Geometric Galois Actions: The inverse Galois problem, moduli spaces, and mapping class groups

Then

Ze2 - (S^ .

s = k(P)s k(V) = k{Tx,^T,)s

x ... xx...xS Ste) it)

= k(Yi,...,Ye) k(Yu...,Yt) =

where the functions in in the theV2 Yi with i E€ M^ are the the elementary elementary symmetric symmetric functions the

T2 Xf with ii EG MM. The Thenormalizer normalizerNw(S) Nw(S)permutes permutesY1, Yi,...,...,1^,which whichproves proves claim in the claim in the thecase case e = 1. 1. It remains —1.1. Now remains to to study study the thecase case e = — Nowwe wecan canorder orderthe theclasses classes C1,..., Gr element of of W) W) such such that C* / C " 1 for 11 < i < jj < ££ Ci,.», C V (using (using an element and C2 Ci / Cj. Then there is a partition partition {1,...,e} {1,...,^} =

MiU...UMsUMs+iU...UMf

with the same properties as as above, above,plus plusthe theproperty property that that C(/x) C(p) =fi C[y)~l with same properties z'. Finally, Finally, C(fi) C(p) = = C(^i)~ only ififss++ 11 < /x Observe that C(p)'l if and only for ptti^v. p < t. Observe then the classes classes C*+i,..., C correspondingly. For For l^ := = |MM| Crr are partitioned correspondingly. we we get get :

I

.5i) x

x 5/. x (Zf2s+1 • St.+1) x ......xx (Z|* • Sit) SS = = Stl x ... ... x St. x

viewed naturally naturally as subgroup subgroup of W W= = Z2 • Si. Thus viewed Thus k(V)s

-= k(T1,...,Tt)S k(Tu...,T£)s = = k(Yu...,Yt)

where the the Yi with i EG MM are functions in the are the the elementary elementary symmetric symmetric functions the Ti (resp., Tf) with i G AfM, for 1 < \x < s (resp., s + I < fi < t). The The o• up to normalizer normalizer N\v(S) Nw(S) permutes permutes 1'1, Yi,..., to sign. Hence Hence the theclaim. claim. ..., Y^ up

Choosing the the group §4. Choosing group G Now we we apply apply the above results with Now with the thefollowing following particular choice choice of of G G and C. andC.

Galois Realizations Galois

147 147

a power (2t—2) §4.1. Let Let nn = r—2 r — 2 = (2£ — 2) >2. > 2.Let Letqbe q be a powerofofthe theprime prime p, and be the finite field with with q elements. and let let lFq ¥q be finite field elements. Let ( j , •••jCr be elements elements = F p(Ci, of the multiplicative multiplicative group group 1^, with with Ci-.-Cr (i.. .(r = 1, ¥lFq q = ..., er), and ((', ...,Cr), C» $7^11for for all all i.z. Let Let ZZbe bethe thesubgroup subgroupof of F* generated generated by by (j, £i> •••,(,, (r,and andset set and set C = (Ci, ...,Cr). Let Let V V be be the the elementary elementary abelian abelian group group F£, and set ( ((i, ..., (r). G =

V•Z, V • Z,

the semi-direct product of of VV and and Z Z (where the semi-direct product (where Z Z acts acts on on V V via via scalar scalar multimultiplication). We write the the elements of G G as as pairs pairs [v, with vv G E V, V, z GE Z. plication). We write elements of [v,z]z]with ^. For zi = 1, ..., r, let C(&) be the conjugacy class of C For 1,..., G consisting consisting of of all all [v, [v,(i], £i], vv G e V. F . Set Set C c = c df The group group G = V • Z embeds embeds naturally as as aa normal normal subgroup subgroup into into

=c =

A0o = V• A V-

and of G C in A0 and the centralizer centralizer of A o is trivial. Via Via conjugation conjugation action action on on G G we we view A0 a group of automorphisms of G. It fixes all conjugacy classes view A as group of automorphisms of G. It fixes all conjugacy classes o a scalar subgroup C(Ci). Let Let M be be aa subgroup subgroup of of ]F, ¥*qview , viewit itasas a scalar subgroupofof GL n (g), and set set A -

V - M.

Consider the natural of the the group Consider the natural action action on on Ni(C^)(A) of group (2) (2)

A0/A A o /A = =

GLn(q)/M

§4.2. Let Br(C) be the §4.2. the group group of of all all Q Q EG Br Br such such that that for for the the permutation permutation it?r== K(Q) we have (^(i), ...,C7r(r)) = (Ci> -,Cr) (equivalently, (G (C7r(l), ..., n(r)) = 7r( i ) , ...,C ..., = ((i, (r) Set £r(() £ r (C) = - Br(() Br(C)nC (G l 7 ...,C (C1, flr.4. The group Br(C) (resp., £P(C) ) ..., Cr) r ) ). ). Set contains the braid group group B^r>) = ker(/s) (resp., the pure pure symplectic symplectic contains the pure braid ker(,c) (resp., braid £(r) = B^ 8(r) nfl £r). braid group group C^ £r). In [V3, §2] (and (and [V5, [V5, Ch. Ch. 9.4.1]) 9.4.1]) the the homomorphism $ c : Br(() [V3, §2] Br(Q —* -> GL n (g) is defined. More precisely, defined. It describes describes the braiding braiding action action on on Ni(C^). More precisely, the image A^ of this homomorphism is a subgroup of GLn(q) (q) with the the followfollowhomomorphism is ing property: each M M << W*the group is the stabilizer in in property: For For each the group A^M/M is the stabilizer q A A0/A (2)) of some some #Br-orbit A o / A = GLn(q)/M (see (see (2)) stabilizr -orbit in Ni(Cc)< >. (The stabilizers of of the the other other Br-orbits because AA0/A i?r-orbits are conjugates conjugates because Q / A acts acts transitively transitively on on these orbits). orbits). See See[V3, [V3,Lemma Lemma2.21. 2.2]. Similarly, the orbits orbits of of the the symplectic symplecticbraid braidgroup groupC4r < Br Ofl Ni(C^) Br on Similarly, the are determined determined by the image image of £ r (C) under $^. Denote this image image by by A^. Then A( Ac < < A c < GLn (g)

Karl Strambach and and Helmut Helmut Völklein Volklein

148

Analogously to [V3, Lemma Lemma 2.2] 2.2] we we conclude conclude that that A^M/M is the stabilizer Analogously stabilizer A an 4-orbit in in A A0/A / A of £ -orbit in Ni^(C )( ). o r c Define £ to be W-rational C^isisW-rational. W-rational.Now Nowwe weapply apply PF-rational ifif and and only only ifif Cc Theorem 2.2. Take Take VP = Pt as in in section section3.3. Then Then C£ = C and W Theorem 2.2. Ve as W are as as C,.r and in section C and and CC = C^ as chosen in the the present section. By section 3. 3. Let Let G chosen in present section. By [V3, Lemma Lemma 2.2(c)] 2.2(c)]condition condition(b) (b) of of the the Theorem Theorem holds. holds. By the [V3, the preceding preceding paragraph condition A^M/M paragraph condition (c) (c) of ofthe the Theorem Theoremholds holdsififthe thegroup group H H = AcM/M is self-normalizing A0/A self-normalizing inin A If C satisfies Cr-i+i = Q with o / A = GLn(q)/M. 1 , then also condition condition (a) of the Theorem holds holds (by the Lemma Lemma fixed e = ± ±1, in §3). Thus we we get

Corollary. W-rational (This (This holds holds e.g. e.g. ifif the the tuple tuple (£i, ..., ...,&) Corollary. Assume Assume £ is W-rational. —1.1. Assume is rational, see see §1). Fix e = 11 or e = — Assume Cr-*+i == Q for for ii = — 1,...,£. 1, ..., 1. Let Let M M be be aa subgroup subgroup of of¥*q. If If H^ H = A^M/M self-normalizing in AcM/M is self-normalizing GLn(q)/M, then H H occurs occursregularly regularlyover over Q. If / / additionally (q)/M, then additionally the centralizer of H in TLn(q)/M is trivial and [Aut(H) [Ant(H) : Irm(H)] pm, of H in Inn(H)] = m, where qq = ptm, has aaGAL-realization GAL-realization over over Q. Q. then H has Here (q) is the group of semi-linear Here TLn(q) semi-linear transformations of the vector vector space space

W%. If we drop the condition condition of W-rationality VF-rationality then A^M/M occurs occurs regularly over has aa GAL-realization GAL-realization over over Qab if Aut(F^) permutes over Q ab ; and it has

In the next give generators generatorsofofthe thegroup groupC4.r. Using these In next section section we we give generators, and basic pairing £ and from [V2], [V2], itit is generators, and the the basic pairing between between<& 4c and $^-i 1c-' from possible The case e == —1 possible to compute the the group group Ac A^ explicitly, explicitly, see see [MSV]. [MSV]. The —1 is is the case of main main interest, since since then A^ is mostly the symplectic symplectic group group Spn(#) or Sp n ( v /g). If q is even even then then PSp n (g) is self-normalizing in PGL n (g), yields GAL-realizations and Corollary applies and the Corollary applies with with M M= = W*This yields GAL-realizations overover q. This Q for the simple simple groups groups Sp n (2 s ) with ii n > 2S+1, s + [MSV]). GAL$ 22 (see [MSV]). Q realizations groups PGL n (#) were found realizations for for certain certain of of the groups (q) were found in (q) and PU n (g) [V3] (using group). [V3] (using the the Artin braid group).

§5. Generators of the symplectic symplectic braid group group

First we specify specify the the base base point point bb.. Let = —I, b2 Let b1 b\ — 62 = —t —^ ++ 1,..., 1,..., b1 bt = = —1 —1 and fer-i+i = —bi for 2 = 1, ...,l.

/ ...IE -b2

-b1

I I

b1

b2 b

2

I

I

Galois Realizations

149 149

Generators ...,Q Artin braid braid group group13,. Br can be specified specified as Qr—i Generators Qi, ..., r _i of the Artin Fix some some ii = 1,...,r— follows (see (see [V5, Ch. 10.1.6]): follows 10.1.6]): Fix l,...,r —1.1. For Forj j =—1,...,r, l,...,r, et be the constant path b3. Let a2(t) (resp., a2+i(t) i¥" 3 ¥" * +1» l flj W be the constant path 6^. Let a;(£) (resp., a» i(t) ) be i i+1, let + ) be a path in in the the complex complex plane with initial point bi (resp., &»+i) and endpoint endpoint travelling counter-clockwise counter-clockwiseon on the the respective respective circle circle with 6i_i_i (resp., 6i), travelling (ai(t), ...,a ..., a,.(t)), center center on the real line. Let Let Qi(t) = (a\(t), a path in O^r\ Note r (t)), a that Qr_i = — Qi. Define Define Qi E€13,.Br ==iri(Or,b) ni(Or,b) totobeberepresented representedby bythe the image of Qi in 0,.. Or. Define P1, ..., P1 E€ Br Define Pi,...,P* B r by by

j

Pi -= QiQr-i,

*i=1,...,t—1 = 1,...,^-1

and

P1=Q1 Pe = Qi These elements elementsare are naturally naturally represented representedby bypaths paths in in the the subvariety subvariety Vt Pt of of These Or. P1 lie in the symplectic braid group group C Crr = iri(Pt, b). We O Hence P1, Pi, ..., ...,P^ symplectic braid 'KiiVt^h). We r. Hence claim:

Proposition. The .., P1 generate Proposition. Theelements elementsP1, Pi, ...,P£ generateC,,. Cr. .

show that that via the thethe elements P2Pigetget Proof. We show the homeomorphism homeomorphism Qt —* —>P,Ve elements identified with the generators identified generators of of ir1(Q1) 7Ti(Qe)given givenbybyBrieskorn Brieskorn[Br]. [Br]. For a moment view W W again again as as acting acting on on C*, C1, in in the the natural representaFor moment view representation of the Coxeter group of type C1. Ce- A Coxeter (generating) system of W consists of the reflections o,, for for ii << I£ acts acts on the points consists reflections ai, ...,(j£, where where cr^ points (ai, ..., a1) E€ Ct + 1) (ai,...,a^) C^ as transposition transposition (i, (z,zj -f 1) on on the the coordinates coordinates (i.e., (i.e., interinterchanging a2 changing a» and a» + i); and at maps maps (ai,...,a*) ...,a^_i, — a^).Take Take (al,...,a,) to to (ai, (ai,...,ati,—at). Q(l) c c = {b1, {&i,..., &^} as base point in Q ^ c C^. ..., b1} as base point in C1. Restricting the we obtain obtain the natural reflection the action actionofofW WtotoIEI! R£ C C C1, C£, we reflection representation of W W in Euclidean space. The fixed point hyperplanes of the representation Euclidean space. reflections in in W decompose into chambers. reflections decompose Re into chambers. The The chamber chamber containing containing c all (ai,..., (ai, ..., a1) with a\ a1 <<... <0.0.The consists of all at) with ... P1). Vt).We We

150

Karl Strambach and and Helmut Helmut Völklein Volklein

b1, b3, ..-,6^), ..., b1), hence hence the line line illustrate the the case case of of P1. Pi. We Wehave havecr1(c) (J\{c) = (b2, (62,61,63, L1 L\ consists of all ((h + b2)/2 + T(h-b2)/2, (h + b2)/2 + + r(b2—bi)/2, T(b2-h)/2, b3, ...M) ((b1+b2)/2 + r(bi—b2)/2, (b1+b2)/2 b3,

,,

rr Ee CC

and we obtain the the path P± by taking rr = eit, we obtain by taking e 2t , 00 < t << ir. n. Clearly, Clearly, the the image image under the map of Pj* under map Q(t) Q^ —>•p(t), V^\ (ai, (ai,...,a^) '—* H-»(ai,'...,at, (ai,\...,a^,—at, — a^,..., — a{)is is ..., —ai) the path corresponds to P1. Pi. path QiQr—i. QiQr-i> Hence P{ corresponds

References Die Fundamentalgruppe des Raumes E. Brieskorn, Brieskorn, Die Fundamentalgruppe des Raumes der regulären regularen komplexen Spiegelungsgruppe, Spiegelungsgruppe,Inv. mv. Math. 12 Orbits einer endlichen endlichen komplexen 12 (1971), 57-61. 57-61. inverse Galois Galois problem problem and and rational [FV] [FV] M.D. Fried and H. Völklein, Volklein, The inverse points on moduli spaces, spaces, Math. Ann. Ann. 290 290(1991), (1991),771-800. 771-800. [MSV] K.Magaard, Magaard,K. K.Strambach Strambachand andH. H.Volklein, Völklein, Finite Finite quotients quotients of of the [MSV] K. pure symplectic braid group, in preparation. symplectic braid preparation. group over over the the rationals, rationals, Math. Ann. [Vi] Galois group Ann. [VI] H. Völklein, Volklein, GLn(g) as Galois 293 (1992), (1992),163-176. 163-176. [V2] [V2] H. Völklein, Volklein, Braid Braid group action action through through GLn(q) and U n (g), and Galois realizations, Israel J. Galois realizations, J. Math. Math.82 82(1993), (1993),405-427. 405-427. Völklein, Braid group action, embedding problems and the groups [V3] [V3] H. Volklein, groups 2 PGL(n,q),PU(n,q2), Forum Math. 6 (1994), 513—535. PGL(n,q),PU{n,q ), Forum Math. 6 (1994), 513-535. [V4] of the Riemann sphere, Israel [V4] H. Völklein, Volklein, Moduli Moduli spaces spaces for for covers covers of Israel J. Math. Math. 85 85(1994), (1994),407-430. 407-430. Groups —- an [V5] [V5] H. Völklein, Volklein, Groups GroupsasasGalois Galois Groups anIntroduction, Introduction,Cambridge Cambridge University Press, Press, 1996. 1996.

[Br] [Br]

Karl Strambach, Universität Universitat Erlangen Erlangen Helmut Völklein, University of of Florida Florida and and Universitat Universitãt Erlangen Volklein, University Erlangen

Applying to the Inverse Applying Modular Towers Towers to Inverse Galois Galois Problem Problem Michael Michael D. Fried and and Yaacov Yaacov Kopeliovich Kopeliovich

Let G be a finite (possibly simple) group, group, and let p be a prime dividing (possibly simple) dividing the order order of G. The The characteristic characteristic finite finite quotients quotients pG of the universal p-Frattini cover of G G are strikingly strikingly similar similar groups. groups. It takes takes an an effort effort realizations for for the Inverse distinguish them them for for finding finding Q regular realizations Inverse to distinguish Galois problem. This This paper paper starts aa program Galois problem. program to to show show one can't realize realize all these of extensions extensions L/Q(x) L/Q(x) with these groups groups as Galois Galois groups groups of with at at most most (fixed) branch branch points. points. Let Let C C be bean anr-tuple r-tupleofofp-regular p-regularconjugacy conjugacy r (fixed) classes of realizations of these groups we use a sequence sequence classes of G. G. To compare realizations of varieties—a Modular Modular Tower—attached Tower—attachedtoto (G,p, (C, p,C). C). The The notation for of varieties—a for is the the kth sequence is is K(*G,C), k k = 0,1,...: 0 , 1 , . . . : U(*G,C) is fcth level level of of this sequence the Modular Modular Tower. Tower. Crucial properties of level level kk translate to to properties properties of the characteristic characteristic modular modular representation representation of of pG. Properties of these these of representations support the the following following statement. representations

Conjecture. For that for for kk >> kr, Foreach eachrrthere there exists exists k,. kr so so that kr, Q Q regular realization of pG requires more more than r branch branch points. points.

For For r = 4 this reduces to showing showing two pieces of geometric information. information. (a) (a) (b) (b)

There is a uniform uniform (with k) bound bound on on the the number number of of absolutely absolutely irreducible components at at the kth irreducible components kth level. level. For kk large C) has no large H(pG,G) no obstructed obstructed components. components.

The of this this paper paper and and [FrK] G = A5, The main main example example of [FrK] isis G A$, p p = 22 and C == C^r with with rr = 44 repetitions of of the the conjugacy conjugacy class class of of 3-cycles. 3-cycles. It allows full full explanation explanation and and illustration of the of obstructed allows the significance significance of components.

§0. Introduction to the the main main problem. problem. [MT]introduced introduced Modular Modular Towers. Towers. This This paper concentrates on exactly one [MT] application of of these these to to the Inverse Galois Problem. Problem. Fix an indeterminate xx application Inverse Galois Support from NSF #DMS-9622928 #DMS-9622928 for forboth both authors; authors; first first author author support from from NSF from Alexander von Humboldt Foundation Foundation and and Institut für der von Humboldt fiir Experimentelle Experimented Mathematik, Mathematik, July July1996. 1996. Correspondence with with John John Thompson, Thompson, after after an Oct. Correspondence Oct. 1996 1996 conference conference at UF, UF, induced induced us us reducing our our Main Main Conjecture Conjecture to a diophantine statement to write §4 reducing statement on on points pointson onModModular Towers. Towers. Conversations Conversations with Bob Guralnick improved improved our confidence confidence with the group group theory paper. Finally, Finally, Elena Elena Black's Black's conference conference talk exclamation, "Do "Do you theory parts of the paper. talk exclamation, you know work with one group time?" inspired inspired the thephilosophy philosophy of of §1.B: §1.B: know how howhard hard itit is to work group at aa time?" classes and and change the group. Fix conjugacy conjugacy classes group.

Michael Fried Fried and Yaacov Michael Yaacov Kopeliovich Kopeliovich

152 152

as a uniforinizing uniformizing parameter for for the projective projective xx line line IF" P 1 = P*. For any field closureofofK. K. The The characteristic characteristic of ofKK isis 0, 0, and and K K K let let Kk be an algebraic K algebraic closure is usually a subfield of the complex numbers C. The symmetric group of usually subfield of the complex numbers C. The symmetric group of degree Sn and subgroup is is An. degree n is S,-, and its its alternating subgroup

of this this paper has The framework framework of has appeared appeared nowhere nowhere else. else. It includes includes an introduction to more results of [Fri] introduction more comprehensive comprehensive results [Frl] and and [FrK]. [FrK]. These These will will quote this paper. quote paper. Modular ModularTowers Towers join join Hurwitz Hurwitz space space constructions constructions and and the universal universal Frattini cover cover of of aa finite finite group. group. App.I App.Ireviews reviews relevant relevant defdefinitions from from [MT]. One crucial crucial point: point: Levels of a Modular initions [MT]. One Levels of Modular Tower Tower aren't Yet, modular homogeneous spaces. spaces. Yet, modular representation representation theory theory pro(usually) homogeneous vides replacement for semisimple Lie vides a replacement for the semisimple Lie group group theory theory often often attached attached to homogeneous spaces.

§0.A. realization §0.A. Basic Notation. Notation.Recall Recallwhat whatisisa aKKregular regular realizationof of aafinite finite group G: G is the Galois group of of an an extension extensionL/K{x) L/K(x) with with L fl K = K. group Galois group LnK K. The main combinatorial data data for for aa regular regular realization realizationisisaa set set of of conjugacy conjugacy classes CC from from G. C. AAplace classes place x'x' ofofC(x) C(x) denotes denotes the thespecialization specialization of the the 1 any place place p of of C C.•LLover elements of of C(x) C(x) associated associatedtotox x'—+ *-» xx'.. Consider Consider any over f be the ramification 1 for x'. Let Let ee = e(p/x e(p/x') x!. ) be ramification index index of of p over over x'. xr. If If e > > 1 some pp over overx\x', call callx'x' aa branch pointofofthe theextension extensionCC• L/C(x). some branch point L/C(x). Then, Then, the completion L at pp embeds completion Lp of L embeds in in the theLaurent Laurentfield fieldC{{(x C{{(x — — x')e}}. Restricting the automorphism automorphism (x (x— — xf)^ '—* H-> exp27"/e(:r — x')* to L gives 1 gp e G. Referencing Referencing only xx', ', and and not not p,p,defines defines Yp gp up to conjugacy conjugacy in in G. G.

This attaches aa unique class C C of of G G to to x'. x'. Thus, unique conjugacy conjugacy class Thus, the the complete complete of branch set j : = {{x1,. # i , . . .. ,x,.} , x r } of branch points of of the the extension extension produces produces aa colleccollection C { C i , . .. .. ,, C conjugacy classes description of of branch branch tion C = {C,,. classesin in G. G. A description Cr} r } of conjugacy cycles for consists of (#i,... ,g G Gr with these these cycles forL/K(x) L/K(x) consists of an an r-tuple r-tuple Qg == (91,. r) E , properties: .

. .

(0.Oa)ggE C, 0igi...gr (0.0a) G C, • • • g=H(g)= = and 1; and r = n(fl) 1; (0.Ob) <(g) (0.0b) G. fl) = G. Here gG E CC means, Here $ means, in insome some order, order, entries entries of of gg are are in inthe theclasses classes C. C. For any regular extension extension L/Q(x) L/Q(x) such such an an r-tuple r-tuplesatisfying satisfying (0.0) (0.0) always always exists. exists. It It isn't, isn't,however, however, unique. unique. The TheNielsen Nielsen class classNi(G, Ni(G, C) of of aa cover cover is is complete collection collection of This the complete of elements elements gg satisfying satisfying the the conditions conditions (0.0). (0.0). This codifies the different covers covers and all descriptions descriptions of of branch branch codifies the relation between different cycles (see App.I). cycles

§0.B. §0.B. The Main Main Conjectures. Conjectures.Let LetCGbebea finite a finitegroup group and and pp aaprime prime C is centerless centerless (has (has no nocenter). center). The The pair pair (G,p) dividing \G\. Also, assume assume G (G,p) dividing Cl. Also,

Modular towers

153 153

produces an an infinite sequence of of characteristic characteristic Frattini covers: produces infinite sequence covers: •••^kpG^kp-1G^---^°pG

(0.1)

= G.

G G §1 .Breminds remindsofofthe thebasics. basics.The Thecharacteristic characteristic homomorphism homomorphism (j>k /k : pG—*—> §1.B Let pC(G) be those has p-group has p-group kernel kernel for for each each nonnegative nonnegative integer integer k. k. Let those conjugacy classes classesof ofGGwhose whoseelements elementshave haveorder orderprime primetotop.p. These Theseare are conjugacy lifts uniquely to the p-regular conjugacy classes of G. Thus, C E p-regular conjugacy classes of G. Thus, E PC(G) lifts uniquely to aa elements with with the the same order. Also, conjugacy class in conjugacy in ^G of elements same order. Also, refer refer to to this this Denote the the cardinality cardinality of of lifted lifted conjugacy conjugacy class class as as C: C: C C eE pC(pG) = PC(G). Denote classesin inC C by by rr = r(C). the conjugacy conjugacy classes :

any Main Conjecture Conjecture0.1. 0.1.Let Letr0rbe anypositive positive integer. integer. 0 be (f.a) (t.a) (f.b) (t.b)

ro) with this property. property. For Forkk> regular realiza= ka(G,p,ro) 3 ka ka = with this > ka, ka, Q Q regular realizawith r branch cycles in requires r > r0. tion ofpG with r branch cycles PC(G) requires r r 0 . property. For k > 3 kb ^ = kt>(G,p,ro) with this property. > kb, k^, Q regular regularrealizarealizakb(G,p, ro) with r$branch branchcycles. cycles. tion ofpG requires requires rr>>r0

notation to consider this over restrict often often to to that It simplifies simplifies notation over Q. So, So, we restrict case. Still, for K finitely case. finitely generated over over Q replacing Q this should hold with ka ka and kb kb depending depending on K. K.

Let Let r be bean aninteger integer and and CCa acollection collectionofofr rconjugacy conjugacy classes classes (possibly (possibly with repetitions) (G). To with repetitions) from from PC(G). Tothis this data data[MT] [MT] canonically canonically attaches attaches aa sequence of of reduced reduced Hurwitz Hurwitz spaces (App.II): sequence (App.II): (0.2)

..• • • ft(£+1G,C)rd ... ÷ n(°pG, C) r d . C) r d -> U{kpC, C) r d_÷->•••-> .

Special Case: Case: G G == kth level level is is the manifold manifold W(*G, C) r d and°VG= = G. Special The kth the dihedral groupof oforder order2p, 2p,ppisisan anodd oddprime primeand andCCisisrr = 44 Dp is the dihedral group of involutions. involutions. Then, (0.2) repetitions of the conjugacy conjugacy class of (0.2) is is the the classical classical sequence of of modular curves: sequence curves: • • • -> Yi(p* +2) -> Y1(pk+1) - + . . . - ) . Y1(p).

This case with the This case agrees agrees with theconclusion conclusion of the Main Main Conjecture. Conjecture. For For this this case, it is easy to case, however, however, it toshow show the thelevels levels have have only only one one (unobstructed) (unobstructed) component. [Fr3, [Fr3, §7] gives gives the the reduction reductionof of(f.b) (t.b)to to (f.a) (t.a) for G = = Dp and for any value value of of r0. r0.

strong; they they translate translate to to For Dp and and r0 ro = 44 (or (or 5) the results results are very very strong; known results of known results of Frey, Frey, Kamienny, Kamienny, Mazur Mazur and andMerrill. Merrill. They They even even give give a statement statement uniform uniform ininpp([DFr, ([DFr,§51 §5] or or [Fr3, §7]). for less less §7]). Conjecture Conjecture 0.1 asks for

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Michael Fried and Yaacov Michael Yaacov Kopeliovich Kopeliovich

realization requires requires at least 6 branch points case: that Q regular Dpk realization in this case: if kk>>kb. kb.(Also, (Also, kb kb here here may may depend on p.) The goal is to generalize the following followingargument. argument. Fix Fixpp and and let let K K be any generalize the any finitely generated extensions extensions over over Q. Q. The genus genus (of the single single component) component) k of y1 goes up up (quickly) (quickly) with with k. Thus, Thus, Faltings' Theorem of Yi(p(uk) ) goes Theorem implies implies the k )/j(pk) K points points Yi(pk)(K) on Y\(p ) are finite, excluding finitely many values of The conjecture conjecture thus comes projective system system pk. The comes to eliminating eliminating a possible possible projective fc+1 §7] handles handles this this when when G = t)k EG Yi(p ) of K rational rational points. points. [Fr3, [Fr3, §7] = Dp and r0 r 0 = 44 (or 5). For For an arbitrary arbitrary finite finite group group G, G, each eachstep stepabove aboveencounters encounters problems. problems. [FrK] reveals main difficulties. difficulties. This [FrK] reveals the main This case case has has G G— = An and and C C = Cy, r repeats of the the conjugacy conjugacy class of 3-cycles. 3-cycles. The The Nielsen Nielsen class Nin,r Ni n?r of r appears often. often. Use the notation ?in,r 3-cycles %n,r for the corresponding corresponding 3-cycles in An appears (inner) Hurwitz Hurwitz space space (App.I). (App.I). Though Though most most groups groups produce produce difficulties difficulties (inner) that don't don't appear appearfor forthe thedihedral dihedralgroup, group,conjecturally conjecturally Modular Modular Towers Towers do do mimic properties of of modular modular curve curve towers. towers. Even, with G = Dp mimic properties Even, however, however, with the conjecture conjecture is open if r0 VQ>>5.5. Here are three accomplishments accomplishments of this paper. paper.

(O.3a)For Forany anyGG itit shows that the (0.3a) shows that the (t.b) (f.b) version version of of Main Main Conjecture Conjecture 0.1 0.1 reduces (f.a): For reduces to showing showing (f-a): For kk large, large, K(pG, C) r d has no Q points. C3r it describes 2 and (0.3b) For For G = A A5, (0.3b) and C = C^ describes progress progress on bounding bounding 5, p = 2 components of the the kih kth level, components of level, kk >0,0,ininthe theModular ModularTower. Tower. (0.3c) It shows (0.3c) shows how how obstracted obstructedcomponents components (§0.C) affect affect the Main Main ConjecConjecture. Further, Further, A5 A$ examples examples support their their disappearance disappearance for for kk large. large.

§0.C. §0.C. Obstructed components components of of aaModular ModularTower. Tower.The Theprojective projective limit of sequence sequence (0.1) universal p-Frattini cover cover PG of limit (0.1) is is the universal of G. §1.C §1.C

explains how similar similar are are all the groups explains how groups kG. produces conclusions conclusions on on regular regular realSubtheme 0.2 Replacing G with pG produces izations of all PG characteristic characteristic quotients.

For fixed fixed r,r, computations suggest there is is a uniform uniform bound on on absolutely absolutely irreducible components of ofU(kG, C) r d . When r=4, r=4, reduced reduced Hurwitz Hurwitz spaces spaces are curves. Bounding kth level Bounding the the number number of of components in the fcth level is necessary Theorem applies. is, the the genus genus of of absolutely absolutely irreto assure Faltings' Theorem applies. That is, ducible components ducible components at at level level kk must must exceed exceed11for forkklarge. large. Still, Still, that that doesn't preclude rational points points at arbitrary preclude rational arbitrary high high levels levels on the most most mysterious mysterious of we call obstructed. components, those we points of %(£+1G, C) r d lie above above aacomponent componentW fl' of Suppose no points ofU(kG, C) r d . We say %' W is We say is obstructed. obstructed. [MT, [MT,§III.D} §111.D] gave gave a big big invariant invariant detecting detecting obob-

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structed components. components. Obstruction Obstruction Lemma Lemma 3.2 3.2 reformulates reformulates this this invariant invariant using modular modular representations. using There is good good news news and and bad bad news in the appearance of There news in of obstructed comcomponents. ponents. The The former former occurs occurs if if there there is is exactly exactly one one component component at level level k and it is is obstructed. Then, Then, (0.3a) (0.3a) has has aapositive positive answer. answer. The The reduced reduced HurHur[Fri] gives witz at level k+1 is empty, so it has witz space space at level A;+l empty, so has no no rational rational points. points. [Frl] gives examples of exactly exactly that that (see examples of (see Thm. Thm. 3.1 3.1 in in §3.A). §3.A). Many cases, cases, however, however,have haveatat least least one one unobstructed unobstructed component. Many component. GenGeneral results describing of a level describing components components of level of a Modular Modular Tower Tower require require conjugacy classesininCC to to appear appear with with high high multiplicity. multiplicity. The The rest rest of this conjugacy classes subsection subsection illustrates this.

Definition Definition 0.3 0.3Consider Consideraacollection collection CCofofconjugacy conjugacy classes classes from from a finite finite group group G. Let Let s0 s0 be aa positive positive integer. integer. Then, Then, C C has has multiplicity multiplicity at at least least s0 so if each conjugacy conjugacy class class in in CC appears appears at least least s0 so times. times.

Suppose CC has Suppose has suitably suitably (explicitly) (explicitly) high high multiplicity multiplicity compared compared to k. k. Then, an of a Conway and Parker result an effective effective version version of Conway and result precisely precisely bounds bounds the components components at level level k. The The following following is a special special case of a result from from [FrK]. Here M(*G) is the Schur Schur multiplier multiplier of of £G. Theorem k. Assume Theorem0.4 0.4Fix Fixa avalue valueofofk. Assume GGisisperfect perfect and andcenterless, centerless, p \ \G\, IGI, all classes in in C C are C has multiplicity Decomall classes are from pC(G) and C multiplicity at least SQ. Decompose pose HipG, C) r d into absolutely absolutely irreducible irreducible (dimension (dimension r—3} r—3) components components Then, there is an explicit k) = = s0 U^ 0 ?i^. Then, explicit so(G, k) so so the the following following hold. hold. (0.4a) tk (0.4a) tk + 1l = \M(kpG)\. (0.4b) Hf0 is unobstructed. unobstructed.

(0.4c) If If C is (0.4c) is aarational rational union union (see (see §1.A^, T~Lfhas fieldof of definitionQ.Q. field definition 0 has (0.4d) %[ is obstructed, obstructed, ii = 11,., . . . ,, tk. tk. .

.

§O.D. Disappearance of of obstructed obstructed components §0.D. Disappearance componentsfor forkklarge. large.Suppose Suppose G has has aa nontrivial nontrivial Schur Schur multiplier multiplier (as (as does An). An). Then, Then, so so does does ^G for all k (Schur (Schur Multipliers Multipliers Result Result 3.3). 3.3). Thus, Thus, Theorem Theorem 0.4 0.4 implies implies obstructed obstructed components appear in %T^r for components appear for r large large (notation (notation from from §0.B). §0.B). The The crucial crucial investigation investigation must must consider consider minimal minimal values values ro r0 = r0 r*o(n, k) of r that produce (n, k) of r produce obstructed components in H^G, C3r) rd , By contrast, a particular Modular Modular Tower considers There are are Tower considerskk large large compared compared to to aa fixed fixed value value of of rr = rr(C). (C). There serious obstructed serious diophantine diophantine troubles for for the Main Conjecture should new obstructed components at each each level. level. components appear at For example, suppose suppose there is is an an s0 SQgiving giving the the following. following. (0.5a) (0.5a) Theorem Theorem 0.4 0.4 holds with SQ independent independent of ofk.k.

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(0.5b) There There is a bound on |M(pG)| independent of k.k. (0.5b) This would produce aa bound bound in in (0.3b), (0.3b), at at least least for for rr large. large. That wouldn't, would produce wouldn't, however, contribute to to the the Main Main Conjecture. Conjecture. Even Even the strongest strongest diophantine however, contribute

conjectures can't eliminate rational points points on conjectures can't eliminate rational on obstructed obstructed components components at high levels levels of a Modular Tower. Tower. Worse Worse yet, (0.5b) doesn't hold. hold. arbitrary high yet, (0.5b)

So, it is a surprise that the case G = A5, 2 and and C C == C^r C3r supports So, it surprise that case G A$, pp = 2 supports the following conjecture. Recall: Recall: A projective variety V following conjecture. projective nonsingular nonsingular variety V is of general type type ifif it's it's canonical canonical bundle bundle isis ample. ample. If If V V is a curve this means V rd C)rd has genus genus at least two. Using Using the the phrase phrase general generaltype type on on H(pG, C) means to some some nonsingular nonsingular compactification compactification of of its components. components. apply it to

Conjecture I IGI and Conjecture 0.5. 0.5. Assume Assume G isis centerless, centerless, pp\\G\ and C are areconjugacy conjugacy classes from from PC(G). classes

Then, there there are are two two constants constantsBB— = B(G, Then, B(G,r)r) and and k0 ko ==

k0(G, ko(G, r) so sothat that for forkk>>k0, k0,the thefollowing following hold. hold. (O.6a) ^(pG, C)rd (0.6a) C) r d has at most most B B absolutely absolutelyirreducible irreduciblecomponents. components. rd (0.6b) Each component C)"' is of general (0.6b) component of H(pG, C) general type. type. rd C)rd are (0.6c) (0.6c) No No components components of ofH(%G,C) are obstructed. obstructed.

The classical connection connection has G = an odd odd prime, prime, and and CC is is rr = 2r 2r'f = Dp, p is an repetitions of the conjugacy class of of involutions involutions for for some someinteger integerr'. r'. Braid repetitions conjugacy class Braid

group action here is easy to group action on Nielsen Nielsen classes classes here to calculate. calculate. As in in App.I, App.I, Ni(G, C) C) correspond correspond to to the orbits of the braid group group Br Br on on Nielsen Nielsen classes classes Ni(G, components of of %(G,C). fl(G, C). [Fr4, showsthere there isis one one orbit. orbit. So components [Fr4, §3] §3] shows So (0.6a) (0.6a) and hold. When 2, §0.B (0.6b) holds. holds. This seems to be (0.6c) hold. Whenr'r' = 2, §0.B notes notes (0.6b) This seems to be unknown higher values values of r'. unknown for for higher of r'. Beyond dihedral groups, groups, evidence evidencefor for (0.6c) (0.6c)isis from from the the case case G G == A5. Beyond dihedral A$. Lemma 3.2 uses uses the the action Lemma 3.2 action of the characteristic characteristic quotient JG on the charcharfor all values kernel kerk ker^ / kerfc+i values of k. A;. Consider Consider this contrast contrast acteristic kernel kerk+l for with with k0 A;o fixed fixed and and r large. large. Then, Then, Theorem Theorem 0.4 0.4 says says 7/(2°A 5 ,C 3 r) rd has has least two two absolutely absolutely irreducible irreducible components; components; one unobstructed and all all at least one unobstructed others obstructed. For rr ==4,4,however, however,we weexpect—a expect—a result result of of [FrK] [PrK] for for others obstructed. For small—no obstructed k small—no obstructed components components for for any any value value of of k. k. There should be aa (ko) of of r for for which there are no obstructed components components minimal value value r'r' == r'rf(k0) jfl in 7^(2^5, C 3 r /) r d for fc0In - Insupport supportofof (0.6c),[FrK] [PrK]makes makesan anexplicit explicit for kk >k0. (0.6c), guess for r'(ko). guess for r'r' = r'(ko). 1. Precise Precise versions versions of of the the main conjecture. §1.

§

Here is the the data for Here is for aa Modular Modular Tower: Tower: a finite finite group G, aa prime prime pp IGI \ \G\ and a collection of r conjugacy collection of conjugacy classes classes C from from PC(G). Recall: is Recall: DDr is the r discriminant locus in projective projective r-space. r-space. We We concentrate concentrate on on realizations realizations

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of a centerless centerless group group G. G. [Fry] equivalent to finding finding aa Q point [FrV] shows shows this is equivalent on an inner inner Hurwitz Hurwitz space space 9L(G, %{G, C) = 1-I(G, H(G, C)in for for some some C. C.See See§O.A §0.A and and for Nielsen Nielsen classes and how (G, C) canonically produces App.I for and how C) canonically produces the moduli Fr \Dr. space H(G, with aa cover cover $ G ,c = * : fl(G, H(G,C) C)—÷ -> P \Dr. Suppose K is is a space fl(G, C) with U{G,C). C).Denote Denotethe thefield fieldgenerated generated field of of definition definitionofofU{G, 'h(G,C) C) and and p G e fl(G, by coordinates coordinates ofofppover (p). In In this this paper paper points points are aregeometric geometric by overKK as as ifK(p). Also, having having coordinates coordinatesfor forppmeans means?-L(G, W(G, C) quasi-projective points. Also, C) is a quasi-projective (even affine) affine) algebraic set set ([Fr2J, ([Pr2], [FrVJ [FrV] or or [V]). [V]). :

§1.A. Nc = N be the least §1.A. Notation for for Q Q moduli spaces. Let Let Afc least common common multiple of the orders of of elements elements in in the the collection collection C. C. The multiple of The group group Z* is the projective limit of the invertible of Z/iV Z/N over projective limit invertible elements elements of over all all positive positive integers integers N. Consider K (a subfield of C). C). A A Galois Galoisextension extensionL/K{x) L/K(x) is is aa K Consider a field K subfield of realizationifif the thefollowing following hold. hold. regular (G, C) realization

(1.la) G(L/Q(x)) and L nfl KK = if, K, and (1.1a) G(L/Q(x)) = G G and (1.lb) points of the x line ramified in L have (1.1b) points of the x line ramified have associated branch branch cycles cycles in in C. C. Covers associated to to such such extensions extensions have have rr branch points. According Covers associated According to to the branch branchcycle cycle argument argument ([Fr2, ([Fr2, before before Thm. Thm.5.1], 5.1],[Fr3}, [Pr3], [V, [V, p. 34]), 34]), there is a first first criterion criterion for such a realization. realization. The The minimal minimal field field containing containing all 02 cyclotomic closure of Q. formulation uses uses the roots of 11 is is Q ^, the cyclotomic Q. The formulation K-cyclotomic group: if-cyclotomic group: C2/C E G(Q HK = {o {a G /Q) ^ V Z* o | a fixes fixes elements elements of of K K fln QCYc}
Each n E G HK HK defines defines an an invertible invertible integer integermodulo modulo TVc- So So C n , all elements of C to the thepower power11, n, makes sense. sense.

Definition 1.1 Call C = (C1 Definition Call C (Ci,..., Cr) C r ) aa K-rational K-rational union union if if C n = C for n each n G HK. Then, CC is a rational union if C = CC for for each each nn eGV. Z*. [MT, p. 161] explains the the meaning of this phrase: 7L(G, C), as a moduli 161] explains meaning of H(G, C), moduli space has has field fieldofofdefinition definitionK.K. Roughly: Roughly: For For any any point point p eG %(G, C) C) the of definition definition of of aa Galois Galois cover cover in in the the Nielsen Nielsen class class representing representing p is least field of 7f (p). The The K-rational If-rational condition condition in the next next result result is is necessary necessary for K(p). for H(G, fl(G, C) to be a moduli space Again, the elementary elementary branch cycle cycle argument space over over K. K. Again, gives this. That That K-rationality if-rationalitysuffices suffices isis tougher—see tougher—see [Fry] [FrV] or or [V, [V, §10.3.2]. gives

K-Branch Cycle Argument if-Branch Argument 1.2 1.2 [Fr2, [Fr2, §5] Assume Assume G G isis aa centerless centerless group. group. The moduli moduli space space H(G, fl(G, C) (with its morphism has (with morphism 4>) field of of definition definition K and only only ifif C C is a K-rational ifif and if-rational union. union. If fl(G, H(G,C)(K) is nonempty, nonempty, then then C) (K) is C is is aaK-rational if-rationalunion unionand and7-((G, %(G, C) C) contains contains an anabsolutely absolutely irreducible irreducible component over Also, if 7/(G, C) produce produce K if regular regular (G, (G, C) component overif. K. Also, K points on 9t(G, realizations.

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For the remainder remainder of of the the paper paper concentrates concentrates on on K K = Q. For simplicity, simplicity, the In Q. In particular, assume assume the conjugacy conjugacy classes classes C from from G is is aa rational rational union. union.

§1.B. Fix conjugacy §1.B. Fix conjugacy classes, classes, change change the thegroup. group.Our OurMain MainConjecConjecand §3 ture divides divides into into cases: cases: (p, (p,Nc) §2 and §3 consider consider the Nc) = 11 and p\ NQ- §2 case (p, (p,Nc) = 1. 1- The Main Conjecture Conjecture reduces reduces to this case case according according to to Theorem Theorem 4.4. Analysis Analysis of components components of W^G, C) as as kk varies varies requires requires ininformation To help, help, we we expand expand here here on on [FrJ, [FrJ, Chap. 20] and and [MT, formation about PG. To Part II]. II].This Thisdiscussion discussion will will continue continue in in [FrK]. [FrK]. The universal —* -» of of G is forfor embedding GG universal p-Frattini j?-Frattini cover cover (J>G '• PG G versal is versal embedding with p-group kernel. The meaning 1 G —+ problems withp-group meaning of versal: Given i\): :AA—*—> G —» 1 f exact, there is a map ip : pG —* —»AAgiving givingthe theobvious obviouscommutative commutativediagram. diagram. Versal, unlike unlike universal, universal, does does not not mean ty1 is unique. If G is perfect, Versal, unique. If perfect, the p-central extension extension of of G G isis an an example example of ofaanatural natural quotient of PG. universal p-central very small small quotient. quotient. We It is, is, however, however, a very We explain this. Let Let ker0 ker0—÷ —> PG—+ -» GGbe bethe thenatural naturalshort shortexact exactsequence. sequence.Then, Then,ker0 ker0isisaa pro-free p-group p-group of of finite finiterank. rank. Define Define ker& kerk inductively: inductively: It It is the closed subpro-free group of PG that [kerk_l, generates. Then, £G == pG/ker^. [kerA;_i,kerjt_i]ker^_ kerk_l] kerk. 1 generates. Follow [FrJ, Chap. Chap. 20] 20] in in calling calling the the minimal minimal number number of of elements elements that that genFollow [FrJ, erate it its its rank. rank. The hypothesis hypothesis (p, (p, JVc) = — 1 1allows allows aaspecial special choice choice of of lift lift ofofentries entriesofof C to Choose lifting tocorresponding corresponding conjugacy conjugacy classes classes of of pG/kerjt kerk = pG. Choose representatives with the the same same orders orders as as their their images in G. G. To representatives with images in To emphasize emphasize this choice keep the the notation notation C for choice of conjugacy conjugacy classes in pG, keep for these these lifted lifted classes. Further, as conjugacy conjugacy classes classes in £G, C isis also also aa rational rational union, union, classes. Further, as inheriting this property from from G. G. The Frattini cover cover property appears appears here. here. Suppose Suppose elements elements of of pG are entries of g' EG C. Also, Also, assume assume g'g' lifts lifts entries entries of of gg G e Ni(G, C). C). Then, entries of Then, ; From the character theory viewpoint—including generalizations (g')) = pG. From the (g viewpoint—including generalizations of rigidity—(^G, C) realizations realizations are much much like (G, C) realizations. :

:

Reminder 1.3. [MT, Lemma 3.6] Suppose ReminderStatement Statement 1.3. [MT, Lemma 3.6] SupposeGGisiscenterless centerless and perfect. Then, so is perfect. Then, so is £G, kk >0.0.InInparticular, particular,aaQQpoint pointon on Ti(pG, C) C) = Hk is equivalent to to giving is equivalent giving a Q regular (£G, C) realization realization [FrV]. [PrV]. Such Such a point automatically gives gives Q regular regular (£G, C) realizations, realizations, from from the images images of the corresponding point in Hj, 00 < jj < k. k. corresponding of this paper isis to One goal of toshow show progress progress on on the thefollowing following problem. problem.

Modular Suppose Modular Tower TowerConjecture Conjecture1.4. 1.4. SupposeGGisisa acenterless centerlessgroup group and and r integer. With k(G, r) suitably is a positive positive integer. With k = k(G,r) suitablylarge largethe thefollowing followinghold. hold.

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(1.2a) For any prime p dividing GI and (1.2a) dividing \G\ and conjugacy classes C supported supported in in k d empty. C(G),U( G,CY (Q) is empty. p p realizations with with at at most most r branch (1.2b) More generally, generally, there are no pG realizations (1.2b) points. points. Theorem (1.2a) (for value of of k(G,r)) k(G, r)) implies Theorem 4.4 shows shows (1.2a) (for some value implies (1.2b) (1.2b) (for (for concentrated on on the the standard a possibly k(G, r)). possibly larger larger value value of of fc(G, r)). [MT] [MT] concentrated standard Hurwitz spaces spaces (as in in [PrV]) [Fry]) arising from from representations representations of the the fundamental fundamental acting on onNielsen Nielsenclasses. classes. App.II App.II explains explains reduced reduced Hurwitz Hurwitz group of group ofPFr \ D Drr acting spaces PSL2(C) Hurwitz spaces. spaces. spaces as a PSL2 (C) quotient quotient of of standard Hurwitz

§1.C. §1.C. Braid action action and andPSL2(C) PSL2(C) quotients. quotients. Let LetGGbebea afinite finite group group one Q with conjugacy regular regular (G, (G,C) C)realization realizationofof conjugacy classes classes C. Consider Consider one This produces infinitely many Q rega group X -> P*. This produces infinitely many Q reggroup by by aa cover cover <\>X:—* Thus, to measure ular with any a E PSL2(Q). ular realizations; realizations; compose compose <> / with any a E PSL2(Q). Thus, to measure , C) C) realizations realizations requires requires counting counting Q Q points points on on reduced reduced Hurwitz Hurwitz spaces spaces C)rd rd = r d These spaces have natural maps { , C) = n k }^LQ. These spaces have natural maps r d .ijrd r d^...->H 7 rL0 • • • -» ni^ ->H k ... o ^# di..Jr. App.II gives gives the the definition definition of of JJr. is aa rational rational union, union, these these spaces spaces r. Since C is

and maps maps have have field field of of definition definition Q. Q. Investigating this tower tower requires requires inforinforination on absolutely for each each level level k. k. We mation absolutely irreducible irreducible components components of of %]^ for We explain more more of of the group theory behind these explain these diophantine diophantine problems. problems. Statement Statement 1.3 1.3 says says a Q regular (p+1 G, C) realization produces a Q (£G, C) realization. It realization. It also also suggests suggests these these groups groups are are similar, similar, for for example, example, when when G G is perfect perfect and and centerless. Lemma 4.1 4.1 adds adds to to this. ItIt characterizes centerless. Lifting Lifting Lemma characterizes cover ip :HH—* —>CGasashaving havingthe thefollowing followingproperties. properties. PG quotients giving a cover (1 .3a) The kernel of ip has has p-group p-groupkernel. kernel. (1.3a) order prime prime to top, (1.3b) For For g' g' = {g[,... ,g'8} eG H H elements elements of of order p, (g') ($') == H H ifif (1.3b) . ,g} and only if (ip{gf)} ==C. G. is aamore moresubtle subtlesimilarity similaritybetween betweenHHand andGCsatisfying satisfying(1.3). (1.3). Brauer's Brauer's Here is Theorem says there as modulesas as there there are Theorem [MT, §II.BJ §11.B] says as many many simple simple IF7, ¥p modules p-regular classes. Therefore, p-regular conjugacy conjugacy classes. Therefore, the two two groups groups have have the the same same simsimmodule ple Fp modules. §3 §3 shows shows the the significance significanceofofan an appearance appearance of of the module 1 (for (for pG) in in the the Loewy Loewy display display of the IF,, ¥p module is the the modulekeik kerk/kerk+i/kerk+j. It is whole whole story story of of obstructed components. Fix r0 r 0 and aa field field K. K. Theorem Theorem4.4 4.4says saysthe thefollowing following are are equivalent. equivalent. . .

(1.4a) realizationswith withatat most most rr0 branch points (1.4a) There are K regular regular ,,kG ^G realizations 0 branch for each each k > 0. for > 0. (1.4b) For some r (1.4b) < rr00 there is is an an r-tuple r-tuple CCofofp-regular p-regularconjugacy conjugacy classes classes of G and for each there are K K regular of each k > > 0, 0, there regular (^G, C) realizations. realizations.

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The Main Conjecture .4b) is impossible when K isis Conjecture reduces reduces to toshowing showing (1 (1.4b) for each each r-tuple C of of G. G. Prova number field for of p-regular conjugacy conjugacy classes classes of ing the Main ing Main Conjecture Conjecture comes comes to considering considering rational points on on reduced reduced Hurwitz spaces spaces (PSL2(C) Hurwitz (PSL2(C) quotients), curves when r0 r 0 = 44 (App.II). Suppose you show show (0.6a) (0.6a) (for (for large large k). k). Faltings' Theorem [Pa] applies ifif Suppose you [Fa] applies (0.6b) (0.6b) holds: holds: the genus genus of the components components at level level kk goes goes up with with k. k. [FrK] [FrK] gives evidence evidence for for this this rise in genus gives genus from from our our special special case. case.

