de Bordeaux 1, 351 cours de la Liberation, F-33405 Talence, France Marc Reversat, Laboratoire D'Analyse sur les Variet~s, Universit6 Paul Sabatier, 118 route de Narbonne, F-31062 Toulouse Cedex, France Alain Salinier, 6, Avenue Montjovis, 87100 Limoges, France Paul J. Sally, Department of Mathematics, University of Chicago, 5734 South University Avenue, Chicago, 60637 Illinois, U.S.A. Roberto Sanchez-Peregrino, Dipartimento di Matematica, Universit~ di Padova, Via Belzoni 7, 35100 Padova, Italy Roberto Scaramuzzi, Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA Wim H. Schikhof, Math. Institut, Katholike Universiteit, Toernooiveld, 6525 ED Nijmegen, The Netherlands Claus Schmidt, Max-Planck-Institut ftir Mathematik, Gottfried-Claren-Stral~e 26, D-5300 Bonn 3, West Germany Steven Sperber, Department of Mathematics, University of Minnesota, Minneapolis, 55455 Minnesota,U.S.A. Harvey Stein, Department of Mathematics, University of California, Berkeley, CA 94770, U.S.A. Francis J. Sullivan, Dipartimento di Matematica, Universit~t di Padova, Via Belzoni 7, 35100 Padova, Italy Marko Tadi~, Department of Mathematics, University of Zagreb, P.O. Box 187, 41001 Zagreb, Yugoslavia Dinesh S. Thakur, School of Mathematics, Institute for Advanced Study, Princeton, 08540 New Jersey, U.S.A. Peter Ullrich, Mathematisches Institut, Universitat Mtinster, Roxeler Sm 64, D-4400 Mtinster, West Germany Bert van der Marel, Mathematisch Instituut, Rijksuniversiteit Groningen, Postbus 800, 9700 AV Groningen, The Netherlands Marius van der Put, Mathematisch Instituut, Rijksuniversiteit Groningen, Postbus 800, 9700 AV Groningen, The Netherlands

382

Lucien van Hamme, Faculty of Applied Sciences, Vrije Universiteit Brussel, Pleinlann 2, B-1050 Brussel, Belgium Guido van Steen, Dept. of Math., Rijksuniv. Centrum Antwerpen, 171 Groenenborgerlaan, B-2020 Antwerpen, Belgium Harm Voskuil, Mat. Instituut, Rijksuniversiteit Groningen, Postbus 800, 9700 AV Groningen, The Netherlands Taoufik Youssefi, Math6matiques et Informatique, Universit6 de Bordeaux I, 351 Cours de la Lib6ration, F-33405 Talence Cedex, France