§1 .D.Projective Projective systems of §1.D. of rational rationalpoints. points.Suppose Supposethe theprogram program of of §1.C works. Then, there are §1.C works. are only only finitely finitely many many Q Q points on some some level level k0 fco of Modular Modular Towers Towers satisfying satisfyingthe the hypotheses. hypotheses. It is, of is, however, however, a formidable formidable task to eliminate eliminate the the possibility possibility of a nontrivial set of Q points at every every level. level. hypothesis (0.6c) (0.6c) enters. enters. It It assumes there are no obstructed This is where where hypothesis assumes there obstructed or beyond. beyond. Then, components at level level k0 k0 or Then, having having points pointsatateach eachlevel level above above k0 k0 produces a projective projective system system of of rational rationalpoints points{Pk {p^E6 %(pG, C) rd }^L 0 . That That +1 rd rd C)rd is, canonical map £ +1 G —* -> kpG inducing -> ft(£G,C) is, the canonical inducing ft(£ G,C) cy' _÷ gives pfc+i'—* «->• Pie. pk-Precluding Precludingprojective projectivesystems systemsisisthe thelast lastleg legfor for rr ==r0r$==4.4. gives Pk+1 To To do so for for any any value value of rr isis aaseparate separateproblem problem independent independent of of other other diophantine considerations. considerations.

Projective 1.5. Assume setup Projective System SystemConjecture Conjecture 1.5. Assumethethe setupfor forthe theModular Modular Tower above above associated to to (G, (G, C), C), with withrrarbitrary arbitraryand andKKfinitely finitelygenerated generated Then, there exists no projective system of K over aa number number field. field. Then, projective system of K rational points on on the the Modular ModularTower. Tower.

§2. Construction of universal §2. Construction universal Frattini Prattini covers. covers. [FrK}expands expandsthe the Normalizer Normalizer Observation Observation of of [MT, [MT,Remark Remark 2.10]. 2.10]. ItIt helps [FrK] decipher decipher the whole whole universal universal p-Frattini cover cover from from representation observaobservations on the original group G. G. Thus, representation representation facts facts about about G translate original group to aa presentation presentation of of the the universal universal p-Frattini cover cover of G, and and so so of of its its infiinfinite string of characteristic characteristic quotients. This This section section presents two preliminary results in this this direction. direction.

§2.A. Relation Relation of of pG to pH with with H H < G. G. Let = Pp == PPG,p §2.A. Let PP = be aa G,p be p-Sylow subgroup subgroupofof G. G. The p-Sylow The notation notation kero(G) kero(G) allows allows simultaneously simultaneously distinguishing tinguishing the characteristic characteristic kernels kernels of the universal universal p-Frattini p-Frattini covers covers for for G and —+ —> map.map. Also,Also, G for and for for aa subgroup subgroup H. H. Use Useq5c <J>Q : PG G the for canonical the canonical denote keri(G)/keri_|_i(G) by Mi(G). M1(G). For For any any subgroup subgroup H H of G, restricting H to to M2(G) Mi(G) gives gives an H module, module, MI(G)H- The rank rank of of any profinite profinite group G is the smallest number of elements elements that that topologically generate it. it. Finally, topologically generate Finally, is t n e is the p-Sylow subgroup = PG,P ^GH-PG.P) P-Sylow subgroup of PG. :

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p-Sylow Frattini Frattini hull hull. Let G be Example 2.1. 2.1. The p-Sylow be any finite group. SupSupLet The free pose Pp has rank rank u. u. The free pro-p pro-p group group on uu generators generators is is pFu. Let NG be the normalizer of NG(P(Pr) be of Pp in G. Then, Then, the the universal universal p-Frattini cover cover P) s [Ri, Th. Th. 3.2]. 3.2]. This a special case of Principle NG (Pr) xN 8NG [Ri, of N G(P p) is pFu x G(Pp)/Pp Since PFU 2.3 (below) by extending extending the the action action of N N= = NG(PP)/PP to to PFU. Since (below) by mapping surjectively to N. is pro-free, pro-free, there there is a subgroup subgroup B of of Aut(pF w) mapping surjectively to N. thatthe thekernel kernelofofthe themap mapBB—÷ —>NNis is a pro-p-group. The essential point isis that a pro-p-group. (Prove (Prove this this by inductively inductively considering consideringthe the characteristic characteristic Frattini Frattini quotients Apply Schur-Zassenhaus Schur-Zassenhaustotosplit splitoff offaacopy copyofofN. N. Thus Thus produce produce of pFu.) Apply [FrJ, Lemma Lemma 20.45]. 20.45].Important Important point: point: Action Action of of N on pFu [FrJ, the action of N on U• is unique unique only only up to conjugation conjugation by an an automorphism of PFU. —*» Ppas Let kerj be the kernel kernel of pFu — asininEx. Ex.2.1. 2.1.For ForEx. Ex.2.1, 2.1,the theaction action on kerg //ker^ extends to to G. G.

of NQ{PP)

Example 2.2. Take G G= = As, A5, 2.2.The Thep-Sylow p-Sylow Frattini Frattini hull hull can can be all of PG. Take and NG(PP) == A4 A* << A5. A$. Then, Then, we wecan canidentify identify kerg as asthe thekernel kernel p— = 22 and 3 or of the universal 2-Frattini of of A5 A5 [MT, Prop. 2.9]. 2.9]. When, When, however, however, p = =3 or 5 (and G G= — A5) As) kerg//ker^ isiscyclic, cyclic, and andnot notananA5 A$module. module.InInparticular, particular, these values values of of pp isn't isn't the kernel —* G. 0• kerg for these kernel of PG -> G.

[FrK] generalizes generalizesthe the observation observationof ofEx. Ex. 2.2. 2.2. Principle [FrK] Principle 2.3 2.3 characterizes characterizes the universal its subgroups. universal p-Frattini cover cover PG of G to relate it to that of its

Subgroup Frattini LetLet GG bebe any profinite FrattiniPrinciple Principle2.3. 2.3. any profinite group group and and p any prime dividing G and dividing IGI. |G|. Then pG is the smallest group that covers G and has its its p-Sylow p-Sylow subgroup pro-free. pro-free. Suppose H << G groups. Then, there Suppose H G are profinite profinite groups. there is is aasurjective surjective pro-p pro-p ker0 (G) —+ (H). Further, is unique group homomorphism homomorphism PG,H PG,H '- kero(G) -> ker0 kero(i^). Further,PG,H PG,H is up composition composition on the left left with with an anautomorphism automorphismofofker0 kero(G). This Thisinduces induces a surjective surjective H GG is is H module module morphism morphism p*G H : M0(G) M0(G) —* —>M0(H). M0(H). Finally, Finally,if if a finite group, group, the the following following are are equivalent: equivalent: :

:

(2.la) PQ H is an isomorphism. (2.1a) (2.lb) (2.1b) dimFp(M0(G)) = dim Fp (M 0 (tf)). (2.lc) = pH. (2.1c) tcl(H)= Proof. First consider Proof. consider the theopening opening characterization characterization of of pG. For G Gany any profiprofitheuniversal universal Frattini Frattini cover cover of of C. G. nite group, a simple property determines the It is is the theminimal minimal projective projective cover cover of of G G [FrJ, [FrJ, Prop. Prop.20.33]. 20.33].AAprofinite profinite group group is projective subgroup of of it is [FrJ, and only only if if each each p-Sylow p-Sylow subgroup is pro-free pro-free [FrJ, projective if and nilpotent group group [FrJ, [FrJ, Prop. 20.44]. Prop. 20.47]. 20.47]. The kernel kernel of of çbQ (f)G isisaa nilpotent 20.44]. So each prime prime pp |IGI. G is is the fiber fiber product product over over GGof of closed closed subgroups subgroups PG for each |G|. —>•CGisisany anyprofinite profinitegroup groupcover coverofofCGforforwhich which pG* Suppose p(j)* : pG* —* :

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Michael Pried Fried and Yaacov Michael Yaacov Kopeliovich Kopeliovich

p-Sylowsubgroup. subgroup. Consider Consider the the fiber fiber product product over over G G of of the has a pro:free pro-free p-Sylow and P>G for for all all primes primes //, p', p' ^ p, Call this this fiber fiber |G|. Call groups PG* and p, dividing IGI. product G*. From the Sylow subgroup characterization of projective groups, G*. From the Sylow of G* G* is is projective. projective. Also, it is is aa minimal minimal projective projective cover cover of G if and only only if if cover of of G G having having a pro-free pro-free p-Sylow. p-Sylow. This This concludes concludes the the pG* is a minimal cover opening opening characterization of PG.

The remainder remainder of the proof proof concentrates concentrates on on universal universal p-Frattini p-Frattini covers. covers. prime p. p. Now consider consider H H< All notation (say, (say, for for ker^) refers refers to the fixed prime PH commuting with the natural surjective maps to to H. The surjective maps Theimage imageofof/3/?isisaaclosed closed subgroup subgroup of PH mapping mapping surjectively surjectivelytoto H. H. Since Since PH isis aa Frattini Frattini cover cover of of H, H, /3 is is surjective. So, So, it induces induces the the surjective surjective map map Pa,H PG,H : kero(G) ker0 (G) —* —>• kero(H) ker0 (H) in the statement. statement. The Therest restfollows follows because because the the 1st 1st characteristic characteristic Frattini Frattini quotient determines the the rank rank of ofaapro-free pro-free p-group. p-group. •

The following §2.B. Decomposing Frattini Frattinicover coverkernels. kernels. The followingisisaaspecial special of [Be, [Be, Exer. Exer. 1, 1, p. p. 11]. 11]. We We fill fillinindetails, details, isolating isolating itit from from its generalities. case of

Indecomposability Lemma kerk(G) module Mk(G) is inIndecomposability Lemma 2.4. 2.4. The The pG/kerjb(G) module M^{G) decomposable. Proof. Let Let M M be be any any G G module. module. Define Define fi(M) to be the kernel kernel of a surjective M with G module homomorphism i/>: P:—* P —> M with P aP projective a projective module, module, minimal minimal for P' isis another for direct direct sum decomposition. decomposition. Suppose Suppose P' another module module with with these these —»P' Pf properties. Then, projectivity projectivity of ofthe thetwo twomodules modulesgives gives maps maps aa: P : P—+ properties. Then, f and /3: p :P'P —* —» F, P, each each commuting commuting with the surjective surjective maps from from P and and P' P' 0/3 and Define /f = a aof3 and the composition composition of / , n times, by fn. Similarly, to M. Define for gn with g the composition /3° a. If n is large, large, then then P = ker(#n) ®Im(^ n ). composition f3oa. Note: Note: ker(#n), being being a summand of of a projective projective module, module, is is also also projective. projective.

Suppose goes to to 00 under under Suppose PP and and P' P' aren't aren'tisomorphic. isomorphic. Then ker(#n) goes the map from from P M. This Thiscontradicts contradicts the the minimality minimality of P, P , and and proves proves P to M. that fi(M) is well defined. Further, well defined. Further, from from Schanuel's Schanuel's Lemma Lemma [Be, [Be, Lemma Lemma 1.5.3], up up to to projective summands 11(M) defined. Similarly, there is 1.5.3], Q(M) is well defined. is the operator of M M in operator 111(M): Q~1(M): the thecokernel cokernel of of the the embedding embedding of in aa minimal minimal injective module. module. In the category category of of finite dimensional modules over group rings, projectives

and and injectives injectives are are the the same same [Be, [Be,Prop. Prop. 1.6.2]. 1.6.2]. Thus, Thus, PP above above is is both both the minimal projective projective covering covering M M and an minimal an injective injective module module containing containing 11(M).

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So, That is, is, up up totoaddition addition ofofaaprojective projective module, module, Q~l(Q(M)) = M. So, is. Apply twice to to the indeM isis indecomposable, indecomposable, if and only only ifif fi(M) is. Apply f2 twice indecomposable module M M= From [MT, [MT, Proj. Indecomp. Indecomp. Lemma Lemma 2.3], 2.3], composable module — lk^. From 2 2 U• So, p2(M) is indecomposable. kerfc/kerfc Q (M) is indecomposable. kerk/kerk+l +i = fi (M). abelian kernel M. Then G acts Consider any cover G with H —> G with abelian kernel M. Then G on acts on Consider cover <\>H :—* M: gg E in G EM Mtoto gmg~x where where # isis any any lift lift of of gg to to H. if. eC G maps maps m

Remark kernels Remark 2.5. 2.5.Frattini Frattini kernelsmay maydecompose. decompose.Let Let M1 Mi and and M2 M2be benonnonisomorphic simple modules modules appearing as respective kernels isomorphic simple kernels for nonsplit nonsplit exextensions G1 and and Gi C2 of of G. G. Consider the fiber fiber product product H H = G1 tensions G\ Consider the G\ XQ G2 and and XG G2 the natural map Frattinicover. cover. map ^1,2 : H -» G. We show G and C2 and G2—+ —>GGare areFrattini Frattinicovers, covers,H1 i?i factors factors surjectively surjectively through through them them both. both. Thus, by the Jordan-Holder Thus, Jordan-Holder Theorem, Theorem, the the composition composition series series of of the the kernel kernel of of H1 H\—* —>>GGincludes includesM1 Mi and andM2. M2.ItItmust musttherefore thereforebebeM1 Mi 0M2. M2. Z/2). Let Two series series of of simple simple groups groups agree agree when when nn = Two = 8: 8: A8 A$ = SL(4, SL(4, Z/2). Let M4 standard 44dimensional dimensional representation representationofofSL(4, SL(4,7L/2). Z/2). [Be] [Be] shows M4 be the standard 2 This gives (A$,M4) 1. This gives a Frattini extension extension of A8 As not not H H2(A8, M4) has has dimension 1. factoring of A A8. of factoring through the universal central extension extension of 8. It is an example of the above. above. • :

Progress on the case §3. Progress case A5 A5 and C = C 3 r. Tile main of [FrK] is the A5 The main example example of [FrK] is A5 Modular Modular Tower Tower associated associated with with r = 4, p = 2,2, and all in ^A5: all conjugacy conjugacy classes classes those of the 3-cycle 3-cycle in 5 : C = C34. C34. Denote the corresponding corresponding reduced reduced Modular Modular Tower Tower (of (of inner inner spaces) spaces) by by d (3.1) ... • • —* • -> ^ ( ^ + 1 i 5C)rd , C) r d -> « ( * i 5 , C) r _÷ -4... —÷ 7i(A5, > «(i4 5C)". , C) r d . This example reveals reveals major major phenomena phenomena that that don't don't occur occur for for modular modular curves. curves.

§3. A. Illustration Illustrationof ofobstructed obstructedcomponents. components. Obstructed Obstructedcomponents components produce both encouraging encouraging and discouraging diophantine First we we produce and discouraging diophantine results. results. First illustrate the former. former. Actual Actual examples examples go go precisely precisely as the outline outline in in §0.C. The proof of the next result C)abs. result explains explains the the variety variety fl(C, W,(G, C) a b s .

Theorem Theorem 3.1. ([Fri]) ([Prl]) Assume Assume ii n >5 > 5 is odd. Then, Then, /H{An,C3n-i)rd(Q) is is dense in H(An, C 3 n-i) r d . Now even. Then, Then, there there are no Now assume assumenn>> 66 is even. regular Q}An,,, C 3 n-i) realizations. realizations. Q regular Galois Further, there is aa Q unramified unramified cover cover9-t(G, U{G,C)u72 C)in -> 7-1(G, U(G, C) a b s , Galois with group Z/2 Itji with withthe thefollowing followingproperties. properties. (3.2a) Each (3.2a) Each ppE<E9-1(C, %(G, C ) a b s corresponds corresponds totoaadegree degreenncover cover
Michael Fried Fried and Yaacov Michael Yaacov Kopeliovich Kopeliovich

164

has (minimal) (3.2b) For pp EG H ( G , C ) a b s , p : X X —* -> P£ ftas (minimal) field field of of definition definition (3.2b) Q(p). Q(p). Its /te Galois Galois closure closure isis defined defined over over Q(p') Q(p') with w^/fc p' p' lying It/in^ over over p. p. a b s (An, (3.2c) (3.2c) pp G n(G, H(G,C)abs C) corresponds correspondstotoa aQ(p) Q(p)regular regular (An, C 3 n-i)realization realization if and only if Q(p') Q(p') = Q(p). For all the set a// n > 5, tfie se£ {p {p G fl(G, U(G, C) abs (Q) with wz%Q(p') Q(p') ^ Q(p)} Q(p)} isisdense. dense. abs For n = 5, even {p G Q(p)} is dense. with Q(p') U(G, C) (Q) with Q(p') 7^ Q(p)} is dense. 7-t(G, e Proof. This example example compares compares the group group theoretic theoretic Hurwitz Hurwitz space space ideas, ideas, equation calculation. Hurwitz with explicit explicit equation Hurwitz space computations are are explicit explicit group theory theory computations. computations.It's It's more more than than aa matter matter of group of taste, for for we are after properties of the the spaces, spaces, not not their their equations. after properties of equations. Other Other examples examples of this are in [Fr3] and [Fr4]. proof has has four are [Pr3] and [Fr4]. The remaining remaining proof four parts, starting starting from from statements relating inner Hurwitz spaces spaces to absolute Hurwitz spaces. spaces. This This Mestre's calculation for n odd, distinguishing lets us focus focus on the effect effect of Mestre's calculation for distinguishing this from from the more niysterious mysterious case case when whennn isis even. even. Note: Note: If Q(p') ^ Q(p) in in (3.2b) hashas (3.2c)), then the the Galois Galois closure closure of of (j)p : :X X—*—> P* (3.2b) (notation as in (3.2c)), group Sn (see below).

Part 1: 1:Absolute Absolute Nielsen Nielsen classes. classes. For For k = = 0, a special case of [Fri] [Frl] shows shows there is is one one i? n -i orbit on Ni(^4n, C 3 «-i). Another Hurwitz space space is handy handy for this problem. It It isis closest closest to to Mestre's Mestre's computations computations [Me] [Me] below. below. for N(C) = For any G < < 5 n , consider the normalizer normalizer Ne,. Nsn (G) of of G G in in 5 n . Let JV(C) elements conjugate conjugate the entries Nsn (G, C) be the subgroup of Nsn (G) whose whose elements G( 1), the the stabilizer stabilizer of of 1 in this degree of C among themselves. We We also use G(l), degree nn representation. Form Form the the natural natural quotient quotient Ni(G, Ni(G,C)/N(C) C)/N(C) = Ni(G, C) a b s . representation. Hurwitz monodromy monodromy group group if n -i = = lri(]Fn_l 7Ti(Pn~1 \ -Dn_i) Then, S n _i, and its Hurwitz abs quotient, act act on on Ni(G, Ni(G,C)abs. C) . Covering space theory produces produces aa sequence sequence of unramified covers unramifled covers (3.3)

in n l pn—l fl(G, C) C) = U(G, C) C)tm —÷ fl(G, C) a b s -> F W(G, -> n(G, ~ \\

Dn^.

abs

As in [Fr2], and [V], interpret H(G, (G, C) C)abs as as a moduli space of equiv[Fr2], [FrV] [FrV] and alence classes of of degree degreenn covers. covers. For For p eG H(G, C) a b s , let 0P : Xp alence classes Xp —* ->- P^ be a representing equivalence here here isis (II.la) (II. la) (in (in App.II). This representing cover. cover. Cover Cover equivalence cover has the following following properties. cover

(3.4a) The TheGalois Galoisclosure closureXX—÷ -> ¥]. of of p represents represents p'p'EG W(G, C)in over overp.p. (3.4b) XX is is the the quotient quotient of of Xk by G(1). (3.4b) G(l). Suppose WJ1 is a connected connected component component of Hin and % abs is the image image of n corresponds to to an an orbit orbit O\ of of B Brr 011 Ni(G, C). on Ni(G,C). Uin iin n ^abs TThen, h e n 7 Uin corresponds The natural natural image image O abs of O[n in Ni(G, Ni(G,C) abs corresponds corresponds to Ulhs. The following set group: following set is is actually a group:

{h E N(C) N(C) I 3QQEG B n _iand g(C) = {he andg 0EG O\nwith with(g)Q (g)Q= =hgh'}. hgh'1}.

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abs on the left of The cover cover U(G, of (3.3) (3.3) is Galois with group group fl(G, C)in -> H{G, 7-t(G, C) c(C)/G. Note: G(C)/G. Note:[MT, [MT,Lemma Lemma 3.3] 3.3] says says G < g(c). G(C).

Using G(C3n-i)/G = 7L/2. Z/2. With With An An << Sn the thestandard standardrepresenrepresenPart 2: Using tation tation of of the the alternating alternating group, group, N(C) N(C) = = Sn. Thus, Thus, G(C) is is a subgroup of Sn/An = 7L/2. Z/2. Also, Also, H(G, fl(G, C) a b s is the moduli moduli space space of of degree n covers with n—i 3-cycles 7i—l 3-cycles as as branch branch cycles. cycles. From From the the Riemann-Hurwitz Riemann-Hurwitz formula, formula, this this isis covers. Further, as a moduli moduli space of genus genus 0 covers. as in in [Fr2], [Fr2], since since the subgroup subgroup ?-l(G, C) a b s is G(l) An(l) is self self normalizing in An, %(G, is a fine moduli space. space. G(1) = An(1) E fl(G, In particular, particular, p G H(G, C) abs (ii0 corresponds corresponds to to aaKKcover cover p : Xp —÷ -> P*. [Frl] on Ni(;4 n ,C 3 »-i). Thus,
g = ((123), (123)—', (145), (145)_i).

Here applying applying Q Q3 (App.I) has has the the same same affect affect on on gg as as conjugating conjugating by by (4 (45). Here 5). 3 (App.I) Apply irreducibility theorem theorem to to the thesequence sequence of of covers covers in in (3.3). (3.3). Apply Hilbert's Hubert's irreducibility This says a dense set of points j: eG Pn~1(Q) have have above above them p' p' eGfl(G, %{G, C)in lying over U(G, C) a b s for for which which Q(p') Q(p') ^ Q(p). Q(p). Excluding Excluding the the statement statement over pp G E fl(G, about Q points, this completes completes the proof. proof.

Part 3: 's calculation. at the 3:Interpreting InterpretingMestre Mestre's calculation. Continue Continue the discussion discussion at beginning of of Part Part 2. Here is is the first beginning 2. Here first place place where where there there is aa distinction distinction between nn even then X,, between even and odd. odd. If (j)p has has odd odd degree, degree, then Xp has a rational rational particular, 0P is is Q(p) Q(p) equivalent equivalent (in sense of la) point over Q(p). Q(p). In particular, (in the sense of (II. (II.la) in App.II) App.II) to aa cover cover P£ —* -> P*. [DFrl, 115] and [DFr2, [DFr2, p. 115] 115] discuss [DFr1, p. 115] am whether this means means aa space space like like T ^ n - i ISis aa ffamily ^y of functions. whether of rational functions. abs x That would would mean, there is a natural map # : 7i^_x xPj —* -> U^-i ^lAlso, rational function function cover cover representing representing p. p. Also, for for any any pp G e H*^_ 1? ^ p is is a rational there is is such such aa map map \I>abs, though we haven't excluded excluded it. it. We don't think there Still, as as in [DFr1, 2], ifif you you add add the the branch points of Still, [DFrl, 2], branch points of the cover cover to the moduli space, you nearly have coordinates for for aa family family of rational rational functions. functions. When n is is odd that that isisexactly exactly what what [Me] [Me] does. does. In In fact, fact, he hegives gives aa dense dense set set of rational points in in W^'n_1.Here Here is how it goes p. 100-101]. is how it goes [Se,[Se, p. 100—101]. a w n ne a Let P(y) = — with the algebraically independent (y ~~ i) ^ ^ Let P(y) nr=i(2/ i algebraically independent indeterindeterminates over If Q(y) Q(y) is a polynomial of degree degreen— n—i, then Q/P maps over Q. Q. If polynomial of 1, then maps 00 to 0. We of Q(y)/P(y) We look look to to choose choose those polynomials polynomials Q Q so the derivative derivative of Q(y)/P(y) 2 is a square (R(y)/P(y))2. square (R(y)/P(y)) . If If Q Q and R are are suitably suitably generic, generic, that means means the cover from has 3-cycles 3-cycles as branch cycles. It is is then then in in the theNielsen Nielsen class class cover from Q/P Q/P has :

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of our our interest. interest. Here of Here are are the the expressions expressions for Q and and R: R: (3.5a) (3.5a)

Q(y)/P(y) = Yl ~ c '/(y -- aai), Q(y)/p(y) «)'

RRIP p

i

==

ci/(y -

where the Q sS satisfy satisfy (3.5b)

a3) == 0 for ] T ^-/(a. -—Oj) for all i.i.

matrix with with ixj i xj entry This is the second second place using ii n odd. The matrix entry l/(a; — ay) and nm xx n. n. Thus, for for for i ^ j and and 00 for for i — = j isis skew-symmetric skew-symmetric and for nn odd odd its its determinant is is 0. 0. So, So, there thereisisaaline lineofofsolutions solutionsfor for the thevector vectorofofc2 c»s.s. m is an absolutely irreducible Q variety. [Frl] [Fri] says says 7/(^4 n ,C 3 n-i) an absolutely irreducible Q variety. So, So, Mestre's Mestre's collection collection of rational functions functions shows shows flabs 7/ abs is a unirational unirational varivariety ([Me] when n is asserts a variant works ([Me] when is odd; odd; [Se, [Se, p. p. 100-1011 100-101] asserts works for for even even

j

too). So So itit gives gives aa Q Q dense dense subset subset of of H a b s . These points are the the image image n, too). in abs of Q points of of n . This is a dense dense set of of pp Ee fl(G, K(G,C) (Q) for for which which of C)abs(Q) [Fr5, Thm. Thm. 5.9] 5.9] shows showsthere there isis aa dense Q(p') = Q(p). When 5, [Fr5, dense set of When nn = 5, Q(p') for which which Q(p') ^ Q(p). ( G , C) abs (Q) for p e€ «1-1(G,

Part 4: 4- For Fornn even, even, W(A n ,C 3 n-i) is obstructed. obstructed. There There is nothing nothing in the the when n isis even. Nielsen Nielsen class class Ni(2^4 n5 C 3 n-i) when even. It goes goes like like this. this. Let An be the —+ the universal universal central central exponent exponent 22 extension extension of of An. Then, \An —> An factors through An -> An. Suppose 20 £E Ni(2^Ln,C). C). The The canonical canonical map map factors lAn — sends 20toto QeG Arn with with II(g) = 1.1. Let Let g0be bethe theimage imageof of 0 —+> An sends transitivity of of Z?n_i on Ni(A n ,C 3 n-i). So, Again use use transitivity in Ni(A n ,C 3 n-i). Again element of Ni(A n ,C 3 n-i) lifts element of Ni(^4 n ,C 3 n-i), then lifts to an element if one element any element element does. On the the other other hand, hand, [MT, [MT, Ex. Ex. 111.12] III. 12] explicitly gives any does. On explicitly gives no lift lifttotoNi(ATh, Ni(^4 n ,C 3 n-i). Thus, Thus, level level one one of the g0 E€ Ni(i4 n ,C 3 n-i) with no Tower for for (An, (An, C 3 n-i) is variety. Modular Tower is an empty variety.

0•

§3.B. Invariants Invariants for §3.B. for obstructed obstructedcomponents. components.Obstructed Obstructedcomponents components and pure group group theory establish establish Modular Modular Tower Tower Conjecture 1.4 1.4 for (An, (An, C3r) C 3 r) with evennn>66and with even and rr == n — — 1.1. [Fri], [Frl], however, however, shows shows aa different different outcome outcome

for the cases cases(n,r), (n,r), rr >> for the

n that Theorem Theorem 3.1 3.1 doesn't doesn't cover. cover. There There are are unobstructed components components in in all alllevels levels of of the the Modular Modular Tower. Tower.

Lemma 3.2 3.2 isis a completely general result result applying applying to to all levels of any Lemma completely general levels of r Modular Tower. with g9 = (gi,. Tower. As in §0.A, §0.A, with ( : H —> is a central : H —* G isGa central 91 extension. order prime to | ker()|, has extension. Then, any any gg EG G of order unique lift lift g has a unique element of H of of the same same order as G. G. Assume Assume ker(4) ker(<^>) has has order order prime prime to an element

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167

r For II(g) H(g) = 11 let s(g) to the orders orders of all entries entries of of g = (gi,.. (#1,... ,gr) G E G '. For s(g) - gr).InInLemma Lemma3.2, 3.2,the theexact exactsequence sequencehas has1 1asaskernel. kernel.Here Here1 1 be II(<7i • • i,.). is the identity module for the action of ^G. module for

Obstruction Obstruction Lemma Lemma3.2. 3.2.Suppose Suppose0Oisisa aBrBrorbit orbitininggEG Ni(pG,C). C).Let Let S(O) S(O) = {II(g) I geG £ +1 G, ge 6CCand and gmod modkerk kerfc==g}. g}. I

if is zs aa union union of0/conjugacy conjugacyclasses classes in in p +1 G- Then, TAen, 11 0 5(0) 5(O) exactly exactly if there exists aa sequence sequenceofofcovers covers This 77MS

with the following properties. properties. (3.6a) The Thekernel kernelofofH2 H2—* -> H1 H1 is w 1. 1. (3.6b) There (3.6b) There exists ezzsfc g* G Ni(#i, C), g* mod kerfc = gg and and s(g*) $^ 0.0. corresponding to 0. Suppose Ho is the Suppose the level level k component component of Hk corresponding O. The The obstructed. previous condition condition holds holds exactly exactly when when Ho is obstructed.

Replace 1 in Proof. / kerk+i. Replace Proof. Use Use induction induction on on the the Loewy Loewy layers layers of of kerk kerfc/ker/fe+i. kernel ininH2 H2—* -> Hi by an an irreducible irreducible module module A A ^ 1. Consider the the kernel 1. Consider Sf of of11(g') n(g') as as g' g' runs runs over over all all allowable allowable lifts lifts of r-tuples gg EG H%with with set 5' 11(g) Here isis the the argument, argument,using usingthat that the the set set SS'f is a II(g) = 1. 1. Then Then SS'f = A. Here braid invariant invariant set. set. That is, one lift. lift. In the orbit, you is, let let g° be one you can braid g° something whose to something whoselift lifthas has the the same same product, product, but but in the braid gf appears side. Now Now form form a lift lift replacing replacing gf by agfa~x = = gfa9i a" 1 . So, on the right side. g 1 lift gives gives Q°a i a" . You You can can do do this this for for any any ii and and any anyaaEGA.A.IfIfAA ^1 1 this lift possible products products of of lifts lifts gives gives g°a with a E€ A A arbitrary. arbitrary. is irreducible, the possible corresponding Nielsen Nielsen class only if A = = 1. So, the corresponding class is empty oniy D•

Schur Multipliers Result Result3.3. 3.3.Assume Assume nn >> 4. 4. For Foreach each kk >0,0, \An has a nontrivial Schur multiplier. multiplier. Generally, Generally,suppose suppose ^G has has aanontrivial nontrivialSchur Schur multiplier. That is, 1 —+ H1 —* — with multiplier. is, there there isis aasequence sequence p +1 G —*—>•H1H\—*—> ^G with 1— >- Hi > H1]Hf generates pG short exact as in Obstruction Obstruction Lemma Lemma 3.2. Then, Then, [kerk, [ker^ifijiff a proper kerk+l. Also, proper closed closed subgroup subgroup H2 Hi ofo/kerfc+i. Also, £ +1 G has hastrivial trivialaction actionon on = kerk+1 kerfc+i /112, /H2, producing aa nontrivial Schur multiplier for M= Proof. [MT, §11.C], §II.CJ,the the exponent exponent 2-part 2-part of the Schur Proof. As is well-known well-known [MT, Schur > multiplier of (ii > 4) is 7L/2. Thus, the statement multiplier An (n Z/2. Thus, statement on the the alternating alternating

groups follows Since kerk ker^ is a proprogroups followsfrom fromthe the general generalinductive inductivestatement. statement. Since free group, of kerk+1. kerfc+i. The action action of of p +1 G on M is is free group, H H2 2 is a proper subgroup of trivial if it is trivial on generators generators of One type type of of generator generator is is vp with of M. M. One V v EG kerk. ker&. Conjugating Conjugating v by by gg EG p+1G gives gives vh vh for for some some hh G Hi. Modulo Modulo

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Michael Pried Fried and Yaacov Michael Yaacov Kopeliovich Kopeliovich

H\[kerfc, H1], Hi], set set hp = 1 for hh e Hi and vh = hv for for v 6 kerfc and hhEe H1. H±. Thus, the the following following hold. hold. mod Hi]. (3.7a) gvPg~l = (vh)P (vh)*>= = v? mod [kerk, H\[ker^/fi]. 1 [vh, v'h'] (3.7b) ^[v, g[v,t;']^v']g1 = [vfc, i/fi'] = [v, [v, v'} t;'] mod mod Hp[kerk, H1]. #x]. (3.7b)

D D

§4. When Whenp|7V §4. piNc.c .

Let Let G be be aafinite finite group group and and let letCCbebea collection a collectionofofconjugacy conjugacy classes classes in G. 1.4 in in the form G. Theorem Theorem 4.4 4.4assumes assumes Modular Modular Tower Tower Conjecture Conjecture 1.4 form (1.2a) holds. holds. From this it concludes Conjecture1.4 1.4isistrue. true. The The core of (1.2a) From this concludes Conjecture core of the proof class that proof starts by by assuming assuming C contains contains at least least one one conjugacy conjugacy class isn't p—regular. then shows, shows, for for kk large, large, lifts lifts of C to j9-regular. ItItthen to conjugacy conjugacy classes classes in in pG can't be aa rational rational union. union. The The gist gist of of this this isisititsuffices suffices to establish establish our major conjectures conjectures when when (p, (p, NQ) — 1.1.

§4.A. Lifting elements of order p.p. §4.A. Lifting map map Continue = =cba(j>o : VG G—>is Gtheiscanonical Continue notation from from §2.A: §2.A: <j> the canonical having kernel ker kero(G,p) = kero(G). ker o (G). having kernel o (G,p) = Lifting Lemma 4.1. The following Lemma 4.1. The followingare areequivalent. equivalent. (4.la) C consists of p-regular conjugacy classes. (4.1a) C consists of p-regular conjugacy classes. (4.lb) For each kk >1, 1, classes from (4.1b) classes from,CCliftliftuniquely uniquelytotoclasses classes pC of the same order order in pG. holds for forkk == 1.1. (4.1c) (4.1b) holds (4.lc) (4.lb) Further, let gQ any any set set of ofgenerators generatorsof of G, G, with with each eachhaving having order order prime prime p. Let to p. Let a : H H—÷ —>GG be beaacover coverofofGGwith withp-group p-group as askernel. kernel.Then, Then, a isisaa Frattini cover cover if and and only only if lifts lifts of of g to to elements elements g of of the the same same order in in H H implies (g) = H. H. Proof. Let Let gg be be in in aa conjugacy conjugacy class class from from C. Apply ApplySchur-Zassenhaus Schur-Zassenhaus [MT, [MT, Intro, to Part Part III] III] to tothe thesequence sequence kero(G,p)

(g).

This shows (4Ja) implies shows (4.1a) implies (4.lb). (4.1b). For divides the the order For the converse, converse, assume assume pp divides order of g. Suppose Suppose gg lifts lifts to k g G pG of of the the same same order, order, for for each each k. k. Then, Then, so sodoes does ga ga with with aathe theorder orderof of g divided by p. The The result result follows follows if we show show lifts lifts of ofggto to higher higher characteristic quotients must increase increase their order order when when gg has has order order exactly exactly p. p. AAp-Sylow p-Sylow k PQ of PG is is aa pro-free pro-freep-group. p-group. The The projective projective limit limit linioo^-fc g isis an an element element Pa the pro-free pro-free group group PQ- Nontrivial profinite groups groups of order order p in the Nontrivial projective projective profinite [FrJ, Cor. Cor. 20.14]. 20.14]. So, So, this this isisimpossible. impossible. have no elements of finite order [FrJ,

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Equivalence of of (4.1a) (4.la) and (4.lb) Equivalence (4.1b)with with(4.lc) (4.1c)follows followsby byshowing showing g, g, of of order Assume PP — = PG lifts to to no no element element of of order order pp in in *G. Assume PQisis aap-Sylow p-Sylow of of GG p, lifts containing be the the pro-free pro-free p-group of Choose containing g. g. Let Fp be of the the same same rank rank as as P. P. Choose a surjective surjective map a : Fp —+ —» P.P. Then, Then,Fp Fpisisthe theuniversal universalp-Frattini p-Frattinicover cover of [FrJ, Chap. Chap. 20]. 20]. Denote Denote its kernel kernel by (P). PQ free is also free and of P [FrJ, by ker kero(P). As o covers is aa surjective surjective homomorphism i\) : :PG PQ commuting —*—>FpFpcommuting covers P, P, there is with the the respective respective maps mapsqS <j> a of and to Hypothesis P. Hypothesis (4.1c) PGPQand FpFp to P. (4.lc) andand of says g lifts to g' g' of of order orderppininPG/([kero(G), PG/([kev0(G), kero(G)] kero(G)]kero(G)P). kero(G)p). Since kero(G) maps surjectively surjectively to to ker kero(P), Since ker kero(G)p maps onto kero(P)p o(G) maps o (P), kero(G)P and [kero(G), kero(G)] mapsonto onto[ker [kero(P), kero(P)]. Thus, ker kero(G)/ keri(G) [kero(G),ker o(G)] maps o (P),ker o (P)]. Thus, o (G)/keri(G) 1 maps surjectivelytotokero(P)/keri(P). kero(P)/keri(P). The image image ofofgg'ininkero(P)/keri(P) kero(P)/keri(P) maps surjectively has order order pp and and it is also has also a lift lift of g. Thus, Thus, aa lift lift of of gg to to something something of of orimplies a lift of it has order p in kero(P)/ keri(P). Assume der p in der in pG implies a lift of it has order p in kero(P)/keri(P). Assume g' g. - Here, Here, apply apply a similar similar argument. argument. Fp/ ker1 (P) of of order order pp maps maps to g. 9' EG Fp/keri(P) Consider the pullback a~l((g)) of This must must map map surjectively surjectively to Consider of (g) in in Fp. This Z p , the universal universal p-Frattini cover of Therefore, an element element of order p p-Frattini cover of (g). Therefore, wouldmap maptotothe the generator generator of of Z/p. Z/p. This in 7L/p2 Z/p 2 would This contradiction contradiction concludes concludes the first first part of of the the lemma. lemma. Now consider condition in the lemma. lemma. IfIfHH—* —>G Gisisa aFrattini Frattini Now consider the the last last condition cover the condition however, itit holds cover condition holds holds by by definition. definition. Suppose, Suppose, however, holds and H —* H —>• GGisisnot notaaFrattini Frattinicover. cover.Then, Then,a aproper propersubgroup subgroupH1 Hi ofofHHmaps maps surjectively to to G. Apply This lifts surjectively Apply Schur-Zassenhaus Schur-Zassenhaus to to H1. H\. This lifts the entries entries of elements gi 91 of of H1 Hi of of the the same same respective respective orders. orders. By Byhypothesis, hypothesis, of g to elements Hi (QI) = H. This contradicts H1 Hi being being a proper subgroup of H. H1>> (g1) D•

§4.B. §4.B. Irrational Irrationalcharacters. characters.Consider Considerthetheuniversal universalp-Frattini p-Frattinicover cover of of let G. For g E G g G G be a lift of g. The p'-order of g is the prime to C be a lift of g. The p'-order of g is the prime to pp C. For e e V part of of the order order of of g. This This subsection subsection discusses discusses the the values values of of the irreducible irreducible characters image kg of characters of ^G on the image of g in in *(?. The The p'-order p'-order of of g equals equals that of relevant example example might might have have An = G n >>55, , pp = = 3, of g. AArelevant and the G, , n 3, and conjugacy given a 3-cycle, 3-cycle, choose choose lifts conjugacy classes classes those those of of 3-cycles. 3-cycles. Then, Then, given lifts to 3-power 3-power orders in kAn. Lift Question Question4.2. 4.2.The Thevalues valuesofofall allirreducible irreducible G G characters characters at g generate a field field Q9. Q^. Similarly, Similarly, let Q^ be the direct direct limit limit of of the the fields fields generated generated by by k k irreducible characters characters of of G. Suppose the values of g at irreducible SupposeGG isisperfect perfect and and pp divides g. Is it possible possible that Q9 Q^ = Q^? divides the the (supernatural) (supernatural) order of kcg. yes, it could quotients are are groups If yes, could be be all all characteristic characteristic PG quotients groups of regular regular Galois extensions extensionsofofQ(x) Q(x) having having aa bounded bounded number number of of branch branch points, yet Galois infinite inertia groups. The infinite The answer, answer, however, however, is "No!" "No!" Call g E£ pG rational k if its image in if G defines defines aa rational rational conjugacy conjugacyclass classfor foreach eachkk> 0. 0.

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Michael Fried and Yaacov Yaacov Kopeliovich Kopeliovich

Lemma 4.3. 4.3.If/ /the theorder orderofofggisisprime prime totop, p, then thenQ9 Qg = suppose gg = Q$. Now suppose (of any any order) defines class in in G. G. Then, (of defines aa rational conjugacy conjugacy class Then, Lift Lift Question Question has a yes answer if and only if there is a rational lift E of g. 4-2 yes only rational lift g £ pG k Proof. Suppose g and g have have p'-order. p'-order. Then, characters of G restricted restricted to Proof. Suppose (> 0.0. each

Assume gg defines defines aa rational rational conjugacy conjugacy class class in in G. G. Suppose Assume Suppose g is a lift lift of has a rational value with Q Q = Qg. This means kg has value in in each each irreducible irreducible g with Then, kgu also also representation of Assumeuu isis prime prime to to the the order order of of kg. Then, of kG. Assume irreducible representation of Values of of takes the same values under each irreducible of pG, Values a conjugacy conjugacy class class on irreducible irreducible characters determine determine the the conjugacy conjugacy class. class. determinesaa rational rational conjugacy Thus, kgu is is conjugate conjugate to kg in in £G, so so kg determines conjugacy k class in G. 0•

Conjecture 1.4 reduces to Modular Tower §4.C. Conjecture 1.4 reduces Tower property (1.2a). (1.2a). Now consider considergg having having order order aa power power of of p. p. Let Now Let kP be the p-Sylow p-Sylow of kG. P= = Pr,. The zeroth characteristic characteristic quotient is G; denote ^P by P Pp. Lemma 4.1 4.1 shows any lift of g to has larger order than does g. shows any lift *P does g.

can be be rational. rational. In In particular, Theorem 4.4. 4.4. Only Onlyp' p1elements elements of PG can particular, let r0 positive integer integer and and let let KK be a number 7 *0 be be any positive number field. There There is an integer integer K, G) with property. Suppose > k0 k0 ko = ko(ro, ko(ro,K,G) with the the following following property. Suppose kk > ko and there there isis a K regular regularrealization realization ofkG. Then, one one of of the the following following holds: (4.2a) there regularrealization realizationcorresponding correspondingto toaapoint pointof ofH(k°G, there is is aaregular for some C with with at most r0 7*0 entries entries from PC(G); or (4.2b) (4.2b) the the regular regular realization has more than r0 r$ branch branch points. points.

C)Td(K)

is a rational and gg is not Proof. Suppose gg e£ PG is rational element element and not aa p'-element. p'-element. Proof. Suppose v Let vv be be the order For any integer integer v,v,gV g is also a rational element element of of PG. Let order of the the image of gg in G. Then, of image of Then,from from Lemma Lemma 4.1 4.1 as as restated restated above, above, gv is a nontrivial element in ker0. nontrivial p-power j9-power element kero. Replace Replace g by by gt). gv. Find the the minimal minimal integer with gg eG ker kernn \\k e r n + i . In particular, particular, gg = 91 gi has nontrivial imiminteger nn with age in kern / n +i. Since kern kern /ker Since ker pro-free pro-p group, any collection collection of n is a pro-free representatives of of the nontrivial cosets of of ker kern representatives n //k e r n + i give topological genkern. Compliment gi with elements erators of ker . g\ #2? • • ,• 19uthat thatfreely freelygenerate generate n kern. gi is not a rational Now, we show g\ rational element element of pG. ker n . Now, .

.

For each each p'-integer p'-integer ra, m, our our rationality assumption For assumption says says there is an an hm hm E£ m for which which g9m = hlZm9 Since ppG/kerm G/ker m is is a finite finite group, group, there there are are pG for mgh^. infinitely infinitely many many p'-powers p'-powersg™ 9mofofggwith withcorresponding correspondinghhm inker kern. Let m' ra' m in n . Let be a nontrivial nontrivial choice choice of of such an integer. integer.

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171

and D D = (ga,. ga). Then, This isis our our setup. setup. Let Let BB = (g) Then, the thegroups groups This (g) and (#2,»• • ,9u)Further, there there exists exists /ihm' E kern B and B and D D freely freely generate generate kern. kern. Further, > G ker with m n with apply [HR]. [HR].This Thissays, says, for foreach eachh'h' eG kern, hm,Bh^ = B. Now Now apply kern, either h! (and B9 = B) or tiB(ti)-x n B B == l. Taking h' ft7== hm' ftm/ violates violates both both h' eG BB (and these conclusions. conclusions. Therefore, Therefore, #g is is not not a rational element. these element. finish the the proof, proof, consider consider the the conclusion on regular regular realizations. realizations. Supconclusion on To finish pose pC is any set of pose of rrconjugacy conjugacy classes classes(not (notnecessarily necessarilyp—regular) p-regular) of pG. AK corresponds to a point of H(^G^C)Td(K) K regular (*G,£C) realization corresponds J (end of of App.I). App.I). Further, Further, this produces produces a K K regular {PG, pC) realization with J the conjugacy conjugacy class classimage image of ^C in in JG, 00 < jj << k. k. There Thereare areonly only finitely finitely pC the many many choices choicesofofp-regular p-regularconjugacy conjugacyclasses classesCCwith withatat most most r$ r0 entries. entries. So, there exists exists some some C C (independent (independentofofk)k)consisting consisting if if (4.2a) (4.2a) holds holds for for each each &, k, there Suppose there are are pG realizations of p-regular p-regular classes. classes. Suppose realizations for for every every kk with with at f branch points, and (4.2a) does not hold with k k . for most most rr0 branch points, and (4.2a) does not hold with k > Then, for 0 0 large k, the the following following hold. . .

,

= (kC1,. (4.3a) and a point point of (4.3a) There exists exists £C = (*Ci,....,CCr) , fc C r ) and with r r0 branch giving a& pG realization realization with r < r$ branchpoints. points. (4.3b) (4.3b) At least one entry of of ^C mod keiv is not p-regular. p-regular. .

not p-regular. p-regular. The Reorder the entries of ^C to assume C1 Ci is not The branch branch cycle cycle be a K-rational union. Let Let gg eG Ci. C1. This argument (§1.A) says ^C must must be if-rational union. puts aa bound r$ on on ck Qk == {gn \(n, (n,NkC) Nk^) == 1}/^G. For For suitably suitablylarge largek, A:, bound of of r0 this contradicts the first D• first part part of of the theproof. proof. I

Remark 4.5. Theorem 4-4- The proof proof of of Theorem Theorem 4.4 4.4 4.5. Effective Effective kk0 0 in Theorem assumes existence assumes existence of kf0 with no p-regular p-regular realizations realizations of of pG for for kk > kf0. Assuming an explicit such k'Q,ititisispossible Assuming possibletotoproduce produceananexplicit explicitk0. &o.The The proof above above returns returns this to the the following. following. Let aa G pFu have have nontrivial nontrivial image image in quotient of pFu. Then, explicit lower lower in the first Frattini quotient Then, we must give an explicit Ck on the number of prime to p powers of a in F /kevk conjugate bound bound Ck on the number of prime to p powers of a in p u kerk conjugate a, where where lirm^oo Ck c& »->oo. oo. The Theargument argumentofof[HR] [HR]can cangive givesuch such aabound. bound. to a,

§Appendix I.I. Nielsen Nielsen classes §Appendix classes and Modular Towers. Towers. This is a quick quick review review of fundamental fundamental definitions definitions from from [MT]. [MT]. Excluding Excluding Theorem 3.1, 3.1, Hurwitz Hurwitzspaces spacesininthis thispaper paper are are the the 7i(G, fl(G, C ) m inner spaces Theorem of [FrV] and [MT]. —> Pj.whose whosebranch branch [MT]. These These parametrize parametrizeGalois Galoiscovers coversXX—÷ cycles fall fall in in the Nielsen class Ni(G, C) C) and have Nielsen class have aa fixed fixed isomorphism isomorphism of of the the automorphism any conjugacy classes automorphism group groupofofXX—* —> P^with withG.G.Let Let £Cbebe any conjugacy classes necessarily p-regular the in *&—not necessarily p-regular classes classesas as in in §0.B—mapping §0.B—mappingtoto C C by the with ker0. kerocanonical quotient with

172

Michael Michael Fried Pried and andYaacov Yaacov Kopeliovich Kopeliovich

The Frattini = G G lifts lifts to to g G £ C . The Frattini covering covering Suppose Suppose&g eE C C with with (g) = means (g) = = kG is So, after the first first level, level, II(9) = property means is automatic. automatic. So, after the = 11 isis the significant significant formula formula for for defining defining the theNielsen Nielsen class class (§0.A). Specifically: Specifically: 1} and II(g) = = 1} Ni(^G,£C) = {g GkpCI g mod ker0 G Ni(G, C) and

is the the kth is A;thlevel level Nielsen Nielsen class. class.

Consider the the free r—1, Consider free group group on on generators generators Qi, ii = l1,.. , . . .., r, — 1 , with these these relations: (I.la) (Lla)

1,. . . , rr—2; QiQi+iQi = Qi+iQiQi+u i« = = 1,..., -2;

(I.lb) (Lib)

and QtQj = QjQi, \i - j \ > 1; and QQj=Q3Q2,

(I.lc) (Lie)

Q1Q2 Qr—iQr—i Qi = QiQ2'-Qr-iQr-i~-Qi = 1. l.

Conditions (Lla) (I.la) and (I.lb) the Artin Artin braid groupBBr. Add (Lie) (I.lc) tto Conditions (Lib) define define the braid group o r. Add get the group H Hrr of degree r, aa quotient The the Hurwitz Hurwitz monodromy monodromy group degree r, quotient of of Br. Br. The Q2s in BBract Qi s in g € Ni(G,C): Ni(G, C): r act on gE (I.ld) (Lid)

(g)Qi = (gv . . . , ^ _ l 7 ftft+i^"1' fti ft+2» • . • •»0r)> i *==1,.. ! , . .- ,. ,r—1. r-1. .

Mod ofGGto to induce inducean an action actionby byH Hr. Mod out by inner inner automorphisms automorphisms of Irrer. Irreducible components of of %(G, h(G, C) correspond ducible components correspond to to orbits orbits of of this this action. action. Braid group extends that that on on Ni(^G,pC) extends on the thelevel level 00Nielsen Nielsen Braid group action action on class. This This produces produces the thecorresponding corresponding sequence sequence of of moduli moduli spaces spaces (1.2) (1.2)

.... . . - » • H(kp+1G,

k +1 p C)

-> H(kpG, kpC) -+ • • • -> H(G, C ) .

center for forkk >0.0. ([FrK] ([FrK] shows showsthis this holds holds if if G G has no no Suppose ^G has no center center and no Z/p center Z / pquotient; quotient; in in particular, particular, ififG Gisisaacenterless centerless perfect perfect group group [MT, Lemma 3.6].) 3.6].) Let K [MT, Lemma K be beaafield field of ofcharacteristic characteristic prime prime to toIGI. \G\. Then, as in [MT, Part III], aa K [MT, Part K point point pp EG H^G^C) gives a a sequence of K K covers covers as (1.3) (1.3)

kX Xp -> kk— ~lXp ->

k

> % —÷ -> P 1 .

Further, iXp -> P 1 gives gives aa K regular regular (JpG,JpC) realization, 0 , . .. .. ,,&k.. realization, jj = 0,. (1.2) is what we When When C C= = ^ C consists of p-regular p-regular conjugacy conjugacy classes, classes, then then (1.2) call a a Modular call Modular Tower Tower as as in in §0.B. §0.B.

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173 173

§Appendix of covers covers of of the the sphere §Appendix II. Equivalence Equivalence of There are two two natural naturalequivalences equivalences of of covers covers of the sphere. sphere.

(II.la) fa : Xi -» —* IN, equivalent if if there there exists exists aa: X1 : Xi—* —>• X2 X2 (ILla) P 1 , i = 1,2, are equivalent ° a with = with 02 ° ot = (j)\. F"1 with with fa oa (II.lb)As IF"1 -» P (Il.lb) As in in (II.la), (ILla), except except there there is is fip : P o a == P o fa. :

:

§App.II.A. fl(G, C) refers §App.ILA. Action of of SL2(C) SL2(C) on Fr\D Below, Below, U fl = U{G, refers to r. \ Dr.

a space in a given Nielsen class classup up to to equivalence equivalence (ILla) (II.la) . space of covers covers in given Nielsen x r The group Sr acts on on the the space space (F" (P ) by permutation of of its its coordinates. coordinates. Sr acts r —* P F'r. gives a natural map map "sr \I/r : (p1)r (P1)'" -» . Consider £ = (x,,...,xr) (#1,... , x r )with with This gives none of the coordinates coordinates equal equal oo. 00. The whose coordinates coordinates are the The point whose are the none m z coefficients of ~~ xi) in coefficients of the the polynomial n i = i ((zz — z represents the image of j: \£r. (If Xi = oo, 00, replace 1.)Also, Also,Wr \£rtakes takesthe thefatfat under Wr. replace the factor zz — X{byby1.) Dr. This This interprets interprets U — pr P r \\ Dr Dr as asthe thespace spaceof ofrr distinct distinct diagonal A r to Dr. urr = 1 unordered points . Thus, Thus, \I>r : U" Ur —* —»Ur Ur isisan anunramified unramifiedGalois Galoiscover cover unordered points in P F". with group Sr. Consider PSL2 (C)as as linear linear fractional fractional transformations transformations acting diagonally Consider PSL2(C) diagonally x r on the r copies (a(xi),. . PSL2(C) and j: G U", U ", j: i-> (a(xi),... ,a(xr)). copies of P . For a Ee PSL2(C) The action of PSL2 (C)isison on the the left, left, commuting commuting with with the coordinate PSL2(C) coordinate perperr mutation action of of SSr. generalizes the mutation action .r The The quotient quotient PSL2(C) PSL2(C) \\UU" = A Arr generalizes A line minus the points points 0,0,1,00. Further, PSL2(C) PSL2(C) \U 1, Further, \ r = JJrr generalizes 00 from from the theory of modular modular curves. curves. It has has the j line minus minus the the point point at oo Thecase caserr==44isiscrucial crucialtotous, us,sosowe wereassure reassure the the complex dimension r — — 3.3. The reader by displaying displaying these these identifications. identifications. Given j: = {x,,x2,x3,x4} up to equivalence (II.la) there {xi,x 2 ,x 3 ,x 4 } up equivalence (ILla) there is aa unique unique 1 degree 2 cover cover Xv -> P ramified exactly at j:. Thus, X X is aa genus genus 11curve; curve; equivalent determines its isomorphism isomorphism class. XT isisequivalent class. Further, Xr its j invariant determines £ PSL2(C) PSL2(C) with identifies to Xf if if and and only only ifif there there exists exists a E with a(y) a(s) = y'. This identifies J4 and the j line minus 00. The natural A line (ramified) cover of the jj line line and minus oo. The natural line (ramified) cover of the is Galois Galois with group S3 [R, [R, 1.59]. 1.59]. Don't confuse confuse this copy copy of S3 5 3 with an an S3 S3 inside the coordinate permutation permutation action action of of S4. S4. inside :

.

Extending PSL2(C) §App.II.B. Extending PSL2(C) action actiontoto7L(G, U{G, C). C). Suppose p G has e H has 1 a representative cover cover 0 p : Xp —* -> F'1. P . Extend the action action of a EG PSL2 PSL2(C) (C) by by G cover in in thethe Nielsen composing (pp with withaatotogive giveaaoo (j)p : Xp—*->F"P,a1 ,new a new G cover Nielsen class. Thus, PSL2(C) C); denote class. PSL2(C) action extends extends to W(pG,C); denote its quotient quotient by rd k rd Cyd. C)rd H(£G,C) . Since Since U{ vG,C) is an an affine affine algebraic algebraic set, so is is W(£G,C) C) is [MFo, Thm. 1.11. 1.1]. These reduced reduced spaces spaces generalize generalize the modular [MFo, the spaces spaces of of modular 1 y,(pk+l) curves Y^p^ ) [MT, [MT, Intro.]. Intro.].

174 174

Michael Fried and Yaacov Michael Yaacov Kopeliovich Kopeliovich

for covers covers up up to to equivalence equivalence (II.lb): (II.lb): So, the spaces Wrd are moduli spaces for reduced Hurwitz Hurwitz spaces. spaces. Many Many have have asked: asked: "Which "Which equivalence equivalence is is more more important?" My ib). important?" My answer: answer: Classical Classical geometers geometers often often like like equivalence equivalence (II. (II.lb). rd (Also, Ar. (Also, it is usual to to use use the thepulback pulbackofofi1jrd H over over A When, however, however, there there r . When, are repetitions repetitions in the the conjugacy conjugacy classes classes C, this this pullback pullback isis inappropriate, inappropriate, often wiping wiping out out the the significant significant arithmetic arithmetic information.) information.) Their justification often rd less than than that of %. is the complex complex dimension of H is 3 less likethat, that, too. For curves when when rr = We like For example, the reduced spaces are curves = 4, allowinguse useofofFaltings' Faltings'Theorem. Theorem.Further, Further,any anyrational rationalpoint pointon on% fl autoallowing matically produces matically produces infinitely infinitely many others others in in its itsPSL2 PSL2(Q) orbit. Allowing this would would defy Still, itit isis equivequivdefy the the finiteness finiteness results results this this paper paper conjectures. conjectures. Still, alence (II.la) supports the the Modular Modular Tower Tower construction. construction. From From that that alence (II.la) that supports construction compatibly reduce all levels levels of the tower tower by by the thePSL2 PSL<2(C) construction we compatibly (C) Conclusion: We We require require both bothequivalences equivalences for for Modular Modular Towers. Towers. action. Conclusion:

References Benson, Representations Representations and andcohomology, D. J.J. Benson, cohomology, I: I: Basic Basic represenrepresentation theory theory of of finite finite groups groups and andassociative associativealgebras, algebras,Cambridge Cambridge studies in Advanced Mathematics 30, Camb. Advanced Mathematics Camb. Univ. Univ. Press, Press, 1991. 1991. [Be2] D. J. Benson, [Be2] Benson, The The Loewy Loewy structures for for the the projective projective indecomposindecomposable modules modules for for A% A8 and and A$ A9 in in characteristic 2, Comm. able Comm. in inAlg. Alg. 11 (1983), (1983),1395—1451. 1395-1451. [DFr1]P.P.Debes Dèbesand and M. M. Fried, Fried, Arithmetic variation [DFrl] variation of of fibers fibers ininfamilies: families: Hurwitz monodromy criteria for rational points, J.J.Crelle Crelle409 409(1990), (1990), 106—137. 106-137. [DFr2]P.P.Debes Dèbesand and M. M. Fried, Fried, Integral Integral specialization [DFr2] specialization of rational function function families, preprint. prepririt. Jan. families, Jan. 1997, 1997, 20 20 pgs. pgs. [Fa] [Fa] G. Faltings, Diophantine Diophantine approximation approximation on abelian varieties Annals Annals of Math. 133 133(1991), (1991),549—576 549-576 M. Fried, Fried, Alternating Alternating groups groups and and lifting lifting invariants, invariants, Preprint Preprint as as of of [Frl] M. 07/01/96. [Fr2} M. Fried, Fields of of Definition Definition of ofFunction Function Fields Fields and and Hurwitz Hurwitz Families [Fr2] 17—82. Algebra5 (1977), 5 (1977), 17-82. and Groups Groups as as Galois Galois Groups, Groups,Comm. Comm.ininAlgebra [Fr3] M. Fried, of J.-P. Serre's [Fr3] Fried, Enhanced Enhanced review review of Serre's Topics Topics in Galois Galois TheTheexamples illustrating braid ridigity, ory, with examples ridigity, Proceedings Proceedings AMS-NSF, AMS-NSF, Summer Conference Conference Cont. Cont. Math series 186, Recent Slimmer Recent Developments Developments in the Inverse (1995), 15—32. InverseGalois GaloisProblem Problem (1995), 15-32.

[Be] [Be]

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[Fr4] M. Fried, Global Global construction construction of of general general exceptional exceptional Covers: Covers: with with motivation for applications to encoding Applications and Algorithms, Algorithms, Cont. Cont Math, Math. 168, 168,G.L. G.L.Mullen Mullenand andP.J. P.J.Shiue, Shiue,editors, editors,1994, 1994,69—100. 69-100. of 3 and 4 branch [Fr5] Fried, Arithmetic Arithmetic of branch point point covers: covers: aa bridge bridge propro[Fr5] M. Fried, vided by noncongruence (Z), in Progress noncongruence subgroups subgroups of of SL2 SL2(Z), ProgressininMatheMathematics 81, 81,Birkhauser, Birkhauser,1990, 1990,77—117. 77-117. [FrJJ M. Fried [FrJ] Fried and M. M. Jarden, Jarden, Field FieldArithmetic, Arithmetic, Ergebnisse Ergebnisse der der MatheMathematik III, III, 11, 11,Springer SpringerVerlag, Verlag,Heidelberg, Heidelberg, 1986. 1986. [FrK] M. page preprint, [FrK] M. Fried Fried and and Y. Y. Kopeliovich, Kopeliovich, A5 A*, Modular Towers, Towers, 30 page 1996. 1996. [FrV] M. Fried and H. problem and and rational [FrV] H. Völklein, Volklein, The inverse inverse Galois Galois problem points on on moduli moduli spaces, spaces,Math. Math.Annalen Annalen290 290(1991), (1991),771—800. 771-800. [MT] M. [MT] M. Fried, Fried, Modular Modular Towers: Towers: Generalizing Generalizing the relation relation between between dihedihedral groups groups and modular modular curves, curves, in in Proceedings Proceedings AMS-NSF AMS-NSF Summer Conference, Cont. Cont. Math series 186, Recent in the InConference, Recent Developments Developments in verse Galois GaloisProblem, Problem,1995, 1995,111-171. 111-171. [HR] W. Herfort Herfort and and L. L. Ribes, Ribes, Torsion Torsion elements elements and and centralizers centralizers in free free products of of profinite profinite groups, J. J. Crelle Crelle358 358(1985), (1985),155-161. 155-161. [Me] J-F. Mestre, [Me] Mestre, Extensions Extensions régulières regulieres de Q(T) de de groupe groupe de de Galois Galois An, J. AIg. Alg. 131 131 (1990), (1990),483-495. 483-495. [MFo] D. D. Mumford Mumfordand and J.J. Fogarty, Fogarty, Geometric Geometric Invariant Invariant Theory, Theory, Ergeb. der [MFo] Math. und und ihrer Math, ihrer Grenzgebiete Grenzgebiete 34, Springer Springer Verlag, Verlag, 2nd 2nd enlarged enlarged edition, 1982. 1982. [Ri] L. Ribes, of profinite profinite groups, groups, Archiv Archivder derMath. Math. 44 [Ri] Ribes, Frattini covers covers of (1985), 390—396. (1985), 390-396. [R] A. Robert, Elliptic [R] Elliptic curves, curves, Lecture Lecture Notes Notes in in Mathematics Mathematics326, 326,SprinSpringer-Verlag, Heidelberg-New York, York, 1973. 1973. [Se] Bartlett and Jones [Se] J.-P. Serre, Topics Topics in Galois Theory, Bartlett Jones Publishers, Publishers, 1992. [V] [V] H. Völklein, Volklein, Groups as Galois Galois Groups, Groups, Cambridge Cambridge Studies Studies in AdAdvanced Mathematics Mathematics 53, Cambridge vanced Cambridge Univ. Press, Press, 1996. 1996. University of of California California at at Irvine, Irvine CA University CA 92717 92717 mfried©math.uci.edu [email protected]

Part Ill. III.Galois Galoisactions actionsand andmapping mappingclass class groups groups

Galois group GQ, GQ, Singularity E7, 2?7, and and Moduli Moduli M3 Ms

Makoto Matsurnoto Matsumoto

Monodromy of of iterated iterated integrals integrals and non-abelian unipotent periMonodromy periods Zdzislaw Zdzislaw Wojtkowiak Wojtkowiak

Galois group GQ, Singularity £7, E7, and Moduli M3 Galois GQ, Singularity M3 Makoto Matsumoto

An Artin group for any any graph, of the usual group is is aa generalization, generalization, defined defined for usual braid braid groups. The E-?Dynkin Dynkindiagram diagramhas hasaanatural naturalsurjection surjectiontotothe the The Artin Artin group group of of E7 mapping curves, defined defined by canonical mapping class class group group of of genus genus three three curves, by sending sending the canonical set of of Dehn Dehn twists, twists,called called Humphries Humphries generators. generators. generators to aa set We first first show show that that this group homomorphism from a geometric geometric mormorWe homomorphism comes from phism phism between between moduli inoduli spaces spacesby bytaking taking TTI. in. Next, we describe the the notion notion of of tangential tangential inorphisms. Next, we describe morphisms. Let U U be be an an affine affine divisors which which are are principal. principal. The variety and and D\,..., D1,.. . , D/ D1 be normal normal crossing crossing divisors The variety defined whenwhen tangential morphism morphism from n •. •. fl• D DjD\ to U to— U -(D1 (DiU.U. .•U•D,) • UisD/) is defined from D\ Di fl. for each each i.i. This we defining ideal of Di for This yields yields a we specify specify the the generator generator of of the defining morphism morphism morphism between between the the algebraic algebraic fundamental fundamental groups. groups. A tangential morphism from a point point Speck Specfc coincides coincides with the the notion notion of of k-rational fc-rational tangential tangential base base from point. Using this, this, we describe the the action Using we describe action of of the absolute absolute Galois Galois group group on the the profinite completion completion of of Artin Artin groups of type An, Bn, C n , Dn, and and E7. E7. (The (The profmite first type Description uses only type gives gives the the Braid Braid group group investigated investigated before.) before.) Description only the action the Belyi's Belyi's coordinates, coordinates, i.e. i.e. the action of the Galois Galois group group on the algebraic algebraic 1 minus 0, 0,1, fundamental F minus fundamental group of IF' 1, 00. we give give an an explicit explicit description description of of the the Galois action on the As a corollary, corollary, we Galois action the profinite completion completion of of the mapping class profinite class group group of of genus genus 33 curve. curve.

Contents

§0. Introduction §0. §1. §1. E7 E7 and and M3 M3 1.1. 1.1. Fundamental group of moduli stack 1.2. generators and and Artin group of E7 1.2. Humphries Humphries generators Ej type 1.3. Variety V(E7) 1.3. V(E7) §2. Tangential Tangential morphisms §2. morphisms 2.1. Tangential morphisms 2.1. morphisms 2.2. Homotopy Homotopy between between tangential morphisms morphisms 2.3. Composition of tangential morphisms 2.3. morphisms 2.4. GAGA 2.5. Example 2.5. Example

The author was supported by by Japan Japan Association Association for for Mathematical Mathematical Sciences Sciences for attendattending the Luminy ing Luminy conference. conference.

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§3. Galois action action on on Artin Artin groups §3. Galois groups 3.1. 3.1. Motivation Motivation 3.2. Tangential base point on on V(E7) V(E?) 3.3. Pj l o o and P j ± l o o 3.4. Galois Galois action on on each each generator generator 3.5. 3.5. The result result on on E7 E7 and and M3 M3 3.6. 3.6. An, Bn, C n , Dn cases

§0. §0. Introduction Introduction This manuscript of three three parts. parts. The first manuscript consists consists of first two two parts are are logically logically independent, and the last part uses independent, and uses the the results results of of the the first first two. two. The first first part of geometric, and and is independent independent of of the Galois §1 is purely purely geometric, The of §1 group. group. We recall recall Brieskorn's Brieskorn's result [5], on the deformation We [5], [6] [6] on deformation of of E7-singularity, ^-singularity, which says that that (1) which says (1) the the smooth smooth locus locus of of the deformation deformation space has the E7 E7 Artin group as the fundamental fundamental group, and (2) (2) on on the the smooth smooth locus, locus, each each fiber fiber is a genus genus three curve. curve. By By (2), (2),we wehave have aaclassifying classifying map map from from the the smooth space to to the moduli stack of of genus genus three three curves. curves. Our smooth deformation deformation space moduli stack first result is is that that by bytaking takingir1 TTIof of this this morphism, morphism, we wehave have aa group group homomorhomomorphism from the the E7 phism from E7 Artin group group to the the mapping mapping class class group group of of genus genus three curve, which maps maps each each generator generator of of the the Artin Artin group curve, which group to the the Humphries' Humphries' Dehn-twist generators generatorsofofthe the mapping mappingclass classgroup. group. This This is is an interesting Dehn-twist interesting coincidence, since the the two sets of generators were invented independently. coincidence, since independently. Then the algebraic Then algebraic meaning meaning of of the Humphries Humphries generators generators are clear, so so we move movetoto the the investigation investigationofofthe the Galois Galoisaction. action. As Asaa preparation, preparation, in in the we second part of §2, we we give givethe the notion notion of oftangential tangential morphisms. morphisms. This This notion notion is is of §2, a slight generalization of the tangential tangential base base point point introduced introducedby byDeligne Deligne[7], [7], developed and effectively used by by Ihara, Ihara, Nakamura, and developed effectively used and the theauthor author[15] [15][17] [17] [22] [18][27] [27](Ihara (Iharaand andNakamura Nakamura even evendealt dealt with with the formal scheme version [22] [18] in [18]). Note that there are also log-scheme theoretic interpretations, which [18]). Note that are also log-scheme theoretic interpretations, fit to motivic motivic fundamental fundamental groups groups[10] [10] [30].) [30].) In the third third part part§3, §3,we we give give a construction construction of of a tangential base base point point for for the deformation space of of the the EE7 singularity. (The construction mimics that deformation space singularity. (The construction mimics that 7 for An in in [17].) seven tangential tangential morphisms morphisms from fromPP'1 minus [17].) We We give give seven minus four four points to the deformation space, then then the Galois action action on on the Artin group points deformation space, group appears for for PF'1 minus three points. As is written in terms of / a , x(a)i which appears a corollary, corollary, the Galois Galois action on on profinite profinite mapping mapping class class group group with with genus genus three is described. also treat treat some other other type of Artin groups, Ani Bn, described. We also Cn, Dn. The motivation of this research Let V motivation of research is as follows. follows. Let V be be aa geometrically geometrically

Galois group GQ, GQ, Singularity Singularity EE7, and Moduli Moduli M3 M3 Galois group 7, and

181 181

connected variety varietyover overthe therational rational number number field fieldQ, Q,and and xx be be a Q-rational connected point. Then, point. Then, we we have have aa group group action action of of the the absolute absolute Galois Galois group group GQ GQ == Gal(Q/Q) of Q on the profinite profinite completion of the fundamental group of V V alg (V,x) Autirj(V(C),x) = GQ —* as a complex complex variety variety [11]: [11]: pVjX : GQ -> Aut7rT(y(C),x) ^ Autir1 Aut7r^(F,x) the coefficient coefficient extension extension V V® ® Q). Q). (here V denotes the 1 Grothendieck conjectured conjectured1 Grothendieck that VV can canbeberecovered recoveredfrom fromPV,X, /9y)X, for for any any with V V "anabelian" variety V. V. Also Alsohe heasserted assertedthe theimportance importance[12] [12]ofofPV,X pv,x with moduli spaces spaces of of (fr) (C)b) isis aa free of two two generators generatorsx,x,y, that a(x),a(y) that free group group of y, that is an an element element of ofthe the profinite profinitecompletion, completion,and andsubstituting substituting x, x, yy in in cr(x), o(x), cr(y) is to some elements in another profinite group makes sense). This was to some elements in another profinite group makes sense). This was proved proved by Ihara and from Drinfeld's Drinfeld's result result by Ihara and the the author author[17], [17], where where the the insight insight comes comes from on the action of the Grothendieck-Teichmüller group on the pro-unipotent on the action of the Grothendieck-Teichmiiller group on the pro-unipotent completed completed braid braid groups groups [9]. [9]. Here we will describe some generalization generalization of of this, this, i.e., i.e., for for V Here we will describe some V deformation deformation space of rational singularity of surfaces (the above case corresponds to the the space of rational singularity of surfaces (the above case corresponds to singularity i-type). Then, singularity of of An_i~type). Then, using using aa tight tightrelation relation(see (seeTheorem Theorem 1.3) 1.3) between the deformation of E7 singularity and the moduli stack between the deformation of E7 singularity and the moduli stack M3 M3 of of genusgenusthree also Looijenga Looijenga [21]), [21]),we weextend extendthe theresult result to to V V= =M M3 (see three curves curves (see (see also 3 (see Theorem 3.1). Theorem 3.1). Note that this this conjecture conjecture for for curves curves has has been been settled. settled. The Thegenus genuszero zerocase casewas was solved solved by H. Nakamura his integrated integrated paper [25]). Tamagawa [33] [33]settled settled this this Nakamura (see (see his [25]). Recently, Recently, A. Tamagawa for affine affine curves curves (also (also finite finite field field case), and S. forthe the proper proper case (he for S. Mochizuki Mochizuki [23}[24] [23][24] for proved p-adic version). version). proved a stronger p-adic 2 After the author's author's talk talk on onthe thegenus genusthree threecase casein inLuminy, Luminy, H. H. Nakamura Nakamura [27] [27] succeeded succeeded in describing describing the Galois Galois action on mapping mapping class class groups with any any genus genus gg by by ?'P 1— — 1 {O, 1,00}. co}.In In the the talk talk at at Luminy, Luminy,the theauthor author depended dependedoon?' {0,1, nP — —{0, {0,±1, ± 1 ,oo} 00}rather rather than the desirable?' 1, oo}. desirable P 1— —{O, {0,1, 00}. Now Now the theresult resultofofthis thismanuscript manuscriptinvolves involvesonly only IF'1 — 1, no), P —{O, {0,1, 00}, and and this thisimprovement improvement comes comes from from the the stimulation stimulation by by the theNakamura's Nakamura's success. The to him him for for informing informing him of of his beautiful beautiful result. result. success. The author is thankful to

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Makoto Matsumoto Makoto

We shall shall exchange exchange the the order order of of the the explanation: explanation: explain explain first about about E7 E7 and M3, and and next about Galois Galois actions. This This isis because because the the former former is is purely purely geometric and independent independent of of the the mess of the the Galois actions, and it seems geometric and mess of Galois actions, seems better for for the the readability. readability.

§1. E7 §1. E7 and andM3 M3

1.1. Fundamental group of moduli stack 1.1. Let Mg^n denote the moduli moduli stack stack of of the genus genus g n-ordered pointed pointed curves curves [8] over Q. . The coarse coarse moduli scheme of Mg,n, M^ n , equipped with C-topology, C-topology, is analytically isomorphicto to the the quotient quotient of the Teichmüller space by by the analytically isomorphic Teichmiiller space mapping class classgroup. group. Then, Then, this analytic has a trivial fundamental mapping analytic space space has fundamental group. However, there is is Oda's theory group. However, there theory of of the the fundamental fundamental group group of of anaanalytic [28]. Roughly Roughly speaking, lytic stacks stacks [28]. speaking, considering consideringthe the fundamental fundamental group group of of n the stacks corresponds corresponds to considering considering the orbifold, and the orbifold, and ^ {Mg^n ®Q C) C) is proved isomorphic to mapping class class group group F ^ n of genus g proved to to be isomorphic to the mapping of the genus oriented Moreover, the monodromy monodromy oriented surface surface fixing fixing nn distinguished distinguished points. points. Moreover, +i — > • M ^ , i.e., the outermonmonrepresentation of the universal fiber M representation of universal fiber 9in —* g n i.e., the outer odromy action of the fundamental group of the base space on that that of of aa fiber, fiber, op coincides canonical morphism morphism Tg^n — coincides with with the canonical —+» Out7r* (E^>n), where E^>n is the n-punctured genus genus gg oriented oriented surface. surface. n 1'g E9g = S^oFor it n= = 0, 0, we simply write M9 Mg = M9,0, M9JQ, T F^o, aand d T, Eg,o. g = r9,0,

1.2. 1.2. Humphries generators and Artin Art in group group of of E7-type i^V-type It is known that the the mapping mapping class class group group I'9 Tg is is generated generated by by the the following following 2g ++ 1 Dehn twists [13], which are are called called the Humphries generators. 2# [13], which generators.

The relation of these Dehn-twist determined by by Wajnryb. Dehn-twist generators was determined There are four four types of of relations (for (for the explicit explicit description, description, see Wajnryb Wajnryb [34]and and Birman's Birman's survey article [3]). [34] [3]).

Galois group GQ, GQ, Singularity Singularity E7, E7, and and Moduli Moduli M3 M3 Galois group (i) (i)

183 183

f D'D, if Braid relations: DD DD' = D'D, Braid relations: if D and and D' Dfhave haveno nointersection. intersection. intersection point point (the number DD'D DD'D = DD'D DD'D if D and D' i)' have have one intersection

of intersections one among among the the above above curves). curves). intersections is is at most one (ii) Monocular (ii) Monocular (or chain) chain) relation, relation, which which is supported supported by by the the five fiveleftleftmost twists. twists. (iii) Lantern relation. (iii) (iv) Coinmutativity (iv) Commutativity with the the hyperbolic hyperbolic involution. involution.

The similaritywith withthe the braid braid group. group. We The relation relation (i) (i) shows shows similarity We write write an intersection diagram diagram FF of of these these 2g 2g ++ 11 twists. twists. Thus, FF has intersection has 2g 2g++11vertices vertices correspondingtoto the the 2g + 11 curves, corresponding curves, and and 2g 2g edges edges corresponding corresponding to the the point. intersection point. The group generated by 2g 2g ++ 11 generators with with relation relation (i) (i)isiscalled called Artin Artin group of of type type F, F, and denoted In general, let G be any denoted by by A(F). A(T). In general, let any graph graph each of of whose whose edge edgeee isis labeled labeled with with non non negative negative integer integer ra me. each the e. Then, the corresponding Artin corresponding Artin group group A(G) A(G) is is defined defined as as the the group group with the vertices vertices as generators generators and the relations relations aba • • • = bab - • • m e +2

m e +2 me+2

for each pair a, b of b, me me is the the of vertices. vertices. Here, Here, e is the edge between between a and b, number assigned for for this this edge, edge, and and ifif there there is no edge edge then we put me number assigned me == 0, i.e., a and and bbcommute. commute. In this terminology, strings is nothing nothing but but the Artin terminology, the braid group of ii n strings group of straight line with n — vertices. straight line with n —1 1 vertices. Brieskorn [5] [5] [6] [6]proved provedthe thefollowing followingprofound profoundtheorem. theorem.IfIfthe the graph graph is is a Brieskorn Dynkin diagram of some some semisimple semisimple Lie Liealgebra algebra over overC, C,then then the the associated group is realized realized as the fundamental fundamental group group of of aa rational rational variety, variety, dedeArtin group of Lie algebra. V(G) has has aa moduli moduli noted here by V(G) V(G) with G the type of algebra. This V(G) is the the smooth smooth fiber fiber locus locus of of the the miniversal miniversal deformation deformation interpretation. It is corresponding to the Dynkin Dynkin diagram. diagram. AcAcspace of of the rational singularity corresponding tually, the Artin Artin group group isis defined defined to describe the fundamental group group of of these these varieties. varieties. The braid group appears for for An-\ singularities, to which the correspondcorresponding rational variety is the configuration configuration space space of of nn points on a plane. plane. Now, we we consider considerthe the case case gg = 3. 3. Then, Now, Then, the the intersection intersection diagram diagram of of Humphries generators coincides with E E7 So, there there is a Humphries generators coincides with diagram. So, 7 Dynkin diagram. surjection from the the Artin Artin group group of of EE7 type to the mapping of surjection from mapping class class group group of 7 type genus 3: 3: A(E7)7)—3* A(E -» F3, r3, by mapping canonical generators generators to to canonical canonical generators. generators. Now, Now, aa natural mapping canonical question arises: does this morphism come from an algebraic question morphism from algebraic map? map? for

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Makoto Matsumoto Makoto

The answer is yes Theorem 1.3 1.3 for for the the precise statement): there The answer is yes (see (see Theorem precise statement): there exists a natural natural family family of of two two pointed pointed genus genus three three curves curves over over V(E7) V(E'7) such such that the classifying map to M3 obtained by forgetting two points induces the classifying map M 3 obtained forgetting induces A(E7) A(E the fundamental fundamental groups. groups. 7) -» F 3 by taking the The rational variety V(E7) V(E?) can can be be described described explicitly explicitly by using the Weyl Weyl 9 group, and the Galois action on (V (E7), x) can be described in terms group, and Galois action n^ (V(E?), x) can be described in of that that on ^1^(^0100^)7 b), for for some some appropriate appropriate tangential base points x, of x, b. b. Thus, by the functoriality can describe Thus, functoriality of of TT^P, we we can describe the Galois Galois action on on I'3. This IV This will will be explained explained in in §3. §3. 1.3. Variety V(E7) V(E7)

V(E7) is aa miniversal miniversal deformation deformation space space of of E7 £7 singularity. singularity. Brieskorn solved conjecture of Grothendieck on the explicit construction solved conjecture of Grothendieck on the explicit construction by Lie group Here we For the [5] of this deformation [5] of deformation space. space. Here we need need this construction. construction. For reference on the reference the following following facts, facts, see see Slodowy Slodowy [31]. [31]. Let $ be a root root system system of of aa semisimple semisimple Lie algebra, algebra, i.e., one of An, Bn, C n , D n , E6, EG, E7, £7, E8, Eg, F4, F4, G2. G2. This This isis aafinite finite subset subset of of aa vector vector space space U set of generates aa finite over Q. The set of reflections reflections generates finite group, called called the the Weyl Weyl over Q. group of of type type <$. 1. For root a E be the reflection group For each each root G $, let H0, Ha be reflection hyperplane hyperplane (i.e., the fixed divisor divisor by the reflection reflection along a) of of the thecomplexified complexified vector vector define as as thethe quotient of of Uc — u0,H0, bya by space Uc := U U <8>QC.C.Let Letusus defineV(4) V($>) quotient — UaH group. This variety V(4) is the action of the Weyl Weyl group. variety V(&) is known known to be aa smooth smooth rational variety theory: the quotient rational variety defined defined over over Q by invariant invariant theory: quotient of of the the affine space spaceby byaa finite finite group groupGG isis an an affine spaceififand and only only ifif GG is is a affine affine space reflection group, group, i.e. i.e. generated reflection generated by by (possibly (possibly nonorthogonal) nonorthogonal) reflections reflections (see (see for for example Theorem 4.2.5 4.2.5 in in [32]). [32]).

Theorem V(c1) is isoTheorem1.1. 1.1.(Brieskorn (Brieskorn[5][5][6]) [6])The Thefundamental fundamental group group of ofV($) isomorphic to the Artin group '1. morphic group of of type < 1>. This This isomorphism isomorphism isisgiven given as asfollows. follows. Take the the fundamental fundamental Weyl chamber CC in U^ as the base point of V(4) Take Weyl chamber base point of V($>) (since the chamber is is simply simply connected, connected, this this makes makessense). sense). For each funda(since mental take aapath pathininUc Uc— — H0, Ha from from this this chamber chamberto tothe thereflection reflection mental root a, take image of C with respect to a, which goes around the divisor H counterimage of C with respect which goes around the divisor H0, a counterclockwise. Then, taking the quotient quotient by by the Weyl Weyl group, group, we we obtain obtain aaclosed closed clockwise. Then, taking corresponding generator o/7rJop(y(),C) to the path in V($>). This This is the corresponding generator of associated with the Artin Artin group group generator associated with aa in the

yields aa usual usualbraid braid group. group. In this The simplest example is An type, which which yields case, with =— 0. 0. If we denote case, U is the set of vectors vectors (x1,x2,. (xi, #2, • • • ? #n+i) with ]T^ Xi If we denote by by ee?;the the i-th z-th unit unit vector vector ( 0 , . . . , 0 , 1 , 0 , . . . , 0), then $ = {e; — ej\l
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inner product product with with any fundamental root is positive, inner fundamental root positive, i.e., (x1,. (#i,... . The reflection associated with with x1 x2 > > > xn+\. The reflection associated withe2e^— —e3ej isis xi > X2 > - and the Weyl group is the symmetric the exchange of x2 and X{ Xj, and Weyl the symmetric group group exchange 5 n + i. The reflection reflection hyperplane hyperplane is the locus locus with with xi = Xj. Thus, Thus, V(An) is is (U (U——the the locus locuswith withx2 Xi==x3Xj for for some somei i ^ j)j) divided divided by by Sn+i, i.e., i.e.,the theconconfiguration space space of of unordered unordered nn + 1 points. Its fundamental figuration fundamental group group is the braid group, i.e., braid i.e., A(An). It is is easy easy to to see see that that the the path pathdescribed described above above corresponds to to the the standard generator of the braid group. corresponds group. In §3, we we describe describe the the Galois Galois action action on on the profinite profinite fundamental fundamental group of some some of of these these varieties. varieties. Each of Each V($) has a natural natural meaning meaning as as aa moduli moduli of singularities singularities as as follows. follows.We Wedon't don't use use the the universality universality in this manuscript, of manuscript, so we we let let the reader reader consult consult the the reference reference for for the the definition definition of of universality universality of these these moduli moduli spaces. spaces. (For the case of F4, G2, see [31].) case of Bn, Cn, [31].)

Theorem system of of A,C,E A, C, E type. type. Then, Theorem 1.2. 1.2. [5] [5]Let Let 1$ be aa root system Then, the germ by the the Weyl Weylgroup group near the origin of U divided by origin isis aaminiversal miniversaldeformation deformation space 1' type. space of of the the rational rationalsingularity singularityofof$ type.That Thatis,is,there thereisisa afamily familyofofcurves curves on the origin origin is curve of $>-typesingularities, singularities, on U, U, such that the fiber fiber at at the is a curve Thequotient quotientofofUaHa satisfying the universal property for the deformation. The by the Weyl Weyl group is the the locus locus with with singular singularfibers. fibers. By the universality of have aa classification classification morphism morphism from from V(4) V($) of M^ n , we have to Mg^n for suitable g, n. If 1 = then it is known that the deformation for n. $ = A n _i, then it is known that the deformation of by the the equation of the singularity singularity is given given by equation y2 y2 = xn + aixn~1 + a,2Xn~2++ . ••+ • -f a n ,i.e., i.e.,the thehyperelliptic hyperellipticcurves. curves.Here Here(a1,. (a x ,... ,a n )is isthe theparameter parameter n Thus, if (b1,. is the the parameter of A A's, then a's a's are the of kn/Sn. Thus, (61,..., bn) is parameter of , then the fundamental fundamental symmetric symmetric polynomials polynomials of of b's. 6's. After After compactification compactification of of the fibers, —* Mg fibers, we have V(An) -> Mg with n = = 2g-\-2. 2g+2. The image is the the hyperelliptic locus of M M9, and itit can can be easily checkedthat that the generators of 7Ti(V(.An)) locus of easily checked g, and to the theDehn Dehntwists twistsalong alongthe thefollowing followingcircles. circles. are mapped to . ,

.

This group homomorphism homomorphism is surjective surjective if and only only ifif gg < 2. For For g = 3, consider an E7 we can consider E7 analogue. analogue.

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Theorem 1.3. deformation of Theorem 1.3.The The deformationofofsingularities singularitiesofofE7 E7 type typegives gives aa family family of two pointed pointed curves curves with with genus genus three. By forgetting the two two points, we we have a classification corresponding group classificationmorphism morphismV(E7) V(E7) —+ —>M3. M 3 . Then, Then,thethe corresponding group homomorphism irj(V(E7)) homomorphism ni(V(E —>I'3Tsisisgiven givenby bymapping mappingthe thegenerators generatorsofofthe the 7)) -4 Artin Humphries generators ofT%. Artin group to the Humphries ofF3.

Proof. The Proof. The E7 E7 singularity singularity isisgiven given by bythe theequation equationy(y2 y(y2 — — x3) x3) == 0.0. ItIt is known is known that aa smooth smooth deformation deformation of of this this singularity singularity becomes becomes a twotwopointed genus three. See See for for example example [31], [31], or, consider a small pointed curve curve with with genus or, consider deformation of of the the one one punctured puncturedcusp cuspy2y2— — x3 ==00totoan anelliptic ellipticcurve. curve. This This yields y(y2 — x3+a) =—0,0,which whichisisthe thethree threepoints pointsnormal normal intersection oneintersection of of onepunctured projective line one-punctured elliptic elliptic curve. By deforming deforming punctured projective line and one-punctured curve. By the three points, points, we we obtain obtain two two pointed pointed genus genus three curves. curves. By By forgetting forgetting the punctures, punctures, we wehave have aasmooth smoothfamily familyFF—* —>•V(E7) V(E7) ofofgenus genusthree threecurves. curves. Now the classifying classifying morphism morphism V(E7) V(E7) —* —> M3. We Weshall shallshow show that that Now we have the the ir1 n 1 gives gives the thecanonical canonicalsurjection surjectionA(E7) A(E7—* ) —> F . F3. 3 Since Since FF—* —>V(E7) V(E7) isisaasmooth smoothfamily familyofofgenus genus33curves, curves,by bythe theuniversality universality -÷>•M9, of M9,1 Mg^i — Mg, we wehave havethe thefollowing followingpullback pullbackdiagram: diagram:

F

i V(E7) V(E7)

M9,1

I Mg. . Mg.

This yields a compatible monodromy map tOP(V(E))

I F9

II

where F9 Tg is the genus g mapping class group. Since Since the the lower lower map map isis injective injective (Dehn-Nielsen'sTheorem), Theorem),ititisisenough enoughtoto show showthat that the image (Dehn-Nielsen's image of each each generator by dromy by the themono monodromy

iri(V(E7)) in(V(E ^ Out(iri(E9)) Outfapg)) 7))-4 coincides withprescribed prescribed Dehn twists. coincides with Dehn twists. First, in of singularities, singularities, itit is is known First, in the the miniversal miniversal deformation deformation of known that the codimension one component of the singular codimension one component of singular locus locus corresponds corresponds to oneonepinched Riemann surface, surface, and and the corresponding is a Dehn pinched Riemann corresponding monodromy monodromy is Dehn

Galois group GQ, GQ, Singularity Singularity EE7, and Moduli Moduli M3 M3 Galois group 7, and

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twist. Thus, each generator generator of of the the Artin Artin group twist. Thus, we we may assume assume each group is mapped mapped to a Dehn twist. Moreover, Moreover, this Dehn twist is is not not trivial, trivial, even even ifif we we pass to to the monodromy in the the first first cohomology cohomology [31]. [31]. Next, we we use use Ishida's Ishida's result result [19]: Let C, CC' be two Next, [19]: Let two Dehn Dehn twists in I'9, Tg, g >2,2,along alongthe thecurves curveswith withsame samenotation. notation.We Wechoose choosethe thecurves curvesfrom from their their homotopy classes classesso sothat that the number homotopy number of the intersections intersections is the minimum. minimum. Then, Ishida's result is: is: (i) C and C' C"commute commute ifif and and only only ifif C C flD C' C == 0,0, if and if CC'C = C'CC (ii) C and C' C" satisfy satisfy the braid braid relation relation CC'C C'CC if and only only if point, Cu UCC'" =one point, (iii) C and C' (iii) C satisfy satisfy no no relation, relation, i.e., i.e., generate generate aafree free group, group,otherwise. otherwise. Now we shall shall complete completethe the proof. proof. We put the names Now we names C0, Co, C1, Ci, C2, G2, C3, C3, C4, C5, C6 for for the the generators generators of of A(E?) A(E7) and give the same C5, Ce same names names to to the the closed closed curves in E9, such that the corresponding Dehn twists are the images curves in E^, such the corresponding Dehn the images of of C1's Gi's in F3. T3. C

0— 0—0------ 0— 0— -o0 C,

C

c,

C C

C,5 C

C, Q

Then, up to homotopy there are only homotopy equivalence equivalence and diffeomorphisms, diffeomorphisms, there

finite possibilities possibilitiesofofC\C1ininT,E9: the one one which which does does not not separate separate E E9, finite g: the p , or, the one which separates Eg E 9 into two pieces pieces with genus gi, #1, g
Next, we consider the image of C2. C2. The Next, consider the image of The tubular tubularneighborhood neighborhood of ofC1, C±, C2, respectively, respectively,isisaatube. tube. Since C1 and and C2 C2 intersect intersect at exactly C2, Since C\ exactly one one point, point, the tubular tubular neighborhood neighborhood ofofC1 C\ UU C2 C2 in in E9 E^ isis the the two two tubes tubes glued glued along along aa square neighborhood neighborhood of of the the crossing crossingpoint. point. This is homeomorphic homeomorphic to to aa torus

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minus aa closed closed disk. disk. Then, Then, the whole surfaceisis aa union union of of this this torus torus and and minus whole surface surface, glued glued at at the boundary homeomorphic the complement complement surface, homeomorphic to a circle. circle. If decomposition, we may If we we redraw redraw the the picture along this decomposition, may assume assume C C2 is as as 2 is follows, homeomorphism. follows,up up to to a homeomorphism.

€3

shall consider consider C3. C3. C3 C3 intersects intersectswith with CC2 at one one point, point, and and does We shall does not 2 at intersect C\. Thus, the tubular tubular neighborhood neighborhood ofofC1 C\ UUC2 C2 U U C3 is is intersect with C1. Thus, the isomorphic genus one are isomorphic to to a genus one surface surface with with two two closed closed boundaries. boundaries. There are two two kinds kinds of of possibilities: possibilities: these these two two closed closed boundaries boundaries are are connected connected in in the complement connected. If If not not complementofofthe the tubular tubular neighborhood neighborhoodinin £E9, or not not connected. p , or connected, choose C4, which connected, then we can't choose which intersects intersects at one point in C3 and the two two boundaries are connected has no intersection intersection with with the the others. Thus, the Similarly, in the complement, so we may may assume assume that that C3 is just just as expected. Similarly, we can are like like the the following following picture. picture. we can prove prove that that C;, i = 1,2,3,4,5,6 1,2,3,4, 5,6 are

Now, Co C0 intersects intersects with only C2, C3, C5, C6, then we Now, only C4. C4. If we we cut cut C1, Ci,C2,C3,C5,C6, we obtain aa tube does not intersect with tube with withseveral several disks disks removed. removed. Since Since C0 Co does with these curves, we conclude conclude that that the only only possibility possibility is D0 DQ or its mirror mirror image image Df0. This completes the proof. proof. 0

§2. Tangential Tangential morphisms §2. morphisms will give giveaarough roughintuitive intuitive explanation explanation on on the the notion of tangential First we will zero dimensional dimensional case case tangential tangential base points. morphisms, which includes as a zero points. Let V* = Spec variety over over aa field field kk C C C, C, and let t1, Spec A A be an affine affine variety £1,..., . , t\ tj be regular . , D1 denote denote the zero regular functions functions on on V*. V*. Let Let D1,.. Di,...,Di zero divisor divisor of t1, . , tj, respectively, and assume assume that that these ti,...,ti, respectively, and these are are normal normal crossing. crossing. Put Z := D1x H fl D D2 fl ... fl D1 and V := — D1 Z := D n • •. n D and V := V* D U D U • • • DhThen, Then,a a ... U U D1. 2 t x U D2 2 . .

. .

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tangential morphism morphism ZZ—i —rVVisisaamorphism morphismfrom from ZZtotoV,F obtained , obtainedbyby"in"inc finitesimally "in the direction" specified finitesimally small perturbation" perturbation"ofofZZ'—f -> V* "in specified by

tl,...,tl. We regard Z, Z, F* as as complex complex analytic analytic spaces spaces via via k C C C. C. Then Then Z is is the the We regard locus in in V* V* defined defined by by £1 t1 = t2 locus t2 = • • •==tj ti= =0.0.We Weperturb perturbZZtotoobtain obtain e,e,with Z', which which is is defined defined by tj t\ = =t2 t2 = = • • •==tj t\= = withinfinitesimally infinitesimallysmall small positive real number e. Since positive Since D^s's are normal crossing, Z' Z' is is homeomorphic homeomorphic V* and 7r*oP(Z) 2 T T ^ Z ' ) -> 7r*op(V). Z' —* to Z, Z,and and we we have have Z Z ^ Z' -> V V C V* Less trivial trivial is the the algebraic algebraic structure, which Less which is explained in the next next section. section. p (Speck, In the case that ZZisis aa point point Speck, Spec k,we wehave have aa section section -K\ (Spec fc,Spec Spec k) which tangential base point. -÷ —> 7Tj g(V,Z),Z), whichis isnothing nothingbut buta k-rational a fc-rational tangential base point. we need need only only the characteristic zero case. However, in In §2, we However, the results in 2.1-2.3, using using Galois Galois categories categoriesare are valid valid for positive positive characteristics. characteristics. We 2.1-2.3, treat these these cases cases in in parallel parallel for for future future reference. reference.

2.1. Tangential bebea afield Tangentialmorphisms. morphisms.Let Letk k fieldofofcharacteristic characteristic p >0.0.We We denote by by N N'7 the set of natural numbers coprime with with p. p. Thus, if kk has the denote numbers coprime characteristic zero, zero, then then N = N'. characteristic N'. Let V be a geometrically connected variety over over k. k. We Let be a geometrically connected variety We denote by by Vfet the category finite etale etale covers coversof ofV. V. We the category of finite We mainly mainly deal deal with with the thecase case V divisor V == V* V* ——D, D,where where V* V*isisaavariety varietyand andDDa a divisorofofV*. V*. IfIfpp>>0, 0, finite covers covers of ofV* V*that that are etale outside we consider V ^ , the category category of finite D and tamely tamely ramified ramified at D. D,When Whenititcannot cannotcause causeconfusion, confusion, we we denote denote simply by Viet is a Galois category. For For fiber functors functors F, F, FF'1 : Vf£ simply byVfet. Vfet. Vf Galois category. et is 71 f Vfet —* —>{Finite {FiniteSets}, Sets},we wedenote denoteiri(V; Tr^V^i F, F') , F the ) the setset of of isomorphisms isomorphismsfrom from Vfet 1 f the functor functor F F to to F F'.. This groupoid. If If F F= F This constitutes constitutes a groupoid. F',, this is nothing but the the algebraic algebraic fundamental fundamental group. group. We shall deal with with the thefollowing following case. case.

Definition Let V be Definition 2.1. 2.1. Let beaaZariski Zariski open opensubscheme subscheme of ofaageometrically geometrically connected variety V* V* over overk. k. Let Let U U— = Spec connected variety SpecAA be bean anaffine affine open opensubscheme subscheme of of V*. V*. We assume assume that there there are are regular regular functions functions t1, £1, t2,. t2, -.., on U, J7, such such . ,tjt\ on that (1) 1) give give normal normal crossing crossing divisors divisors (each (each denoted denotedby by (1) U = =0 (1 < ii < I) 0 (1 D{) on U and and (2) (2) VV flf) U is the complement complement of of these these divisors divisors in in U. U. Let Let Z Z be the intersection of Di, i = — 1,2, 1,2,...,/. . , 1. The above . , t1,ti, Z)Z)is iscalled above datum datum (V, (V,V*, V*,U, U,t1,. £1,..., calleda atangential tangentialmorphism morphism fromZtoV,denotedbyr:Z—7V. from Z to V, denoted by r : Z -7 V. (I.e., r : Dx n • • • Pi Dt -r U - (Di U ...uD1)cV) • • • U A ) C V\) This This tangential morphism morphism is is called called k-rational. fc-rational. Let Let L L be bean analgebraic algebraic extension extension field fieldofofA:. k. A A tangential tangential morphism morphism to to V®L V® L is is called called an an Irrational L-rational tangential morphism. morphism. .

. .

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If Z is k, then If is aa point point Spec Specfc, then rr : :ZZ—r —7VVisiscalled calledk-rational k-rationaltangential tangential (base) (base) point. point A tangential morphism an exact morphism induces induces an exact functor functor from from Vfet to to Zfet, and and hence a morphism between fundamental groupoids, hence groupoids, as as follows. follows. of U U along alongZ, Z, i.e. i.e. Spec Spec AA := := Spec lim«_nGN Let U be the completion of where/jz where z = (t1,. ( i i , . . .. ,, it1) i ) i is s the ideal ideal defining defining Z. Z. Let us denote by Z{t) Z{t} := t1} the spectrum of Let denote by := Z{t1,. Z{t\,...,. , t{\ .

.

A{t1

. . . ,

U

t,}

NEW

system of of JV-th N-th roots A. ZZ is where £*' denotes a compatible compatible system roots of U E € A. identified with the closed subscheme of Z{t} defined by t1 = ... = 0. 0. identified with closed subscheme of Z{t} defined by t\ = • • • t1 = t\= = When Z is Spec Speck, of V, V, Z{t} = Spec When fc,by the regularity regularity of Spec k{t1,. &{£i,..., t1}, £j}, which k. is the ring ring of of Puiseux Puiseux series series over over A;. There —* V*. V". There is a natural morphism morphism Z{t} Z{t} ->> V* obtained as asZ{t} Z{t}—÷ -> U U -+ of this map is contained in any Zariski of Z Z The image of Zariski open neighborhood neighborhood of in F*. Let Let W be beaafinite finite etale etale cover cover of V. Then Then we we take take the thenormal normal closure closure of V* with respect to W, and call with respect call it it W*. In In aa neighborhood neighborhood of Z in in V*, the possibly possibly ramified ramified locus locus isist1 t\ • • t1 • t\= =0.0 .We Wepullback pullbackW* W* along Z{t} along Z{t} —*—> V*, and denote the fiber fiber product product by by W*Iz{t}. W*|z{t}- Take Take the the normal normal closure closure of of Z{t} with respect respect to to the the generic generic fiber fiber of the fiber fiber product, denote it Z{t} product, and denote N(W\z{t1,...M})' is finite finite etale over over Z{t1,. Z{£i,... .,fy}. follows ,t1}. This follows t1}). Then this is from Abhyankar's Abhyankar's Lemma for for Z{t1,. Z\t\,... . ,,t{\ —> (IT, U, since since the theramification ramification isis from t1 } —÷ preserved by completion. Then Then we we obtain obtain an an etale etale cover cover of Z by base change, which is denoted by rJ (W). et which is denoted by Tet(W). have constructed constructed a functor functor r^ Tetet : Vjet + >•Ziet. We have Vfet — Zfet. This Thisfunctor functorpreserves preserves es n the fiber product, since the generic fiber of Wjz{t1 not does fiber product, since the generic fiber of W\z{tlt...,ti} ° t change t,} d ° after after taking taking normalization. normalization. By Byuniqueness uniqueness of ofnormalization, normalization, taking taking fiber fiber products commutes with normalization. This This functor functor maps maps direct direct sum sum to to didirect sum and epimorphisms for the the same same reason. reason. Thus, by epimorphisms to epimorphism, epimorphism, for SGA1 [11], [11],Tf r;et SGA1 functor between between Galois Galois categories, categories, which which yields yields et is an exact functor f irj(r) :: irj(Z; F, F')) —* a morphism morphism between between groupoids groupoids TTI(T) iri(Z]F,F —» iri(V; T T I ( Vr(F), ; T ( F ) ,r(F')), T(F')), where F, F' are and r(F) r(F) isisthe where F,F' are any any fiber fiber functors functors of of Zfet, and the fiber fiber functor functor F o Tfet : Viet Vfet—+ —>{Finite {FiniteSets}. Sets}.Now Nowwe weproved: proved: . .

,

.

.

:

:

Proposition 2.1. Proposition 2.1. For aatangential tangentialmorphism morphism Tr : Z —r V,V',we wehave havean an associated exact +> induces exact functor ret r^et :Vf€t Vfet — Zfetwhich , which inducesa homomorphism a homomorphism Zfet, of groupoids iri(r) F, F') —* 7n(r) : :iri(Z; 7n(Z;F,F') ->iri(V;r(F),r(F')). ^{V\ :

:

AssumeZZ = Speck. Assume = Spec A:. In In this this case, case, rr isiscalled called aak-rational k-rational tangential tangential point. point. This yields yields aagroup grouphomomorphism homomorphism ir1(Speck,x) ?ri(Spec k,x) —* —>- iri(V,r(x)), 7ri(V,r(x)),

Galois Galois group group GQ, GQ, Singularity Singularity £7, E7, and and Moduli Moduli M3 M3

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any geometric geometricpoint. point. This This is is a section to the ci —* where xx : :Spec Specfi -> Spec kk is any section to structural morphism (Speck, structural morphismiri(V, TTI(V,r(x)) T(X))—+ -> 7Ti(Spec fc,x). ItIt isiseasy easy to to check check this at at Galois category considered Galois category level. level. This This isis nothing nothing but but a tangential base point considered [7] [15]. [15]. in [7] ir1(Spec fc, k, x) x) on 7Ti(V; iri(V; r(x), This yields an action of G^ Gk = 7Ti(Spec r(x),F) F)for forany anyfiber fiber fimctor left by by ao e€ Gk functor FF from from left G^ : 7'yE 6 iri(V;r(x),F) 7ri(V;r(x),F)i-+ 4 7 0 7Ti(r)(cr~1), and similarly an action from 'y '—* similarly from left on on iri(V;F,r(x)) Ki{V; F, T(X))byby 7 I-» TTI(T)( ?ri(Spec Here, Here, f is is the the k-rational fc-rationalbase base point, point, which which is is obtained obtained by by extending extending the the cocoefficient in the the construction construction of ofTf 1etet. efficient in k, x) acts on 7TI(V;T(X)^T iri(V; r(x),/(X)). r'(x)). IfIf By 'y7 '—* •->> 7ri(r/)(cr)77Ti(r)(cr)~1, iri(Spec 7Ti(Specfc,x) ; r == r', (V, r(x)). This T , then this this isis an an automorphism automorphism of of ir1 KI(V,T(X)). This representation representation Gal(k/k) Gal(k/k)—* —> Aut7ri(F, r(x)) isiswhat whatwe wewant wanttotostudy. study. —

2.2. Homotopy morphisms. LetLet F, G Homotopybetween betweentangential tangential morphisms. F, :GC: —* C -> VV be be two exact functors functors from from aa Galois Galois category category to to another. another. We We call an isomorphic natural transformation from FF tto G aa homotopy fromFF to to G. If the target transformation from oG homotopy from target category V V is {Finite sets}, then F F and and GGare are fiber fiber functors, functors, and and aa homotopy homotopy between fiber functors functors is is nothing nothing but an element between fiber element of of iri(C; 7Ti(C; F, C). G). This This isisan an analogy to pathisisaahomotopy homotopy between between two two points. points. analogy to that aa path Let T,T' r, T':: ZZ —r V be Z{t} — = Let —rV be two two tangential tangential morphisms, morphisms, which which comes comes from Z{t} V*, Z{t1,.. . t1} —* Z{s} = Z{si,. s1} —* respectively. Z{h, ...,*/}-»• V*, Z{s} = Z{su ..., si) -> V*, ,

.

. ,

Definition Definition 2.2. 2.2.An Aninfinitesimal infinitesimal homotopy homotopy from from r to to r'rfisisan anisomorphism isomorphism Z{t} 4 Z{s}, V*,and andmapping mapping Z{t} —> Z{s},commuting commuting with with the the morphisms morphisms totoV*, thethe divisors (ti) to (si) for each i. divisors (ti) (si) for each z. Proposition AnAn infinitesimal homotopy yields a homotopy Proposition2.2. 2.2. infinitesimal homotopy yields a homotopyfrom fromTf*et rjet t0 T to 'f*ef

Proof. Proof. Let Let W W be beaafinite finiteetale etalecover cover of of V. V. The Thenormalizations normalizations on on Z{t} Z{t} and on on Z{s} Z{s} are arecanonically canonically isomorphic, isomorphic, since since they they coincide coincide outside outside the the normal crossing divisor. divisor. Thus, —÷ Thus, we we have a functorial isomorphism Txet(W) —> Tf*et(W), (W), which yields a homotopy. homotopy. 0

This proposition proposition says says that that the This the tangential tangential morphisms morphisms essentially essentially depend only on the choice of the the basis of of normal normal cotangent cotangent bundle of Z in choice of in V*. For example, example, t{ »-> ^(1 —— tj)~1does doesnot notchange changethe themorphism. morphism.This Thisproposition proposition is useful when we we compare compare two two distinct distinct tangential tangential base points useful when points which which are are automorphism of of V. V. mapped each other by an automorphism

ti)'

Makoto Matsumoto

192 192

Another interesting interesting example example isis Q{t} Q{t} -» FQ1OO and andQ{—t} Q{-£}—+ -» Poioo- These homotopic. If we add e2ni^N for two are not infinitesimally infinitesimally homotopic. for all N, iV, then then we we b Qab{_t} have an infinitesimal have infinitesimal homotopy Q°6{*} -> Q* {-t} by t^N -> elri/Nel/N. e^e1^. This expresses the fact fact that aa small 0i to iö expresses the small half half circle circle from from 01 10 yields yields a path, on which GQC*& acts trivially. This This path path realizes realizes the theinfinitesimal infinitesimal homotopy. homotopy. By taking the conjugate conjugate by this path, we we may identify identify the action of GQab Ofl on 06 tangential base points. If we change the base field to If we these two Q ' -rational 6 6 Q" , then for any Q° {C*} with (£ aa root of of unity. We We can regard regard Qab, then this is true for as Spec Spec Qlb homotopically moves moves along infinitesimally small small circle circle along an infinitesimally this as around zero. zero. A A similar similar thing can can be be said said about about Q{t} Q{t} and Q{at}, Q{a£}, with with a a around number. There Thereexists existsaahomotopy homotopyififwe we"restrict" "restrict"totoIJNENQ[ah/N1. rational number.

2.3. Composition ususconsider Composition of oftangential tangentialmorphisms. morphisms.LetLet considerthe thediadiagram

V

ZI

B

f are tangential tangential morphisms which are composable in Here, rT and rTI are in the the followfollowing sense. sense. The vertical inclusions are open open immersions, immersions, and and the horizontal ing inclusions are arrows are are immersions. arrows immersions. There There is is an an open open affine affine subscheme subscheme U of of V**, on which there there are are functions functionst\,..., ti,. . .,ti,si,.. which U, S\,..., s m . S1,. . . ,Sm, (1) (V, (V, F**,U, [/,t1, * i., . . .,, t1, U, si,..., 5 mZ') , Z')gives givesa atangential tangentialmorphism morphismZ'Z1 —r f f V, which denoted by by 'r'r'. by ti = which is symbolically symbolically denoted rr . Thus, Thus, Z' Z is is defined defined by U,[/,and complement 0 in • • •== titi==Sisi== • •=• = sm= = 0 in andV Vfl U C\isUthe is the complementofof . .

ti...tiSi...Sm0lflU. t\.. .tiSi.. .Sm = 0 in U. (2) (V, V*, UHfl V*,ti,...,t V*,tj,. .hZ) . ,tj, Z) gives (2) (V, V*,U gives a tangential tangential morphism morphism Tr : Z Z -r V. (3) Z* is the of U U defined by tit1 = ... ... = the closed closed subscheme subscheme of defined by = t1 ti = 0. 0. 1 (Z, (Z,Z*,Z*,si,..., sm, Z') gives gives a tangential tangential morphism morphism T' r' : Z —rZ.Z. si,. . . ,Sm, Proposition There is is a canonical homotopy Proposition2.3. 2.3. There a canonical homotopybetween betweentwo twofunctors functors 0 T;et,

Vfet rTfet •: V fet -»• Z'fet.

Singularity E?, E7, and Moduli Galois group GQ, Singularity Moduli M3

193 193

Proof. Let W be Proof. be aafinite finite etale etale cover cover of V. rjfet is defined defined by the thefollowing following diagram Tf*et(W) W N(WIz{t}) () -N(W\ *W Z{t})

joj_

Z

Z{t}

•V*

where •0 denotes the fiber product, and B3 denotes: where denotes: to totake takethe thefiber fiberproduct product at the by and take normalization normalization of the base scheme scheme at the generic fiber (denoted (denoted by NO). N()). ItItisiseasy easy to tosee seethat that this thisdouble-square double-square diagram diagram extends extends to to

'

N(r} N(ret(W)) et(W))

to

TV

D

Z*

Z*{t} •Z*{t}

By uniqueness of normalization, we I z{t}) ~^ we have have aa unique unique morphism morphism N(W N(W\z{t}) Z*{t}. N(W\z+{t}) which which is is aapullback pullback along alongZ{t} Z{t}—* ->- Z*{t}. Then, by by pulling pulling N(WIz.{t}) back along of along ZZ—* -» Z{t}, we have r*fet(W). N(rfet(W)) is the the pullback pullback of N(WIz*{t}) restricted to to Z, and N(W\z*{t}) ifif restricted and hence hence on on all all Z*. Z*. Next Next we we consider consider

jo

N(T N(Tfet(W)IZI{t}) /et(W)\z.{t})

D

Z'

Z'{t}

z*

and

(TT%tW

—+N(T JV(Yfet(T'V)Izl{t,8}) fet(W)\Z.{ttS}) D

Z'

Z'{t, s} •Z'{t,s}

Z*{t}

There morphism from from the the upper upper diagram canonical morphism diagram to to the thelower lower There exists exists a canonical one extending the the pullback pullback Z* Z* —> Z*{t} theright rightend. end.This Thisisisbecause because {t} atatthe when restricted locus t / 0 of of Z'{t}, Zf{t}, E3 is just pullback, and when restricted to to the locus just aa pullback, normalization is unique. unique. Now Nowwe we have have aacanonical canonical morphism morphism r?et o° the normalization T wnicn 1 1 fet(W) —÷ -* (TT')*fet(W) which is a pullback pullback of of the the identity identityZ'Z—+ -> Z , hence Z', isomorphism. an isomorphism.

§

Makoto Matsumoto

194

assume A k; totobe 2.4. GAGA. In In this section, we assume beaasubfield subfield ofofCCand anddenote denote by k the algebraic algebraic closure closure in C.

morphism without without completion. completion.In Inthe the construction construction of of r^ Tet 2.4-1° Tangential morphism et

the completion of U U along along ZZ is actually unnecessary. For For a finite in 2.1, the completion of actually unnecessary. finite etale cover W of of V, of ZZ as in the etale cover W V, we obtain aa finite finite etale etale cover cover r^et{W) of the following diagram: diagram: following T;et(W)

-N(W\ N(WIul/N) Ul/N)

Z

ul/N

W* W"

101 zj U

defined by Ui/N is defined Z

0

Djo N,

A1,

where the right t1), A1 rightdown downarrow arrowUU—* —>• A1A1 isisgiven given by by (ti,. (£i,...,fy), A* 4—>•A1A1is is 1 a common multiple of of thethe ramification of of given by ti *->• i—+ sfwith withN N a common multiple ramificationindex index each divisor corresponding to U = 0, 0, and 0Oisisthe theorigin originof of the the affine affine space. space. since by by Abhyankar's Abhyankar's Lemma Lemma N(W) N(W) is This gives gives the same same rjfet, since is etale etale over over andZZ—* -> UI/N factors Ui/N and factors through Z{t}. .

. ,

of a tangential point. For 2.4-2. Analytification Analytification of For an an algebraic algebraic variety variety V V over over an fc, k, we wedenote denote by by VV the thebase baseextension extension totokkCC C,C, and andby by V the thecomplex complex analytic variety corresponding corresponding to V. V. It is is known that the thecategory category of of unramified unramified finite covers covers of Van is canonically category of of finite finite etale etale covers covers of of VV (SGA1 (SGA1[11]). [11]). ically equivalent equivalent to to the category an For a point For point Q of V , there is the the associated associated fiber functor functor FQ :: Vf™t—*—>• another point point R, i?, the homotopical homotopical lift {Finite sets}. sets}. For another lift of of aa path path in Van from yields an isomorphic isomorphic natural transformation transformation from from FQ to FR. FR. from Q to R yields This gives gives Q, R) —* iri(V; FQ, FR),

becomesan an isomorphism isomorphismififwe wecomplete completethe theleft left set set to which becomes where N N runs over the finite index index normal subgroups subgroups of of 7r1op(Van; Q, R)/N, R)/N, where \

Q).

Galois group GQ, GQ, Singularity Singularity £7, E7, and and Moduli Moduli M3 M3 Galois group

195

In the the case case of of aa geometric geometric tangential tangential point, point, there there isis no nocorresponding corresponding So, we we content content ourselves topological point point on Van. So, topological ourselves with giving an isomorisomorphism between between the fiber fiber functor functor coming comingfrom fromaatangential tangentialpoint point and and another functor coming coming from from a geometric geometric point. fiber functor Let U9 : Spec k —7VVbebea ageometric geometrictangential tangentialpoint. point.Thus, Thus,wewehave haveanan immersion FVccF * , open affine subset U C V*, fl U == UU——DDwhere immersion affine subset V*, with V Pi where , t1 G E r(U, D is the union union of of the the zeros zeros of of t1,. ti,.... ,t\ F([/, OJJ), which which are normally normally Let P be by £1 t1 = ... ... = t,£/ = 0. Since crossing. Let be the the point point on on U U defined defined by local coordinate coordinateatat P, F, and . ,£/) t1) gives U = 00 are normal normal crossing, crossing, (t1, (£1,... gives aa local and induces an an analytic analytic isomorphism from an an open open neighborhood neighborhoodMp AlpofofPP in induces isomorphism from an uan to U to an anopen open ball. ball. Fix a point point Q Q in in this neighborhood, neighborhood, and and identify identify by these coordinates. Let A1p,1/N be the inverse Q — (ci,... these coordinates. Let Afp^i/N C U*? be the inverse = (c1,. ,cj) C N Q an image open ball. ball. image of of Np Alp C c U . This is again an open Fix a compatible for N N EG N. Fix compatible system system of of c\/N G fc, 1 < i < 1,/, for N. Put E Ic, QN := (c , . . . , Cj ) G Np^i/Nwe can specify an isomorphic Then, we can specify an isomorphic x e Alp,1/N. QN natural transformation from from r^et to FQ, FQ,as asfollows. follows. Let V, V*, Ubebeasasinin 2.1, 2.1, IfW be be a finite of V, V, and Ui/N, Let V,V*,U finite etale etale cover cover of N(WIul,N) be as in 2.4.1. Then, the finite set FQ(W) Then, FQ(W) is the fiber fiber of of W -> V above Q Q Ge FV,, which on bijects to FQN(N(Wlul,N)) by C/ U. Since N(WIUl/N)an Ui/N -4 bijects FQN(N(W\UI/N)) ^ ^ U. JV(W|^ ^ ^ v 1/N ) —÷ is etale, the inverse image of Alp,i/N is isomorphic to the disjoint union of the inverse image of MP^/N is isomorphic to the disjoint union of copies of and 9*fet == Fp(N(WIuj,N)) copies of Alp,1/N. MpA/N. Now Now both FQN(N(WIul,N)) FQ^(N(W\Ul/N)) ^(^(^1^,^)) biject to to this this set the copies. It is to see see that that this biject set of of the copies. It is not not difficult difficult to this defines defines aa natural transformation U —÷ FQ. natural transformation 0 —> FQ. Now we wehave havefixed fixedaa"path" "path" from from 99 to to FQ. FQ. There Now There is is aa canonical canonical way way to to (1 < < i <<1) choose Q and choose and QN. QN- We We take c2 c» (1 I) to to be be small small enough enough positive positive real real numbers. We numbers. We choose choose the c/ to be all all positive positive real. (This (This choice choice is then f unique.) Let Q' . c) be be another another such such point. point. It is unique.) Q = = (ci,. (c' l 7 ..., c[) is easy to see that defined the natural transformation transformation FQ FQ —>FQI FQ< definedbybya apath pathfrom fromQ QtotoQ'Q'in in the positive real region makes the following diagram commute. following .

.

.

,

. .

.

.

,

. ,

<9**

I

99*

IFQI

f (This is because because the the positive real path between positive real between Q and Q beQ' lifts to that be-

tween Q'N in Ui/N.) Ui/N.) tween QN QN and Q'N Definition 2.3. 2.3.We Weidentify identify these thesefiber fiberfunctors functors FQ FQfor for any any such such Q, Q, and an denote it simply by simply by 9 .

Makoto Matsumoto

196

2.4-3. Compatibility. We We consider considerthe the case case where where kk = = kk and and Z' Z' is isaageget2.4.3. Compatibility. ometric and ometric point point in in2.3. 2.3. Let r : ZZ —i —r V V be be aa tangential tangential morphism, morphism, and 89:: Z' = = Spec Spec k —7ZZbebea ageometric geometrictangential tangentialpoint, point,such suchthat thatwe wehave haveaa composite tangential tangential point pointrO T6: :Spec Speckk —rV,V,asasinin2.3. 2.3.We Wedenote denote by by PPthe the corresponding geometric point of == 0. 0. of V** to t1 ti = = • - •= = ti= =8j si= = = sm ...-=•Sm f assume another another such such 99': that we wehave havea a tangential We assume :Spec Spec k —rZZexists, exists, so so that tangential f morphism r6 rO'. would like liketo to understand understand the the upper morphism morphism . Then, we would morphism in

9an 9!an)—

n{op(Zan', 6an, e'an

- 7r[op{Van; (r6)an,

(TO')™

iri(V;rGi,r92)

iri(Z;Oi,92)

By definition definition of of 0\ and the the composability composability of of r91, rd\, we we have have aacommutative commutative diagram Q

Z

(UnV*)t,1/N (unv*)t,i/N

I

i

Z:,1/N

Z

•u* Ut,i/N t,l/N

V**

p

.V* V

QN

Speckk P = Spec

V*)1/N etc. isis defined Here, (Unn V*)I/N Here, (U defined by by

(UnV*)t,l/N (unv*) til/N—^u UflV D

u

Ut,1/N

At

Z:,i/N

o•

tt

N

A'

Un

z* uSjtA/N 0•

0 s,t S,t

Ss

N

•

At+m

N

At+m.

a is point in Z in chosen P is is defined defined by by sj si ==... • •=• 8m = sm= = t\= = • t1• •=ti 0.—Q0.isQ a point Z chosen to analytify coordinates are are ((Si,.. = (ci,..., (ci,. . ,Cm). analytify #i, whose whose coordinates s i , . . . ,,Sm) sm) = c m ). Since we fix 1/NO fixthe theN-th iV-throots rootsofof c^'s,we wehave haveQN QNononZ Z* 1*N. Let W be aa finite of V. V. Take Let finite etale etale cover cover of Take N ATasasa acommon commonmultiple multipleofofall all ramification indices (W) isisidentified indices over V**. F**. Then, Then, 9* 6*o oT*r*(W) identifiedwith withthe thefiber fiber .

.

Galois group group GQ, GQ, Singularity Singularity £E7, and Moduli Moduli M3 Galois 7 , and

197

of a suitable etale cover of cover of Z*X,N on the point point P. P . This Thisisisidentified identified with with

P in the fiber fiber on P in N{W\Ustl/N)

-> Ua,t,1/N, where

—÷

P=

—* U8ytii/N Speck Spec k ->

denote the the functor functor taking taking the the fiber fiber on on P by defined by by 5s = t£ = 0. 0. We denote is defined Fp, which is identified identified with (y9)*. (T0)*. coordinate is small positive realreal numbers: Q e V**anwhose , whose coordinate is small positive numbers: We fix fix Q

(Si,...

=

( s i , . . . ,Sm, , s m , t t1,. i , . ., . . , , t t1) i ) = ((c1,. c i , . . . .. i,Cm, C m , dd1, i,..

Then we we have

QN QN

.

. , d1). d{).

. ,

-Q

U

U8,t,1/N

For #', 0', we have construction to obtain Fp> ,FQ» ,Fp,,FQ,. The The have a similar similar construction coordinates for 0' is given by si,. , but we take the common t1,.. . , £j coordinates for 0' given by s[,..., s'm, but we the common £1,..., and and dd1,...,d1. i , . . . , d/. What we we want to to see see is is the thefollowing. following. Let Let -y 7 be a topological topological path from from an to Then, which morphism can for the question Q to Q' in Z . which morphism can be substituted the question Q Q' mark in the the diagram diagram below, below, in order to to make make the thediagram diagram commute? commute? .

1' Fp

.

rFnQ

=

'•-

.1' FQI

=

II

1' Fp>

II

-

F

Q

Fr,. FP..

We can regard Fp, FQ, respectively,as as aa functor functor to take a fiber FP,FQ,FP,FQ, respectively, on P,QN,P,QN -> respectively, in in the the etale map N(W\Ustl/N) —+ on P,QN,P,QN E€ U8itii/N, respectively, an Us,t,i/N- Let Let Afp Mp cC U be be the the open open ball ball neighborhood neighborhood of P and , P,, and U3t,1/N. be its its inverse 2.4.2. Then, Then, PP = P P,, QN inverse image imagebybyU8,t,1/N Usjyi/N—* ~^ U, U, as in 2.4.2. QA = 1/N n 1/N 1/N A/N 1/N 1/N / 1/N l/N ',' / ; / l/AT 1/N il/Nx nx (c1 ,.. , 0, . ,. , Cm , 0) and QN = (c1 ,.. , d1 7 ) are all —

(Ci

.

,...,Cm

. . ,

.

, 0 , . . . , 0) and QN = {ci

.

,...,Crk

,d/

.

,...,d

) are

on.A/p,N. Since 1/P,N is simply connected, all the functors Fp, zFQ and on Afp,N- Since A/p,iv is simply connected, all the functors Fp, FQ and are identified. are identified. Similar things hold for 0'. Thus, the problem is reduced to obtaining Similar things hold for 0'. Thus, the problem is reduced to obtaining

F FQ

FQ

^

^

FQ,

Makoto Matsumoto

198

We may may consider consider these these four four functors functors as taking fibers in N(WIutl/N) We N(W\ut 1/N) —f —> on tne Ut,i/N on the points RN, RN, R'N, where where Ut,i/N points RN,R'N,RN,R'N, i/N

—

i/N

RN ( c i , . . • , c m , 0 , . . .,O), . , 0 ) , RN RN == (c1,.. ( c i , . . .,Cm,di . , c m , d / ,., . .. .,d1 . , d / )) RN = (Ci,...,Cm,O,. .

in the (s, (s, t) coordinate, coordinate, and /N N R'jq = (ci, . - . , 4 , 0 , . . . , 0), JJ'N = (ci,..., 4 , dJ ,..., d)' ) R'N —+ in the (s',t) in (s',£) coordinate coordinate (note (note that N(W\ut l/N) — > £^,I/JV isisetale etaleon onthese these points). f an What we need need to do is to make What make a path from from Q to Q lifts Q' in F , which lifts f Ut,i/N to a path from RN to R R'N, in Ut from RN N, such such that the the square square of paths ti/N to

RN R N

IR'N

RN RN

*~R'N R'N

homotopically trivial trivial in U Uj,i/N for any N. is homotopically N. t^/N for . ti)JO < ti Let Ze be the the inverse inverse image image of of {(t1, {(£i,..., ti)\0 t\ < —+ Q, Q' —* given (d1,. . d1). (di,..., d/). Now the the following following is clear. clear. . . ,

.

.

. ,

.

. ,

. ,

Proposition above the£fteimage 'y inin Proposition2.4. 2.4.The The above'y'7'is zs imageof0/7 a n GFan)-...÷ 7r^ p (Z a ^00an, ,0' a r 7-> ^R.tloP(van; ( F ^ ; (r0)an,

{rO1)^.

given by the translation of In sum, sum, the the above above homomorphism homomorphism 'y 7 •-> 7' of 'y7 y' is given from . , £/ tj (the parameters giving tangential from Z to outside, by perturbing t1,. t±,..., morphism morphism rr : Z —rV) V)totosmall smallpositive positive real, real, so sothat thatthe theperturbation perturbationisis homotopically trivial in inZ6. Ze. .

Proof. Take Take the theinverse inverseimage imageofofZZ— - Ze by (Un (U n V*)t,i,N V*)tA/N -> Un U DV*, V*. The The , connected component with ^ ,.. .,t] > 0 contains both RN and if^, connected component > 0 contains and RN, isomorphic totoZZ—— Ze. If we addZZtotothis thiscomponent, component,then thenititbecomes becomes and isomorphic isomorphic to to Ze, and 'y' 7' lifts from RN i?;v to R'N, which which makes the isomorphic lifts to a path from the square of paths homotopically trivial on on (U fln V*) V*)tl,N. homotopically trivial ^/ . 0 t N .

Identifiable tangential points. 2.4-4- Identifiable points. If two two geometric geometric tangential points points have an infinitesimal homotopy, and and the germs of the positive have infinitesimal homotopy, positive real real regions regions

Galois group group GQ, GQ, Singularity Singularity £7, E7, and and Moduli Moduli M3 M3 Galois

199

coincide,then then we wecan canidentify identifythem themasasanalytic analytictangential tangentialpoints, points,too. too. In In coincide, this case, we say say these two tangential points points are are identifiable, identifiable, and identify identify through the the canonical canonical isomorphisms. isomorphisms.

2.5. 2.5. Example. Example. line minus minus three points. For 2.5.1. Projective line For V V == PQ1OO = Spec SpecQ[£, 1/(1—Q[t, 1/(1 1 we take P' as V*. Six Q-rational tangential points are considered in £)], we take P as V*. Six Q-rational tangential points are considered in t)1, l/t, —l/t, which were denoted by by ol, 01, Ooo, fb, 10, Ihara t —— 1, 1,l/t, Ihara [15] [15]t,t,—t, —t,11— —t,£,£ — l/t, which were denoted respectively. The The symmetric symmetric group group S3 S3 acts acts on these points. ool, 00O, loo respectively. points. Precisely saying, we need need canonical canonical infinitesimal infinitesimal homotopies. homotopies. For Precisely saying, For example example the image oo is 11—- l/t l/t = (t —- 1)/t, image ofof11—- t by 00 *+ 00 1)/*, but Q{(t Q{(t — - 1)/t} l)/t} and and Q{t Q{t— —1}1}are arecanonically canonicallyinfinitesimally infinitesimallyhomotopic homotopicbyby(t (t——1)/t l)/t'—+K>(t (t— — 1)22 + (t(t— 1)/(1 -f (t (t —1)) - 1)) = (t —1)— - 1) - (t(t—- I) - I) 3 — , so we may identify identify 1)i(1 + these these two. two. We Wewrite write r01 T01 : Q Q —i —r Pjioo for f° r the ^e corresponding corresponding Q-rational Q-rational tangential base point, and and T5I the geometric tangential point given given by Q <->• r two generators generatorsx,y x,y e€ 7Ti(Poioo' oi)- We have aa path path pp e€ Q. We We have have two We have Q. An element element acEGQ ^iCoiooJ roi^ rr10). io)- An e GQacts actsononppbybyp p'—* «->- rioi^proiia)'1 == r where is an element of free profinite Pf f
0o

0o o0

1

p

1

j>

0

p

1

2.5.2. Let a be 2.5.2. Another example. example. Let be aa rational rational number. number. We We consider consider the the (s, t)-plane minus minus aa divisor: divisor: VV — = Spec k[s, t]t]— (s,i)-plane Specfc[s, —D, D,where where DDisisthe theunion unionofof fiveprime prime divisors: divisors: s = 0, t == 0, the following following five 0, t( == 1,1, st st == a,a,s(l s(l——t)t)=—1.1.

Makoto Matsumoto

200

I

mr's zI Z,

00

There are two two horizontal horizontal Pjioo? namely, namely, tt = 0, 0, tt = = 1, 1, and one one vertivertical Poioo> s = 0. Z2,2 ,Z Z3, For Z1 0. We name name these theseZ1, Zi,Z respectively. For Z\ we we 3 , respectively. choose choose t for for the the tangential tangentialmorphism, morphism,for forZ2 Z 2we wechoose choose11— —t,£,and andfor forZ3 Z3 we choose choose s. s. Then, Then, we we have have three three tangential tangential morphisms morphisms r; : Z; —7 V. —p V. S ?4V lsis, The composition forr rr1, Q{s, composition of tangential tangential morphisms morphisms Q—^PJICXD"^^ i Q{ ^}t} f° i> — t}t} for Q{as, 11 — for r2, r2 ,and andQ{t, Q{£,s}s} for for T3.r 3 . The The asasfor forr2r 2comes comesfrom from that that e ta Z2 is isomorphic take as as as the the coordinate. coordinate. The image of isomorphic to PQICXD if wwe ^e as image of p:'- T01== Q{0—*->Tj0 rio== Q{1—- t}t}ininZ3 Z 3bybyr3r 3isisa ahomotopy homotopyfrom fromQ{s, Q{s,t}t}toto V Q{s, 11 — —t}. t}. Thus, Thus,unless unlessao 7^1,1,we wehave have to tofix fix aapath path(or (orhomotopy) homotopy)from from theother other Q{as,1 — 1— t} patch to patch the geometric tangential point Q{s, 11 ——t}t}totothe t} to Z 3 and Thisisisdone done by by aahomotopy homotopy si/N s^N K-> e-^°^a^N(as)l^N from Z3 and Zi. This from Q{s} to Q{as}, l . Here we choose the argument of log a to be ni if aa a named to Here we choose the argument of loga to be -in if (/ ) is a homotopy Q{s, 1 — £} -> Q{as, 1 — t} compatible with is negative. r 2 a is a homotopy is negative. r2(la) 1 —t} —+ Q{as, 1 —t} compatible with . Let us define Xa(&) £ Z Galois action. We write down Galois action on l Galois1 action. We write down Galois action on a Let us define Xa(°) - i a / a ( 7 - i :: Q{5 } _^ Q { 5 } i is by aid '") = e2WNxa(
(i)51/N

This coincides coincides with

. Now, set r1(x)

Yi V\ := ri(y) X3 := r3(x)

r3(y) :

Y2 := The Yi and and #x3, y3are aresame sameasas that that of The actions of of cn a E€ GQ GQ on Xi, #1,2/1 of 3,2/3 since r1 and r3 since T\ r 3 are are tangential tangential morphism morphism over over Q: Q: a : Xi 1—> x- ,yi

Galois group GQ, GQ, Singularity Singularity £7, E7, and and Moduli M3 Galois group Moduli Ms

201 201

l need the action for ii = 1,3. For For ii = 2, we need f<j(xi,yi)~ yf^fa(xi,yi) y2) for action on on1a Zo. fa(Xz, substitution, wewe have We have have cr cr:: pp'—+ \->pfa(x, pfa(x, y)y)and ando a :ta la »->• laxXa^aKByBy substitution, have .t

I—*

i

\—1 x(o): i X2

and 2/2 -> / a ( X 3 , W 3 ) - ^ a ( * V * ^

Y2

This example shows that that topologically identical objects objects may example shows topologically identical may give give rise to to different Galois actions. actions. Even a scaling in the embedding of PQ1OO influences different Galois influences on the Galois Galois action. action. §3. Galois Galois action on Artin groups groups §3. 3.1. Motivation Let V be aa geometrically geometrically connected variety over over the rational rational number numberfield field Q, and x be a Q-rational Q-rational point point of of V. V. Then, Then,we wehave have Galois Galois representations representations Q, on the fundamental fundamental group group of of V: V: —f Pv, Autiti(V,x), PV,xx : Gal(Q/Q) -> :

(V, X) x) denote denote the profinite completion of the topological where where TTI(V, topological fundamenfundamen-

x). We tal group group 7r*op(y(C),x). We may may use use Q-rational Q-rational tangential tangential base base point point as x. To obtain obtain an an approximation of the the Galois Galoisgroup, group,itit isis aa natural idea to To approximation of consider all V, and (V, x), x), one for for each and consider consider systems systemsof ofelements elementsof ofAutif1 Aut7fi(V, V, which commutes with with any anyalgebraic algebraichomomorphism homomorphismVV—+ —>WWdefined definedover over Q. QWe may may start start with We with some some restricted restricted families families of of varieties varieties and and restricted restricted families of ofmorphisms. morphisms. Grothendieck-Teichmiiller Grothendieck-Teichmüller group defined families defined by Drinfel'd Drinfel'd [9] the one and M0,5, and some natural morphisms [9] isis the one obtained obtained from from M0,4 M0,4 and M0,5, and some natural morphisms (see (see Ihara Ihara [16]). [16]). with all It would It would be be also also natural natural to to consider consider the the families families of of Mg^n with all g, g, n. n. One open open problem is whether whether aa new One problem is new relation relation occurs occurs or or not not ifif we we consider consider not only = 0. For gg = = 2, to prove that the 0. For gg > > 0, 0, not only gg = 2, itit is is easy easy to prove that the GrothendieckGrothendieckTeichmüller group acts on M2, and hence no new relations will be be obtained. obtained. Teichmiiller group acts on M2, and hence no new relations will So, the first nontrivial example is g So, the first nontrivial example is g = —3.3. The Theauthor authorknows knowsnothing nothingon onthis. this. The The author author also also suggests suggests the the possibility possibility to to consider consider the the family family of of V($) with root with root systems systems 1, <$,with with some some (still (still ambiguous) ambiguous) suitable suitable morphisms. morphisms. One of the reasons is that these varieties have a moduli interpretation interpretation of of One of the reasons is that these varieties have a moduli the deformation spaces of singularities with finite finite group group action action [31]. [31].

Makoto Matsumoto

202

This idea comes from aa trial to find comes from find aa geometric geometric meaning meaning of an interestinteresting research ing research by P. P. Lochak Lochak and and L. L.Schneps Schneps [20]. [20]. There, There, they they treated the the Bra, n> 3, with Brn the braid group of n-strands families , n > 3, with Br the braid group of n-strands families of groups Arn, Br n n consisting of of braids braids that that connects the first and Arn the subgroup subgroup of Brn consisting connects the first point with the first first point, i.e., i.e., the the inverse inverse image of first of the stabilizer stabilizer of of the the first -> Sn. Their main is: if we consider consider element the morphism morphismBrn Brn—* main result is: element by the Arn, Br rc > 3 and suitable group homomorphisms, then we can characterBrn n n> 3 and suitable group homomorphisms, then we can characterize ize Grothendieck-Teichmuller Grothendieck-Teichmüllergroup groupas as the the automorphism automorphism group of a tower of 3,44 suffices.) suffices.) of these these groups. groups. (In (In fact, fact, nn — = 3,

The author just notes notes that that this thissomewhat somewhat strange strange group group Arn is the funfundamental group of V(Bn) V(Bn) = V(Cn). The author does not see what are the the corresponding algebraic algebraic homomorphisms homomorphisms to to those group corresponding group homomorphisms homomorphisms considered in in [20]. [20]. We shall write down in terms We shall down the Galois Galois action action on on 7tiV($), in terms of of that on on Pj l o o for $ = An, Bn, Cn, Dn or E7. £ 7 . The The author author did did not not study study E6, £ 6 , E8, £8, case was was treated treated in [17] and J5, B, C, D G2- Since Since An case [17] and D basically basically follow follow F4 and G2. F4 from this, we first the most most nontrivial nontrivial case case E7. £7. from first treat the

3.2. 3.2. Tangential Tangential base base point point on onV(E7). V(E7). 3.2.1. 3.2.1. E7 £7 root root system. system. An An E7 £7 root root system system isisconstructed constructed asasfollows. follows. ConCone m i ^ •> sider seven vectors e2 — 63, e$ — e4i e4 — e^, e$ — e§, e& + 67, — -^ X^=i mA8, e i , ... .. ,, e8 eg denote vectors. Let Let UUbe bethe theseven sevendimensional dimensional where e1,. denote the the unit vectors. 8 spanned these seven seven vectors vectors are are funfunsubspace of A A8 spanned by by these these vectors. vectors. Then, these £7 root system. Let Let U(E7) ^(£7) denote denote UUminus minus the thereflection reflection damental roots of E7 x e s( i + ee8) s) of hyperplanes. If If we we use use the the coordinate coordinate Yll=2xiei + x8(ei hyperplanes. of U, U, then then the corresponding corresponding Weyl Weyl Chamber Chamber C isis —x6 — x6 <<x7x7 <<x6x6 <<x5x5 << • •<• < x2, x8 < — —\{x2 + — X7)By Bya ageneral generalresult resultofofBrieskorn Brieskorn[6], [6],the thefollowing following x8 ... -++x7) are the generators of E7 £7 Artin Art in group. group.

000000

0o

0

x7

0

x6

0

x,

0

x4

0 x,

0 x2

(i=2,3 6) (i=2,3,-.,6)

x, - (x 2 +x,+...+x 7 )

Galois group GQ, GQ, Singularity Singularity EE7, and Moduli Moduli M3 M3 Galois group 7, and

203 203

of the the configuration of the the points points give give aa path In this figure, figure, the moves moves of configuration of U(E7)(C) to TiC, T^C, where where Ti is an element element of of the the Weyl Weyl group group in U(E7) (C) from from C C to T{. After taking taking the quotient quotient by by the the Weyl Weyl group, group, these these to Ti. corresponding to with itit Dynkin Dynkin diagram diagram as as give desired elements elements of 7rJop(V(£V), C), with give the the desired follows: tcf

0— O—0-——— 0— 0—-o 0 t X

'r X

22

'rX 4

33

r X 5 5

X

T 77

8

3.2.2. Blow Blow up. up. Let A 7 1 denote seven seven dimensional dimensional affine affine space with coordicoordian WeWe shall define nates t2,. nates t2, •. • ,• t8, 5^85and d A7a;that thatwith withcoordinates coordinatesx2,. x2, . •. •, x8. • >#8shall define 7 7 A ^ —+ —> A X , which whichinduces induces .

U(E7)r

s.A7t=V*

II

U(E7) U(E 7)

c

that the upper morphism Q-rational tangential point (U(E7), Y so that morphism gives aa Q-rational (U(E7), V*, Y, tt2,..., tg, SpecQ),for fora asuitable suitableopen openaffine affine Y YC C V*. 2,..., t8,SpecQ),

A7t —+ —>• A7Xisisdefined definedasasfollows: follows:

x2 =

t2

x3 = t2t3

x4 = x5 = x6 = x7 =

t2t3t4

t2t3t4t5

t2t3t4t5t6 t2t3t4t5t6t7 1

+ ^2^3^4 t2t3t4 4+ t2t3t4t5 + t2t3t4t5t6 X8 == -—- ( £(t2 2 ++ tt2t3 M 3 M 5 42^3 + + t2t3t4t5t6t7 + 2t2t3t4t5t6(t7 + 1)t8). 4- t2t3t4t5t6t7 + 2t2t3tAt5t6(t7 + l)t 8 ). These parameters are chosen so that on U(E7) we have the following inverse These parameters are chosen so that on U(E7) we have the following inverse

Makoto Matsumoto Makoto

204 204

map: map:

t2 == x2 *2 X2 tta x3/x2 3 = x3/x2 t4 = x4/x3 £4 x4/x3 t5 = x5/x4 t6 = tt7 = 7 =

x7/x6 X7/XQ

t8 ts = ~(x2 + x3 x3 + • •.++ x7 x7 + 2x8)/2(x6 + x7). The removed divisors are Xi±Xj

=0

(2
7

V^ ±Xf -f 2xs number of of minus minus isis even.) even.) 2x8 = 0 (the number i=2

are: in A7 x. The corresponding corresponding divisors divisors in A7* are:

_
7

] T ± t 2 • • 'U = ^ t i=2

2

--U + 2*2 • -t6{l + t7)ts,

2=2

the lower lower identity identity is is even. even. We denote denote by D the where the number of minus in the set of the prime divisors divisors contained in the the above-defined above-defined locus in A7*. So, D contains divisors t{ = 0, (2 < i < 8, 8, i ^ 7). After removing removing these, contains the divisors these, there (0,..., 0).This Thisisiseasy easytotosee seefor forthe the are no prime divisors divisors on (t2, (£2, .• • •,t8)^s)==(0,. . 0). former type of divisors. For the latter, let k be the minimum such that the former type of divisors. For the latter, let minimum such that the minus sign then minus sign isis chosen chosenininthe the left left side side of ofthe the second secondequation. equation. IfIf kk < 5, then —t22 - - • tk ±± t2 t2 • • • tk+i ±" ±=•t2•• t2 holds. Both sides cansides can —1 • • = ttk • • • tk +1 • £fc+i H holds. Both + 2 2 t2 = 0). Thus we be divided by the first term term (since (since we we already removed U 0). we get t8 with —2 == tk+1 tk+i xx f(t), f(t), where wheref(t) f(t) isisa apolynomial polynomialinintk+2,. £fc+2, • - , -,ts withconstant constant term term 2 or 0. These These do do not not lie lie at at the the origin. origin. IfIf Ick = = 6, 6, then automatically automatically the minus corresponding • t7.Then, Then,thethe correspondingdivisor divisorafter after minus sign sign isis chosen chosen for for t2 • •. t7. divided by t2. is —1— t7 = 1 +2(1 i.e., t2 ••• t6 -l-t7 1 +t7 +t7 + 2(1+t7)t8, +t7)t8, i.e., (1+t7)(1 (l+* 7 )(l +t5) + *8)== 0.0. If k does not exist, then all all signs signs are plus, and and we we have (1 + t7)t8 t7)t$ = 0. Since 8) are normal Since t2 t{ = — 0 (2 < i < 8) normal crossing, crossing, we defined defined a tangential . ,t8,SpecQ) morphism SpecQ withY SpecQ—i -r U(E7) by by (U(E7),V*,Y,t2,. (U(E7), V*,Y,t 2 ,...,t ,SpecQ) with Y the the 8 complement 7}. complement in V* F* of of D D —{t2 - {U = 012 0|2 < i < 8, i^ 7}. Note that t7 = 00 is not aa removed removed divisor, and and thus thuswe weneed need some some modifimodifit7 = cation for the definition of tangential morphism. this is easy: easy: just definition of morphism. However, However, this do not take but simply take t? , but simply tt7 definition of the associated associated functor functor 7 in the definition . . ,

. ,

. .

. .

.

i

.

Galois group GQ, GQ, Singularity Singularity £7, E7, and and Moduli Moduli M3 M3 Galois group

205

between the etale etale site site in in 2.1. 2.1. Then Then itit is is easy easyto tosee seethat that (£1,..., (t,,. . . ,t6, between the t6, ct7, £g) for same tangential morphism. We denote this tangential any c G Q gives the morphism. denote tangential E point by 0, 0 , as asorigin. origin. Now we we have have seven seven tangential morphisms Now morphisms fi{ (2 (2 < i < 8), which which correcorrespond to tg). It is not difficult difficult to show that this this gives gives spond to (£1,..., (t,,. . ti-i, £i+i,..., ta). a tangential morphism morphism from from A' A1— —{0, {0,±1} ±1}totoU(E7) U(E7)for forpj, //»,1 1 ... ,^8 constant tk+, x /(£), 1(t), where term . , 0, —1,0,.. —1,0,.. ..,0), 0), but but not term 2 or or 0. 0. These These may may pass pass through through (0,0, (0,0,.. .,0, not through (0,0, . , 0, 1,0,. . .,0).) 0).) Thus, through (0,0,.. .,0,1,0,.. Thus, we can define define a tangential tangential point point (£2, • • •, tj-u tj—- 1, fy+i,..., ts)ononU(E7). U(E7). Fort7, =0(1< For t7, all the divisors on (0,0,..., 0, -1,0) arearet7+1 t7 + 1 ==0, 0, U = 0 (1 < alithedivisorson (0,0,...,0,—1,0) Again aa trouble may lie in —2 —2 = i <8,i 8, i ^ 7). 7). Again trouble may lie in = tk+, £fc+i xx 1(t), /(£), but but this this divisor divisor exists for for k k < 5. 5. Thus Thus we . , te,t t6, t7 1, t8). exists wehave havea atangential tangentialpoint point(t2,. (t 2 ,..., + 1, £s)7 . . ,

.

.

.

. .

. .

. .

,

. ,

i

.

3.3. Pj l o o and P j ± l o o . consider a double cover We consider cover

A' —{0,±1}--+A' 4 A1—{0,1} -{0,1} given by t »->• t2. t2. Then, Then, TTJ^A 1 ——{0,{0,±1}, ±1},0i)01)hashas three generators shown three generators as as shown below. 1

w

U

V

2 — It is easy easy to to see seethat that u = xx2, , v = y, w = x lyx in the inclusion ^ 1 {0,±1},01) {0,±l},01) -> Trf^A -— {O,1},0i). {0, l},01). Let p denote denote the path path from from 01 01 to to 10 iO

Makoto Matsumoto

206

1) interval the(0, (0,1) intervalofofA1 A1— —{0, {0,±1}. ±1}.An Anelement elemento aEGGQ GQacts actsononthis thispath path on the 1 pga(u, v, w). w). Here, 1}), we by by pp'—+ ^ pgaiu, Here, gg(u, <7a(u, v, v, w) is a proword. proword. In Inir,(A' TTI(A— —{0, {0,1}), we 2 1 f o r an tw0 have ga(u,v1w) = = fff{x,y). Thus, ^ ( f , ^ " ^ ) = fA^v) for any Y two elements £, r/ij in any any profinite group. group. define elements elements of groupoids groupoids in the We also define the figure figure below. below.

-1 -1

P

'

it/

P

00

We notation. For an odd We use the following following notation. odd number number x, we we denote by by rX rx the following following path. 0 0o

counterclockwise. By passing Here, Here, the point goes goes around (x— (x~ 1)/2 l)/2 times counterclockwise. to the theprofinite profinite completion, completion, we wedefine define rX rx for x%E£ Zx. Zx.

3.4. 3.4. Galois Galois action on each generator. generator. 3.4.1. Ti rj (i = 2,3,4,5). Let i be a number 3.4-1number 2 << i <5. < 5.Then, Then,roughly, roughly, r; is realized by rp in by moving moving ^ + i in the the figure figure below, i.e., rp in the theprevious previous section. section. (See 3.2.1 for for a global . global picture. If U+i moves moves as below, below, then £i+i,... ,^7 (See and Xi+\ is reversed move proportionally, proportionally, and the the ordering ordering of the points xi and move the other other orderings orderings are are stable.) stable.) and the . .

o 0 o

1

>-

t3,. • • • >£t?U+2,. • ,•t8) Let /ii+i,t be the tangential tangential morphism morphism given given by (t2, (£2?^3> • ?^s) 1 from A' — ±1} to U(E7). Then, /ii+i,t(rp) is a topological path on {0, from A — {0, ± 1 } to U(E7). Then, /ii + i^(rp) is a topological path on U(E7), with tjt3 = e for jj / i+1, i+ 1, and U(E-j), and tHi ^ + imoves moves along along rp. Since Since fe is is positive small moves around around the small positive real small real real number, number, x7, £7,x6, # 6 , .. . . ,,x;_|_2 Xi+2 moves number and and don't change their order. to the right to number change their order. Only Only Xi+i moves moves to to x2, X{, .

. ,

.

. .

and other other Xj x3 (j (j <
Galois group group GQ, GQ, Singularity Singularity EE7, and Moduli Moduli M3 M3 Galois 7, and

207 207

If we contract contract the the fundamental fundamental Weyl Weyl chaniber chamber (since (since it isis simply simply conconnected), then image of of this path is is surely surely T;+I after U(E7) U(E7) divided divided by by nected), then the image Weyl group. However, to correct Galois Galois action, action, we we need need to to the Weyl group. However, to obtain the correct connect the connect the end end (tangential) (tangential) point point of of this this path path to the tangential base point TiO. (Remember (Remember that Ti is the element of the Weyl Weyl group group corresponding corresponding to ri.) tO Ti.) 4 for T. Ti exchanges exchanges Xi and £»+!. Let us assume assume 22 < i < 4 fora awhile. while.Then Then the coordinates . s 8 , with s2 ^ 3 ,..., s2 = t2, t2, s3 S3 = t3,. £3,..., Si = coordinates giving giving T10 TiO isiss2, s 2 ,53,. = £i+i*t+2jSi+3 = U+3,--<>S8= =t8.^8-(This (Thiscan can tiU+i,Si+i 5i+1 = l/ti+i,si+2 be calculated calculated by by exchanging exchanging Xi+i and Xi in the the defining defining identities identities of t's. £'s. For this case case soon soon later.) later.) We For i = 5, 5, sg 7^ t8, t$, and we we shall consider consider this We denote denote ±1} totoU(E7) by £*i+i,s the tangential tangential morphism morphism from from A' A1— —{0, {0,±l} U(E?)given givenby by s TheThe endend tangential point of /ij+,,t(rp) is given (82, (52,S3 S3,..., Si, Si_|_27 • • •s8). 1 s)tangential point of /x^+i^(rp) is given by (t2, ^3, •. • •, U, *i+—i 1, - 1, ti+2, • t8). • •, is). (t2, t3,. t2(1 + + (ti+1 -— 1)), 1 —- s i + i = Since Si -= ^(1 - {ti+1 —- 1)/(1 1)/(1 + {ti+1 —- 1)), 1)), s i + 2 == this tangential point ^ + 2 (1 + H- (£»+!—- 1)), 1)),8i+3 5 i + 3 == ^ + 3 , . . . , 5 8 == t8, t8, this tangential pointisisidenidenwe move from tifiable tifiable with (82,53,. (s 2 , s 3 ,. • .•,'Si, Si,11—— Si+i, Si+2> • ? Now $8)- Now we move 5i 5i+2, . • •se). + i from tangential point given byby (52, s3,. . se). 11— —i etoto6,e,then thenwe wereturn returntotothethe tangential point given (52,53,..., s 8 ). Thus, the path from the tangential point (ti) to (si) is obtained by the Thus, the r» from the tangential point (ti) (si) is obtained the composition of the following three paths: U+\: e to 11— composition of following three — e(on (onthe theinterval interval :1 ——c etoto6 e(on (0,1)), 11 —6 to 1+6 —etol + e (rotate (rotatecounterclockwise), counterclockwise), Si+i:l (onthe theinterval interval (0,1)). Here, the end point of the first path is identified with the starting (0,1)). Here, path is identified starting point of the last path point path via viainfinitesimal infinitesimal homotopy. homotopy. Then, n = = ^+ M (p~ 1 )/i;+i,t(r)/^+ M (p). .

,

.

. .

.

. ,

. ,

.

,

. .

,

.

.

.

,

. .

,

. ,

Thus, the image image of this element element by by a eG GQ GQ isis /ii+i,a(&r(ifc, v, w)~1p~1)(jLi+i^t(rx{a))iii+1^t(pg(7(u,

v, w)). w)).

We define This path connects define & by & := fj,i+ij(x). This connects the the fundamental fundamental Weyl chamber chamber to to another another chamber, chamber, and and can be Weyl be considered considered as an an element element of 7n(F(£ ir,(V(Er)), where V(E7) = U(E7)/W(E7). )), where V(E ) U(E )/W(E ). of 7 7 r 7 One can show show that

=

=

A difficulty in the the proof proof is is in the jump along chambers. The element difficulty in along Weyl chambers. element £i in the Artin Artin group group yields yields an an element element of of the the Weyl Weyl group, group, denoted denoted by by JQ, which which is given given explicitly explicitly by

(x,,. . (xi,...,

.

,

x8) H^ (x (x2,. . ±x7, x$) Xi, - x i + i , -Xi + 2 , •. •,, -—x6, x 6 , ±x x$) 2 ,..., 7 , x8) .

. .

Makoto Matsumoto

208 208

Here, Here, ± of x7 x-j is chosen chosen so sothat that the number of minus minus isis even. even. The The tangential Here t7*7== 00 point XiO is given by (*1? t2, ...,*», -ti+i,ti+2, • •.,±t7,t8). •, ±£7, *s)- Here does not not ramify, ramify, so so ±± does the last identity i^rf^i — does does not affect. affect. To show show the we have have to to show show that that the move Hi+ij(w), for example, example, we move of £;+i in the next next /11+i,t(W), figure figure .

--l1

0 0

0

identifying by X» the Weyl coincide coincide with withT12, rf, after identifying Weyl chamber chamber containing containing XiO with that containing containing 0. O.But, But,explicitly explicitlyX2 Xi exchanges exchanges vv and and the the above above path, and and hence hence both both give give rf. Since we have Since rf (
with both & and r?. Since fa(x,y) y) is in the Now, &+1 commutes commutes with the comcomNow, T mutator, we we have have fa{Z>i,T?) = f<j{r)i,r2), i)i where where m = f i f ^ . This This path path along xx at the given by moving r/f is is given moving £j+i along the same same time time moving moving ti+2 along movesand and other other Xj x3 are are fixed. x1. Thus oniy only a:i+i moves fixed. We We have have aa formula formula x'. ThisThis cancan be proved by chasing r]i = Ti+lTi+2 Ti+iTi+2 • • T4T5(T6T7)T5T4• • T4T5(T6T7)T5T4 • • • Ti+2n+i. be proved by chasing the Weyl Weyl chambers. chambers. we need need a small modification, since the the s-coordinate For ii = 5, 5, we For modification, since s-coordinate is then 1 k 6 = t^" , 57 = t6t7, but

—(x2 2x8)/2(x5 x7) S8 = -{x + ..• • x7 • x+7 42x 8 )/2(x+ aj7)= =(x6(x+6 4- x7)t8/(x5 2 45 +

=

t6(1

+ t7)t8/(1

-fx7)

+ t6t7).

Here we need need to check check the existence existence of of an an infinitesimal infinitesimal homotopy homotopy between between s (t2,.. 87,«7»88). only difference (£2, • • ,•,t5,h, t6t6 —- 1,1,t7, £7,t8) ts)and and(82,. (s 2 ,.• . •,85,1 •, «5» —1 ~ 56? s)-The The only difference s8, and is sg, .

S8 58

1))t7) = t8(1 + (t6 t 8 (l + t7)(1 «7)(1+4-(t6 (*6— - 1))/(1 1))/(1 + (1 (14(*6— " 1))*7)

shows the the infinitesimal homotopy. The remaining arguments follow shows infinitesimal homotopy. follow in the same way. way. Thus, we we have have ci:

where % rj == Tri+lTi+2 for 2 < i < 5, 5, where • • •T4T5T6T7T5T4•• T4T5T6T7T5T4 • • • Ti+2Ti+i i+iTi+2

Galois group GQ, Singularity Singularity E7, Galois 2?7, and Moduli Moduli M3

209

3.4.2. 3.4-2. r6. TQ. T6 TQ isisthe theimage imageof of rp rp by by /x7>t,similarly similarly to to the theabove. above.T6 T 6 exchanges exchanges 5 x6 . . ,555 — = XQ and andx7 X7and andmaps mapst-coordinates ^-coordinatestotos-coordinates s-coordinates by by §2== t2, ^2?---? t8. Now the situation is completely analogous £ 5 ,sS6 £ 6 £ 7 ,s = lit7, l / £ 7 , s 8 = ^8- Now the situation is completely analogous t5, s77 = 6 = t6t7, to the the previous previous section, section, and and we we have have .

~-i>

=

T6 =

and cr

,9(u) , p7,3(v),

: r6

,8(w))

w 1L7,t(V), P7,t(W)). /JL 7yt(v), 7,t( ))-

A difference from the the previous previous section section isis that that we don't have for difference from have aa good good £»+i for iz = 6. 6. Thus we we need We have u = = 11 in iri(A' 7Ti(A1—{±1}). —{±1}). need aa direct direct computation. computation. We However, fi7^{w) 1i7,t(w)requires requiressome somecalculation. calculation.This Thisisisthe the path /JL7J(V) = 1L7,t(V) = TQ. However, in U(E7) U(E7)shown shown in inthe the figure figure below. below.

rj

-(x + ... +x - x6 - x )/2 V

5

2

2

5

6

77

-x,

-(x + x3+ ... + x?) 12 2

7

3

This This is actually actually (r7r8r7)2. {T7T$T7)2. This can canbe bechecked checked by byaastraightforward straightforward chasing of the the Weyl Weyl Chambers Chamberswhere wherethis thispath path passes passes over. over. It It is left chasing of left to to the the reader. with both both rr7 and r8, by the braid reader. Note Note that that (r7r8r7)2 (T 7 T8T 7 ) 2 commute commute with and rs, braid 7 relations. Thus, we have r6

(r7rsr7)2).

i—*

Now r6 commutes Now TQ commutes with with both both r 2 and and(r7i-8r7)2, (T 7 T8T 7 ) 2 , hence hence ga cancels cancels out. out. Thus Thus we have have T6

3.4.3. isthe the most most complicated complicated element. element. If 3.4-3. r7 T7 and r8. r 8 . rT7 If we move move tt7 7 is 7 along

r'p' r'p1 (see (see 3.3), 3.3), then thenxi's x^'smove moveasasfollows. follows. -(x + ... +x+x 5 - xxx ft- x)/2 —x6

x7

-(x2 + x + ... + x ) /2 v

2

3

3

7 J

i

this r7, x8 to the To make make this r 7 , we we have have to move move x$ theleft. left. This can be be done done as This can follows. Let s be the image of t by T7. T., maps —x77 and and x7X7 F—* follows. Let the image of t by T 7 . T 7 maps x6 XQ \-t —x \-> —x6. —XQ.

Makoto Matsumoto

210

Thus we have have ss2 —t6t7, Thus t2,...,s.5 , = £t5,5 ,sS66 = = —x7/x5 -x7/x5 = -t sS77 = = X6/X7 x6/x7 = 2 = t2,. 6t7, —X7+2X8)/(—X6—X7) = —(t8+i). l/t7, S8 s$ = - 11/2 / 2 • (x2 + • • • + x—x6 x x + 2xg)/(-x xy) = - ( t 8 + 1). lIt7, 5 6 7 6 .

Now, rrr7 is Now, is realized realized by rT7 = Ii%i8(<x)v>7,8'(p''~1)v>7it(r'pl). 7 =

morphism coming from (82, s3,. . —(s8+1)), Here, fj,7jS>is isthe thetangential tangential morphism coming from (52, S3,..., s&, ~( the following and a is the following path: . ,

/o

^

o

-1

^

0

\

1 That is, x~ x1p'r'' TheThe rightright mostmost uz7,t(r'p') moves x2,.x2, .-.,x8 as as That is, p~1r/in ~1 3.3. in 3.3. H>7,t{r'p') moves • ,xs .

f 1 3.4.2, ^7 /17,s'(p'') in the figure in 3.4.2, changes the theratio, ratio, then thenp8,8(a) /xg)S(a) unwind unwind x8 xs iS'(p ~ ) changes to get r7. r7. For For these these three three paths, paths, the theend endtangential tangential point point isisidentifiable, identifiable, via an infinitesimal homotopy,with withthe thestart start tangential tangential point point of of the the next next path. infinitesimal homotopy, The end of the first first path isis (t2, (t2,..., —(£7 ++ 1), of the the The . , ^6, t6, —(t7 1),t$), t8), and and the start of 5 second (s2,..., , S6, sg, s7 57 ++1,1,—(88 ~( 8 + !))• These are are identifiable. identifiable. The second path isis (82, 1)). These next point is is (s2,. (s2, ...,57, — (s$-\-1)),which which isiscompatible compatible with with both both fi7^siand and . —(88+1)), . .

. .

.

. ,

/L8,s. M8,s-

Now Now the the Galois Galois action action isisdescribed describedasasfollows: follows:

o(r7) =

(cr(p''))1L7,t(cr(r'p')).

We shall shall calculate calculate these these three terms from We from the the right. right. Now Now

= where the exchange of w w and v comes from the relation between between p and p'. p'. We have P7,t(V) 1*7,t(v)== T l /17,t(W) = M7,t(p'-VV) /17,t(P'r2P) = M7,tM = (T7r8r7)2. (r 7 r 8 r 7 ) 2 .

Then

7,t(rpp 1r1 r / xW-V (l,«;,«)) w, v)) IL7,t(0(rp')) = = IM7,t(rVp'»7,tWr'p')) ff
7 S 7

The The middle middle term term is is 1L7,s'(9a(1,VJ,tL)1P'') 1*7,8* = 1L8,8(cr)' .

w,

Now,^we wehave have

= = (r7rsr7)2.

6

Galois group GQ, GQ, Singularity Singularity EE7, and Moduli Moduli M3 M3 Galois group 7, and

211 211

and the The former The former is proved proved using using the commutativity commutativity of of r6 r 6 with with T7, T7,rr8, 8, and latter is is proved proved by drawing drawing a picture. Then Then we we have have =

(T7TST7)2,

The first term term isis obtained obtained asasfollows. follows. (T [Ou J=— X

o(a)

J (T \

p

iP

= x_

P

)

x2)x

.

Then we have

= and

= r.

Thus, Thus,

r).

= Gathering all together, we we have have —X(u)+l

T7 i—+r8

(r8—1T72 T8,

2 7ä )

(r8—1 r72 T8)

—x(o)+I 2

(a)

x

(r7r8r7)2, (r7r8r7)2,

Thus we get r7

—x(o)+l

(r8—1 2

r82 ) (r8—1r72 r8)

2

(r7r8r7)2 ,

(r7r8T7)2,

r6,rr7 are commutative, cancels out, out, and we commutative, go cancels we have the Since (r7T8T7)2, (^rgry) 2 , rg, 7 are following result: following T7

—X(c)+1

T8

2 (r8—1 T7 r8, r82 ) (r8—1r72 r8)

2

r7 (r7r8r7) X

—1

3 simplify3 To simplify this complicated expression, we we need 2-cycle 2-cycle and 3-cycle 3-cycle defindefining relations of of Grothendieck-Teichmiiller Grothendieck-Teichmiiller group group[9] [9][15] [15] [16]. y, zz be be elements elements of of aa profinite profinite group group with with xyz xyz = Let x, y, = 1. 1. Then, (I): (I)if<,(x,y)f
version of ofthis this manuscript, manuscript, the the author author stopped stopped the calculation In the previous previous version calculation here, and the above above expression expression was the final final form. form. The Thefollowing following simplification simplification was was given given by by who got got her insight from the L. Schneps, Schneps, who the comparison comparison with with Nakamura's Nakamura'sresult result[271. [27].

Makoto Matsumoto

212

m m (II):f.7(z,x)zmf.7(y, z)ymf.7(x,y)xm = 11 (II):f U(x,y)xm = a(z,x)z f(r(y,z)y x 1—1 Z. Set temporarily only only in in this this paragraph paragraph xx = = T£, where m m = ^~ gE Z. hold, where = T^T^2T^X and x = r^1yr7 hold. hold. Put Then z = {xy)~x = Put y = T^XT7TS. Then and 2 (rrr8rr)2. This is in the center of the group generated by r7 and c = (T7T8T7) . in the center of the group generated by r 7 and'r8, T$, holds. and zc = r | holds. Now, of r7 T7 becomes becomes Now, the the image of

x-mfa(y,x)y-mT7cm x

y

= f.7(z,x)z rn fcy(y,z)r7c rn (z, X)Zm

=

z)cm

= f.7(z,

=

z)

(')

= f.7(x,z) —'r7x(c) =

The last step used used f.7(x,z) fa(x,z) = = fa(x,TjC~l) = = fa(x^r^). follows from from This follows that f
a : r8 3.5. The Theresult result on onE7 Ej and andM3. M3. 3.5. In summary, summary, we we have have proved proved the thefollowing: following:

Theorem 3.1. a e£ GQ Theorem 3.1. Gq acts on on the the profinite profinite Artin Artin group group n^9(V(E7), 0) O) as as follows: follows:

a:

'—*

for22 < i <5, for < 5, Tf5 I-+ r r7 7 '—+ ^ / r 7 )

r

7

/o- ( r 8 5 r 7 ) >

and '—*

where where r 2 , . . .. ,,

rg are arethe £/iefollowing following generators generatorsof ofE7 E7Artin Artingroup, group,

Galois group group GQ, GQ, Singularity Singularity£7, E7, and and Moduli Moduli M3 Galois

213

0— 0—0--—— 0— 0— -o0 t

-t 2

and 11i and Tji ==

-C 1

4

t

-C X ¶

7

K

Ti+lTi+2

If we take the image If image of of 0OininV(E7) V(E?)—* —>M3, Ais, then thenwe weget getthe thesame sameaction action where each generator for r2,. . , T8, r 2 ,..., T$, where generatorisisaaDehn DehnTwist Twistasasfollows. follows. .

cases.

3.6. An,Bn,Cn,Dn

3.6.1. An. This case is exactly exactly same as the action on on profinite profinite braid groups groups given unit vectors vectors of of the the given in Ihara-Matsumoto [17]. [17]. Let e1,. e i , ... . , e n + i be the unit n ++ 1-dimensional vector space. space. We 1-dimensional vector We consider consider the the n-dimensional n-dimensional subspace subspace x2,... , U := := {(x1, {(xi,X2,...,x + x2-{+ ... +h^n+i = O}. 0}. n+i)\xi The root root system systemisise2ei— —e3ej (1(1 < i i / jj x2 > x3 > " -> {(xi,..., £ n +i)|xi > x2 > x3 > > £ n +i}. There are n walls of this Weyl chamber, which are are given by x1 Xi+2>> • •>• > #n+i chamber, which given by x\ > > • • •>> Xi== Xi+i>>x+2 = 1,1,2,..., 2, . n), n),corresponding corresponding to to the the i-th z-thfundamental fundamental root. root.We Wedenote denoteby by (i — complement of the union union of of the the reflection reflection hyperplanes. hyperplanes. U(An) the complement The corresponding corresponding Artin group group is is generated generated by by Ti, 7^, 11 < i < n. n. . ,

.

. ,

. . ,

o

o

o

o

X xn+I

X

X

x,

O

o

X

x,

Makoto Matsumoto Matsumoto

214

These are actually actually the the standard of the the braid These are standard generators generators of braid group, after after divided by the Weyl to the symmetric as the vided by Weyl group isomorphic isomorphic to symmetric group 5 n +i as permutation of coordinates. permutation coordinates. The The elements elements rji are necessary necessary to write write down down Galois action. action. the Galois (2 < i < n), The "suitable" parameters are U := (xi—xn+i)/(xi-i—xn+i) The (2 < ti := (xi — Then, we we get get t1 — £ n +i). We see easily that Xi —xxn+\ == t\t2 •••£*.Then, the same as as in in 3.2.2: the sameconstruction construction 3.2.2: C/(A n )c

II

U(An) r U(An)

-A n t = -y*

j

The tangential and get get a Q-rational Q-rational tangential tangential point point ((t,,. and t i , ... .. ,, tn) on U(An). The re area easily checked to be from morphisms given givenby by (£i,..., (t,,. . , £», ti+2, •. .•, • ? ^n) morphisms easily checked to be from PQ1OO to U(An). U(An). Similar Similar arguments arguments to those those in in 3.4.1 3.4.1 show show that the the Galois Galois action is given given by .

n -> 1


with 1/n r]n — = 1.

These cases cases can can be be reduced to the An 3.6.2. 5 n ? C n . These ^.^.^. reduced to An case. case. The Bn < ii <
.

The generators generators of the corresponding corresponding Artin Art in group are r »(1 ( l<< ii << nn — — 1)1) and rjn-i1 as as in the the following following figure. figure.

Galois group group GQ, GQ, Singularity Singularity£7, E7, and and Moduli Moduli M3 M3 Galois o

00

0 o S

0 o

••• ^ j

n

Sn

CL

jo

s

s

"O

0O

0o

...

S,

O

...

S

t h-1

215

n-1

i+1

l

Ii

wewe we take take the the parameters t1 If we t\ = — s1, si,t2 £2== s2/s1,. S2/S1, •. • •> *n then = = s n /s n _i,then have a Q-rational tangential tangentialpoint pointon onU(Z/2)' U^l2^ — — Uifa, and andhence henceon on y ( S n ) . similar argument with An case shows shows that that the generators are mapped by: A similar by: . ,

Vn-l r£<$ tin—i H> tin_i

n -* Umrfr^Mmrf)

(1<j<
(1

j

3.6.3. 3.0.3. D n . The root system system is is {±ei ± ejil ej\l < i ^ j < n}. n}. The The vector vector space space spanned by roots minus minus reflection reflection planes planes is: is: U U= = {(xi,...,x {(x1,. ,xn)IX2±X3 O}. spanned )\xi±Xj ^ 0}. n n 1 The Weyl group is (Z/2) ~ x 5S,,, which isis the the set set of of permutation permutation matrices, n , which coefficients0,0,±1, ±1, where wherethe the number number of of —1 —i is even. It with coefficients It isis easy easy to to see see that that t)\t2 = s1 s\ • • • s n , Si ^ Sj},bybyputting putting the quotient quotient U(z12)' £/(z/2)'1 is {(^i,..., {(Si,.. sn, t)jt2 Si := := x2 Xi and := x\ • • -xn. Then Thenwe wedivide dividethis thisspace spacebyby Sn. The Theresult result and t := 2 n n x n 2 n — xx is {(u1,.. {(ui,...,un,tin,t)1t2 ,t)\t un, x -u + u2x ... + (-l) un has no == Un, multiple root}. This This isis aadouble double cover cover of of the the configuration configuration space space of of ordered ordered n points, points, which which ramifies ramifies only along the locus locus un = 0. 0. .

.

.

.

Thus, we we obtain n\(V(Dn) — — {un =0}) 0}) as asthe thesubgroup subgroupofofindex indextwo twoof of which is is the kernel of the composition 7Ti(V(Bn)), which composition Tt\(V{Bn)) —* -+ Z -> Z/2 Z —* 1, r» i-> (1 << i < n — 1)). By (the left left arrow arrow maps mapstin—i ryn_i i-> 1, Byvan vanKamKami—p 0 (1 / pen's theorem, theorem, K\(V{Dn)) is quotient by by normal normal subgroup subgroup generated generated is its quotient by the square by square of the path path which which goes goes around around the the divisor divisor un = 0. Thus, Thus, T)n-i = 11 is the the newly newly added added relation, relation, and iri(V(Dn)) 7Ti(V(Dn)) isis generated generated by by Ti,..., r n _i, r'n_l := ?)n_iTn_ilJn_i. ryn _irn _i7/n_i. It can be checked checked that Tn_i r n _i and r'n_x T1,.. commute, and and the group is isomorphic to the Artin group commute, isomorphic to group of of Dn type. type.

Makoto Matsumoto

216 216

x

«

t..

xt

tn.1

By the compatibility of the Galois action with an algebraic algebraic morphism, we have have Tb_i ri ^i—+ / „ ( * , T?)-1T?^UVi,

r?)

(1 < 'i» < 11—2). n - 2).

Note Note that r/» = r i + ir i + 2 • • • r n _ 2 r n _ir4_ 1 r n _2 • Tj.f2Tj44. • • ri+2Tt+iAcknowledgment. The main main idea idea of of this this research research comes comes from from the joint joint work with Y. Ihara, work with Ihara, who who constantly constantly encourages encourages me me by by giving giving invaluable invaluable T. Oda Oda gave gave me a motivation to study Galois actions on mapping comments. T. class groups. As for related with with the the deformation deformation of of singularities, singularities, groups. for the result related am indebted indebted to to P. P. Slodowy, Slodowy, K. Saito, and J. J. Matsuzawa Matsuzawa who taught me me I am much when RIMS. S. Kitano, A. A. Ishida Ishida and and other other much when II was was in in RIMS. S. Morita, Morita, T. T. Kitano, at the theRiemann RiemannSurface Surfaceworkshop workshopininHokkaido Hokkaido University University topologists I met at informed me topology of A. Tamagawa Tamagawa helped helped me me on on informed me of of the topo]ogy of Dehn Dehn twists. A. notion of of tangential tangential morphisms. morphisms. H, H. Nakamura Nakamura kindly kindly informed informed me of the notion of beautiful result on the Galois Galois action on the mapping mapping class class groups. His His his beautiful L. Schneps Schneps kindly kindly stimulation enables enables me to get rid of FQ ±1OO in my result. L. showed me simplification of expression in Theorem 3.1, using using the the showed me the simplification of the expression two cycle relations relations of of the the Grothendieck-Teichmüller Grothendieck-Teichmiiller group two and and three cycle group in in a nice celebrating, because the original original complicated complicated expression expression nice way. way. This is celebrating, is dramatically shortened to a familiar expression. The anonymous referee dramatically shortened a familiar expression. The anonymous referee showed showed me meaa number number of ofimportant important improvements. improvements. I would would also also like liketoto thank thank all friends working working on all my friends on algebraic algebraic fundamental fundamental groups, groups, in in particular particular P. Lochak Lochak and L. Schneps, who inspire me constantly. Lastly, I would would like like to to dedicate dedicate this research Lastly, research to the memory memory of of categorist categorist Yoneda, who whomotivated motivatedme metoto start start mathematics mathematics when when II was was a student N. Yoneda, in a computer department. department.

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Department of of Mathematics, Mathematics, Keio Keio University University 3-14-1 Hiyoshi Hiyoshi Kohoku-ku Kohoku-ku 223 223 Japan 3-14-1 e-mail: matumoto©math. keio.ac.jp e-mail: [email protected]

Monodromy of Iterated Iterated Integrals Monodromy of Integrals and

Non-abelian Unipotent Unipotent Periods Non-abelian Periods Zdzislaw Wojtkowiak Zdzislaw

Contents Introduction. Introduction. 1. Canonical connection connection with with logarithmic singularities. 1. Canonical —* 2. The Gauss-Manin associated with the morphism 2. Gauss-Manin connection connection associated morphism d X ^W of cosimplicial schemes. 1 3. Homotopy relativetangential tangentialbase basepoints pointson onPJP'(C) 3. Homotopy relative (C) \ \{ai,. { a i , . . . , an_|_i}. 1 4. Generators of 4. Generators ofiri(1P1(C) TTI(P (C) \\{ai,. {ai,.. . ,a n + i},x). 1 Monodromyofofiterated iteratedintegrals integralson onPP'(C) 5. 5. Monodromy (C) \ {a1,.. { a i , . . . , aaa+i}. n +i}. 6. 6. Calculations. Configuration spaces. 7. Configuration Z/5-cycle relation. 8. 8. The Drinfeld-Ihara Drinfeld-Ihara Z/5-cycle 9. Subgroups of of the groups of automorphisms. 9. Subgroups 10. Torsors. 10. Torsors associated associated to to non-abelian non-abelian unipotent periods. 11. Torsors periods. 12. Torsors associated associated to to the canonical 12. Torsors canonical unipotent unipotent connection connection with logalogarithmic singularities. 13. 13. Partial information information about cDR(P1(C) GDR^1^) \\ {0, {0,1, oo}). 14. On the group 14. On group cDR GDR for pointed pointed projective projective lines lines and and for for configuration configuration spaces. spaces. A.1. Malcev A.I. Malcev completion. completion. A.2. The Thetorsor torsorand andthe thegroup groupcorresponding corresponding to to the the Drinfeld Drinfeld -Ihara relation. relation. 0. 0.

.

. .

. ,

§0. Introduction. §0. 0.1. Let X be be aa smooth smooth projective projective algebraic algebraic variety variety defined defined over over aa number number field k.k. Let Let ao:: kk ^'—* C be an embedding. We x C. Let Let X(C) X(C) be We set Xc = X X xC. the set with its complex set of of C-points C-points of of Xc with complex topology. topology. There is is aa canonical canonical isomorphism Pcornp

0 CC—Z - ^ HlR{X) ®C

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Zdzislaw Zdzislaw Wojtkowiak

between between Betti (singular) (singular) cohomology cohomology and andalgebraic algebraic De DeRham Rhamcohomology. cohomology. where {u^} defined by equations u^ = The period period matrix (pij) is defined ^Pji&j, = } and fa} are are bases of of H%R(X) and and H%(X(C)). groups. The main motifundamental groups. In this paper we we study periods periods for fundamental vation vation of the paper is is the theconstruction construction considered considered in in [D2]. [D2]. We We describe describe it briefly. First we recall briefly. recall that there there isisaacanonical canonical way way of ofassociating associating aa class class 2 p := H%(X(Q) 0®Q{(2iri)r) anda aclass classC1DR(Z) clDR(Z) cl Q((2iri)P) and C1B(Z) in H B (X(Q)(p) := B(Z) in in H^R(X) with with an an algebraic algebraic cycle cycle Z on on X X of ofcodimension codimension p. The Theclass class dirnZ multiple of the Poincaré CIB(Z) multiple Poincare dual dual of ofthe the homology homology class class CIB (Z) is is a (2Txi) determined Z. determined by Z. We also recall recall that that T is We also is a G-torsor if T is is equipped equipped with aa free free transitive C T. T. We We say that SS isis aa subtorsor subtorsor of T if there there is a action action of of G. G. Let S C subgroup H of G such that S is an H-torsor under the natural action subgroup H G such that 5 an i7-torsor under the natural actionof ofH. H. Let us assume that X Gm(Q) X isis an an abelian abelian variety. variety. Let LetVv E€ G on Q(2iri) Q(2TTZ) m (Q) act on Then the thegroup group GL(HQ(X(C))) G m acts acts on as multiplication multiplicationbybyi/" v',1 .Then xX Gm 7n HB(X(C) )(n) for any andm m because because HB(X(C)m) can can be beexpressed expressed any r, n and via HA(X(C)). H^(X(C)). Let LetG Gbe bethe thesubgroup subgroup of of GL(HA(X(C))) GL(HlB(X(C))) xxGm G mfixing fixing all 171 form CIB CIB(Z), . tensors of the form (Z),ZZ an an algebraic algebraic cycle cycleon on some someXxm. k-algebras to to sets sets such such that that an be aa functor functor from from fc-algebras an element element of P(A) P{A) Let P be is an AA —* CLB(Z) 01 0 1 an isomorphism isomorphism pp :HA(X(C)) HB(X(C)) 0 0 —>• H^R(X) ®A <&mapping A mapping CIB{Z)

to C1DR(Z) for all all algebraic algebraic cycles cycles ZZ on all CIDR(Z) 00 11 for all Xn.

The Theisomorphism isomorphism Pcompbelongs belongstotoP(C). P(C).The Thefunctor functorPP is represented represented by by an algebraic algebraic variety variety Pcomp over overfc, k, denoted denoted also also by P. P. Let Let Gk Gk '-= GGxxkkbe beobtained obtained from from GGby byanan extension of scalars. The The group group Gk Gk(A) of A-points ^4-points of of Gk -- acts extension (A) - the the group of P(A) by by composition composition and P becomes becomes a Gk-torsor. Gfc-torsor. It It is is aasubtorsor subtorsor of on P(A) the GL(HA(X(C)) GL(H^(X(C)) 00k)-torsor A:)-torsorIso(HA(X(C)) Iso{H^(X(Q)00k, k.H^X)). over k of of the the torsor torsor Iso(HB(X Let T be the smallest subtorsor defined defined over (X(C)) (C)) Ok, asaa C-point C-point and and let 0fc,iJ^ H (X)), which which contains Pcomp pComP as let GQ be be the thecorrecorresponding over k) of (X (C)) 00 k). Observe sponding subgroup subgroup (defined (defined over of GL(HB(X(C)) Observe that T CC PPbecause both are subtorsors defined over k, both contain because both are subtorsors defined overfc,both containPcomp pcomp as aa C-point C-point and and T T isisthe thesmallest smallest one one with with these these properties. properties. Let Z(pcomp) denote the the k-Zariski in Iso(if^(X(C) Iso(HA(X(C) 0 k-Zariski closure closure of Pcomp pComP in Let Z(pComp) denote k, HJJR(X)) i.e. the smallest Zariski closed subset defined fc, H^R(X)) i.e. the smallest Zariski closed subset definedover overk,fc,which which contains Vcomp asaaC-point. C-point. Then we pComp as we have have Z(pcomp) CP Z(pComp) C TCTCP and QcG. QcG.

In order order to tocalculate calculate the thedimension dimension of Z(pcomp) one needs needs to know know the transcendence degree of of the the field field generated generated by by the the periods periods p^ Pij over transcendence degree over k. In

Monodromy of of iterated iterated integrals Monodromy

221 221

[D2} (Remark 1.8.) [D2] (Remark 1.8.) Deligne Deligne asks asks whether Z(pcomp) Z(pcomp) = P. He mentioned mentioned has shown shown itit when when X X is that Chudnowsky Chudnowsky has is an an elliptic elliptic curve curve with with complex complex multiplication. On the other hand to calculate calculate T and and G Q seems to be an easier easier task. The The requirement that that TT be a subtorsor requirement subtorsor of lso(H^(X(C)) ® k, k,H})R(X)) is very strong and usually weak information information about about the periods Pij usually only relatively relatively weak pij necessary to to calculate calculate T T and Q. We give obvious example. example. If If X X= is necessary give an obvious = PQ is the projective line over Q and Pcomp p c o m p :: # 1 ( P Q ( C ) ) ® 0 C C A H]yR(F^) ® 0 CC then in order order to to show show that thatZ(pcomp) Z(pcomp) is is isomorphic isomorphic to PQ \\ {0, {0, oo} oo} we must must know that that 2iri know 2iri is transcendental. Observe that that the group Observe group Gm G m has has no non-trivial algebraic subgroups other Notice that that for than the the finite finite groups groups /i^ of roots of of unity. unity. Notice for any rational rational number a / 0 the set of — aa==00isis number of all roots of an equation xk — //^-torsor a aILk-torsor and these are the oniy only /x^-torsors. Hence already 2ni is and these pk-torsors. Hence already the the fact fact that 27ri is not not a th implies that T T isis isomorphic isomorphic to fc -root of a rational number for any k eeNN implies P^ \ {0,oo} and g = Gm. andG=Gm. groups. In this paper we we construct analogues analogues of T and Q for fundamental groups. of PP and G. On the other hand hand we we have no analogue analogue of G.

Let us briefly discussthe the content contentofofthe the paper. paper. Let us 0.2. Let briefly discuss us assume assume that X is of finite X is aasmooth smoothquasi-projective quasi-projective geometrically geometrically connected connected scheme scheme of finite type over over a number number field field k. k. Let Let x be aa k-point fc-point of X. X. InIn[Wi] [Wl](see (see also also [D4]) we defined defined affine affineconnected connectedpro-unipotent pro-unipotent group schemes over k and [D4]) we schemes over De Rham Rham fundamental fundamental group and Q respectively: irfR(X,x)x) -- the the algebraic algebraic De x) -- the Betti fundamental (of 7rf (X(C),x) fundamental group. We We have also the inclusion inclusion (of Q-points into C-points) Q-points C-points) $ x : iriB(X(C),x)(Q) Trf (X(Q,x)(Q) -> 7rfR(X, such that that the theinduced induced homomorphism homomorphism on on C-points C-points 7rP R (X,

is an isomorphism isomorphism (see [Wi] [Wl] Theorem 7.4.). 7.4.). The affine affine pro-algebraic scheme over k

Iso := := Iso(7rf Iso(Rf(X(C),x) Iso (X(Q,ar) x

k,ir?R(X,x))

Let Z(
= Z(
Zdzislaw Wojtkowiak Wojtkowiak

222

The reader can can compare compare this this question question with with the thequestion questionofofDeligne Deligne in in [D2] [D2] (Remark 1.8) 1.8) and and Conjecture says that any (Remark Conjecture 1 in [An] [An] which which says any polynomial polynomial relation cofficients in for the flth n t h cohomology of relation with cofficients in Q between periods periods for smooth projective projective algebraic algebraic variety variety defined defined over over Q Q comes comes from algebraic algebraic a smooth also [G]). [G]). cycles (see also We shall shall denote denote by cDR(X) x)). GBR{X) the image image of of GDR(<£X) in Ont(7rfR(X,x)). Our aim is to calculate the group gDR(X) QDR(X) or at least to to get get some some information information about this this group. group. equivalenttotothe thecalculation calculation The calculation of the homomorphism homomorphism V(C). We denote by by 9X : in(V(C),x) -> ir(V) w(V) the the monodromy monodromy ir(V) —* iri(V(C),x) —* homomorphism (This is is the monodromy monodromy homomorphism homomorphism homomorphism at the base point x. (This of all all iterated integrals on V.) Let us us take take the vector 01 of V.) Let 0i as a tangential base point. Then Then we we have: have: .

:

be the Theorem A. A. LetS0, Let So, Si and and S©0 S^ be the elements elements O/TTI(P 1 (C)\{0, 1,00}, 1, oo},01) ot) shown shown in the the following following figure (loops around 0, 11 and andoo oorespectively, respectively,with with monodromy homomorphism S0 Si . - Soo= =1).1^.Then Thenthethe monodromy homomorphism So .- S1

iri(P'(C) \ {0, 1, oo}, 01) —÷ ir(V) is given by

= (—2iri)X =

.

(—2iri)Y •<(X,Y) .

Z)Z) . iriX = (-niX) • {alliX,Z))1 Z))~x. •(—2iri)Z (-2ni)Z • aJ?(X, • mX .

where ZZ=-X-Y = —x — Y and 7r(V)). where and aJ?(X,Y)Y)E €(ir(V), (n{V), n(V)).

Monodromy of iterated integrals Monodromy of integrals

223

so0

The element ctoi(X, Y) Y) E£ ir(V) TT(V) isis the the same same one one which which appears appearsinin[D5] [D5] and and Y) satisfies in [Dr]. [Dr]. In In section section 88 we we give give a proof proof that the theelement element a}fl(X,Y) satisfies Z/5-cycle relation. relation. Our Our proof analogous to to the proof proof the Drinfeld-Ihara Drinfeld-Ihara Z/5-cycle proof is analogous in [12] [12] and relations in in [D5}. [D5]. It andititisissimilar similartotothe the proof proofofofthe theZ/2 Z/2 and and Z/3 relations It is different [Dr]. In In section 7 we discuss discuss the monodromy monodromy different from from the the proof proof in in [Dr]. configuration spaces. (stated somesomeof iterated integrals on configuration spaces. The main result (stated what imprecisely) imprecisely) is: is: the monodromy monodromy representation representation of iterated integrals on a configuration configuration space space of ofnn points points in in C C is is determined determined up up to conjugacy conjugacy by by the homomorphism monodromy homomorphism iri(IP'(C) \\{O, {0, 1,

ot) -4 ir(V). n(V).

Let us now describe the the main main results results of the the second part of the paper. Let now describe second part paper. One shows that there is an affine pro-algebraic pro-unipotent group scheme One shows that is an affine pro-algebraic group scheme H(V) II(V) over Q such that the the group group of of C-points C-points of of 11(V) H(V) is ir(V). ir(V). Let Let us usset set T(C) := {(x,y,z) e )3\ T(C):={(x,y,z)e{AV)) 3a G C*, x = aX, y

z ~ aZ, xx.y.z=O} • y • z = 0}

and Aut*(11(V))(C)

{f e

f(X) = cxX, e C*, f(X) aX, f(Y)

f(Z) aZ}. f(Z)~aZ}.

(Here w means ir(V)), whereas means conjugation conjugation by an an element element of of(ir(V), (n(V),ir(V)), whereas ~ means conjugation by by an an element element of of n(V).) ir(V).) One means conjugation One shows shows that 'T(C) T(C) (resp. (resp. Aut*(11(V))(C) set(resp. (resp. a group) group) of of C-points C-points of of an anaffine affine proproAut*(n(V))(C) )) is aa set algebraic group scheme) scheme) over over , which T r (resp. by Aut* (11(V)). The group Aut* (11(V)) is a subgroup of the group (resp. by Aut*(lI(V )). The group Ant* (TL(V)) is a subgroup of the group of of 11(V). MoreoverTT is Let of automorphisms automorphisms of II(V). Moreover is an an Aut*(lI(V))-torsor. Let r 0_4 : iri(V(C), 7Ti(V (C), ~ot) —>ir(V) TT(V)be bethe themonodromy monodromyhomomorphism homomorphismfrom fromTheoTheorem A. Observe Observe that that the triple

E T(C).

to be bethe thesmallest smallest subtorsor subtorsor defined defined over over Q Q of of T such T{6_±) to that the C-point of Let the triple triple (0_^(SO), 0-^CSi), O^S^)) is aa C-point of T{6,). T(01). Let ) be the corresponding corresponding group. We We have: We We define

224 224

Zdzislaw Wojtkowiak Zdzislaw

Theorem affine pro-algebraic group schemes . The affine pro-algebraic group schemes TheoremB.BThe and G(6_±) are are isomorphic. isomorphic. and

1, oo}) GDR{^Q\ {O, \ {0,1, oo})

We explicitly. Let Letus usset setir'n':=:=(rr(V), (^(V), We calculate calculate aa quotient quotient of of G(6—±) G(01) explicitly. ir(V)), TT(V)), ir" TT" := := (ir',ir') (T^TT')and andir2(V) n2(V) := :=ir(V)/ir". ir(V)/ir". ir2(V) TT2(V) isisalso also aagroup group of ofCCr points of of an affine affine pro-unipotent pro-algebraic pro-algebraic group group scheme scheme H2(V) n2(V ) defined denned over regard ir2(V) as a Lie algebra. also regard n2(V) as algebra. The Theelements elements X, X, Y Y over Q. We can also := == and {YX)Xi~1Yi-1 and : = (...(Y,X)X)...X)Y)...Y) ( . . . (Y,X)X).. .X)Y).. .Y) ii = = 1,2,3...,j 1 , 2 , 3 . ..J form a linear as a Lie 1, 2,3,... form 1,2,3,... linear topological topological basis of of n2(V) considered considered as algebra. Let w u be be an anelement element of of this this Lie Liealgebra. algebra. If the thepolynomials polynomials ex exp( and 1 ++ Yl Fn^rnXnYrn are P( E fi,jxiYJ) and areequal equalthen thenwe weset set n=1,m=1

i=1,j=1

(w,exp(...)):=w+ n= 1,m= 1 n=l,m=l

Let us us set set := {(x, y, z) E x=aX, x'y.z=O} 3a e C*, rr = a l , y « a7, z ~ aZ, x • y • z = 0} and Aut*([12(V))(C) := Aut*(n 2 (V))(C):=

czX, aZ}. aY, 1(Z)f{Z)~aZ). {/ f(X)= = aX, f(Y) f(Y)*aY, {f EG Aut(7T20O)| 3ae€ C*, f(X) As before before there are (112(V)), whose are affine affine pro-algebraic schemes schemes T2 and Aut* Aut*(Il2(V)), sets of C-points are (112(V)) is is a group scheme 72 are describe describe above, above. Aut* Aut*(Il2(V)) scheme and T r is an an Aut*(112(V))_torsor, Aut*(Il2(V ))-torsor. Let Let 06 :: iri(V, 7Ti(V, "oi) —* —>•ir2(V) n2(V)be bethe thecomposition composition of 6>_, with the ir2(V). thequotient quotientmap mapir(V) n(V)—* -> ir The triple triple (0(S0), (0(SO), 0(S1), 0(Si), 2{V). The

We define define T(6) T(9) to to be the smallest subtorsor defined 0{Soo))e T2(C). We defined over over Q

of T72such such that that the thetriple triple(9(S0), (6{S0), 0(S1), fl(Si), 0(Soo)) is a C-point C-point of T(0). T(0). Let G(0) be be the thecorresponding corresponding subgroup subgroupof ofAut*(112(V)). Aut*(n2(V)). Then Then we we have: have:

Theorem Theaffine affinepro-algebraic pro-algebraicgroup groupscheme schemeG(0) G{6)isisisomorphic isomorphicto to TheoremC.C.The a quotient of quotient £DR(PQ \ {0,1, oo}). Let C : Gm Aut*([12(V)) begiven givenby by C(t)(X) C(t)(X) = Let G m —* -> Aut*(n = tX, C(t)(Y) C(t)(Y) = = tY tY. 2 (F)) be Let H H be bethe theimage imageof ofC. C.The Thenext nexttheorem theoremgives givesaadescription descriptionofofthe thetorsor torsor T(0) and and the thegroup group G(0). G(0).

Theorem given Theorem D. D.i,) %) The Thetorsor torsorT(0)(C) T(0)(C)is is givenbyby (2k—i)!

exp laX,(aY, (aY,exp(f; £' k=1 i+j=2k 1

fcl

i i 1j 2,jf c1

'

/

'

M

Monodromy of iterated iterated integrals Monodromy of

E E

(k-i)! ((1 - e(k))

225

+ e(k)

k=3 k odd odd

E C,, —((2k) - C ( 2 * ) == r2k r 2fc

• (2Ri)2k (27ri) 2fc ,

and E(k)=O ((k)EQ, and e ( * ) = 0 if C(*)eQ,

V 6i, fc _i E € C, C, V

Ck cfc = —((k)

e(k)=1 if e(Jfc) = 1 */

correspondinggroup groupG(9)(C) G(6)(C)is isgiven givenbyby nj JTiecorresponding ii) The

f(Y) == {/ f(X) ==tX, tX,f(Y) f 6 Aut*(7r2(F)) I f(X) { (k—i)!

(tY,exp(

(i —

Ck

(1

tk)

k=3

k-odd fc-odd * > l , i > l

with with t eEC*, /31,k-1 /?!,*_! Ee c}.

iii) Let G be the smallest subgroup of Aut* (112(V)) m7 6e t/ie smallest closed closedaffine affinepro-algebraic pro-algebraic subgroup o/Aut* overQ, Q, which which contains contains G(0) G(O) and and H. H. Then defined over Then G(C) = = {f E Aut*(1r2(V)) f(X) = tX, 1(Y) =

(tY, exp( {tY, exp

E C}. with t EG C , /3i,k—1 A,^! G C}. k=3 k

odd

i+j=k

Corollary E.E.The groupG(O) G{6)contains containsthe thegroup groupHHif ifand andonly onlyififallallnumnumCorollary Thegroup bers ((2k 2,... are C(2fc + 1) 1) for kk = = 1, 1,2,... areirrational. irrational. Aut*(H(V)) be given Let CC::Gm by C(t)(*) C(t)(X) == *X, tX, C(t)(Y) G m—* -> Aut*(n(F)) given by C(*)(y) == tY. *V.Let Let r 91 be the image of C. C. Let Let G be the smallest smallest subgroup subgroup of of Aut*ll(V) Aut*II(V )defined defined % be over and 7-1. LetTTbe be aa subtorsor subtorsor of of T7 defined over Q which which contains contains G{6^) and %. Let defined over Q, whose whose corresponding correspondinggroup groupisisGGand and such such that that the triple over Q, triple

is is

a C-point C-point of T. T. Let Z be the Let the Q-Zariski Q-Zariski closure closure of the triple

(01(s0), in Y. T. Then Then we we have have

ZC T. Z c T(61) CC T.

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Zdzislaw Zdzisiaw Wojtkowiak

Question. Is Is itit true truethat that Z= = T{9^) = = T? T? section 15 we westate state some some conjectures conjecturesrelating relatingthe the group group GDR(X) cDR(X) to to the In section the Galois representations representations on fundamental Galois fundamental groups. Deligne in in [D4] [D4]considers considersthe the motivic motivic Galois Galoisgroup group of ofthe the tannakian catDeligne cat1). The group egory of of mixed mixed motives motives generated generated by by the Tate motive egory motive Q( Q(l). group oo}) from from this this paper paper should ^cDR(P'(C) D R O ^ Q \ {o, {0, 1, l,oo}) should be a quotient quotient of of the thegroup group consideredby byDeligne. Deligne.Perhaps Perhapsthey they are are even even equal. equal. In this paper considered paper there there motivic interpretation interpretation of cDR (X). We are also not discussing is missing missing aa motivic GBR(X). discussing structures on the fundamental the mixed mixed Hodge Hodge structures fundamental groups, groups, though though it would would be very very natural We hope to study these these topics in a future future natural to do this here. We paper.

Acknowledgments. Acknowledgments.I would I wouldlike liketotothank thankvery verymuch muchProfessor Professor Deligne, Deligne, who one-form considered the case case of who once once showed showed me me the the one-form consideredin in Section Section 11 in the C \ {O, {0,1}. are due to Professor Y. Ihara for showing me his proof Thanks are due to Professor Y. Ihara for showing me his proof 1 }. 5-cycle relation, of the 5-cycle relation, which which helped helped us us to find find analogous analogous one one for for unipotent periods. II would would like like to express express thanks to to Professor Professor Hubbuck Hubbuck for his his inviperiods. to Aberdeen, Aberdeen, where where Section Section 8 was written and and where where in in May May 1993 1993 I tation to gave seminar 5-cycle relation periods. gave seminar talks on 5-cycle relation for unipotent periods. I would Ihara for his would like like to thank thank very very much much Professor Professor Y. Ihara his invitation invitation to Kyoto, Kyoto, and and Professors Professors Oda Oda and andMatsumoto Matsumoto and andTamagawa Tamagawa for for useful useful discussions and comments comments during during my seminar seminar talks. discussions and Thanks once invited invited me me to to write write a Thanks are due due to to Professor Professor L. Lewin, Lewin, who once chapter in in the the book book on on polylogarithms polylogarithms and and suggested suggested also including including results on monodromy very much monodromy of iterated iterated integrals. integrals. This encouraged encouraged me very much to continue continue to work work on this this subject subject (see (seepreprints preprints [W2] [W2]and and[W3] [W3]of ofwhich which some some parts are are included included in the the present paper). Finally, II would would like like to to thank thank Leila Schneps and and Pierre Pierre Lochak Finally, Leila Schneps Lochak for their help in the the editing editing of of the paper paper and andtheir theirencouragements, encouragements, and andthe thereferees referees for their comments, remarks and interest. for comments, and interest. This paper isis aarevised revised version version of of [W5]. [W5].

Monodromy of of Iterated Iterated Integrals Monodromy

227

§1. Canonical Canonical connection connection with with logarithmic singularities. §1. singularities. In this this section section we we introduce introduce the the canonical canonical connection connection with with logarithmic logarithmic singularities. singularities. The results results of of this this section section are are well-known well-known and are are due due to to AoAoand others. Our inoto, Chen, moto, Chen, Deligne, Deligne,Ham Hain and Our approach approach isisslightly slightly different different from We consider consider the the one-form one-form of of the the canonical canonical unipotent from the one one in in [HZ]. [HZ]. We connection as a connection form of a principal G-bundle, where where G G is is isomorisomorphic to (a (a quotient quotient of) of) TT®R(X,X).x). We found found this approach useful useful studying functional equations equations of of iterated iterated integrals integrals (see functional (see [W4]). [W4]). It is also well well adapted to study the the monodromy. monodromy. We We also also use use itit in in work work in in progress progress concerning concerning a generalization of the the Zagier Zagier conjecture conjecture on on polylogarithms polylogarithms to to arbitrary arbitrary itergeneralization of iterated integrals.

Connection on on a trivial principal 1.0. Connection principal C-bundle. G-bundle. Let Let C G be be aa Lie Lie group. Let us consider the principal G-bundle group. consider G-bundle 1.0.1. 1.0.1.

p:VxG-#V. p:VxG->V.

Let 80 be the canonical one form form on on G C i.e. i.e. the Let canonical one the left left invariant invariant one-form one-form with valuesininthe theLie Liealgebra algebra&gofofGGsuch suchthat that6e6e(X) forXX G E g. Let values (X) —=XX for Let pr::VV xx G pr G —>•GGbe beaaprojection. projection. Then Thenthe theone-form one-formpr*9 pr*0isisaaconnection connection form on on the the principal C-bundle 1.0.1. 1.0.1. Let form principal G-bundle Let w E G Q'(V) ^ ( V ) ® gg be a one one form form on V with values on V VxG values in 9. g. Let Let a) be a one-form one-form on G vanishing vanishing on vertical vectors and and such such that vectors i) &(v,e)(Y) — where (v,e) (v,e) EGV V xxC, G,eeisisthe theneutral neutralelement element of of = wv(pr*Y), where G and Y E T(v,e)(V X C). ii) ii) ti(Vyg)((Rg)*Y) = — a^{9~1)^{vye)0ir)'> where where R9 Rg is is the the right right action action of ofggon on VxG. V xG. It follows from [L] [L](Theoreme, (Théorème,p.p. 68) 68) that that pr*9 pr*0 + uj isis the follows from the connection connection form form of aa certain certain connection on the the principal 1.0.1. If w of connection on principal G-bundle G-bundle 1.0.1. a; is integrable integrable then pr*0 pr*9 + & CJ isisalso also integrable. integrable. Abusing Abusingthe thenotation notationwe weshall shallalso alsodenote denote pr*0 +-j- UJbybywUJ the one-form one-form pr*Q shall a connection form andand wewe shall saysay thatthat w isa;aisconnection form on on the principal principal CG- bundle bundle 1.0.1. 1.0.1. 1.1. Let Let XX be bea asmooth smoothprojective projective scheme scheme of of finite finite type over over a field field k of characteristic zero. Let Let D be in X X and characteristic zero. be aa divisor divisor with with normal normal crossings crossings in and let let V == X V X \\ D. D. Let Let

A*(V) := I'(X,ç(logD)) be the differential differential algebra of of global sections of the algebraic algebraic De De Rham Rham comcomplex plex on X X with with logarithmic logarithmic singularities singularities along along D. D. It follows follows from from [Dl] [Dl] Corollaire 3.2.14 that that each each element element of of A*(V) A*(V) isis closed closedand and the the natural map Corollaire 3.2.14 map A*(V) -> H^R(V) is injective. A*{V) injective.

Zdzislaw Wojtkowiak Zdzislaw

228

1 (A1(V)) We shall denote by (V)) the exterior We byA2 A2(A exterior product of of the the vector vector space space A1(V) with itself itself and and by A\V) A'(V) A A (V) with A A'(V) Al(V) the theimage image of of A2(A'(V)) A 2 ^ 1 ^ ) ) ininA2(V). A2(V). l Let H(V) H(V) := Let := (Al(V))* (A^V))* and andR(V) R(V) :=:=(A1(V) (A^V) AAAl(V))* A {V))* be be dual dual vector vector 2 X 1 1 spaces. The A'(V) spaces. Themap mapA2(A1(V)) A (A (F))—p -> A'(V) A ^) AA ^ ) induces induces aamap mapR(V) R(V)—+ -> 2 A2(H(V)). (H(V)) be over jfkc on A (H(V)). Let Lie (H(V)) be the the free free Lie Lie algebra over on H(V). H(V). Observe Observe terms of of Lie Lie(H(V)). (H(V)). Let (R(V)) (R(V)) be the that R(V) R(V) isis contained contained in degree degree 2 terms the Lie ideal ideal generated generated by by R(V). We Lie We set X

Lie (H(V))/(R(V))

Lie (V)

and n (V)/I'nLie (V)).. L(V) := lim n (Lie (V)/r L(V) Lie (V))

n

Definition 1.1.1. 1.1.1. The The Lie Lie algebra L(V) equipped equipped with the the multiplication multiplication given by by the Baker-Campbell-Hausdorff given Baker-Campbell-HausdorfF formula formula is aa group. group. We Weshall shall denote denote this group by ir(V). TT(V).

Observe that ir(V) Observe that TT(V) is is aaset setofofk-points fc-pointsofofananaffine affinepro-algebraic pro-algebraic proprounipotent group group defined defined over over k. The The Lie Lie algebra of ir(V) ft(V) can can be beidentified identified with L(V). that H(V) can with L(V). Observe Observe that canbe beregarded regarded as as included included in L(V). L(V). Let T[H(V)] be the tensor algebra over over kk on onH(V) H(V) and and let (R(V)) (R(V)) be the ideal of of T[H(V)] T H v T[H(V)] generated by R(V). R(V). Let be T[H(V)] generated by Let Q(V) Q(V) := T[H(V)]/(R(V)) l ( )}/(R(V)) be the the quotient quotient algebra and let Q(V) be its completion with respect to the augmentation algebra be its completion respect to the augmentation ideal k), i.e. ideal /I := ker (Q(V) (Q(V)—* ->fc), i.e. Q(V) := Urn (Q(V)/in). (Q(V)/lnY n

Definition denote P(V) group Definition 1.1.2. 1.1.2.WeWe denotebyby P(V)thethe groupofofinvertible invertibleelements elements in in 1. Q(V), whose whose constant constant term term equals equals 1.

Observe that the vector vectorspace spaceH(V) H(V) can can be be regarded the Observe that the regarded as as included included in in the algebra of of P(V). algebra identify the the elements elements of of L(V) L(V) with can be of of We identify with Lie elements elements (which (which can length) in in P(V). P(V). The infinite length) The exponential exponential series series defines defines an an injective injective homohomomorphism exp: ir(V) —* exp : n(V) -> P(V). P(V). Lie Lie

r The inverse exp(ir(V)) of P(V) P(V) and it is inverse of exp is defined defined on the the subgroup subgroup exp(?r(V )) of given by by the the formula formula 2 — 1)2 logz ._ 1)— logz == (z -1) - y1/2(z ++ 2(z -1)

011 the Lie Lie algebra algebra L(V) on V with values values in in the (and in in L(V) (and P(V)) in the the following following way: way: of P(V))

We define aa one one form form wy Wv We define the algebra the Lie Lie algebra

'/3(z y3(z —- 1 ) 3 —- 1/4(z y^z —-1)4 ++ .. . . .

Monodromy of of Iterated Iterated Integrals Monodromy

229

1.1.3. Definition 1.1.3.

The form corresponds to to the identity form WV uy corresponds identity homomorhomomorphism id^i(^) idAl(V) under under the natural natural isomorphism isomorphism 1 l 0 (Al(V))* A'(V) A'(V)®H(V) =A (V)®H{V) = {V) ® ( A

Horn(A'(V),A'(V))).

(see [D4] [D4] and and [H] for for the definition definition of of the theform formWV). uy).

Lemma Lemma1.2. 1.2.(see (see[D4], [D4],[H]) [H])The Theone-form one-formWV ujy isisintegrable. integrable.

Proof. The The facts facts that that Lie(V) Lie(V) isisaaquotient quotientofofaafree free Lie Lie algebra algebra by by the the ideal ideal (R(V)) and (R(V)) and that thateach eachelement elementofofA' A1(V) (Vr)isisclosed closedimply implyintegrability. integrability.

Let us assume assume that that kk is complex numbers numbersC. C. Then Then V(C) Let is the the field field of complex V^(C) is a complex variety with with the the standard complex topology. topology. Abusing Abusing the the notation complex variety we shall shall denote denote it by V. V. Let us us consider consider the principal principal bundles bundles

(1) Vxir(V)—*V. (1) VxP(V)—*Vand V x P ( F ) - > V a n d (2) (2) V x ir(V) -> V. We equip equip them with the We the connection connection given given by bythe theone-form one-formWV. ujy. Let us us fix fix aa path 'y of the the trivial 7 on V V from from x to to zo. ZQ.We We define define a germ germ of fiat flat sections sections of / bundle (1) ZQ\ Ay(z;x, y) G P{V) (resp. (resp. Ly(z;x,/y) G (1) (resp. (resp. (2)) near nearzO: e P(V) E Av(z;x,'y) TT(V)) ir(V)) to be be aa function function of of zz moving moving in in aasimply simply connected connected neighbourhood neighbourhood of z0 ZQ with with values values in P(V) P(V) (resp. (resp.ir(V)). TT(V)). We Werequire requireitittototake takethe thevalue value1 1 (resp. 0) at xx along along the the fiat flat section section over over the the path path 'y. 7. (resp.

Definition 1.3. Definition 1.3. homomorphism

Let xx eG V 7Ti(V,X)bebea aloop. loop.We Wedefine define aa Let V and and let let aa eGiri(V,x)

9x,V : ir,(V,x) —* P(V) -> P(V)

(resp. 0x,V 6xy :: ir,(V,x) —* ir(V))

by the formula formula (resp. 6xy{a) := Lv(a(1);x,a)) := Av(a(1);x,a) (l);x,a) (resp.

and we call it the the monodromy monodromy homomorphism homomorphism of the form form WV ujy atat the base point x. (The point (The homomorphism homomorphism 6xy we shall shall also denote by 0X.)

Proposition 1.4. VVand letlet'y7bebea asmooth Proposition 1.4.Let Letx1,x2 x\,X2 e G and smoothpath path from from x1 x\ to x2. path 'y 7gives this X2>The The path givesananidentification identificationofofn-i(V,x,) TTI(V,x\) with withiri(V,x2). TTI(V,X2)>Via Via this identifi cation the monodromy homomorphisms °xj,V and identification the monodromy homomorphisms 0Xly and 0X2y of the form form ujy are conjugate. WV are conjugate. Proof. This This isis aa property property of ofaaconnection connection on on aaprincipal principal fibre fibre bundle. bundle.

^

The next result is is aa special special case case of of a result result of of Chen Chen (see (see [Ch2], [Ch2], Theorem 2.1.1).

Zdzislaw Wojtkowiak Wojtkowiak

230

Proposition Proposition1.5. 1.5.If IfA'(V) A1^) —+ —>Hj,R(V) HpR(V) isisananisomorphism isomorphismthen thenthe themonmonodromy x) —+ induces isomorphism odromy homomorphism homomorphismOx,v 6xy : 'ir,(V, fti(V,x) —>ir(V) TT(V) inducesanan isomorphismofof

the Malcev C-completion of TTI(V,X) iri(V,x) into C-completion of into ir(V). TT(V).

then H'(V, C) Remark. If the map map A'(V) A1^) —* -> Hj,R(V) H\)R{V) is an isomorphism isomorphism then Hl(V,C) has a Hodge Hodge type (1,1). (1,1)- Examples Examples of of such varieties are Zariski open subsets of simply simply connected connected smooth projective project ive varieties. 1,2) Let Xi (for 2) be smooth, projective projective schemes schemes of finite finite type over over k. k. Let X1 (for ii = 1, Let Di D be crossingsininXi X (for 2). Let Let be divisors divisors with normal crossings (for ii = 1, 1,2). Let V1 Vi = X\D1 Xi\Di 1 (for i = 1,2) and : X\—* —>• X2be beaamorphism morphism such (D2) =—D,. D\. and let let /f Xi suchthat thatf~f'(D2) (for A'(V1). Let /„ : H(V,) be the Then induces f* /* : A'(V2) Al{V2) —+ -> A^Vi). Jf(Vi)—+ -> H(V2) if(F 2 ) be Then /f induces dual map. This This map map induces induces group group homomorphisms homomorphisms ir(Vi) —* /* : P(V,) P(Vi)—* -> P(V2) P(V2) and /* : 7r(Vi) -> ir(V2).

Lemma 1.6. 1.6. We Wehave have

(idøf*)(wvi)=(f*®id)(wv2). {id 0 U){wVl) = (/* ® id)(u>v2).

The lemma from the definition lemma follows follows from definition of of ujyi as as idi A1 (vi).

Corollary 1.7. 1.7. We Wehave have (z; x, 7)) = Av2(f(z);f(x),f('y)) KV2 (f(z); f(x), and

/,(L f(x), f('y)). f*(Lv1 (z; x, x, 7)) 'y)) = LV2 (f(z); (1(z); f(x), V l (z; . of AA'1^).(V). Let Let X X,,.. Let w1, a>i,..., un be a basis basis of i , . . . , X n be the dual dual basis basis of H(V). Let C{{X,, . be the algebra of formal power series of H(V). Let C{{Xi,... ,X n }} algebra of formal power series in in non-commuting variables X\,..., X,,. . Xn. Let C{{X,,.. , Xn}}* be the multinon-commuting variables Let C{{Xi,..., multiplicative group of of these these power power series series whose whoseconstant constant term term is is 1. Then the plicative group the algebra Q(V) is a quotient of C{{X,,. . by the ideal generated algebra Q(V) quotient C{{Xi,... ,X n }} ideal generated by R(V). The group P(V) is a quotient group of C{{X,,. . R(V). group P(V) is a quotient group of C{{XU..., X n }}*. If If a(X,,. a ( X i , . . . , X n ) is aa formal formal power power series series in in non-commuting non-commuting variables variables X,,. in P(V). X i , . . . , Xn, we shall also denote by a ( X i , . . . ,, Xn) its image image in P(V). . . ,

. ,

. . ,

.

,

.

. ,

.

.

.

. ,

,

.

Proposition Proposition1.8. 1.8.Let Let-y7bebeaapath pathon onXX from from xx totoz.z.We Wehave have i)

Av(z; x, ) =

1

+

f

.

.

,

. .

.

E

P(V).

Monodromy of Iterated Iterated Integrals Monodromy of

231 231

(The summation summation is over all non-commutative non-commutative monomials monomials in in variables variables Xi,. X\,..., interated integrals along thethe path Wy.) Xn, the interated integralsare arecalculated calculated along path 7.) ii) llj

Lv(z;x,y) ±j\f yZ\ X^'Y) = - log(Av(z;x,'y)).

Proof. We . ,X n }}* —+ Proof. We equip equip the the principal principal bundle bundle VV xxC{{X1, C{{Xi,... ->• V with . . ,

n

the the connection connection given given by by aa one-form one-form >^ <*>*0® -X"t-Using Usingthe thedefinitions definitionsinin1.0 1.0 we calculate calculate the the horizontal subspaces and and then the we horizontal subspaces the system system of of differential differential equations for for horizontal horizontal sections. sections. We We get get that horizontal equations horizontal sections sections are are given given by f w. zz^{zl(z,1 + y { { - i f WiXi •. ... -Xi)

that exp Hence the point point i)i)follows. follows. Observe Observe that exp : ir(V) K(V) —+ —>P(V) P(V)identifies identifies l l WVuv eGA'(V) (P(V)). Hence thethe point UJV A {V)®0Lie Lie(ir(V)) (n(V))with with A (V)0 Lie ® Lie (P(V)). Hence point WV eGA'(V) ii) follows. follows. 0

The Gauss-Manin connection associated associated with with the morphism §2. The Gauss-Manin connection morphism _^ —÷ XdA[i] oof cosimplicialschemes. schemes. f cosimplicial

XA[i]

2.1. We review review here heresome someconstructions constructionsfrom from[Wl]. [Wi]. Let Let /I be a unit interval and let X be be an anarc-connected arc-connected and andlocally locally arc-connected arc-connected topological topological space. space. Let pp: :X' -> X xx X X be be given given by byp(W) p(u) = = (W(0),W(1)). (CJ(0),O;(1)). This is a fibration. fibration. X1—* Applying local Applying the the connected connected component componentfunctor functor to to each each fibre fibre we we get get a local system of of sets sets pp: :PP—+ -» X the fibre fibre over over (x, x) x X is X x X, X, such that the x) G eX X X TTI (X, x) and and the the restriction restriction to to X X xx {x} {x} is is aa universal universal covering covering space space of of X. X. iri(X, We want analogous construction construction in algebraic algebraic geometry. geometry. Let us us We want to do an analogous notice the inclusion inclusion of of simplicial simplicial sets notice that the

induces a morphism of of cosimplicial cosimplicial spaces spaces —+

realization isispp :X'X1—+ —>XXx xX.X.WeWe useuse thisthis observation observation whose geometric geometric realization to construct an an analogue analogue of of the local local system system p: p :PP—+ -» X X xx X in algebraic geometry. Let V be aa smooth smooth quasi-projective quasi-projective scheme scheme over over a field k of characteristic zero. The zero. The inclusion inclusion of of simplicial simplicial sets 9A[1] °->> A[l] induces aamorphism morphismof of cosimplicial schemes

232

Zdzislaw Zdzislaw Wojtkowiak

Assumethat that k is numbersC. C.As Asin in the the case case of aa Assume is the thefield field of of complex complex numbers morphism of S, the morphism of algebraic algebraic varieties varieties XX—+ -» 5, thecohomology cohomology of the thefibres fibres Hl(pm ~ 11(x,y);C) (x, ?/);C) form form aalocal local system system of ofcomplex complex vector vector spaces. spaces. Imitating the construction construction of of Katz and and Oda Oda(see (see[KO]) [KO]) we we equipped equipped the the sheaves sheaves Hl(tRplQ*,vA[1]y,vdA[1],) of the relative De Rham cohomology cohomology with the inconnection dc such such that the the horizontal horizontal sections sections form form aa local local syssystegrable connection (Rpl is the thecomponent-wise component-wise derived tem described described above. above. (Rp derived functor functor of ppi and t is the thefunctor functor which which associates associates a total total complex complex to toaabicomplex.) bicomplex.) and H°(pm—'(x, ~l{x, x), x),Q) thering ring of ofpolynomial polynomial functions functions on on the theMalcev Malcev rais the raH°(p' Q) is tional completion completion of the fundamental group TTI(V, X). The representation of of the fundamental group iri(V,x). The representation of 1 7Ti(F,x) TTI(V,X) (x,x);Q) is induced by iri(V, x) x iri(V, x) on on the the vector vector space space H°(p* H°(p ~ '(x, x); Q) is (bijections x ir1(V,x) of ir1(V,x))

(bijections of TTI(V,X))

= a.g.f3'. given by (p(a, f3)(g) /3)(g) = a-g-f}-1. Therefore Therefore the sheaf sheaf H°(tRp H°(tRplCl*vA[1])/(vdA[1]))

the connection connection dc is is an ananalogue analogue of ofthe the local localsystem systemPP—* —> equipped with the inalgebraic algebraic geometry. geometry. This This suggests suggests the thefollowing following definitions definitions of the X x X in Betti and De Rham fundamental fundamental groups. We We set

:= Spec o(p*-1(x,x);Q). nf(V,x)~SpecH If V is aa smooth smooth quasi-projective quasi-projective scheme scheme over over aa field k of characteristic zero zero and xx is ofofVVthen isaak-point fe-point thenwe weset: set: R R.DR(vx) := Spec Spec H° (p' 1(x,x)). n? (V,x) := ^fax)). DR

2.2. 2.2.

Let V We assume assume additionally additionallythat that A'(V) Let V be besuch such as asin in1.1. 1.1. We (V).

2.2.1. The inclusion inclusion of of complexes complexes (C-%A1(V)-^A1{V)

X '—p A1(V) AA (V) -> 0) 0) *->

induces an isomorphism isomorphism

H° := H^BaviC-^A^V^A^V) H°

l A1(V) AA {V)—* -> 0)) « H°DR{pm '(x,x)). ~l{x,x)).

(The bar construction construction is is chosen chosen to to be becompatible compatible with with the the inclusion inclusion into into 1 the total total complex complex of of the De De Rham Rham complex complex of pm ~'(x, (x, x). x). The Theisomorphism isomorphism l X means that to calculate HpR(pm ~ 1'(x,x)) only A A'(V) A'(V)A means (x, x)) we we need need only {V) and A {V) A A1(V).) A\V).)

Monodromy of of Iterated Iterated Integrals Monodromy

233 233

(Al(V))*, T[H(V)} T[H(V)] is is the tensor We H(V) = (A^V))*, tensor algebra algebra on We recall recall that that H(V) H(V), R(V) and Q(V) H{V), R(V) := := (A'(V) (A^V) AAAl(V))* A^V))* isis contained contained in A2H(V) A2H(V) and Q(V) := T H v T[H(V)]/(R(v)) i ( )}/(R(V))- Let Let us us set set

(Q(V)/Jn)*,

Q(V)*

where /I is the augmentation augmentation ideal ideal of Recall that where of Q(V). Q(V). Recall Q(V) urn (Q(V)/jn Q(V) : = U 4— n

\

and P(V) P(V) C C Q(V). 2.2.2. 2.2.2. Notice that the the symmetric symmetric algebra algebra on on Q(V)* Q(V)* is is the the algebra algebra of of poiypolynomial functions functions on Q(V). Q(V). 1 Let us fix a basis u)\,..., un of A A'(V). (V). Let Let Xi,..., Xn be the dual basis of H(V). Then of H(V). Then T[H(V)] T[H(V)] = C{X1,.. C{XU..., Xn} is the polynomial polynomial algebra algebra on * ... • Xik)* denotes the Let (X21 non-commuting . ,n. non-commuting variables X1, Xi,...,X (X^ • ... denotes the ) linear on T[H(V)} T[H(V)] which linear form form on which to f/ ininT[H(V)} T[H(V)] associates associates its its coefficient coefficient at Let T* be the the the monomial monomialX21 X^ • ... • Xi,,. Xik. Let the vector vector space space generated generated by all all {Xh -...-Xik)* including including 1*. 1*. Observe Observe that Q(V)* Q{V)* isis aa subspace subspace of of T*. T*. We We equipped T* with a shuffle multiplication and a Hopf algebra structure structure given shuffle multiplication given q)* = . by b y (X21 ( X i , •...• X i p ) * • ( X i p + 1 ..... X i p J q ) * = £ ( X i a W •...• Xia(p+q)Y . ,

.

. .

.

. .

.

.

.

. . .

. .

.

.

a€Sh(p,q)

k

and (Xh •... • XihY .

-+ J2(Xh

—÷

. .

00 (Xil+1 . .. . . . Xiky.

. .... . . Xity

. .

The shuffle The shuffle

1=0

multiplication and the Hopf Hopf algebra structure on on T* T* induce induce the the analogous analogous structures on on Q(V)*. Q(V)*. H° H° isisalso alsoequipped equipped with with aashuffle shuffle multiplication multiplication and and structure which on the form by the same a Hopf Hopf algebra algebra structure which on form level level are given given by same formulas. Lemma 2.2.3. 2.2.3.There Thereisisananisomorphism isomorphismofofHopf Hopfalgebras algebras

given w1, —* (X^ (X2, .... given by iyw ^ ®0... . . .®Wjk ® ujik -> •... •

Xik)*.

Proof. We We identify identify the linear linear form form (X^ • ... .• Xik)*)* with the thevector vector(X21 (X^ ® 0 1 = il®...®uJi Observe that H° asubspace of COA1(V)O ... 0 Xik )* =Lj . Observe that H° is a subspace of C 0 A (V) 0 ®. k 1 1 1 = T*. (A^V) 0 ^ O O ) 0 (A {V)®A {V) O-A ^)) © ••• = r*. Both algebras algebras H° and Q(V ) * have a natural grading. and Q(V)* grading. The The part part of of H° H° inindegree degree 22 is is equal equal to to 1 x ker(A1(V) ker(A (^) 0 A1(V) AX(V) -> A'(V) A1^) A AA (y)). This of Q(V)* Q(F)* in A1(V)). This is is also also the the part of degree and Q(V)*are Q(F)*are isomorphic isomorphic in in each each degree. degree. degree 2. 2. One One checks checks that that H° and .

.

Zdzislaw Wojtkowiak Wojtkowiak

234 234

Let Sym(T*) Sym(T*) be the the symmetric symmetric algebra on T*, T*. Let Sh(V) Sh(V) be be the the ideal ideal in in Sym(T*) Sym(T*) generated generated by by the the shuffle shuffle product relations relations

(xiY -...... • xipY - (xip+1 ...•...Xjp+q)* • xip+q)*= =

2 ^ ( \ ( 1 ) • • • • • xia{p+q))*.


Then Sym,(T*) Syrn(T*) divided by Sh(V) Sh(V) isis isomorphic isomorphic to to T* T* equipped equipped with with shuffle shuffle

multiplication.

2.2.4. This This implies that the symmetric implies that symmetric algebra algebra on Q(V)* divided divided by the * equipped with the ideal Sh(V) Sh(V) nflQ(V)* Q(V) * is isomorphic to to Q(V) Q(F)*equipped the shuffle shuffle multiplication. Aig (TT(V)) (ir(V)) be the algebra of polynomial polynomial functions functions on on n(V). ir(V). It It follows Let Alg from from a theorem of of R. Ree Ree (see (see [R] [R] Theorem 2.5) that aaformal formal power power series series P(V) is in P(V) C in the image image of of exp : ir(V) ir(V)—* -> P(V) C Q(V) Q(V) ifif and and only only ifif its its f/ ee P(V) coefficients (i.e. the coefficients of some lifting to C{{X1,.. . satisfy coefficients (i.e. the coefficients of some lifting to C{{Xi,..., Xn}}) shuffle product relations. relations. Therefore 2.2.2 and and 2.2.4 2.2.4 imply imply that shuffle product Therefore 2.2.2 ,

P(V) exp ::ir(V) exp TT(V)—* -> P(V) induces an isomorphism of Hopf algebras induces 2.2.5 2.2.5

R :: Q(F)*-^ Alg (ir(V)). Alg(n(V)).

Combining 2.2.1, 2.2.1, Lemma 2.2.3 and 2.2.5 Combining 2.2.5 we we get:

Lemma 2.3. Lemma 2.3. isomorphic. isomorphic.

algebras H^R{pm~1{x,x)) The Hopf algebras

and andAlg(ir(V)) Alg(7r(V)) are are

Corollary 2.4. 2.4.

The group (ir(V))) are group schemes schemes TT^R(V^X)x) and and Spec(Alg Spec (Alg (7r(V))) >R isomorphic. x) —* isomorphic. We We denote denote by by uu :: 7rf (V,x) —> Spec(Alg Spec (Alg (ir(V))) (TT(V))) the the isomorisomorphism induced by isomorphisms isomorphisms 2.2.1, 2.2.1, 2.2.3 and phism and 2.2.5. 2.2.5.

2.5. 2.5.

Let Let kk be bethe thefield field ofofcomplex complex numbers numbers C. C. We We shall shall assume assume that

A'(V) In section considered the the principal principal 7r(F)-bundle ir(V)-bundle equipped equipped with with the section 1 we considered integrable connection toy We shall relate it to the objects described described integrable connectiongiven givenbybyWV. in 2.1. The group ir(V) acts on Alg (ir(V)) on the left by the formula group acts on Alg(7r(V)) left by the formula a) g(f)(x) := f(x . g) a) 9(f)(x):=f(x.g)

l b) g(f)(x):= := f(gf(g'x) b) g(f)(x) x)

We form form the the associated We associated vector bundle V x ir(V) V xxAlg Alg(ir(V)) (n(V)) ztVx n(V) xx Alg Alg(ir(V)) (n(V))—* -+ V ir(V) 7T(V)

Monodromy of Iterated Iterated Integrals Monodromy of

235

and and we we equip equip it with with the theconnection connection induced induced by by WV. uy. We We shall shall denote denote f m Let tl1is connection connectionbybyu'wç, in the case a) and by 'WV in the case b). in the case a) and ujy the case b). Let this v tyVA[i])/(vd±[i])be k ethe thealgebraic algebraicDe DeRham Rhamcomplex complexofofsmooth smoothrelative relativedifdifA A ferentials on V M (i. . on each V W»). (i.e. each e ferentials be the Let H% (respectively x ^ ° ) be the restriction restriction of of the the vector vector bundle bundle x. Both to X X xx {x} {x} (resp. {x} {x} x X.) Bothvector vectorbundles bundles H°(tRplQ*,vA[l])/,vdA[1])) are equipped with the induced connection, which we also denote by dc. equipped induced connection, which dcOur main result in in this this section section isisthe thefollowing following theorem. theorem.

Then the algebraic Theorem 2.6. Theorem 2.6. Let Let V V be be as in in 2.1. 2.1. Then algebraicvector vector bundles bundles equipped with equipped withconnections connections and

(H°x,dc)

{VxAlg(x(V))->V,ui'v)

algebraicvector vectorbundles bundlesequipped equippedwith with are isomorphic isomorphic in the the category categoryof ofalgebraic algebraic integrable algebraic integrableconnections. connections.

Remark. (xW°,dc) and (V (ir(V)) —* 'WV) are (V xxAlg A\g(n(V)) —>V,V/uy) arealso alsoisomorphic. isomorphic. Proof. We We shall calculate horizontal sections of the connection connection dc on % := recall the the construction of dc- Following Katz #°(£i?p*£l? yA[1])/ , vaA[1] A We recall and Oda (see defined by (see [KO]) [KO]) the filtration {F^}i on £lyn+2 is defined ® (pn)*fiVxV -> ^ i W n + 2 := image image(il^ +2 ®

+ 2

).

The filtrations filtrations {F^i for for nn = 0,1,2,... 0,1,2,...give giveaafiltration filtration F2 Fi of QyA[1]. The connection dc is the connecting homomorphismofofthe thefunctor functorH°(tRpl( H°(tRp( )) connecting homomorphism )) exact sequence sequence of of complexes complexes on VrAt1l for the short exact F°/F2 _>. F°/Fi —+ ^ o. 00 _> F1/F2 Fl/F2 _+ F°/F2 0. n+2 wehave have On V we

i) t)

ii) F°/F2

F°/F1 <

Observe thatOy ov 00 fi^>n ® Observe that ® Ov O V C F°/F2 Let us assume assumethat that 5^ >Wii 0. .®tc;im e€ A(V)®m u;^ .. ^(V)®771isisclosed closed in in the thetotal totalcomcom.

1 plex of fi*(p* ~1'(x,x)). (a:, x)).ItsItscohomology cohomologyclass classccbelongs belongs to HpR(pm ~'(x,x)), (x, x)), the fibre fibre of W over over (x, (x, x) x) eGVVxxV.V. We face two two problems. problems. We We must must extend extend cc to to a continuous section of of % fl We face continuous section and we must show that this section is horizontal. Let Tot (n —* 7 := Tot(n -> show this section is horizontal. T Qv Ov 0®1l*(Vn) Q*(Fn) 0ODV) Oy) be bethe thetotal totalcomplex complexofofthe thebicomplex bicomplex(p,(p,q)g)—f -> Ov Oy 0®

236

Zdzislaw Zdzislaw Wojtkowiak

top(Vq) ® Oy. The element element u^ ® ® ... ® ® u^m can be interpreted interpreted as aa global global Ov• The continuous section section of of T. T. One that the continuous One verifies verifies that the class class of of the the element element .

.®wi X

O
X

e ov

®

is a horizontal section of flan• Uan. (flan (Uan := U get: fl ®ov O^1). Hence we get: :

2.7. Let (a,f3) (x,x)). Then (a,/?) E € iri(V 7Ti(V xXV, V,(x,a;)). Then the therepresentation representation of of iri(V TTI(V x X V,(x,x)) V, (x,x)) on onthe thevector vector space space HjjR(pm ~1'(x,x);Q) (x,x);Q) isisgiven given by

(a, /3) /3) : u)ix (8)... ® 0 vim -> :

.

.

^ , . . . , a;ifc )

0
O
(f

By 2.2.5 the the Hopf Hopf algebra Q(V)* Q(V)* isis isomorphic isomorphic to to Aig Alg (ir(V)). (7r(y)). Hence Hence we we shall consider u>fvon follows onthe theassociated associatedbundle bundleVVxxQ(V)* Q(F)*—*—>V. • V.It It followsfrom from Proposition 1.9.i) that the themonodromy monodromy representation representation isisgiven given by by mi^x)

... • Xik)* -+ ^ ( X , , •... . . . • Xit) 3 a :: (Xt, •... .

(j

.

for the the connection connection u' wç,, and by by v, and

a:

... Xjk)*

...

. ,

for the for the connection connection 'WV. 'uy. . HO4Q(V)* defined —+ The isomorphism isomorphism H°^>Q(V)* denned by wix 0. ®... . gives gives an an isomorphism isomorphism of monodromy monodromy representations of holomorphic vector bundles (H°x,dc)an and (V and (V xxQ(V)* Q(V)*—+ -> F,o;^) an (resp. (resp. (xn°,dc)an an an (V xx Q(V)* V,IwV)an) the point point xxeV. E V. (( ) ) is Q(F)* -> V/a;v) ) atat the is an analytic analytic object object corresponding to to an an algebraic algebraic one one). corresponding ). Observe that that both connections Observe connections are regular. regular. One One can can check check this directly directly for the connection u'v. The The connection connection dc dH(F,x)(C). Let aa eGiri(V,x). TTI(V^X).There Thereisisa acanonical canonicalmap mapiri(V,x) ni(V,x) —+ —>

Monodromy of Iterated Iterated Integrals Monodromy of

237

TT^R(V,

x)(C) (the (the monodromy monodromy homomorphism) homomorphism) and and ititfollows follows from 2.7 that the value value of c on the image image of of a is equal to ^2 fa_1 u^,..., uJim.

Homotopy relative tangential base points on §3. Homotopy on P1(C)\{ai,. P 1 (C)\{ai, • • , •, a n +i}. 3.1. In this section we define a tangential base point on on the the complex complex projective line minus a finite number of of points. points. 1 ..., n + i}. A tangential base point Definition 3.1.1. Let P1(C) {ai, Definition Let X X =P (C) \\ {a x, ...,a on X X is on is aa non-zero non-zero tangent vector vector iiv at at one one of of the the missing missing points points a*. We shall shall denote denote by by vii (or (or by by v) v) the the tangential tangential base base point point defined defined by bythe the tangent tangent vector if. v. identify the the tangent space at a2 If C\ {ai,..., If XX = C\{ai, ..., an} we identify a^ with the vector vector +ii. space C and the tangent vector vector iiv at a2 a* with with the thevector vector aix , ,where where xx == a2di+v. 1 1 The complex complex projective projectiveline lineP1 P (C) (C)has hastwo twocanonical canonicalcharts chartsC C—+ —>P1(C), P (C), z —+ z -> [z [z : 1] and and CC—* —>P'(C), PX(C),z z—*—> [1 [1 z]: which z] which allow allow to to identify identify tangent tangent vectors vectors to P1(C) P 1 (C) with tangent tangent vectors vectors to to C. C. Let say that aa tangential A; be beaasubfield subfield of C. We We say tangential base point if v at aa^isisa a Let k ii vE G k.k. tangential base k-point fc-pointifif a2 ai E G kkand anda2a^+ 4:

:

Definition 3.1.2. 3.1.2. Let X = P'(C) {ai, ...,a n +i} and let v be a tangentangenDefinition Let X PX(C) \\ {a u ..., tial base point missing point X starting starting from from the the tial base point (at (at the missing point a*). a2). A A path path on X tangential base point point ifvis smooth path pathininIF" PX(C), satisfying satisfying the the following following is aa smooth conditions: i) the 1) is contained contained in in X; the image image of of the open open interval interval (0, (0,1) ii) the tangent ii) tangent vector vector to the path path at at a2 a^ isis equal equal to to if. v. We say saythat that aa path path 7'y ends endsatat aa tangential tangentialbase basepoint pointv ifif ifthe thepath path7"'y'1 We -1 (7 (£) := 'y(l 7(1 — —t))t))starts startsfrom fromthe thetangential tangentialbase basepoint pointif.v. (C)starts starts from from vif or or ends ends at at w ill or or starts starts from P 1 (C) from ifv and ends ends If a path 'y7 on IF" at w and if 'y((O, 7((0,1)) 1))CCXXthen then we wesay saythat that 7'y isis aa path on X. X. Definition 3.1.3. homotopy between between two two paths paths -y and 7' -y'on onXX starting 3.1.3. A homotopy 7 and from smooth homotopy homotopy H H8 between the the paths from a tangential base point if v is aa smooth 8 between (C) satisfying the 7y and 'y' 7' on P1 PX(C) the following following conditions: conditions: i) H8((0,1)) Xf for o r ssE G [[0,1]; 0,l]; # s ( ( 0 , l ) )Cc X ii) the tangent the path H8 ii) tangent vector vector of of the Hs at the the starting starting point point isis ifv for s E G [0, [0,1], 1]. between two two paths paths 7'y and and 7' 'y' which which end end at at a tangential base A homotopy homotopy between base 1 point vif isisaahomotopy homotopybetween between paths andj -y'' point thethe paths 7"-y' and ' ~ l which which start at ii. v. According to this this definition, definition, the the paths a, According to a, /3 f3 on on CC\\ {0} {0} shown shown in in the thefolfollowing figure figure are are not homotopic. lowing homotopic.

Zdzislaw Wojtkowiak Zdzislaw

238

w

Definition 3.1.4. Let 'y be a path starting 7 be starting from from aa tangential tangential base base point point vil (a tangent vector at a^) and ending at at the thesame sametangential tangentialbase base point v. point 'US. We say say that 7 is null homotopic if there is a smooth smooth homotopy homotopy H3 Hs between between 7'y and map into into a^ such such that: that: and the constant map i) H3 i) Hs is a path on X X starting starting from from v and ending ending at v for for s < < 1; ii) H1 ii) Hi is the constant map into a^. If 7 is a path on on X X we wedenote denote by by [7] [7] the the homotopy homotopy class class of of 7. 7. Suppose Suppose that aa path (p on on XX ends endsatataatangential tangentialbase basepoint point v and and aa path path ipon on XX path ço from ii. v. We We define define aa composition composition of of homotopy homotopy classes classes [u']. [ip] • [>00be besmall. small. Let Let (fi-£ := := y?|[o,i-e] that ip and rp coincide assume that <^(1 —— e) == ip{e).We Wedefine define the the and ^ e := ^|[e,i]- We can can assume composition ?/>e • ^ i _(with (with a necessary composition of paths ip •
n+1

1 X = XU of X X and the tangent Let X XUU |J (T tangent spaces spaces (Taj(]N(C))\{0}) a. (P (C))\{0}) be a sum of

i=l

minus zero zero at at the missing points. We define defineaagroupoid groupoidPP over overJXx xXXi nin the missing points. X x following way. If (x, y) E X then is the set of homotopy classesof of following way. (x, G X x X then P^y is the set of homotopyclasses paths on X and ending ending atat y. y. We We set set PP = U,x X starting from from xx and y)exxx^^y (#, 2/)partial We define -> X x X by by p(Px,3/) = (x, define aa projection projectionpp :: PP -÷ y). The partial —>kXx xkXinto intoa groupoid a groupoid composition of homotopy classes classes makes makes p ::PP—* composition over XX xxX. X. The The fibre p~1(x,x) is is fundamental over fibreover over(x,x), (x,x),TT^X^X) iri(X,x):= :=p'(x,x) thethefundamental Observe that thatxxcan canbe beaatangential tangentialbase base base point point xx EG X. Observe group of of XX at aa base point ii. v. 3.2. We (z; ii, is aa tangential base point. We shall shall construct construct functions functions Ax Ax{z; v, 7) if vii is Let us us set set XX = C\{ai,..., XQ eGCC and andlet let6#: :[0, [0,1] —>> a;+ + t • Let 1] 3 t t—+ C\{ai,.. , an}. Let x0 a (x0 (#0— ~~a) i) be be an aninterval intervaljoining joining a2 ai and and x0. XQ.Let Let 7 be be aa path pathfrom from a2 a^totozzE£XX (not passing passing through tangent to toS5in in a.a*.We Weassume assume (not through any any a^, ak, fc k = 1,.. .71) n) tangent .

.

Monodromy of of Iterated Iterated Integrals Monodromy

239

that in in aa small small neighbourhood neighbourhood of a* the path 7 coincides with 6. 5. We identify identify v3= — XQ - cii with a tangent vector vector to to C C in in a^. Let w1 u)\ = — ^z^7? • • • > ^n =— ztzan • We set

-i-,..

Wai,. .

if ax / Li.

. ,

J a,i

from e to z, im(8) be near a*. Let 7^ be a part of Let e £e im(S) of 'y 7 from 2, and let let 8e S£ be a part of of 86 from from £ to x0. :co- We We set set

-\, \

r

dt dt

f

dt dt

t—

:

A

= }™ lim IJ

t — a2

a i i tr(*,...,*)(*) w

if

and •

•

:= /

, Aa $(i,.. .,i,ak+i.. .

Jatn

.

.,ai)(t)ujaa/+l

. ,

if a^-f 1 7^ z, andl and / >k+1. k + 1.

Lemma L e m m a 3.2.1. 3.2.1. The T/ieintegrals integrals Aaii#(aii

exist and and they • • • ^fcX-z) exis^ t/iey are anaana•

,

lytic, rnultivalued functions on on X. /2/^'c, multivalued functions X.

Proof. Proof. Assume Assume that at ^ i for for t << II and ai+i = i. The function function g(z) := f*. (jjai,..., uJai isisanalytic analyticmultivalued multivaluedononXXU U{a2} {a^}and andvanishes vanishes in in a. a^. Hence the integral g\{z) := J^. ^(^)^z^: exists, the function g\(z) is analytic Hence the integral g1(z) := f g1(z) •

,

XU U {a;} multivalued on X {a2 }and andvanishes vanishesinina^ a2.Hence Henceby byinduction induction we we get get that that A a i ,v(^i,...,, an)(jz) exists and it isis analytic, analytic, multivalued multivalued on onXX UU {a2}. {ai}. Aa1,v(ai,. Assume now now that that at = <1I and — i for for tt < and a/+i 7^ i. Without Without loss loss of of generality generality we can can assume assume that that a2 Observe that a^ = 00 and x0 XQ = 1. 1. Observe . .

m

lim

f (logz)mdz =

Zdzislaw Wojtkowiak Zdzislaw

240

where where /% are rational numbers. numbers. The The function function zq(logz)p for q and p positive positive

integers, integers, is analytic multivalued multivalued on X, X, continuous continuous on on any any small small cone cone with with a vertex in a; (0 in this case) and itit vanishes vanishes in a^. The function function j ^ - for Hence the the on any any sufficiently sufficiently small neighbourhood neighbourhood of a*. Hence j 7^ ii is bounded on integral lim im urn

ff

.

,

j)(t))

dt t — a3

is an analytic inultivalued multivalued function function on V, V, continuous continuous and univalued univalued on any small cone with a vertex vertex in in a* and it vanishes vanishes in a^. This also implies implies that the last integral integral exists. exists. 0 G

We recall that that X1,. us set We recall X x , . . . , Xn are duals of 7^7, • • • > jz^-LetLet us set . . ,

. 'y) = 11 + $^(-l)*A o < | i r (ai,...,a*) Ax(z; ... .• Xai. AxOz; ff,7) (*)*<,, .•... .

,

Observe depends on on v, ii, i.e. i.e. on x0. (2; iT, #,7)y)depends XQ. Observe that Ax Ax(z;

Lemma 3.2.2. 3.2.2. The The map map

Xxx P(X) X 3 z -> (z,A X e X P(X) (z,Ax(z;ii,y)) x(z;v,j)) is horizontal horizontal with with respect respect toto UJX • Proof. We We have have dA . a

.

= Aa,,ti(ai, .

. .

system of of differthe functions functions (—l)fcAa<)j?(ai, • • • »«fc)(^)satisfy satisfythethe system differential defining horizontal horizontalsections sectionsofofthe the principal principal bundle bundle XX x ential equations equations defining P(X) —>XXequipped equippedwith withthetheconnection connectionwx. LUX0 P(X) -+ Hence

. .

,

It remains (z; iT, remains to to define define functions functions Ax Ax(z; #,7)'y)ififa^a == oo 00 or all ai are different different x from oo. Let Let /f :: Y -> X = P (C)\{ai,... , a n + i } be a from 00. Y = C\{6i,... ,bn} —*

regular regular map of the form form ^ ^ with det det (

J ^ 0. 0. Let Let yy E GY Y and and let let

-y be aa path on 7 be on Y Y from from y2/totoz.z.Then Thenit itfollows follows from from corollary corollary 1.7 1.7 that that if ifyeY. y V. We We shall shall use this fact fact to to /*(A y (z;y,7)) = A Ax(f(z);f(y),f(7)) x(f(zyj(y)J(j))

define Ax (z\ (z; v, ii, 7) -y)where whereviTisisa atangent tangent vector vector to P 1 (C) in a2 a^ and 'y 7 is a path path

from iT.v. from a2 a,i totoz,z,which whichisistangent tangenttoto We set

(3.2.2.1.)

Ax(z; iT, 'y) := /•(Ay(/- 1 (z);/,- 1 (fl),/ A^(z;t;,7) f1ey))).

Monodrorny of Iterated Iterated Integrals Monodromy of

241 241

not depend depend on on the the choice choiceofof/f.. Thus It isis clear (z; ii, does clear that thatAx Ax(z;v^) does not Thus we we have the the following following lemma. lemma. 1 Lemma 3.2.3. Let Let X X = P P1(C) (C) \\ {ai,... {ai,.. ,a n +i} and let v be a tangential .

base point. point. Then Thenthe themap map X

z —*

e Xx P(X)

is horizontal respecttotoWX. UJX • horizontalwith withrespect logAx (z; v, ii, 7). -y).IfIfwe weare aredealing dealingwith with only oniy one space (z; ii, We set Lx{z\ v, y) 7) := \ogAx{z\ X we il, 'y) we shall usually usually omit the subscript subscript X X and andwe weshall shallwrite writeA(z; A(z;v, 7) and L(z;v,7), or Ail(z;'y) A^(z;y) and Lil(z;'y), L#(z;7), or or even even Ail(z) A${z) and Lil(z). L${z). L(z;ii,'y), or The constructions from from 3.1 3.1 and and 3.2 3.2 are are summarized summarized in inthe thefollowing following propoproposition. sition.

Proposition 3.2.4. X.The Thefunctions functions Ax(z;v,'y) Ax{z\v,'y) and and Lx(z;v,'y) Lx(z;v^y) Proposition 3.2.4.Let LetvvEGX. depend only on the homotopy class class of ofj'y in Pvz . §4. Generators of 7Ti(P 1 (C)\{ai,... ,a n + i},x).

Let X X = ]P1(C)\{aj,. . let xx G E X. a n + i} and let X. Let Let vl be beaatangent tangentvector vector Let P 1 (C)\{ai,..., is the following element in a&. Then the loop around at the base point loop a& at the base point v*k is the following element iri(X, v*k)(see 5^. of of?ri(X, (seepicture). picture). . ,

svi vi

Now we weshall shalldescribe describe how howto tochoose choosegenerators generatorsof ofTTI iri(X, (X, x). Let Letus uschoose choose a tangent vector v{ G Tai (P^Q) \ {0} for each i = l , 2 , . . . , n + l . i) If ofpaths pathsFr = {7i}"J"11 on X X from If x eGXXthen thenwe wechoose choose a family family of from x to each each ^ such that any any two two paths paths do do not not intersect intersect and and no no path path self self intersects. The in such such a way intersects. The indices indices are chosen chosen in way that when when we we make a small circle circle around around xx in the direction, starting from small the opposite opposite clockwise clockwise direction, from 7j, we 71, we meet 72,73,. 72,73,...,, 7n+io eac ii) If xx = v{ family of of paths paths F = {7i}^J21 from each v1 then we fr°m xx tto h ii) If we choose choose aa family such that making a small circle around a1 in the opposite clockwise Vi such making a small circle around a\ in the opposite clockwise direction, starting starting from from 72 72 we we meet meet 'y3,. 73,... ,,7 n +i. iii) We element Si S: in in TTI(X, iri(X, X) x) to the path iii) We associate the following following element path 7i. 7*.We We move move along 7i, 7z, we wemake make aa small small loop ioop in in the the opposite clockwise direction around clockwise direction . .

. .

Zdzislaw Wojtkowiak Wojtkowiak

242 242

a2 and and we wereturn return to x along 7" 1 . In the case ii) the element di element S1 S\ is the loop loop

sSv-i. v-x.

We can can choose any order order of of missing points to to construct iv) We iv) choose any missing points construct the the elements elements

S1,...,Sn+I. The following following lemma isis obvious. obvious.

Lemma 4.1. S1,S2, 52,..., areare generators of o/7Ti(X, iri(X,x). X). WeWe 4.1. The Theelements elements Si, • • • > ^n+i generators have = 1. •.. . S2 S1 have 5 n +i •... • S2 * S\ — 1. .

Definition iri(X, x) Definition4.2. 4.2.The Theordered orderedsequence sequence (S1,. ( £ 1 , . . . , S^+i) of elements of TTI(X, X) obtained from from the the family family of of paths paths FI' as in i) ,iv) will be called a sequence obtained i),...,iv) sequence of geometric generators of of TTI(X, iri(X, X) x) associated geometric generators associated to to F. F. . .

,

..

§5. Monodromy of of iterated iterated integrals on P 1 (C)\{ai,..., (C)\{ai,. a n +i} §5. Monodromy . . ,

Let XX = PIpi1 (C)\{ai,. . Let (C)\{ai,..., a n +i}. We want to compare compare functions functions Av(z) for different base points. Let x3 6 EX different Let x1, £1, x2, £2, #3 X and and let let z0 ZQE€ X. X. Let Let62Sifor fori i==1,1,2,3 2,3 1 be a path on X from be from Xi to z0. z$. Let Let us us set set 7^ := 5" o J^. .

Proposition 5.1. Let us continue (z) along Proposition 5.1. Let continue each each function function AXi(z) along Si to the point z0. ZQ. There There exist exist elements elements 0^.(7*^) EG P(X) P(X) such suchthat that ('yjj) =

(z) .

(z)

for for all all zz in in aasmall small neighbourhood neighbourhood of of zz0. The elements elements a%i.(/yij) satisfy satisfy the the 0. The following following relations relations

=

1,

('yjj) = .

aXk(yjk)

1,

=

(Observe that the ('yj3)depend depend on on the the choice of paths 'yjj.) the elements elements a^. (7^) choice of 7i?.)

Proof. The ('ye,)follows followsfrom fromthe thefact factthat that the AXi(z)'s The existence existence of a** (7^) are (z)'s are horizontal sections. The fir^t two two relations relations are are obvious. obvious. The The last relation horizontal sections. relation follows from the equalities follows from equalities Ax.(z) .• a** (7^) == AXj(z), AXj(z) . • axi(
If path on X If -y 7 isis aa path X from from vv to to w w then then we we define define .

=

where Av(z) and Aw(z) are continued along along y. 7.

0 0^(7) by the formula formula

Monodromy of of Iterated Iterated Integrals Monodromy

243

Proposition Proposition 5.2. 5.2. Let v*kEGTaklF1(C)\{O}. TafcP1(C)\{0}. Let Let Sv-kbebea aloop looparound around ak 1 based (P1(C))\{O} (see figure at the beginning of section TheThe basedatat v*Ek Tak G Tak (P (C))\{0} (see figure at the beginning of section 4)monodromy of Av-k (z) Sv-.k by by (z) along along Sv-*isk given is given A — I'

A

•

(

\

e —2iriXk

Proof. The Proof. The monodromy monodromy of Av-k(km)(z) := Av~(fc, &,...,,k)(z) k)(z) along along S,5L S^ is .

. .

In

.1

given by Sv-*k : A,* (*")(*)—* -> Av-k(km)(z)+ + £ K-h(km-l)(z)l=%±. LetLet given :

•

be a path on 7'y be on X X from from v*ktotoz zand andletlet ai ^ k.fc.We Wemust mustanalytically analytically n continue the function continue function Av-k (fc , a i , . . . , a7,)(z) ap)(z) along the composition composition 'y7 • 5 ^ . can be made arbitrarily The contribution contribution along along Sv-k can arbitrarily small. small. However However the function Av-k(ktm) that the changesafter after the the tour along function (km)(z)(z)changes along Sv-k. This implies implies that m (km,aj,.. monodromy of Av-k (A: , a i , . . . , ap)(z) along along Sv-i isis given given by by . .

.

,

. ,

Sk /(

Av^(fc

o

•\j -\l

, 1!

1=1

for for a1 a\ / k. The The forniula formula for for the the monodroiny monodromy of Ay*(z) along 5 ^ follows follows k (z)along from this. from this. 0

Let XE E T^P^QXJO} Ta.P1(C)\{O}for for ii = = 1,2,...,n+ x GX. X. Let Let us uschoose choose v{ G 1,2,..., n + 1. Let of geometric geometric generators generators of of TTI(X, iri(X, X) x) associated ((S1,.. 5 i , . . . , Sn+i) be a sequence sequence of I' = {ji}^i (if (if xx G E X) to F X) or or I'F == {7i}^ 2 1 (if (^ xx is *s aa tangential tangentialbase basepoint, point,i.e. i.e. xx=vi),whereFisafamilyofpathsonXfromxtot3fori= = tJi), where F is a family of paths on X from x to t^ for i = 1,2,..., 1,2,...,n+1 n+1 or ii==2, or 2 ,... . .., ,n+las n + 1 as in section 4. 4. . ,

Theorem Theorem5.3. 5.3.The Themonodromy monodromyof ofthe the function function Ax(z) along the loop Sk Sk isis given given by by Sk :

—*

Sk : Ax{z) -»• Ax(z) • (al*^))'1

• e-2«iX* •

af{lk).

Proof. It follows from Proposition 5.1 that Proof. It follows from Proposition 5.1 that = .

for z in the (t). This the small small neighbourhood neighbourhood of of some some 7fc(£). This equality equality isis preserved preserved

after after the inonodromy monodromy transformation along along Sk, S&, hence hence we we have (*2) (*2)

{Ax(z))s» = {kv-k{z))s» •

af{lk),

Zdzislaw Wojtkowiak Wojtkowiak

244 244

Sk denotes the function function ( ) after after the the monodromy monodromy transformatransformawhere (( )) denotes tion along Observe that that (Av-k(z))Sk = (Av-k(z))Strk. It follows follows from from along Sy>k. Observe Proposition 5.2 that Proposition 5.2

(Av-k(z)) =s*=Av-h(z).e-2*iX*.

(*3) (* 3)

Substituting (*3) then substituting (*i) for Av~k(z) yields and then yields the (*3) in (*2) and formula for for (Ax(z))Sk. 0

Corollary 5.4. Corollary 5.4.The Themonodromy monodromyof of the the function function Lx(z) along the ioop loop Sk Sk isis by given by Sk Sk ::

where where

Lx{z) —* -> Lx(z) • KHTfc))"1 • ( .

af{^k) == \og{af{^k)).

immediately from from Proposition Proposition 1.8. Proof. The The corollary corollary follows follows immediately 1.8. ii).

0

Definition 5.5. 5.5. Let Letxx E€ t.X.Define Definea ahomomorphism homomorphism

P(X) (resp.^, x : irj(X,x) TTipT,*)—* -> P(X) 7n(X,x)—* -> ir(X)) n(X)) e6x,x x,x : iri(X,x) by the formula formula

= =

1 v_•

—1

*iX

af (7fc) (resp. (resp.

9XtX(Sk)

.

called the the monodromy monodromy homomorphism homomorphismofofthe theform formux wxatat the the base base point point it is called x. Observe that xx can can be a tangential base point. point. If If xx E X then the Observe that tangential base €X the homohomomorphisms from from Definitions Definitions 1.3 and 5.5 5.5 coincide. coincide. Let v,v' E TTI(X, V) and and Let v,v' £X X and and let let 7 be be a path on on X X from from v to v'. v'. If If irj(X,v) iri(X, v') are identified via the path 'y, then the homomorphisms 9v,X and 7Ti(X, v') are identified via the path 7, then the homomorphisms 6v,x and #v',x are conjugate. conjugate.

Proposition 5.6. 5.6. Let Let

regular bijective bijectivemap. map. Then Then for any path be a regular for any v, v,ww EG X, X , any path 'y 7 from to zz from v to and any and any path path 5 S from from v to to w w we we have have

v, UAx(z;v,'Y))=A Ay(f(z); 1(v),1)) f(-y)) Y(f(z);f(v)J(

and

Monodromy of of Iterated Iterated Integrals Monodromy f

/

245 245

f(v)

— af(W)

f(v) := f*(v) if vv is aa tangent tangent vector). vector). (notation: 1(v) Proof. The The proposition proposition follows follows from from the the definitions definitions of of Ax Kx{z\v^) (z; v, 'y) and and a^ for tangent vectors, for vectors, and and from from Corollary Corollary 1.7. 1.7. 0

j6. Calculations. §6. Calculations. Let V V = P 1 (C)\{0,1, oo}. The The forms forms & and ^ form form a basis of A'(V). Ax{y). Let X and Y Y := ( ^ ) * be the dual dual basis basis of of (A^V))*. Set Set Z := Let X := (&y and —X — — Y. Y. The The group group P(V) P(V) isisthe thegroup groupofofinvertible invertiblepower power series series in in nonnoncommuting variables equal to 1. 1. commuting variablesXX and and YY with with constant constant term equal Fix a path 71 = interval [0, 1]from from01 01toto 10. iö. It follows [0,1] follows from from Proposition 5.1 the path 71 we have 5.1 that along along the (0)

Y) =

Let 1(z) = 1l — — z. Let f(z) z. ItItfollows follows from from Proposition Proposition 5.6 5.6 that that

= (We omit omit the arrow over vectors, when it does (We does not not cause cause confusion.) confusion.) Proposition 5.1 implies implies that (1)

=

1.

Observe and f*(Y) = X. X. Hence we get Deligne's formula Observe that /*(X) = Y and formula (2)

Y) . X) == 1.\. al\{X,Y).al\{Y,X)

(The proof of (2) given here essentially essentially repeats that that ofofDeligne Deligne (see[D5]).) (see[D5]).) Let us us fix fix aa path path'Y,c, 7 ^ = interval [0, [0, e] arc from from eetoto—e — epassing passingthrough through e] 4+ arc (i) e •+einterval [—e, (-i) + interval [-e,oo] 00]from from iii Ol to to 00O where where ee >>00(see (seefigure). figure). Yl

Let S0 0), Si So (around 0), Si (around (around 1) 1) and and SQQ (around (around oc) 00) be be geometric geometric generators of of iri(V, 01) associated associated to to the family generators TTI(V, 01) family {71,700} (see ( s e e picture on picture on page 5). Then Then we wehave have S0 So .- S1 Si - 5 ^ == 1.1.

246

Zdzislaw Wojtkowiak

Theorem 6.1. Theorem 6.1.The Themonodromy monodromy of A^z) is is given given by by the the following following formuformulas: las: —* A ol (z) • e SSoo :: A ol (z) -> (3) S1 5i : :A0-1(z) A ol (z)—-> Aol(*) • « ( X , r ) ) " 1 • e^2^Y • aJ?(X, F), iX —+ A ol (z) • e~« Soo : A ol (z) -> • « ( X , Z))" 1 • e<-a*«>z • < ( XZ) , Z) :

from TheTheProof. The The formulas formulas for for the themonodromy monodromy along along 5o So and andS1 Si follows follows from needs some explanation. By Theorem orem 5.3. The monodromy along orem S^ some explanation. 1 5.3 it is given by ( z ). > ^ ° ((X, X , Y))' Y))-.e(_2lTi)Z. -^- 2 ™^by the the formula 5oo : A ol (s) (z) -> A o l (z) Y) = Y) • a81'°(X, Y). One a%?(X,Y). By Proposition Proposition 5.1 5.1 a^°(X,Y) = a%£(X,Y) a$°(X,Y). One Y). By f(z) = calculates using using 3.2.3 3.2.3that that a(Jf(X,F) a8?°(X,Y) = eniX. Let /(z) We have calculates ^ f j . We have = Z. It It follows follows from (X) = X and and /*(y) (Y) = Z. from Proposition Proposition 5.6 that :

=

oJ?(X, Z) =

Hence we get get the the formula Hence we formula describing the the monodromy monodromy along along SQQ.

§

We recall that that the We recall the Lie Lie algebra algebra L(V) is is the thecompletion completion of of the the free free Lie algebra over C C on two generators X and algebra over generators X and Y. Y. The group group ir(V) n(V) is is L(V) equipped with the themultiplication multiplication given given by bythe theBaker-Campbell-Hausdorff Baker-Campbell-Hausdorff equipped formula. Let us set a(X, a(X,Y)Y) := a$(X,Y)Y) := log(oJ2(X,y)). follows Y)). It follows z s ven from Corollary the monodromy monodromy of L o l((z)) is ^ given gi by by the thefollowing following from Corollary 5.4 that that the formulas: :

—*

.

(—2iri)X,

(4) a(X, F ) " 1 •(—2iri)Y (-2?ri)y • c*(X,Y), y), (4) Si Si:: L ol (z) -> —+ L ol (z) • a(X, (—2iri)Z . - o:(A, Z)Z) . • (7^^jA. oo : LQ^(Z) —>• ^ o i v / . *(—iri)X v ^ v ^ - ' ^ v ^ ^ A) ' \—*'R%)Zi .

.

.

:

Observe that Theorem introduction) Observe that TheoremAA(in(inthe the introduction)follows followsfrom from the theformulas formulas (4). Let (4). Letus uscompute compute the thecoefficients coefficients of of a^(X,Y) and a(X,Y). If w is a monomial in X and Y we denote by a(w) the thecoefficient coefficient of agi(X, Y) at at w. w. Let X be bethe thefirst firstLie Liebasis basis element element and and let let Y Y be bethe thesecond second element. element. We choose choose a basis of a free free Lie algebra on X X and andYYasasinin[MKS] [MKS]pages pages324-325. 324-325. If w is is an anelement element of ofthis thisbasis, basis,let letcE(w) a(w) be be the coefficient of w Y). the coefficient w of a(X, a(X, Y). n n Observe Observe that the thecoefficients coefficients a(X ) and and a(Y ) vanish. vanish. We We assume assume that zz approaches the formula approaches 1 in the formula (0). Then Then ititfollows follows from from the the formula formula (0), (0), the the formula formula (1) and the the formula formula (2) that that = (-l)"C(n + 1), (5) a(XnY) = (6) a(XZY3) =

f

a(YnX) = (-l) n + 1 C(n + 1)

Monodromy of Iterated Iterated Integrals Monodromy of

247 247

and i j i j a(XZY3) + a(Y2X3) a(X Y )+a(Y X ) = 0. 0.

w i-times.) (If one-form then UJ1 := O;,CJ, ...,u> i-times.) Let Letususset set ctij := (If a; w is aa one-form X)Xz—1Y3—l), wheretwX2YJ .)X)Y) .)Y). ((...((w, X)X) )X)Y).. a((y, X ) ^ * - 1 ^ - 1 ) , where i ^ y * -= ((... ((w, X)X).. . ^ . CComomparing the the coefficients coefficients at atY3XZ Y^X1 and and X*y^' in in a(X, Y) Y) and and in in log(aJ;J(X,Y)) we get = (_1)ila(YJX2) = (_1)2a(XiYi). .

. .

.

It follows fromthe theequality equalitya^Y^)+0,^X3) a(XZYJ)+a(YZXi) = follows from = 00 that aitj = ajyi. Observe { j j This last equality that we also have = 0. 0. This we also have a{X Y ) + + (-1)*+*a(Y X*) = equality can also be be obtained obtained from formula (1.6.2) (1.6.2) in [Chi]. also from formula 7 (ir', ir') and ir2(V) (ir(V), ir(V)), ir" Let us us set Let set ir' nf := (ir(V),n(V)), *" := (TT',^) TT2(V) := ^(V)/^ . (V, 01) —* It follows from (3) that the follows from themonodromy monodromy homomorphism homomorphism 6^ : ir1 TTI(V,01) —> ir2(V) is given by :

SSoQ —*(—2iri)X, ->(Si —+(—2iri)Y + [—2iriY, Si -^(-27rz)y + [-2?rzy, a(X, Y)]

(7) (7)

= = (-2ni)Y++

£ (27ri)a i=O,j=O i=O,j=O

Let us set F(z) = Let us set F{z) =

f

The formula (los(

^ z ) ) "'. The formula =

(which can be proved by induction) implies that = by induction) implies that (which can be proved f (log(1 — (8)

a n + h m ={ L) y > 71.TH. 6.2. Suppose that 1 for i = l,...,k— land (8)

2. Then we have

6.2. Suppose that rti > 1 for i = 1,..., k — 1 and n^ > 2. Then we have oo .

1

.XTh21YXTh11Y) =

(i1 + j2)n2

.

.

These numbers are values values of the functions functions (i1 + j2)82 ..

.

+ ... +

)Sk

.

(i1 + . . . +

Zdzislaw Wojtkowiak Zdzislaw

248

at z = 1 and ( s i , . . . , sk) = (n x ,..., nk). atz=land(sl,...,sk)=(nl,...,nk). §7. Configuration spaces. spaces. and V V' = the Let V V= = V1(Q\{au...,an+1} and = P ^ Q Y K , . . . , < + 1 } . If If the Let / x') are sequences (a, x) := (ai,.. x) and (a', x') . sequences (a,x) := ( a i , . . . ,a n + i,x) and (a^x') := (a^,... ,a^ , + 1 ,x ) close then then the groups iri(V, x') are close TTI(V, x) X) and and ir1(V', TTI(V, a;') are canonically canonically isomorphic. We We shall study how how the themonodromy monodromy homomorphisms homomorphisms 9XiQr:=:=6X,V 6xy and &nd #x',a' ,a' :=: = ,v' from 55 9x>y> fromsections sections1 1and and .

. ,

ir(V)

iri(V,x)

Ox,a:

II

iri(V',x')

—*

depend depend on aa and a'. a'. topological space space we set T? = {(t {(t1,.. If T is is a topological G T n |It2 U ^ t3 ^ ifif ii ^ u . . . , £„) e 1 jj}. } . Let Let Wn := C™. The space space A (TVn) of global holomorphic holomorphic one-forms one-forms dZj=dZj on Wn with logarithmic singularities is spanned by by ujij — d ^~^ Z j for for i,ji,j e d b e their {1, m} and <j. formal duals. duals. We set { 1 ,2,. 2 , ... . ,, n} and ii < j . Let Let Xy = ( ^ l f i ) )* * be their formal set 2 0. In A (Wn) we have Xji = Xij and and X« = 0. .

Vki A CJ^ 4- cjjfc A A LUki H- cJij A Wjk u ; ^ = 00

for z, i, j,j , kA;different. different. These Theseare arethe theonly onlyrelations relationsbetween betweenthe the2-forms 2-forms ujijAü.'jk (this result is is due to to V.I.Arnold). V.I.Arnold). Dualizing the Dualizing the map map

A\A1(Wn))-+A1{Wn)AAA1(Wn) we that R(Wn) is is generated by we find find that generated by i, j, k different [Xij,XiX2k 4different + Xjk] X3k] with with i,j,k k

and

with ii,j,k,l (see [C]). [Xij^Xki] with , j , k,l different different (see [C]). abase x= E W n +i be a = ( x i , . . . , xn, x n + i ) G base point. point. Let Letp2 pi :: W n + i —+ -> Wn (i = 11,., . . . ,, nii -h + 1) be a projection projection pj(Zi,. p i ( z i , ... . , z n + 1 ) = (zi,.. ( z 1 ? . . . ,, zh . . . ,

Let Let

.

.

.

,

.

let x(i)

:= (x1,

.

.

.

,

xi+i,. .

and let let V(i,x) F(i, x) := p " 1 (x(i)) = C\{xj,.

. ,

.

Monodromy of Iterated Iterated Integrals Monodromy of

249 249

V(i,x) — —* be given by k2(z) = (2 (z means means z is isomitted). omitted). Let Let kki : V(i,x) >> W n + i be given by ki(z) = :

(x1,. (#1,..., Xi-i, z, Xi+i,..., x n +i). The inclusion inclusion k2 ki induces induces

: P(V(i,x)) —+ and

: ir(V(i,x)) One

calculates calculates that

(ki)*

_dzi) = dz /dzi-dzj\ dz (dzi-dzj\ . , . , . _dzi) == 0 if I ^ iandj and (ki)* = and % and j / i. zi—zi z—xj \ zi—zi Zi — Zj / Z — Xj \ Z\ — Zj J

This implies 11 implies that (ki)*(Xj) = Xij for for j = — 1,2,...,i— 1,2,..., % -1,i+ 1, i +1,...,n+ 1,..., n + follows from the form of the where X is the formal dual of It where X/ It follows from the form of 3 is the formal dual of jz^~z—xj theLie Lie subalgebra subalgebra of of Lie(Wn) generated generated by generators of of R(Wn) that the by X^i,.. .Xij-i, Xiyi+i,... ,X i ? n + i is is free free and it is isaaLie Lie ideal ideal of of Lie(Wn). and it .

. . . ,

.

Hence injective and its image, (ki)*(n(V(i, x))) is a a normal Hence the map (/?»)* is injective subgroup of 7r(Wn+i). Let and x' x' = = (xi,. E Wn+i and Let x = = (a?i,...,a: n ,x n+ i) G (xi,... , ^ , x ^ + 1 ) eG W n +i. Let + l,x) 1,x) and V' =: + 1,x'). V := V(n V(n + and V =: V(n + l,x'). We Wechoose choose aafamily family Let us set V of non-intersecting x1 to to x[,..., xc,. . xn to x^ non-intersecting paths pathsFYi, 7 1 , . . . , 7 n , 7n+i in C from from #1 and xn_|_i to to x'nJtl. We shall identify identify 7Ti(V,a;n+i) and Ki(V,xfn+1) in in the the following way. Observe Observe that that 'y following way. 7= = (71,..., 7 n , 7 n +i) is aa path in in Wn+i from ('yi,. s x to to x'. x'. The Theidentification identification isomorphism isomorphism 'y. 7. : 7ri(F,a;n+i) — —+> 7i"i(^',^n+i)isi unique isomorphism isomorphism making the following following diagram commute the unique .

. ,

. .

. ,

.

:

^

7Ti(W n+ i,x)

i 7# 11,1

\

(TI?

lrflv ,xn+l) F

/

induced by by the the path 'y 7 in aa standard way). way). is induced

Proposition 7.1. After the identification of 7.1. After identification of of the the fundamental fundamental groups groups of V == C\{xi, V C\{xi,... ,xn} and V' V= = C\{xc,, C^x^,.. .,xfn} via via 'y, 7, the the monodromy monodromy homohomo. . . ,

.

. ,

morphisms 0Xu+ltv ' *i(V,xn+1)

-> ir(V) ir(V) and and 0x>n+1,v : ^i(V',x'n+1)

—*

:

4-+R(V') n(V) =ir(V) n(V)

are conjugated conjugatedby byan anelement elementofofthe thegroup group 7r(Wn+i).(The (Thegroup group (kn+i)*n(V) is a normal subgroup of so normal subgroup ofTt(Wn+i) so 7r(Wn-|_i)acts actson onir(V) K{V)bybyconjugation.) conjugation.)

Zdzislaw Zdzislaw Wojtkowiak

250

Proof. The The result result isisaaconsequence consequence of of the the following following commutative diagram: diagram: * ^ £

TT(V)

(3)

(1)

(4)

9,

/

1

(k7(2) ir(V') = ir(V) 1S a

by the the element It where c ^ (a;';a;,7) is a conjugation conjugation by element Lwn+I(xf]x,j). It followsfrom fromProposition Proposition1.4 1.4that that the the square square (1) commutes. follows commutes. Corollary Corollary 1.7 1.7 implies that that (2) and (3) implies (3) commutes. commutes. The The square square (4) (4) commutes commutes by the conconstruction. 0

Corollary 7.2. Let V(i) := := V(i,x). V(i,x). Let Corollary Let x = (xi,.. .,x n +i) E€ Wn+i. Set Se£ V(i) Let a23 be beaageometric geometricgenerator generator ofni(V(i),Xi), of iri(V(i), xi), which dij which is aa ioop loop around around the the point point x3. Let its image in 7Ti(Wn+i,a;). Then 0x,wn+1(Aij) is conjugate to Xj. Let A23 Aij be its (—27ri)Xij in the the group group 7r(Wn+i). from Theorem 5.3 that ^Xi,y(i)(^ij) is conjugated Proof. It follows follows from Theorem 5.3 conjugated to to (—27ri)Xijininthe thegroup groupir(V(i)). n(V(i)). Hence Hencethe thecorollary corollaryfollows followsfrom fromCorollary Corollary 1.7. 0 Now we shall shall study study the the relation between Now we between the monodromy monodromy representations for for the the configuration configuration spaces spaces (P 1 (C)\{0, l,oo})? and ( P ^ C ^ O , l,oo})™. We shall use use the the Ihara result (see We shall (see [Ii] [II] The Injectivity Injectivity Theorem Theorem (i)). (i)). 3 Let us set set Yn = (IP"(C)) (P^C))" and yn = (IP'(C)\{O, (P^qXfO, l,oo})^" . The group group PGL2 (C) acts diagonally on Yn and yn = Yn/PGL2(C). (C). The PGL2{C) The identification identification is given given by by the the map (0, 1,00, —* (xi, . , x n _ 3 ). Let ^ : (0,1, oo, xi, xux2x2,. ,..., . x n _ 3 ) -> (xi, X2, x2,..., —* the composition of the map (x11. —+> Xk_1,Xfc+i,..., Xk+1,.. x n ) — W n _i -> 3^n be composition map ( x i , . . . ,, Xfc_i, (x1,. . .,Xfc_i,oo,Xfc+i,.. —+ yn. The map (xi,.. and the the projection projection Yn —t , xkl, 00, xk+1,. . .,x n ) and —* H(yn). Let ipk induces (fc)* : ff(W n _i) -> Let us us set set X^- = {i/>k)*{xij) = )* where (We use the same notation for E ,

. .

. .

.

.

.

,

,

:

where X^ = ( ^.Z^ )* ^ ^(W^n-i)- (We use the same notation for Xij EG if(W n _i) and and its itsimage imageinin H(yn). Notice Notice also also that that ititcan canbe beshown shown by computation does not depend on the choice computation that Xij in H(yn) does choice of ipk-)

Monodromy of Iterated Iterated Integrals Monodromy of

251 251

Let A23 Corollary 7.2. 7.2. We We shall also denote denote the Aij €e fl"i(Wn_i, x) be as in Corollary image of A^ in yn by by A23. A^.

Corollary conjugatetoto {—2ni)Xij in in 7r{yn). Corollary 7.3. 7.3.The Theelement element 0yyn {Aij) is conjugate fromCorollary Corollary7.2 7.2and and the the commutative commutative diagram diagram Proof. Proof. This follows follows from ^x)

—+ ->

7T(W n _i)

—*

0 Let Aut(7r(3^n)) be the the group group of of algebraic algebraic automorphisms automorphisms of aa complex complex Aut*(lr(yn)) pro-algebraic group 7r(yn). Let Aut*(7r(3^ a subgroup subgroup of of Aut(7r(3^n)) n)) be a

defined denned in the the following following way: way:

Aut*(lr(yn)) Aut*(7r(yn)) = {f {/ E Aut(7r(y n ))|3a/ €e C*,f(X23) a conjugate to".) (~ means "is conjugate to".) Let us set Let Tn{Q =={ipee Hom(7ri(;yn,y);7r(;yn))|3av E€ C* (A23 {A^ EE 7Ti(yn,2/)

Corollary 7.3.). 7.3.). The group Aut(7r(Xi)) acts freely are as in Corollary freely

We indicate indicate arguments argumentsthat that the action is transitive. The on Tn{C). We The homohomon morphism tp e£ T {C) (C) factors factors through through the theMalcev MalcevC-completion C-completion of ofni{yn, y)y) andand are thenthen the comand it becomes becomes an isomorphism isomorphism Out*(7r(J n_i)) is injective for nn>> 5. injective for 5.

ker(Out*(7r(yfl)) Proof. Let Outl(7r(;yn)) := ker(Out*(7r(y C*), where N{f) N(f) = ) , where af . n)) 4 C = cr1.

algebra of of Out*(7r(3^ Out*(lr(yn)) is the the Lie Lie algebra algebra of of special special derivations derivations of The Lie algebra n)) is modulo inner inner derivations. derivations. The Lie L{yn) modulo Lie version version is proved proved in [Ii] [II] section section unipotent algebraic groups and nilpotent Lie 4. The The categories categories of complex complex unipotent algebraic groups Lie algebras of of finite finite dimension dimension over over CC are are equivalent. equivalent. The group algebras group is pro-unipotent. Hence Hence the Lie Lie version version implies implies the result for for and then also also for Out*(lr(yn)). Out*{7r{yn)).

252 252

Zdzislaw Wojtkowiak Wojtkowiak

—÷ The surjective surjective homomorphisms homomorphisms (Pn+i)* (pn_j_i)* : /K\(ynjti^y) — > K\(yn,y') and and : —* induce a morphism of torsors {Pn+i)* ^(Xi+i) —t ^(iVn) induce a morphism of torsors (Pn+i)* :

—*

compatible with Out*(7r(3^ Out*(lr(yn+i)) compatible with n+1)) -> Out*(7r(yn)).

Lemma 7.5. 7.5.The T/iecanonical canonicalmorphism morphism of of torsors torsors tn+1(C) — —*> • £n(C) isisinmjective for for n > 4. jective immediately from from Proposition 7.4. This follows follows immediately 7.4.

Corollary 7.6. Corollary 7.6.The Tftemonodromy monodromyhomomorphism homomorphism 0yyn : 7Ti(^n,2/) —»Tr^n) y) —* is zs determined determined (up to conjugacy conjugacy by an element of 7r(yn)) by by the homomorhomomorphism 9y,',y4 y') —+ iyA : ir1(Y4, 7riCy4,2/') -^ ir(Y4). TTQ^). Observe Observe that ^,3; n EG tTh(C) t n (C) and ^',^4 is the image image of 6yyn under n 4 the canonical morphism £ (C) —+ -> £t4(C). (C). Let a := ((ai,. a i , . . . , a n , an+i) a n + i ) be beaasequence sequence of of n+ n +11 different different points pointsininIP"(C) P 1 (C) 1 := PIN(C)\{aj,. . a n , a n + i } . The vector space H(Va) and let Va Va := (C)\{ai,..., H(Va) is spanned by X2 Xi := ( ^ - —- i z t r r ) * i* == 1,...,n. 1,.. •, n. Let Letususset set Xn+1 :=:= - E? = 1 Xt. by Let Afc denote geometric generator Let Ak denote a geometric generator of of V Vg, whichisisaaloop looparound around a&. ak. Let Let a, which us set Proof. Proof.

. .

,

.

Ta(C) Ta(C) := {f {/ € Hom(7n(Vra,x);7r(Vra))|3a/ eGC*,VAk C*,V^lfc f(Ak) f(Ak)

Assume that that a = (a^)^ 1 is such such that that a1 1, a3 Assume ai = 0, 0, a2 = 1, as = = 00. The The fibration fibration Vaa — V

>yn+2 —

>yn+l

(Va(pn+2) (Va =

11

realizes ^(V ir(Va) as a normal realizes normal subgroup subgroup ofoflr(Yfl+2) ^(^+2) ((fcn+2)*(Xj) == a) as Hence the group acts on Ta(C); Hence 7r(3^n_|_2) Ta(C); let

ta(C) U Q := T a (C)/7r(^ n+ 2). Observe Observe that any any 7r(J;n+2)-conjugate of Xi>n+2 is in the the image image of ofir(Va). ir(Va). Hence the restriction map Hence the (kn+2)* : tn+2(C) -»• t0(C) to(C) given by byf /—÷ —> /|7n(VOJa:)is^sdefined. defined. We Weset set

Ta(C) r a (C) := im im

—+

ta(C)).

Monodromy of Iterated Iterated Integrals Monodromy of

253 253

Observe that the diagram Observe that diagram (kn+2y )

Ipr t4(C)

(fc 4 ).

——^

Tofifoo (C)

1 1, commutes where the map pr\ pr1 is induced by the the inclusion Va Va ^'+ PP'(C)\{O, (C)\{0,1, 1 oo}. The map map (£4)* is bijective because because Y4 y4 = P'(C)\{O, P (C)\{0, l,oo}. oo}. 1, oo}. Lemma 7.5 surjective by the implies that is injective. injective. The The map map (fcn+2)* is surjective implies that the map pr is construction. construction. Hence Hence both maps, (kn+2)* (kn+2)* and and pr\ pr1 are injective. injective. Therefore Therefore the following following result. we have proved proved the

Proposition 7.7. i)i)The The 7r(yn+2)-conJugacy class class of the monodromy homoProposition 7.7. 7Ti(3^n+2,x)—* —>• 7r(3^n_}_2)isisdetermined determinedby byits itsrestriction restriction morphism 0Xiyn+2 : irl(Yfl+2,x) to7n(V to iri(Va,x'). a,x'). class of the monodromy ii) The The fK{ynjt2)-conjugacy class monodromy homomorphism homomorphism Oxya : is determined determined by by the the monodromy monodromy homomorphism homomorphism 'iri(Va,x) —* Oxv,P1(C)\{O,1,oo}

:

in

ir(IF"(C)\{O, 1, oo}).

(IF'1 (C)\{O, 1, oo}, x') ,l,oo},x')

§8. The The Drinfeld-Ihara §8. Drinfeld-Ihara ^/5-cycle relation. relation. In this section show that that the section we show the element element which which describes the the monodromy monodromy 1 of all iterated (C) \ {0,1,00} iterated integrals integralson onIF" P (C) satisfies the the Drinfeld-Ihara Drinfeld-Ihara re{ 0, 1, oo} satisfies lation. lation.

8.1. 8.1. Introduction Introduction We We recall recall that Yn = = (IN(C)) (Fl(Q)" and and yn = = (P^QUO, l,oo})"~ ) " ~ 3 . Let X (C) be three different points and let a, b, c eE P IF" a,b,c (C) be three different points and let (paib,c(z) = = Y^L ' f ^ c * .

The map ^ 4 j 5 :: Y5 by Th map yy5 given i bby Y5 —f —>))5 given $4,5(£l, x2,

x3, x4, x5) =

I

CPx1,xz,xs (x5))

induces a bijection bijection (p4,5:

—*

Y5.

The group E5 action ofof E5£5On £5 acts on Y5 by bypermutations. permutations.The The action onV5 I5 induces induces an an action of £5 on ^5- The map a o : Y5 ^5—+ —>•3)5, ^5,o(s,t) (r(s, t) = f | ^ , i jcorresponds corresponds = to the the permutation permutation&a of ofY5 I5 given given by

&(x,,x2,x3,x4,x5) = (x2,x3,x4,x5,x1).

Zdzislaw Wojtkowiak Zdzislaw

254

Observe that that the points Observe

i\

(Vi— 1 2

-y/S-1

1

-\/5

2

'

i\)EY5

fixed by by a. are fixed a. and The one-forms ^ , ^ 3 , f, f , j-^ j - ^ p ^ f generate generate A1(Y5) A\y5) and ^3, T1and andAN be their their formal duals. There Sj, T0, Let 5o, Let 50,^1, To, Xi T be formal duals. There are aretwo twonon-trivial non-trivial relations in inA2(Y5): A2(y5):

I

dt ds — dt ds —dtdt dsds——dtdt dsds ds dt —A—+----A — A—+ —A —+ — A—, A —, s—t s—t s5 tt tt s—t s—t s s and

ds — ds dt dt ds —dtdt dsds——dtdt dsds + s—i 5—1 tt—1 — 1 t—1 t — 1 s—t S — t s—t S — t s—i5 — 1 Elementary computations of linear linear algebra algebra imply imply that that the Elementary computations thesubspace subspace R(Y5) 2 of H(y^)® H(y5)®2 is generated by A / \

~

—

_|_ -\~

—

A / \

•

I T"

;

A / \

i=O,1; [S^Tj + ^ i V ] i=O,1; i = 0,l; i=O,1; [Ti,Si] + [TuN] < = 0,l; and and [S1,To] [SUTO]

[S0,T1] [5 0 ,Ti]

where [A,B] = A® B - B ® A. where[A,B]=A®B-B®A.

We recall recall that that P(Y5) We P(y$) is is aa multiplicative multiplicative group group of of the thealgebra algebra of of formal formal power series in non-commuting variables So, Si, To, T1and and N N divided by the 5o, Si, TQ, T\ the ideal generated by byR(Y5). i?(3^s). The principal fibre fibre bundle bundle

Y5xP(Y5)-*Y5 y 5 x p(y6) -+ y5 we equipped equipped with the theintegrable integrable connection connection given given by by the theone one form form Wy5

dt

-

ids—dt rds-dt +(\ s—t \ s-t

dt\ dt\ dt\

ti

—) t) tJ

(_

Ar

, ds ^ s5

a

t

5

ds ds

-s—iI

— T1 shall write w instead of of Wy5. where T^ = —T0 —To — T\——N.N.More Moresimply, simply,wewe shall write u instead

Integration of 8.2. Integration ofwu

Monodromy of of Iterated Iterated Integrals Monodromy

255

Recall that on 1, 00} oo} we have onIF'(C)\{O, P 1 (C)\{0,1, 8.2.0

A00-1(z) A ^ ( z ) • a^(Y,Z)Z) == AlSo(z)

(see (see Proposition Proposition 5.1). 5.1).

Please notice that series in in Y Y and that we we are are considering considering a^(Y, Z) as a power power series < Z. Thus be able able to to substitute substitute Y Y and and Z in Z. Thus we we will will be in a ^)(Y1Z) Z)by byother otherpairs pairs of letters. We shall shall denote denote the generators X, Y Y and and ZZ from from the thebeginning beginningof ofsection section 6 by T0 To + N, T1 T\ and and T00. T^. The inonodromy monodromy of of Ao^1(z) is is given given by: by: (around co) 00) :: Ao^1(z) —* —> Ao^1(z) • e~27rlTo°, (around (around1): 1)A00-1(z) : A ^ ( z )—+ -> A ^ ( z ) • a ^ ^ , ^ ) • e " 2 ^ 1 • (a^fT^Too))- 1 .

(see Theorem 5.3). 5.3).

It follows from the the definition of horizontal horizontal sections sections starting starting from from a tanfollows from definition of tangential base base point, and also also from from the identity identity dz zz—1 — 1

dz z

dz zz

dz

dz zz

z—1 z— 1

°°

that asymptotically asymptotically at at oo 00 and and 1,1,we we have have respectively respectively

8.2.1

A -,(«) ~ e ^

dt

T

T

_( r- dflT

t T ~ ) = 11 urn (A lira (A—1(z) 00-, (z) • ee ~ ^ i T )/ °°)

i.e. i.e.

z—>oo z> 1 V ° °

1V

y

/

and

dt\T

dt

dt i.e. lim

e

—

00

dt\

—

)=

1

Z

z>1 z>l

where ee is is a small positive real number. number. Let Let Pe = (e, (e, 1 + e) 6 ^5 where positive real Let section of ofu;wsuch suchthat thatAp ApjPe) Aps((s,t); path) be be aa horizontal horizontal section 1. Let Let 'y7 e(Pe) = 1. be aa path from PE P£ to too-(P6) cr(Pe) = (e, 1/e) which is e) on the pathinin3)5 y5 from 1/i) which is constant constant (= e) first coordinate. coordinate. Assuming Assuming 5s == constant constant (= e) we have ApJE,z)

= A1g0(z).

Hence for small small positive positive e, we asymptotically have Hence for have 8.2.2

i'

al±E lao loo

e = 0

i'

dt

dt\\T

Zdzislaw Wojtkowiak Zdzislaw

256

from 8.2.0, 8.2.0, 8.2.1 8.2.1and and 8.2.2 8.2.2and and Proposition Proposition 5.1 5.1 that that for It follows follows from for small small positive £, we we have dz\

I

8.2.3

P

.W«);7)

(

't —1

o

t

Then = =1 because thethe path Let p = cr4('y) <J4(7) •
= Acy4(p)(Pe;04(7)) K4{Ps)(Pe;a4(j)) • Aas{Ps)(a4(Pe);a3(7)) 1 = APe(Pe,p) = • K{Pc)(
which follows from from Corollary Corollary 1.7 1.7 implies implies that that which follows

1=

.

.

.

.

L

where L = Apjo(P6);y). where L Ape(cr(Pe);7). Let

L = e(filk

. .

It follows from 8.2.3 8.2.3 that follows from 1 £=0

The factors e< XT (( A-I— ~ f ))M=<^)) and a n de(J" e ( / > * ) ^ can c a nbe b e put put The factors together in the product product 0\J(L) •... •LL because because T^ == cr^(Ti) commutes commutes with with After the calculations cr*(Ti) = 5o and a*(Too) = S1. Si. After calculations we get .

1+£ ffl+C

J-

. . .

1/£

dt dt" dt dt\ f dt ç-_____)+J —=—log(1+e). t—1 t /

1

the same argument for N, T1 for S1, Si, S1 Si + T1 Ti ++ iV, T\ and S0 So and and passing passing to the getget thelimit limite e—* —>0 we 0 we Repeating Repeating

.

.

.

a

(T1, can be be written written in the form formula can form where a = a ^ ( T i , Too). The last formula

a(S1 +T1 TX + N,So) • aiSuTj a(S 1 +

a(So,Si) • a(Too,S1+T1 + T+N). a(SQ,Si) • a(TuT^= 1 + N)-

1

Iterated Integrals Monodromy of of Iterated Integrals

257 257

a*(S0) = = T ^ , a* (Si) = = Si Si++T1 Ti++N, W, a*(r 0 ) ==N, N, ^ ( T i )==SoS and o and — S1 — N. = 3)5 be = ^4,5(21,22,23,24,00). Let Let ^5 : Ct -—*• 3^5 be given given by by ^5(21,22,23,24) =

since since

:

Let Let

(dzi

(A;A -• be be formal formal duals duals of of ( —£I r-^2-) J

J

\

z

z

i

j

= = S1 + = S0,

—

S1

. Then Then we wehave have

) ij —

— T1 — N, -Si-To-Ti-N,

T0

ip5*(Ai2) = So T1 + N,

ip5*(Au) = So,

(A23) = S0 + T0 + N,

^5*{^23) = So + TO + N, = Si, ^5.(^24) = N. ^5.(^34) = - 5 0 — - SS1i -— N.

= $4t5(oo, z2,z3,z4, =

by il>i(z2,z3,z4,z5) Using Vi : C* —* ->• ^5 given by get get :

z5) we we

N, ^1*^23) = 5 0 + T o ++ iV, =

= T1, N, (A34) = =-So— Si S1 — <M>l34) - JV, = —T0 ^.(^35) = - ^ —- T1 ^ -— ^N,,

= N.

^1*^45) = ^ .

Set

e = 1,5; then X15

:=

T0. Hence finally we get a formula

Set X^ := tpet(Aij) s = 1,5; then X15 = To. Hence finally we get a formula 8.2.4 8.2.4

a(X13,X14).a(X24,X25).a(X35,X13).a(X14,X24).a(X25,X35)

a(Xi3,Xi4)-a(X24,X25)-a(X35,X13)-a(Xu,X24)-a(X25,X35)

=

1.

= 1.

—*> 3^5given If we use use 42,4 $2,4 :: C^ — givenby by(1)2,4(0,8, $2,4(0? s, 1,t, 1, ^, 00)==(s,t) (5, ^)and andrepeat repeatthe the 's names calculations in in y5 we get the the same same formula formula as asbefore, before, but butthe the X^'s names of 5o, S S1,... arenow nowdifferent different and andthe theresulting resulting formula formula is: i , . . .are

8.2.5 a(X15,X12).a(X23,X34).a(X45,X15).a(X12,X23).a(X34,X45) a(X 15 , X12) • a(X 23 , X34) • a(X 45 , Xi 5 ) • a(X i 2 , X23) • a(X 3 is exactly the theformula formula which which appears appears in in[12, [12, p. p.106]. 106]. This is

Proposition 8.3. Proposition 8.3.For Forany anypermutation permutation a of of five five letters we we have i)

.

.

.

ii)

a(X,,(lS),

=

1,

=

1,

.

.

.

Zdzislaw Zdzislaw Wojtkowiak

258 258

where a(ij) a(ij) == a(i)a(j). where o~(i)o~(j). Proof. This This follows follows from 8.2.4, 8.2.5 8.2.5 and Corollary Corollary 1.7. 1.7.

0

Remark. formulas Remark.The The formulasofofProposition Proposition8.3 8.3 take take place place in in the group group P(y5). If we apply log we obtain formulas in the group ir(Y5). In the the group 7r(y5). In thesequel sequel we we apply we obtain formulas shall work in the the group group7r(Y5). TT^S)-

We finish finish this section section with aa formula formula from from which which the the Deligne Deligne Z/3.cycle ^/3-cycle relation can be obtained. The The proof proof isis an animitation imitationofofDeligne's Deligne'sproof. proof.

Proposition 8.4. 8.4. Let Let aa := := log log a. In Inthe thegroup groupir(Y5) 7r(3^)we wehave have i) i) X35) (—iriX25) = -7riX a(X23, ( —iriX23)a(X35, X23) ( —iriX35)a(X25, a(X23, X25) X25)(-niX X23)(-7riX X35)(-7riX = 23)a(X35, 35)a(X25, 25) 14c and and

ii) (iriX23)a(X35, X23) (iriX35)a(X25, = —iriX14. a(X23, X25) X2b)(niX X23)(niX XX35)(iriX25) ii) a(X23, 23)a(X36i 35)a(X25, 35)(7riX2b) = -7riX 14 .

Proof. We Proof. We prove prove only only i) i) since since ii) ii) is is similar. similar. Let &(x1,x2,x3,x4,x5) 5(x\,x2,x3,X4i,xr)) = (x1,x5,x2,x4,x3). (#i, £5, £ 2 , £4, £3). Then the the induced inducedmap mapa:a3)5 : y$—* -* 3^5 isis given given by by cr(s,t) a(s, t) ==

( ¥ • ^T. ¥ )

and

Let PP_=(r,1—r) and °2M = fe I^)- Let ~ = ^ 1 - ') and P+ =

(r, 1 + r) r) where where rr isispositive positive and and small, small. Let LetQ. Q_ ==(—r, (—r, 11——r)r)and and Q+Q+= = (—r, 1-hr). Let 7 be a path from P+ = (r,1+r)toa2(Q_)=(r,1/r),which (r, 1-hr) to (r,1 (r, 1++ re i(v?+7r))and andlet let5'S" path [0, [0, lr] TT] 3 y? —÷ —> (—r, (—r, 1 +-h re l ^ +7r ^). Let Letus usconsider consider the thecomposition composition p = = f f f 2 2 2 / 2 f oS' oa(7)oa(S)oa2(7')oa2(S')o'yoS. If a(j )oa(S )oa (j)oa (S)o1 oS'oa(-f)oa{S)oa (j )oa (S )o1oS. a(7')oa(S')0a2('y)oa2(S)oy' we integrate integrate the form along this this path and we form w u along and pass pass to tothe thelimit limitasasr r—* —>0 0wewe square of obtain the square of the left left hand hand side side of of the equality equality in ini). i). Let aa be direction around around (0,0) (0,0) in beaaloop loopininthe theopposite oppositeclockwise clockwise direction in the the plane t ) e CC22 \ as as + + fit = 0}. The integration integration of of the the form form w u along along plane P P = {(5, {(s, t) gives (-27rz)(S a gives (—2iri)(So+N+To) = (—2iri)X23. (-27ri)X23. In In the the model model of of Y*/PGL2(C) in in o + iV + To) = 1 5 which the the subspace subspace{(x1, {(xi,xx2, ,x x3, ,x±,x§) x4, x5) \ x\ = = x4} X4} of (P (P'(C))5 (C)) degenerates degenerates 2 3 point (for (for example example for for $2,5(0, s, 1, 1, 00, t)t) == (s, (5,t)), £)), the thepath pathppisishomohomoto aa point I

loop around one one of the points (0,0), (0,0),(1, (1,1) or(oc, (00,00) the plane plane topic to aa loop 1) or oo) in the passing corresponding point (1,1) or (oo, (00,00) passing through through the corresponding point (0,0), (1, 1) or oo) (the point (1, 1) in in the the case of of the model $2,5)- Hence the square (1,1) square of the left hand side side also(—2iri) (—2ni) .• X14. X14. 0 expression i)i)isisalso of the expression

Corollary 8.5. 8.5.For Forany anypermutation permutation aaofoffive five letters letters 1,2,3,4, 1,2,3,4,55 we we have 1 formulas ii') which are formulas ) and ii'), ii1), which are obtained obtained from from formulas formulas i)i) and and ii) ii) by byreplacreplaco'(2), a(3), ing indices indices 1,2,3,4,5 by by aa(1), (I), o~(2), 0£<7(i))*=i,...,5.The The

Monodromy of Iterated Iterated Integrals Monodromy of

259

3)5 —> which implies ®® id)w map cia : J^ satisfies(a* (a* id)u= =(id(id which implies induced map 0 ® <J*)U>, 3)5y$satisfies formulas i') and ii'). formulas i') ii'). 0 :

Remark 1.1.We Remark Wehave haveX23 X23++X25 X2*>+ +X35 X35 ==X14 X14 ininthe theLie Liealgebra algebraLie(Ys). Lie(^5 ).IfIf we set X14 X14 = 00 then the the formulas formulas i) and and ii) ii) reduce reduce to to the theDeligne Deligne formula formula [D5]). (see [D51).

f(z) == jz^- Let us return to Remark 2. Let Let f(z) to the the notation notation from from section section 6. 6. (X) = Y Y and (Y) = Z. Hence we get We have f+(X) and /*(T) =

=

power series series in in YY and and Z.) Therefore (We consider the element a ^ as a power Therefore we we have a(X, Y).

1] be be a path The element is the Let Let pp = [0, [0,1] path from from 01 01 to to 10. 10. The element a)1(X, a^(X,Y) Y) is by Ihara Ihara is result of from Ol ol to 10. The element result of integration integration from element considered considered by the result of the the Galois Galoisaction actionononthe thepath pathppcomposed composedwith withthethepath pathp~ p'.l. result of Both elements elements satisfy satisfythe the Z/2, Z/2, Z/3 and and Z/5 Z/5 cycle cycle relations. relations. The The analogy analogy is is we consider considertorsors torsors corresponding corresponding to to the the homomorphism more striking when we 1, oo}) oo}) and and the associated 1, oo}, 01) —÷ {0,1, associated %i : ^iri(1P1(C) ( P ^ C ) \ {0, {0,l,oo},0l) -> ir(IP'(C) ^(P^C) \ {0, groups (see Appendix A2).

Subgroups of of the groups of automorphisms. §9. Subgroups automorphisms. summarize here here the the notation notation which will be be used used in in the the rest rest of of the 9.0. We summarize which will paper. Let kA;be beaafield field of of characteristic zero. zero. We We say say that that XX isisan analgebraic algebraicvariety variety defined over kk ifif X X is k. If A defined over is an an algebraic algebraic scheme scheme over over Spec Specfc. A is aa k-algebra, fc-algebra, we setXA XA:=:=XX xx Spec A. We Wedenote denoteby by X(A) X(A) the the set of we set Spec A. of A-points A-points of Speck Specfc

X. that X(A) X(A) == XA{A). say that XA(A). We X. Observe Observe that We say that GG isisan anaffine affine algebraic algebraic

(resp. pro-algebraic) overfc, k, if if G G is algebraic (resp. (resp. pro-algebraic) group group defined defined over is an an affine affine algebraic (resp. pro-algebraic) group scheme over Spec k. pro-algebraic) group scheme over Spec k. Let pro-nilpotent) Lie algebra over over aa field Let L L be be aa nilpotent nilpotent (resp. (resp. pro-nilpotent) Lie algebra field kk of of characteristic zero. We equip the Lie algebra L with a multiplication characteristic zero. We equip the Lie algebra L with a multiplication given given by formula. We We denote denote by by nir the the obtained by the the Baker-Campbell-Hausdorff Baker-Campbell-Hausdorff formula. obtained group. group The is the group of k-points in a connected affine group. The group n is the group of fc-points in a connected affine algebraic algebraic unipotent unipotent (resp. (resp. pro-algebraic pro-algebraic pro-unipotent) pro-unipotent) group group scheme scheme over over k, A;,which which we denote by by II. H. Let Let Rbe R be a&k-algebra. We we denote fc-algebra. Wedenote denotebyby11(R) II(i?)the thegroup groupofofRitpoints of H. Let us set 11R H x SpecR. Observe that 11(R) = HR(R). points of II. Let us set UR := U Speck x SpecR. Observe that U(R) = UR(R). Speck

Zdzislaw Zdzislaw Wojtkowiak

260 260

Let V be as in section 1.1. 1.1. We We shall assume that that A'(V) ^41(V)—+ —> HpR(V) isisanan isomorphism. In 1.1 we defined definedthe thegroup groupTT(V). ir(V). It It is easy easy to see that ir(V) isomorphism. 1.1 we TT(V) is the group of k-points of a connected affine pro-unipotent pro-algebraic is the group of fc-points of a connected affine pro-unipotent pro-algebraic over k. We denote by 11(V) group scheme scheme over II(V) the corresponding group group scheme. scheme. (11(V)is is Spec Specof ofthe the algebra algebra of k-valued k-valued functions functions on on ir(V).) (II(y) n(V).) 1, oo}. oo}. We We recall recallthat that Lie(V) Let V = 14 \ {0, = PQ {0,1, Lie(Vr) is the free free Lie Lie algebra over Y = on X = ( ^ ) * and Y = ( ^ j ) * and that ZZ:=:=—x -X — - Y. We recall recall that Q on L(V) is is the completion completion of Lie(V) Lie(Vr) with to the filtration filtration induced induced by L(V) with respect respect to 1 the lower lower central V := [L(V), [L(V),L(V)], [L'.LL'] ] and and central series. series. Let Let us set L' L(V)], L" L" := [L', L2(V) := := L(V)/L". L(V)/L"- We We denote denote by byH2(V) ^ ( F )the thegroup group scheme scheme over over SpecQ SpecQ L2(V) corresponding to (V). Let n (V) = U (V)(C). corresponding to L L2(V). us set ir2(V) = H2(V)(C). 2 2 2 l Notation. We set set wX WXZY2 N o t a t i o n . Let Let w wE G LL2(V). We Y^ : =:= ( . (...(w,X) . . (w,X). 2{V).

If /(X, f(X, Y) = 11 +4-

>^2

a

.X)Y) ..X)Y)

. .

.. ...Y). Y). .

n 7n

anmXnYm nmX Y is is aa power power series, series, then we we set set

TL 1 ,ni 1 n=l,m=l

(w, f(X, Y)) (wJ(X,Y)):=w+ w+ 1 nn=1 = l ,,m= 77i=l

= wXi+aY^b in L2(V). 9.0.1. Observe Observe that (wXiY^)XaYb = L2(V). This implies that the for plies that the elements elements X, Y Y and and(Y, (Y,X)Xi~1Yj-1 for ii = = 1,1,2,3,..., 2, 3,. form a linear topological basis of L2(V). j = 1, 2 , 3 , . . . form L2(V). 1,2,3,... Let F2 (Lie(V)) be be the Lie Let F2(Lie(V)) Lie ideal ideal of of L(V) L(V) topologically topologically generated generated by byLie Lie brackets brackets in X and and Y, F , which which contain contain X at atleast least twice. twice. Let Let us usset set L2(V) L2(V) := L(V)/F We denote denote by byH2 II 2the thecorresponding corresponding group group scheme scheme over over L(V)/F2(L(V)). We 2(L(V)). SpecQ. TT2 :— II2 (C). IfIfwe Spec Q. To To simplify simplify notation notation let us set 7r2 := 112(C). wereplace replaceX X by Y Y in the above definition definition then the corresponding corresponding group of C-points C-points we denote i by w2. The elements elements X, and (YX)Yi (resp. ) ii = = 0,1,2,... X, Y and (resp. (YX)X (YX)X1) 2 2 form topological linear of n (resp. m ). We We denote denote by G the form a topological linear basis of w2). Gm m the multiplicative group scheme scheme Spec(Q[t, t" ]). t 1']). 9.1. Let us set set Aut*(1r2(V)) := {/ {f EG Aut(7r2(y))|3 af EGC*,f(X) Aut*(7r C*,/(X)== afX, 2(V)) := f(Y) » a / y, f(Z) - afZ} (« is a conjugation by by an an element element of of(ir2(V), (TT 2 (F), ir2(V)) TT2(V)) and ~ is a conjugation conjugation 2 ir2(V)). p: 7r2(V) w2 be the natural projection. The by an element of TT (V)). Let p : n (V) -> ZA7 be natural projection. The 2 2 2 Gm(C) map pp induces induces p* :: Aut*(1r2(V)) Aut*(7r2(V))—+ -> Aut(ti7 ). Let CC : G ->• Aut(tti2) Aut(tu2) m (C) -÷ tY. We shall investigate be given by by C*(X) = tX, ^X, Ct(Y) = tF. investigate liftings liftings of of C to to Aut*(1r2(V)). Aut*(7T (V)). 2

Monodromy of Iterated Iterated Integrals Monodromy of

261 261

Let Proposition 9.1. 9.1. i)%) Let4 $ Gm(C) : G m (C) —*-»Aut*(1r2(V)) Aut*(7r2(l0)be begiven given by by $t(X) = tX oo

=

Y

J^

n=2

—

*

where where

c

i,j €E C. C.

t+i=n * > 1, j > 1

1 is T/aen $ 25 aa homomorphism. homomorphism. Then ii) All homomorphisms $f' of Gm(C) such thatp*o$ that All homomorphisms o/G m£o Aut*(1r2(V)) Aut*(7r2(V)) s^cft = m (C) into C and $t(X) = tX £X are are of of this this form. form. C and iii) All Hi) All one-dimensional subgroups subgroups G G of of Aut*(^(V)) which which are areprojected projected onto C(G C(Gm(C)) by the map and preserve the subgroup of ir2(V) onto (C)) by the map p* preserve the subgroup of 7r2(V) m generated by by X X are are of ofthis this form. form. iv) The group G is is defined defined over aa subfield subfield k of of C ifif and andonly only ififall allcoefcoefficients Cij are in ficients in k. k.

Proof. The uses 9.0.1. 9.0.1. To To Proof. The point point i)i)isisaastraightforward straightforward verification verification if one one uses N—i N-l

show point point ii) we can assume that &t(Y) = = (tY,exp( (tY,exp( ^ show assume that

n=2

tt2)XZY2

+

£

fijtyXiY*)).

i+jN i>lj>1 and * , ( * t ( y ) ) we we get get fid(t)

— ]T} ci,j(l ~

i+j=n i 1 1

Comparing coefficients coefficients of $ t

= f^s) + ^ / , , , - W . This im+ tNfid(s) 4= This imHence = Cij (I plies fij(t) = Hence the the point point ii) ii) is proved. proved. If aa subgroup subgroup of Aut* (ir2(V)) is is one one dimensional dimensional then then the are algebraic of Aut*(?r2(V)) thecoefficients coefficients fij are algebraic functions of of t.t. They because then then the difunctions They cannot cannot be bemultivalued multivalued functions functions because mension of ofthe the subgroup subgroup would wouldbe begreater greater than than 1. Hence mension Hence fi,3 fij are Laurent Laurent polynomialsof oft.t. Now Now the the point point iii) follows fromthe the proof proofof ofii). ii). The The last polynomials follows from point is isobvious. obvious. <) — tN). — tN).

In §13 we we shall shall also also need needresults results about about the subgroups subgroups of Aut(7r 2 / r n+i 7r 2) and Aut(ir2). Let C' and Aut(7r 2 ). Let C :: Gm(C) G m ( C )—+ -> Aut(7r 2 / r27r 2) be be given given by by C[(X) = n tX, Let (p )* : Aut(7r 2 / r n +27r2 ) —+ tX,C[{Y) = tY. Let -> Aut(7r 2 / r27r 2) and p* : = tY. and Ps Aut(ir2)2)—+ be induced Aut(7r —>• Aut(7r 2 / r27r 2) be induced by by projections projections of 7r2/rn+2n2 and ir2 ?r2 onto 7T 2 / r 2 7 r 2. Onto Corollary 9.2. All Allone one dimensional dimensional subgroups subgroups C G of of Aut(7r2/rn+27r2) (resp. (resp. 2 n Aut(ir2)) Aut(?r )) which which the the map map (p )* (resp. (resp. Ps) p*) projects projects onto (C)) of onto C'(G C'(Gm(C)) are of m the form form G= = {ft eAut(7T 2 / r n +27r2 ) (resp.Aut(ir2)) (resp.Aut(7r 2 )) ft(X) = tX, tX, n(resp.oo)

ft(Y) = tY+ + ft(Y) =

J2

c2(t — °i(t - tti+1l)((YX)Yi_i) )((YX)Yi~1) with with t E C5}.

Zdzislaw Wojtkowiak Zdzislaw

262

The group group G G is defined over a subfield k of C if and only The defined over subfield kofCif only if all all Ci 's's are are in in k, k.

We shall shall denote denote the the subgroup subgroup G G of We of Aut(7T2/r»+27r2) (resp. Aut(ir2)) Aut(?r 2 )) conconsidered in in the corollary by G(ci, c2,. . , cn) (resp. G((ci)g 1 ) = sidered C2,..., = G(ci, c2,. C2,..., cn,...)).

Proposition 9.3. Proposition 9.3.Suppose Suppose that that for for each each nnwe we have have aa point point (c»j) i+j=n G lv,.. . ,, ZJ? of Cn~1 and aa sequence sequence Z™,... of linear linear forms forms on on Cn~1. Let Let Vn be be aaset set zeros of of common common zeros of lv,... Z™,..., ZJJ. Let us us set set LL := {(Z™,..., ZJJ )nEN} )n€N} and o,ndcc := := .

)nEN}.

Let i) Let G(c,L) :={f G(c,L) {/ eG AnffaiV)) f(X)= =tX,tX,

i)

f(Y)

= n=2

e

E C*, teC*,

(c23(1 —ta)

C,v

n

i+j=n

v

=

.

O}.

Then G(c,L) Then G(c,L) isisaasubgroup subgroup of ofAut*(1r2(V)). Ant* ( Any Any subgroup subgroup G G of of Aut*(1r2(V)), Aut*(7T2(V)), which which preserves preserves the the subgroup subgroup of of ir2(V) 7T2(V) generated generated by by X X and andwhose whose projection projection into into Aut(w2) Aut(tu 2 ) isC(Gm(C)) C(G m (C)) is of of this form. form. 1 Two subgroups subgroupsG(c, G(c,L)L)and andG(c', G(c',LL') coincide if if and and only for each Two ) coincide only if for each

ii)

iii) Hi)

+ VI n (*;)*££ + +Vn = = (dii)gi+ h i1j2' The The group G(c, G(c,L) L) is is defined defined over overthe the subfield subfieldkkofofQC ifif and and only only ifif for each over k. k. eachnnthe theaffine affinespace space (cij) i+j=n+ 4- Vis is defined over n defined

iv)

The point Proof. Proof. The point i) i) is aastandard standard checking. checking. Let Let G be as as in in the the point point 2 : G —* Aut(zzi2)). ii). Let G1 Any element f E G1 is of thethe ii). Gi := ker(p* : G —> Aut(o7 )). Any element / € G\ is of form 1(X) = X, f(Y) form f(X) f(Y) ==(Y, (y, exp(> exp( g n=2

£

f^X^)).

thatf /—* -> Observe that

i+j=n

i1,j>1

t>i,i>i

N

G1G\ —*-> J2 ((fij) i+j=n)n=2,3 )n=2 ,..., 3 N Ndefines definesa ahomomorphism homomorphism i

^

1

-^

1

' '

n=2 n=2

'

S Ga(C) Ga(C)ofof i+j=n

G1 (C). Hence G\ into a direct sum sum of of additive groups groups Ga G a (C). Hence there are only only linear linear relations between various / i / s . The group G (C) by G is is an an extension extension of of Gm G m (C) the pro-unipotent pro-unipotent group group G1, Gi, hence hence there there isis aa lifting lifting C1 C\ : Gm(C) G m (C) —>GGofofC.C. 1 a If we calculate the the coefficients coefficients of Cit C\t oo ff o Ci^ we we get that itit can can only only be be linear relations between between the thecoefficients coefficients (/t,n-t)*=i,...,n-i- Hence for each n n of linear forms forms lv,. Z",..., we have have a finite number of . Z£ on C n - 1 such such that . ,

G1

= {f 11(X) = X,f(Y) = n—2

i1,j1

Monodromy of of Iterated Iterated Integrals Monodromy e C,v 0ij G C,V

VV n \
263

liWhn-l,p2,n-2,

• • •) = 0}.

ne group Ci(Gm(C)) is as Let LL = Let = {(/?,...,JJ n )nGi\r}- TThe group Ci(G as in in Proposition Proposition m (C)) is Observe that that G1 9.1.i) for 9.1.i) for some some sequences sequences (c™) z+j=n , n = 22,3 , 3 , , . . . Observe G\ oo . .

.

i>

l,j>1

Ci(Gm(C)) = G(c,L). Hence for dimensional reasons G = G(c,L). The Ci(G The m (C)) = G(c,L). Hence for dimensional reasons G — G(c,L). points iii) and iv) are evident. points iii) and iv) are evident. <0 Let {ei)il1 be a sequence such that that e» G {O, 1} for for each each z.i. Let (c^)?^ be sequence such e {0,1} of complex complex numbers. numbers. For For ii = 1,1,2,...}, 2, . .}, we a sequence sequence of we set .

=G G(c1, ^ie^Zi) = ( C l , cc2,... |e 1 ,eE2,...) / = /t, ( e i A )£i €e Aut(7r2) 2 , . . . El, 2 , . . . ) : = {{f

f(X) = t X, f(Y) == t • Y Y +

+ (1 —

with teC,fteC e C. Replacing oo by nii in the above definition gives a subgroup of of Aut(7r2/rn+27r2), denoted by by G G(ci,. denoted ( c i , ... . , c n | e i , . . . , en). Let S((ei)f=1) be a number number of €i equal to 1. 1. Then Then dimG(ci,..., cn\eu ... ,en) = *((ei)? = i) + 1. .

2 Corollary 9.4. Aut(ir2)), 9.4. All All subgroups subgroups of Aut(7r2/r«+27r2) (resp. Aut(7r )), which which

(resp. p4 the map (p n )* (resp. p*) projects projects onto onto C'(Gm(C)) C / (G m (C)) and andwhich which preserve preserve the the 2 subgroup of ir ir2 generated generated by byXX are are oftheform of the form G G(ci,. subgroup ( c i , . . . ,, c n | e i ,. . .. .. , ,e n ) (resp. (^res^. The group is defined over a subfield of C C if and only G((ci)g 1 |(ei)g : 1 )). The defined over subfield kk of only if all all numbers numbers (1(1— if — £i)ci are areinink.k. We shall denote denote by by Go G0 the the subgroup subgroup {f{ft We shall € Aut(7r2/rn+27r2) ft(X) = = t e .

.

tX, tY, t ee C*} tX, fft(Y) C*} of of Aut(7r2/rn+27r2). Let 6 : CC—÷ -> {O,1} {0,1} be aa = tY,t t(Y) map defined 5(z) = 11 if z ± 0. defined by 5(0) 6(0) = 0, 0,6(z) 2 Lemma 9.5. be siicft such that that fo(X) f0(X) = aoX,fo(Y) 9.5. Let Le^ /o G aoX,fo(Y) = E Aut(7r /rn+27r2) 6e

n

Let G G be thesmallest smallestclosed closed algebraic subgroup a0Y + -f ^ ^ ( ( y X ) y * " 1 ) . Let be the algebraic subgroup i=l

of such that that Go G0 Cc G E G. of Aut(7r2/rn+27r2) such G and and fo f0 G G. Then Then

Proof. Let xt(X) = xt(Y) = tY tY and let fifi = ffo0o° (Xao)'1- Then Proof. Let Xt(^) = tX tX and xt(^) and let f1(X) = X and h(Y) f1(Y) = Y+J2 0SdQ1((YX)Yi-1). Let G1 X and Gi C G be aa subsubgroup consistingofofhhsuch suchthat that ft(X) h(X) = X, h(Y) h(Y) = Y + group consisting + fl

f3i((YX)Yi~l).

Zdzislaw Zdzislaw Wojtkowiak Wojtkowiak

264

Assume that for {Id} because becausef\fi e G1. for some some i,i, f3f ^ 0. Then Then G1 G\ ^ {Id} G\. Assume that The to a subgroup it is The subgroup subgroup G1 G\ is isomorphic isomorphic to subgroup of ( G a ( C ) ) n , hence it given by aa finite number of of linear linearforms. forms. Let Let kk — = xt given by finite number (Xt)" 1 - Then Xt °o h °o (xtY1. k(X) k(Y) = Y Let Z(a?i,...,x l(xi,.. n ) be one of y + f2tif3i((YX)Yi~1). Let k(X) = X, Jfe(y) 2=1

linear linear forms forms defining defining G1. G i .Then Thenl(tf31,t2/32,. l(t(31,t2p2,. .• .,tnpn) == 00 implies implies 1/ = 0.0. Hence the th, non-zero, linearly Hence the /?i, /?2? • • • ,j8n»ififthey theyare are non-zero,they theyareare linearlyindepenindependent. Therefore dent. Therefore G = G ( 0 , 0 , . . . , o|<J(/3j),..., <*(#1)). <> .

.

. ,

C Aut(7r 2/r"+27r2) be the smallest Corollary 9.6. Le£ G C smallest closed closed algebraic algebraic 2 subgroup that the subgroups G0and andGG(ci,...,cn) such that subgroups Go (ci,... ,cn) subgroup of Aut(n /^+2^2) are contained in G. G. Then contained in Then G =

G(0,...,0\5(c1),5(c2),...,6(cn)).

Proof. any element /f of G(ci,..., G(ci,. . , cn ) such such that that f(X) 1(X) = aX and Proof. One One takes takes any .

§10. Torsors. Let G be a group. We say that a set T is a G-torsor if T is is equipped with a free free transitive action action of G. subset S 5 C T is is a subtorsor of G. We say say that that a subset T if if there subgroup H C C G such such that natural action action of H on on S is is there is a subgroup that the natural free free and and transitive. Main example. example.Let LetG1 Giand andG2 G2be betwo twogroups. groups. Assume Assume that G1 G\ and and G2 G2 are isomorphic. are isomorphic. Then Then the the set set of ofisomorphisms isomorphisms from from G1 G\ to G2, G2, which which we G2), is an Aut(G2)-torsor. denote by Iso(Gi, Iso(Gi,G2), Aut(G2)-torsor.

For any non-empty non-empty subset subset S C Iso(Gi, Iso(Gi, G2), G2), the intersection intersection of of all subtorsubtorsors of Iso(Gi, G2), which contain S, G2), which sors Iso(Gi,G2), which contain 5, is is aasubtorsor subtorsorofofIso(G1, Iso(Gi,G2), which we denote by T(S). we T(S). 10.1. 10.1.

Unipotent affine groups and and torsors. affine algebraic algebraic groups

Lemma ananaffine L e m m a 10.1.1. 10.1.1.Let LetGGbebe affineunipotent unipotent algebraic algebraicgroup group over over aafield field kk of characteristic zero. zero. Then group Aut(G) Aut(G) over of characteristic Then there there is an an affine affine algebraic algebraic group over k such such that that for for any anyk-algebra k-algebra A A we wehave have

Aut(G)(A) = Aut(GA). Let unipotent algebraic groups over over k. k. Then G\ and G2 G2 be be two affine affine unipotent algebraic groups Then there there Let G1 is aa smooth variety Iso(Gi,G2) Iso(Gi, G2)over overk,k, such such that that for for any is smooth affine affine algebraic algebraic variety any k-algebra k-algebra A we we have have Iso(Gi, G2)(A) = ISo(G1A, G2A).

Monodromy of of Iterated Iterated Integrals Monodromy

265

Proof. Let Let gg be be the the Lie Lie algebra of of G. Let Let us us equip equip gg with with the the group grouplaw law given given by the Baker-Campbell-Hausdorff Baker-Campbell-Hausdorff formula. The The exponential exponentialmap mapexp: exp:g g—+ -> of affine affine algebraic algebraic groups. groups. The group automorphisms isomorphism of automorphisms G is an isomorphism of g coincide coincide with with the the automorphisms automorphisms of of the Lie algebra g. One One can can easily easily give k[GL(g)]. One One constructs constructs Iso(Gi, G2) in give an ideal defining defining AUtLie(9) AutLie(s) inin k[GL(g)]. Iso(Gi,G2) in a similar similar way. way. 0

We say say that an variety TT over over A k: isisaaG-torsor, an affine affine algebraic algebraic variety G-torsor, ififthere thereisis a morphism morphism TTxxGG—* —>T Tover overSpec Speck,k,which whichdefines definesa afree freetransitive transitiveaction action of o f G oonn TT (see (see [S] [S] page 149). 149). Let S5 be of TT and and let H be aa closed closed subvariety subvariety of if be be aaclosed closed subgroup subgroup of of G. G. We say that that SS is or a subtorsor of TT if We say is an an H-subtorsor i7-subtorsor or subtorsor of if S S is is an an H-torsor if-torsor under under the natural action action of of H. H.

Lemma 10.1.2. T be Let T\ T1 be an H1-subtorsor T 10.1.2. Let Let T be aa G-torsor. G-torsor. Let Hi-subtorsor of T and let T2 be be an H2-subtorsor H2-subtorsor of of T. Assume Assume that thatT1 T\ flD7'2 T2 / 0.0. Then Then the the intersection T1 fl T2 is an T\ D an H1 H\ flDH2-subtorsor H2-subtorsorofofT.T.

G1G\and 10.1.3. Main Mainexample. example.LetLet andG2 G2bebetwo twounipotent unipotentaffine affine algebraic algebraic groups over k. that there —*> (^2)^ of k. Assume Assume that there isis an anisomorphism isomorphism (Gi)^ — algebraic groups. Then G2) isisananAut(G2 )algebraic groups. Then the thealgebraic algebraicvariety varietyIso(Gi, Iso(Gi,G2) Aut(G2)torsor, if if we we equip equip Iso(Gi, Iso(G1,G2) C2)with with the the obvious obvious action action of Aut(C?2). Aut(G2). Let k CC CCbe e : Let bea asubfield subfield ofofthe thefield fieldofofcomplex complex numbers numbers C. C. Let Let O G1(C) isomorphism. Then e is C-point of Iso(Gi, G2). We Gi(C) —÷ -> G (C) be an isomorphism. Then 9 is a Iso(G , G ). G2(C) 2 x 2 denote by closure of of © e in Iso(Gi,G2) Iso(Gi, C2) i.e. i.e. the smallest by Z(Q) the k-Zariski A;-Zariski closure smallest algebraic subset subset of of Iso(Gi,G Iso(Gi, C2) definedover over&,k,which whichcontains contains00 as as a Calgebraic 2 ) defined point. The Theunipotent unipotentaffine affine algebraic algebraic group group G. Gi isis isomorphic isomorphic as an algebraic algebraic variety over over k to the the affine affine space space A™, hence hence 06 can canbebeviewed viewedasasaaC-point C-point (9?;j)i
ej,j. Lemma 10.1.3. The Lemma Thefield field k(0) k(Q)does doesnot notdepend dependon on the thechoice choice of of isomorisomorphisms phisms G2 Gi « A™ and the transcendental transcendental degree of of the the field field k(0) k(Q) over overkkisis

equal to to the the dimension dimension of of Z(e). Z(Q). Proof. This follows follows from Lemma 1.7 1.7 in in [D2]. [D2].

0

Definition-Proposition LetLetT(Q) T(0) be be thethe intersection ofofallallsubtorDefinition-Proposition10.1.4. 10.1.4. intersection subtorsors TT defined over kk o/Iso(Gi,G2), of Iso(G1,G2),which whichcontain contain©0 as as a C-point. Then defined over Then T(0) Iso(Gi,G2) for some G(0) C Aut(G2). T(Q) isisaaG(0)-subtorsor G(0)-subtorsor ofoflso(G G ) for some G(0) C Aut(G ). u 2 2 Proof. The The intersection intersection of of aafamily family of of algebraic algebraic varieties varieties coincides coincides with an an intersection of of a finite number of of them. them. Hence Henceititfollows follows from from Lemma Lemma 10.1.2 10.1.2

Zdzislaw Wojtkowiak

266 266

that T(e) T(Q)exists existsand andititisisunique. unique.The Thegroup groupisisalso also unique unique because because it is an an intersection of the corresponding corresponding subgroups subgroups of of Aut(G2). Aut(G2). 0

Lemma 10.1.5. 10.1.5.Let LetGGbebea aunipotent unipotentaffine affinealgebraic algebraicgroup groupover overk. k. Then Then Aut(G) isisan of o/Aut(G Aut(Gab)abby a unipotent anextension extensionof ofan an algebraic algebraicsubgroup subgroup ) by a unipotent affine algebraic anan extension ofof ananalgebraic algebraicgroup. group.Hence Hencethe thegroup group G(©)is is extension algebraic a unipotent affine algebraic group. subgroup ofof Aut(G ab) by subgroup by a unipotent affine algebraic group.

Proof. Let be the filtration Let gg be be the theLie Liealgebra algebra of of G and and let let (Pg); be filtration of of g by the lower central series. lower central series. Any Any automorphism automorphism of the the Lie Lie algebra algebra gg preserves preserves the filtration andthe theinduced induced automorphism automorphism of of I y / r i + 1 gg is determined by filtration and F'g/ç2g the automorphism automorphism çjf of gab g ab = = F Bfr2g- Hence Hence AutLie(g) AutLie(#) is extension of is an extension of 3h closed subgroup subgroup of of GL(g ) by aa unipotent unipotent group. group. The Thelemma lemmafollows follows a closed from the the identification identification of of Aut Aut(G) AutLie(s) exponential map from (G) with AutLie (g)by by the the exponential map exp : g -^ G.

0

(orG Lemma 10.1.6. 10.1.6. Assume Assume thatG(e) that G(0) isisan anextension extension of ofGGm (orG such such that m 1 H'(Gal(k/k),G)= a aunipotent unipotentaffine affinealgebraic algebraicgroup groupN, N.Then Then T(Q) H (Gsl(k/k),G)= 0)0)byby has aa k-point. k-point. from [S] [SIProposition Proposition4.1 4.1that that ^(Gal^/fc),TV) H' (Gal(k/k), N) = 0. Proof. It follows follows from 0. It followsfrom from[S] [S]Proposition Proposition2.2 2.2and andthe the assumption assumptionof of the the lemma lemma that follows H'(Gal(k/k),G(e)) = 0. 0. Prop. 1.1 1.1 of of [S] [S] implies implies that T(S)(k) / 0. 0 0. (G) be a filtration series. Let Let P(G) filtration of a group G by the the lower lower central central series. (i) G i+1 us set The isomorphism induces G2(C) us set G := G/pi+1G. /r G- The isomorphism 6e ::G,(C) Gi(C)—* -> G 2(C) induces isomorphisms 9 ( i ) :: G^(C) —+ Let kk < -> G^(C). Let < i.i. The Theprojections projections Gf ->÷ {k) for j = =1,2 1,2 induce induce G for k —* pi:Iso(G^\G(^)-^lso(G[ \G{2k))

Lemma_10.1.7. Lemma 10.1.7. We Wehave have z(e(k)), i)

iii) Hi)

W

=

and p(2)l : Aut(G^) -> —+ Aut(G^).

ii) pUne^)) == r(e(fe)), a)

()

Proof. In In the the point point i) i) (

k-Zariski closure ) means the thefc-Zariski closure and andwe we omit omit its itsproof proof because wedo donot notneed needthis thisfact factlater. later. Let Let us us set set pp = p\ and p' p' = because we = p(2)L. p(2)lk* f Observe that the p'(G(B(0)) Observe that the image image p (G(@^)) of of the thegroup group G(@^) by the the morphism morphism p' is a closed subgroup of defined over k. This implies p' is a closed subgroup of Aut(G;T ) defined over &. This implies that p(T(0W)) is a closed closed subvariety of lso(G[k\G(2k)) and and aa p'(G(e^))-torsor defined over 9 ^ as sowe wehave have defined over k. k. This torsor contains e(k) as a C-point, so

T(e(k))

C

and

C

Monodromy of of Iterated Iterated Integrals Monodromy

267

Let Let P (resp. (resp. P') P1)be bethe theprojection projection pp (resp. (resp. p') p')restricted restricted to toT(e(2)) T ( 0 ^ ) (resp. (resp. G(0«)). Then P" 1 ^©***)) is aa PF_l(G(e(k)))_torsor P / - 1 (G(9^)))-torsor defined defined over over fc, which contain 0(j) which contain 0 « as as a C-point. C-point. Hence Hence we get p - 1 ( T ( e ^ ) ) = T(e(z)) T(©W) = T(e<*>) T(e(k)) and This implies that p(T(0«)) = and P ' - ^ G i e ' * ' ) ) = G(0W) This implies that = G(e(k)).

10.2. AfRne Affine pro-algebraic pro-algebraic pro-unipotent pro-unipotent groups and torsors. 10.2. torsors. limG(z), where 10.2.1 assumethat that GC = limG^, 10.2.1 We We assume where the groups groups G ^ are affine affine

unipotent iinipotent algebraic algebraic groups over over k and the the morphisms morphisms GW —+ —>GU) G^)are arealso also iJtl G/çi+1G. = G/Y G- Finally over k. We further that G^ = over k. We assume assume further Finally we assume assume that the Lie algebra g of of G G isis finitely finitely presented, presented, i.e. i.e. that for for ii big big enough enough the 1 1 of degree less than i + 1 does not number of relations defining g/fi+lg number of relations defining O/r*" " ^, degree less than 1 does not depend on on i.i.

10.2.2. The Thecondition condition that that GGisisfinitely finitely presented presented implies implies that for for ii big big enough the morphisms Aut(G^+1^) — —*> enough • Aut(G^) are aresurjective. surjective. We Weset set

Aut(G) := :=limAut(G (i) ). Similarly, ifif Gi G1 and and G2 G2 satisfy satisfy 10.2.1 10.2.1 and and ifif there there is an isomorphism (Gi)*. —+ —> pro-algebraic pro-unipotent pro-unipotent groups, groups, then then the {G2)k of affine affine pro-algebraic the morphisms morphisms are surjective Iso(Gi~ ,G2 ) —* —> Iso(Gi ,G^ )are surjective for fori ibig bigenough. enough.We We set Iso(G1, G2) := limIso(Gi JGJ ). Iso(Gi,G2) i

Observe G2) is an Aut(G2)-torsor defined Observe that Iso(G1, Iso(Gi,G2) defined over over k. k.

10.2.3. 10.2.3. Examples of groups satisfying satisfying 10.2.1. 10.2.1. 1) Let V over k. k. Let us set F bbee aa smooth smooth algebraic algebraic variety variety over

C :=

'(x,x)))

(see §2 §2 and and [Wl]). [Wi]). Then (see Then GGsatisfies satisfies 10.2.1. 10.2.1. 2) Let V be as in 1.1. Then the group be as in 1.1. Then the groupscheme schemeH(V) H(V) satisfies satisfies 10.2.1. 10.2.1.

Lemma 10.2.4. 10.2.4.Let LetCGbebeasasinin10.2.1. 10.2.1.The Theaffine affinegroup groupscheme scheme Aut(G) Aut(G) is an extension extension of of aa closed closedsubgroup subgroupofofCL(Gab) GL(Gah) bybyananaffine affinepro-unipotent pro-unipotent pro-algebraic group. pro-algebraic group. This lemma follows from Lemma Lemma 10.1.5 10.1.5 and and the definition follows from definition of Aut(G). Aut(G).

268 268

Zdzislaw Wojtkowiak Wojtkowiak

10.2.5 Let Gi G1 and G2 ê :: G be an G1(C) G2(C) G2 satisfy satisfy 10.2.1. 10.2.1. Let Let O -> G an X(C) —* 2(C) be Then 6e = I i m 9 ( i \ where e(2) —+ G^(C) are isomorphism. Then 9 ( i ) :: G^(C) -> i

isornorphisms forall alli.i. Let Let us isomorphisms for us set set

T(e) := limT(9 (i )), G(9) := UrnG(6 w ). Z(@) := := lnnZ(6 ( i ) ), T(0) i

i

i

Then T(9) is a G(e)-torsor. G(9)-torsor.

§11. Torsors Torsors associated associated to to non-abelian non-abelian unipotent unipotent periods. §11. Let X X be be aa smooth smooth quasi-projective quasi-projective algebraic algebraic variety variety defined defined over over a number field k. that XX has aa k-point k. Assume Assume that fc-point x. x. Let us us fix fix an an embedding embedding c In [Wi] k —* -> C. In [Wl] §7 §7 (see also also 2.0 2.0 in this paper) paper) we we have have constructed constructed affine affine pro-algebraic pro-unipotent finitely presented pro-algebraic pro-unipotent presented group group schemes schemes 7rf (X(X(C),x) (C), x) and ?rPR(X, x) over Q and Spec k respectively. over Spec SpecQ We set

x)

:= :=

x).

Then we we have: have: (n) and n? irf(X(Q,x) =lim7r?(X(C),z) =

=

[Wi] Proposition 7.5 In [Wl] 7.5 we we have also constructed aa homomorphism homomorphism (called (called Bock [Wi]) B o a i nin[ W l])

such that the such the induced induced map map on on C-points, C-points,

is an isomorphism. We We have have cpx == lim<^", where where y?" : 7r^(X(C),a;)(n^(C) —+ —>• :

n

induced by For each each nn we x)(n\C) is is induced by cox. (px. For we have have an an Aut(7rPR(X, x)^)n := n x torsor Iso'2 torsor Iso := Iso(7rf (X(C),x)W x Jb, 7rf^(X,x)( )). Applying Applying the the conconstruction from §10 to the isomorphism we get a struction from §10 to the isomorphism x) isisaasubgroup subgroupofof Aut(?rPR(X, x)^). We We set set and and TTP R (X,

Monodromy of of Iterated Iterated Integrals Monodromy

269

We have projections pn : Aut(7r? R (A»< n >) x)(n) Aut(7rPR(X, x)

x) (n)))

and R p: —* Out(7rP (X,z)). p : Aut(7rP R(X,x)) -> is the image Definition The group Definition 11.1. 11.1. i) The group Q{X,x)W is image of

ii) ii)

G((p%)inin

The group Q(X,x) x) is is the the image image of of G((px) in Out(?rPR(X, x)). x)).

Let xx and k-points ofofX. X. Let beaa path path in in X(C) X(C) from Let and yy be two two fc-points Let -y 7 be from xx to

y). The Then-y7 induces induces an an isomorphism isomorphism c 7 : TTP R (XC,X) y. Then x) —» 7rPR(Xc,y). The induced of outer outer automorphisms groups induced isomorphism isomorphism of groups :

(c 7 ), : Out(7rP R (X c ,z)) —+ -> Out(7rPR(Xc,
does not depend depend on on the thechoice choice of of 'y7 and and gives gives the thecanonical canonical identification. identification. We need this k. Let us this identification identification over over Spec Specfc. us consider consider the the morphism morphism 1 —÷ XdA[i] of of cosimplicial cosimplicial spaces. spaces. Spec Spec (H^R(p— (x,y))} p. . XA[i] _^ n))) isis R y)-torsor (see a ?rP (X, 2/)-torsor (see [Wi] [Wl] Section Section 3). ItItfollows follows from from [S] [S] Proposition 4.1 4.1 R R y)-torsor is trivial that any any 7rP (X, y)^-toTSOi is trivial, hence any 7rP (X, ?/)-torsor (inverse limit of of surjective surjective maps maps of of sets sets (k-points) (inverse limit (A:-points) is is always always non non empty). empty). 1 k-point 77ofofSpec Any fc-point Spec ( ^ D R ^ * " ^ ? a))) / ) ) ) determines determinesananisomorphism isomorphism

The isomorphism is unique unique up up to conjugation The isomorphism is conjugation by by elements elements of of Hence the induced isomorphism Hence isomorphism (c,), : Out(7T?R(X,z)) ->

nfR(X,y). y).

Out{nfR(X,y))

canonical, in in other other words words itit does not depend on k-point of is canonical, on the the choice choice of a fc-point of

Spec ((HgR(P51(X, s))). Spec ^ ( p - " 1 ^ ,
^

7r?R

I induced by by the the path path 7. Observe where c 7 and c1 c7 are induced Observe that c7 = c,1 cv ooconj(#), conj (g), where 7ij7 is (HgR((x; where is a k-point fc-point of Spec Spec (H^ x, n))) y))j and andconj(g) conj(#)isisa aconjugation conjugation R((X; x, x)(C). This implies by an element element gg E€ 7rfR(X,x)(C). implies the the following following result.

Zdzislaw Wojtkowiak Zdzislaw

270

n y)(fl) Proposition 11.2. 11.2. The Thegroups groups g(X, Q(X,x)^ and g(X, Q(X,y)( ^ coincide under The groups groupsQ(X,x) g(X, x) and Q(X,y)y) coincide the canonical canonical isomorphism (c^)*. The under the the canonical canonical isomorphism isomorphism (c^)*. We We shall denote denote these these groups groups by by respectively. GDR(X) respectively. and cDR(X)

§12.

Torsors associated to to the Torsors associated the canonical canonical unipotent unipotent connection connection

with logarithmic singularities. singularities. 12.0. Let 12.0. Let V V be be as asin insection section 1.1. 1.1. We Weshall shall assume assume that thatA'(V) A1^) —* -» HLR(V) H]^R(V) is an isomorphism. We assume assume that that V isomorphism. We V has a k-point fc-pointx. x. We recall from from Corollary 2.4 that that we we have an isomorphism uu :: Trf^V, x) x) —+ —> 11(V) := Spec(Alg7r(V))). Spec(Algir(V))). Let u(C) ::
and bDR

irf(V,x)(C).

: iri(V(C),x)

Let ^ a;^ ®. representing a class c in if£ R(/9 # ~ 1(x, #)). 0 •.. . 0 ^ f c be a cocycle cocycle representing R (We Spec (flS R (p— ^ ^ x ) ) ) . ) We can (We recall recall from from §2 §2 that that n? (V,x) := Spec one-forms (see (see 2.2.1). 2.2.1). Let Let a TTI(V(C),X). The assume that all all Uit are one-forms assume a Ee iri(V(C),x). The value of c on on bDR(a) is given by the the following following formula: formula:

12.1. 12.1.

:= c(bDR(a)):=Y
J

. . , (see 2.8). -->«>ik (see2.8).

in the same way. way. We want want to The homomorphisms 6Q and 6^ are defined defined in calculate G(<^x)-torsor T((px) associated associated to the comparison homomorphism comparison homomorphism .÷

Lemma 12.2. Lemma 12.2. The Tfte G(ipx)-torsor T(tpx) is isomorphic isomorphic to G(u(C) G(u(C) oo ipx)T(u(C) o ipx). torsor T(u(C) fromthe thefact fact that that ua :: Proof. It follows follows from k. *.

11(V) TT^R(V^X)x) —* -> Ii{V)

is defined defined over over

0

We shall relate relate the G(u(C) oo (px)-toTSOTT(u(C) We shall T(u(C)o o cpxto ) tothe the monodromy monodromy H(V)(C) of the homomorphism 0x,V 9xy :: iri(V(C),x) 7Ti(V(C),:r)—+ -> II(V)(C) the form formWV UJV from §1. §1. Lemma 12.3. 12.3. We Wehave have

u(C) obDR

Monodromy of of Iterated Iterated Integrals Monodromy

271 271

—*k —>®. Proof. The The isomorphism isomorphism u is is induced by Uix®. ®.. .<8>u)i (X^ ®.. .(&Xik)* . .®Xzk) 2.4.). The value (see corollary corollary 2.4.). value of the class class cc on on bDR(a) is given given by 12.1. value of of £(* i f c , . . . , c ^ The value (see Definition follows from Definition 1.3 1.3 and and Proposition Proposition 1.8). 1.8). It follows from [Chi] [Chi] (1.6.2) (1.6.2) that (see .

f

It follows froni the definition follows from definition of of the the homomorphisms homomorphisms bDR bDR and b^ that LB — LDR

iriB(V(C),x)(Q) R be a k-algebra. Let R fc-algebra. Let Let iR iR : Trf (V(Q,ar)(Q)—+ -> Trf ( ^ ( C ) , ^ ) ^ ) be the inclusion of Q-points Q-points into into ii-points. R-points. We the inclusion of We have have iC ic °o bq = b^. We set :

=ZR0bQB.

define a functor Tv on k-algebras in the following way: We define a functor Xy on fc-algebras in the following way: r 7r1(V(C),x) —* H(V)(R) IV(R) := := {/ -> II(V )(/2) | 3 an isomorphism Iv(R) {f : TTI(F(C),X)

/ : **(V(Q,x)(R)

—*rI(V)(R), -> U(V)(R), fobBR = / } .

Observethat that /f is determinedby by/.f. The The functor functor Xy Iv is Observe is uniquely uniquely determined is represented represented k, which we also denote by by an affine affine pro-algebraic scheme overfc, by Xy. Moreover Xy Iv is over is an an AutH(V)-torsor, AutII(V)-torsor. Observe Observethat that0x,V 6xy ::iri(V(C),x) TTI(V(C),Z)—* -> H(V)(C) n(V)(C) is a C-point of of 1v Xybecause because6x,V 6xy = = (u(C) (u(C) oo ipx) o b^. over kk of of Xy Iv such We denote by T{0xy) the smallest smallest subtorsor subtorsor defined defined over such that 0x,v 6xy EG T(6xy)(C) (i.e. (i.e. the theintersection intersection of ofall allsubtorsors subtorsors of of 1v Xydefined defined over fc, k, which over which contain contain 0x,V 6xy as aa C-point). C-point). The corresponding corresponding group group we we denote by G(9xy).

Lemma 12.4. The Lemma The G(9xy)-torsor T(6xy) ipx)-torsor T(u(C) oo ipx).

is isomorphic to the G(u(C) ao

First we notice that that the Aut([I(V)-torsor Iv isisisomorphic Proof. First we notice Aut(II(V)-torsor Xy isomorphic to the the Aut(ll(V)r) torsor x) xx k,k,H(V)). 11(V)). The isomorphism Aut(II(V torsorIso(ir13(V(C), Iso(?rf (V(C),a;) isomorphism is is given given by by /f —* -> /f.. The equality 9x,V 8xy = (u(C) (u(C) oo ipx) oo bB implies the lemma. lemma. 0 Corollary Corollary12.5. 12.5.Let LetVVbebea projective a projectiveline lineover overkkminus minusa afinite finitenumber numberofof k-points. Let Let S be be aaloop loopon onV(C) V(C) around aroundaamissing missingpoint. point.Then Thenthe theelement element is conjugated bDR(S) e K?R{V,V){C) conjugatedtoto s~2ni, where wheres se e 7r?R(V,v)(k). Proof. ItIt follows follows from from Lemma 12.3 and Corollary 5.4. 5.4.

0

12.6. Let Let V be be aaprojective projective line line P£ minus minus a finite finite number number of of k-points. fc-points. If x is If is aa tangential tangential base base k-point fc-point iiv then then the thegroups groupsin8 ?rf and andinf)R ?rfH are not defined in in [Wl]. [Wi]. We defined We set set iniB(V(C),ii) ir?(V(C),v) := (lrl(V(C),ff))Q (iri{V(Q>v)h -Malcev -Malcev rational iri(V(C) , ii). completion of 7ri(V(C), v). For For the Q-algebra R, the map map 6f : ini(V(C), TTI(V(C), V) —+ -> (lrl(V(C),ii))Q(R) n?(V(C),v)(R) is the natural naturalmap mapini(V(C),ii) ni(V(C),v) —+ -> (ni(V(C),v)) (see Q(R)

272

Zdzislaw Zdzislaw Wojtkowiak

ii) —* §A.l). The monodromy monodromy homomorphism homomorphism 0$y :• iri(V(C), fti{V(C),v) —»11(V)(C) Il(V)(C) isis 1), hence the Malcev Malcev C-completion C-completion (see (see §A.l), hence there is is an an isomorphism isomorphism (p$ : iriB(V(C), o ?rf (F(C), ii)(C) v){C) —÷ -> 11(7) (C) such such that (fyob£ 0€y. 11(V)(C) == Otl,v• The G(0ity)-toT8OTT(Oil,v) T(9$y) isisdefined definedasasabove. above.We Wesetsetalso also ?rfR(V, ii)v) := [1(V). The isomorphism H(V). isomorphism u is the identity and and the theG(Oji,v)-torsor G(^y)-torsor T(6^y) is isomorphic to G(^)-torsor T{{p$). isomorphic

j13. Partial information {O, l,oo}). 1, oo}). §13. information about GDR(PQ \\ {0, 13.1. oo}and andlet letvvbe be aa Q-point Q-point of of VV or or a tangential 13.1. Let Let V V = PQ \ {O, {0,1,1,oo} tangential Q-point of V. base Q-point V.For Forany anyQ-algebra Q-algebraRRwe weset: set: 3 'T(R) := {(x,y,z) T(R):={(x,y,z)€(H(V)(R)) \ 3a eR*,

x = aX,

y « aY, z ~ aZ, x-y-z

= 0}

and

Auf (n(V))(ii):= E

E

R*, 1(X) =

1(Y)

aY, 'S-' cxZ}. aY, 1(Z)f(Z)~aZ}.

The functors (11(V)) are represented functors TT and andAut* Aut*(II(V)) represented by by affine affine pro-algebraic pro-algebraic schemes Aut*(II(Vr)). Aut*(11(V)) Aut*(II(Vr)) is is schemes over over Q, Q, which which we we also also denote denote by by TT and Aut*(11(V)). an affine pro-algebraic group group scheme, scheme,aasubgroup subgroupofofAut(II(V)). Aut(H(V)). Moreover Moreover TT affine pro-algebraic is an Aut*(II(Vr))-torsor. (Let (Let (x, (x,y,z) y, z) and and (x',y',z') (xf, y', zf)bebeininT(7Z). T(T^). The The map map be be extended to an of the Lie Lie algebra, x —* —>x',x', y —* y— > can y' can extended to automorphism an automorphism of the algebra, y' which is also an automorphism automorphism of of the the group group 11(V) U(V)(R). Hence the action action isis which (R). Hence transitive.) If we replace in the above replace the group group 11(V) U(V) in above definitions definitions by H2(V) n 2 (F) (resp. (resp. [12), II 2 ), then we obtain an an Aut*(112(V))_torsor Aut*(n2(Vr))-torsor T2 (resp. Aut*(112)_torsor Aut*(II2)-torsor T 2 ). Let us fix generators 5Q, S'1 iri(V(C) , v) which are loops loops around 5^ and S^ of TTI(V(C), V) which f 0, such that S 0 • S[ • S'^ ==1.1. 0,11 and oo respectively, respectively, such

Lemma 13.1.1. 13.1.1. The The Aut* (U(V))-torsorT is aa subtorsor subtorsor of of the the Aut(11(V))Aut(U(V))torsor Xy. torsorlv. Proof. Let (x,y,z) 11(V)(R) be given Proof. Let (x,y,z) EG '1(R). T(R). Let ni(V(C),v) —* -> U(V)(R) given Let /f : iri(V(C),v) :

by f(S'Q) = x,

iri(V(C),v) —+ f(S[) = y. The homomorphism homomorphism 6f : KI(V(C),V) -> iif(V(C), iri(V(C), v) Trf (F(C), v)(R) isis the the Malcev Malcev R-completion ^-completion of TTI(V(C), V) (see (see §A.1), §A.l), hence there is an isomorphism isomorphism /f: :Rf(V(C),v)(R) 7rf (V(C), v)(i?)—+ -^ 11(V)(R) II(Vr)(i2) such such that that /ofeg = /. Therefore ff £l 0 (R). E Iv v(R). :

iri(V(C),v) —+ Let 0V : 7Ti(V(C),v) -> H(V)(C) n(V)(C) be the monodromy monodromy homomorphism homomorphism of the one Observethat that the triple (0v(SfQ), 0v(S[), ^ ( S ^ ) ) £E '1(C). one form form WV. uv- Observe T(C). Let T' (Or) be the smallest subtorsor defined over Q of '1 such that the Tf(9 ) be the smallest subtorsor defined over Q of T such that the triple v

Monodromy of of Iterated Iterated Integrals Monodromy

273 273

e Y(C) (01/OSo), ev(S[), OviS'n)) G T(C) isisaaC-point C-point of of T'(0V). Let Let G'(0V) be be the the corresponding group. group. We recall recall that in corresponding in§12 §12 we we defined defined G(0v)-torsor T(6v). Lemma The G(0v)-torsorT(0v) and the G'(0v)-torsorT'(0v) are Lemma 13.1.2. The subtorsor of of Xy 1v via a subtorsor equal if we identify T with a viathe themap map from from Lemma Lemma 13.1.1.

both torsors and We shall shall use use the the same notation T(0V) and We and G(0V) for both torsors and both groups. groups. 7r1(V(C), ~ot)—*—> 13.2. We We recall recall that that the themonodromy monodromy homomorphism homomorphism 0_A> : 7Ti(V(C), givenby by H(V)(C) is given Tl(V)(C)

= (—2iri)X 0^(SO) = (-2ni)X

=

Y))11 .• (—2iri)Y (-2ni)Y . <(X,Y)y)

= (—iriX)

.

.

Z))11. (—2iri)Z. aJ?(X, Z) Z).

where aJ?(X,F) e6 (ir(V),ir(V)) (n(V),ir{V)) and and 5 0 ,5i,5oo are as as in inthe the figure figure in in0.2. 0.2. E T(C). Y(C). Hence we get get the G(0^)The triple (0^(S o ),0_£(Si),0^(Soo)) ^ Hence we torsor T(0__A.), aa subtorsor of of Y. T.

Proof of of Theorem BBfrom Proof from the theintroduction. introduction. Let Let vu be beaaQ-point Q-point of ofV. V. It follows from Lemmas Lemmas 12.2 12.2and and 12.4 12.4 that that the follows from thegroup groupschemes schemes G(tpv) and and is induced induced by by the the isomorphism isomorphism uu:: G(0V) are isomorphic. isomorphic. The isomorphism isomorphism is TTP^V, V) v) -¥ —+ n(V). H(V). Hence the images of the groups G((pv) and G(0V) in the groups of outer automorphisms automorphisms are also also isomorphic. isomorphic. The The homomorphisms homomorphisms groups 0V and 0_A, are conjugate; conjugate; hence hence the the images images of of groups G(0V) and and G(0_^) G(01) in r r Out(ll(V)) is Out(II(V )) are are equal. equal. Observe Observe that the the map map Aut*(H(V)) Aut*(II(V )) -÷ ->• Out(H(V)) O\it(U(V)) is injective. areare isomorphic. injective. Hence Hence cDR(V) fon(V) and G(01) G(0—±) isomorphic. 0 iri(V(C), ot) it) ^ H2(C)) 13.3. Let Let 00::iri(V(C), 7n(V(C), it) ot)—+ -> nH2(V)(C) (resp.02 :: TTI(V(Q, n 2 (C)) 2 (V)(Q (resp.02 be the composition composition of of 0_^ with the the projection projection on onH2(V)(C) n2(Vr)(C) (resp.H2(C)). (resp.n2(C)). The triple (0(5o),0(5i)1°0(5oo)) ee T2(C) (resp. (02(5O),02(51),02(5OO)) €e T 2 (C)). Repeating the theconstruction construction from from 13.1 13.1 we we get getaaG(0)-torsor G(0)-torsor T(O) T(0) 2 T(92)), (resp. G(92)-torsor G(02)-torsor T(0 )), a subtorsor subtorsor of ofAut*(H2(V))_torsor Aut*(n 2 (^))-torsor 7 T2 (resp. Aut*(n2(C))-torsor T 2 ). The projections projections p:p 11(V) : U(V)—+ -> nH2(V) p 2 :: 11(V) II(Vr)—* -^ 112 n 2 induce homohomo2 (F) and p2 morphisms of of group group schemes schemes Aut*(11(V))

2 Aut*(112(V)) and (p2)* : Aut*(H(V)) (112) Aut*(n(F))-> Aut* Aut*(n )

and morphisms of torsors compatible with homomorphisms homomorphisms of of group schemes

P*:T^T2

and and (p2)* : T-> T' 2 .

Zdzislaw Wojtkowiak Zdzislaw

274 13.3.1. We Wehave have Lemma 13.3.1.

G(9)

T(O),

and

= T(92),

>)) = G(02). G(e2).

This lemma lemma follows follows from Lemma 10.1.7. 10.1.7. Observe that the the group group G(9) G(9) isis the the image of the the group group G(Q—±). G(01). This Observe that image of This implies Theorem C from the introduction. implies Theorem C from the introduction.

Corollary Corollary 13.3.2. 13.3.2.

The group to a quotient of group G(02) G(62) is isomorphic isomorphic to quotient of

gDR(P1(C) \ {0, 1, oo}).

13.4. Below we shall shall calculate calculate the G(62)-torsor Below we G(02)-torsor T(62). T(0 2 ). We We have have fl2 II2 = = 2 n 2 2 ofT lirn (n /T + II ). If in definitions lim definitions of T and and Aut*(H(V)) Aut*(II(V)) in in13.1 13.1 we we replace replace the group then we get get an Aut*(n 2 /r n + 2 n 2 )-torsor group 11(V) U(V) by n 2 /r n + 2 II 2 , then 2 2 T . Let 6 n be the the composition compositionofof92 0 with with the the projection projection onto onto II 2 /r n + 2 II 2 (C) 2 2 2 = n (C)/r"+ II (C). The triple triple (e n (5 0 ),e n (5i),e n (Soo)) e€ 7^(C). By definition Tn := T(Qn) is the smallest subtorsor of of T2 defined defined over Q which C-point and Gn = G(On ) is the corresponding group, contains the triple as a C-point a subgroup subgroup defined defined over over Q Q of Aut* A u t * ( n 2 / r n + 2 112). n 2 ) . Let Let p^ : T2+ —+ x and pTh+l Autt p^1 : A u t *(( n) 2 / r n + 1 + 2 n 2 ) -^ Aut* Aut*(n 2 /r n + 2 n(2 )) be induced induced by the the follows from from Lemma Lemma 10.1.7 10.1.7 that projection n 2 / r n + 1 + 2 I I 2 -> n 2 / r n + 2 n 2 . It follows

13.4.1.

PZ+1(Tn+1) = Tn

and P£ + 1 (G n + 1 ) = G n ,

We recall recall that by by definition definition T(02) T(02) := := limT n and G(92) G{62) := limG n . nn

nn

We recall that the the group group of of Q-points (II 2 /r n+2 II 2 )(<2) is a Lie algebra over Q equipped equippedwith withthe themultiplication multiplicationgiven givenby bythe theBaker-Campbell-Hausdorff Baker-Campbell-HausdorfT formula. A linear basis of of this this Lie Lie algebra algebraisis given givenby byX,X, Y, Y, (Y, X),..., formula. linear basis (Y,X),..., (Y, (Y,X)Yn~1. 2 by 02(So) —÷ II 112(C) The homomorphism homomorphism 62 : iri(V(C), 7Ti(F(C), ot) -> (C) is given given by 92(S0) = (—2iri)X {-2m)X and U(S1) 0{Si) = {-2m)Y + § {2-Ki)C>{k){{YX)Yk-1) (see §6). k=2

0. Calculations Calculations of of T0 To and and G0. Go-

Monodromy of Iterated Iterated Integrals Monodromy of

275 275

Observe that (2iri) e0(s1) = We = (—2iri)X (-2ni)X and and 60(51) = (—2iri)Y. (-2ni)Y. Observe {2m) We have have 60(50) e0(s0) = 2,3... . . This not afc-th k-th root is not root of of aa rational rational number number for for any any kk == 1,1,2,3... This implies implies that T0(C) T0(C) -= {aX,aY {aX,aY | a eG C*} C*}and andG0(C) G0(C) =- {fa {fa I fa(X)==aX, aX, fa(Y)== 2 2 2 aY C*}CCAut(112(C)/F2H2(C)). Aut(n (C)/r n (C)). aY I aaeGC*} 1. Calculations Calculations of of T1 Xi and and G1. Gi. = (—2iri)Y, (—2iri)X and have 6i(5 e1(So) = We have ) = {-2m')X and e1(s1) 0i(Si) = (-27ri)y. Hence we get 0 I

T1(C) =

{aX,aYIaa e C * }

and Gl(C) = {/a I / a W = <*X, /a 0 0 = ^

I * G C*}.

2. Calculations Calculations of of T2 T2 and C2. G2. (—2iri)X, (—2iri)Y + (2iri)((2)((YX)Y). We have 0e2(S0) = We have (5 ) = (-2ni)X, 602(S1) (2ni)C(2)((YX)Y). 2 O 2 (5i) = (-2ni)Y 3 Observe that (2iri)((2) = ^(-2TTZ) . This implies that Observe that (2TTZ)C(2) = T { a l , a 7 + ^ a 3 ( ( y i ) y ) | aa EGC*} C* } and and T2(C) = {aX,aY+ 2(C) = G OLX,fa(Y) fa(Y)==aY aY |aaEeC*}. C*}. G2(C) = {/a I fa(X) ==aX, 2(C) = 3. Calculations Calculations of of T3 T3 and and G3. G3. We have have 6e3(s0) = (—2iri)Y+ = (—2iri)X, (-2TTZ)X, 6e3(s1) (-27ri)y + (27ri)((2)((YX)Y) + 3(5I) = 3 (5 0 ) = Assume that that dimG dimG33 = dimT3 = 1.1. Then ititfollows +(27rz)C(3)((rX)F2). Assume dimT3 = follows from 13.4.1 13.4.1 and G3 = = G(O, G(0,0,c ) where c e Q. T3 from and Corollary Corollary 9.2 9.2 that that G3 0, c3) where c3 e 3 3 Q. As 7'3 3 one-dimensional variety we have (C) = {aX, aY + -^a ((YX)Y) + is a one-dimensional have T T3(C) {aX, 3 2 P3((YX)Y ) I aa e C*, /33fteC, 0} for for some some Laurent Laurentpolynomial polynomial C, p(a,/?3) ==o} p(x,y) to each each value value of of aa corresponds corresponds exactly Observe that that to p(x, y) € e Q[x, ^,2/]y]. Observe value /33(a), because 1'3 T3 is is aa G3-torsor. G3-torsor. Hence one value Hence /3 /33(a) = p(a) where 3(a) = p(x) E6 Q[x, Q[x, i ] . If If (aX,aY+ (aX1aY+^a3((YX)Y)^p(a)((YX)Y2) G T3(C), then then e T3(C), 2 ((at)X, ((aOX,(aOy+^(aO 3 ((yX)y)+(p(a)^ 4 + C 3 ^-^ 4 )a)((yX)y )) EG T3(C). T3(C). ((YX)Y2)) 4 Hence p(at) = p(a)t4 + c3(t The Hence p(at) = p(a)t + c3(t — — t4)a. tA)a. Therefore Therefore p(x) p(x) == ax4 ax 4 ++c3x. c3x. The 4 equality a(—2iri)4 + equality (2iri)((3) (27rz)C(3) = = a(—2TT2') + c3(—2iri) C3(-2TT2') implies c3 = =— C(3). implies aa = = 0 0 and c3 —C(3).

we get Therefore we

dimG33 = 11 if and only if dimG if £(3) Q. ((3) GeQ. Therefore ((3) ^ Q (one knows knowsthat that £(3) ((3) is irrational) then dimG3 Therefore if C(3) dimG3 = 2, T3(C) T3(C) = = {aX,aY+ { a X , a F + ±a3((YX)Y)

+/? 3 ((FX)F 2 ) | aa EG C*, /%EGC} C} +i33((YX)Y2)

Zdzislaw Wojtkowiak Wojtkowiak

276 and

C3 (0,0,0 | 0,0,1). 0,0,1). G3 = (0,0,01 n. Calculations of Tn and Gn. n. We have 9 n (S 0 ) = (—2iri)X, {-2ni)X, n-l

+

=

+ k=2

Assume that Assume Gn-\

0, C3, 0, C5, 0, C7, . = G(0, G(0,0, C3,0, C5,0, C7,... | 0,0,63,0,65,0,67,. 0,0,63,0,£5,0,67,...) .

.

Corollary 9.4), where = c4 == ,.. . . .==C2k c2k = .... .. = = 0, C2fc+i = = Corollary 9.4), where c1 c\ = c2 c2 = and —((2k + 1) and 62k+I = 0 0 and 62k+1 = 1 -C(2fc 4-1) e2k+1 0 if ((2k C(2fc ++ 1) 1) EGQ, Q, and C2k+1 c2fc+i == 0 and e2k+1 = 1ifif (see

((2k+1) -1) g Q. Then the torsor to the group G n _i is given by corresponding to torsor T n _i corresponding n-l

= {aX, k=2,k-pair n-l

((i -— £fcHa + ek/3k)((YX)Yk_l) ((1 ((YX)Yk-1) | a € C , Vk VA; ft E€ c}. ,

J

k=3,k-impair n+1 Assumethat that nn = 2p. 2p. Then C(n)(27ri) = r n (-27ri) Assume , rn eE Q. Q. This n equality, the fact fact that (2ni) £ Q Q and the property 13.4.1 13.4.1 imply imply that equality, the

Gn = G(0, 0, C3, 0, C5, . . . , Cn-i, 0 I 0, 0,ff3,0, . . . , £„_!, 0)

and

= {ax, k=2,k-even

((1 k=3,k-odd fc=3,fc-odd

—

fc 1 ((YX)Yk_l) + 6kflk) ekfa) ((yX)y - ) | a e6 C*, V ft EGc}. c}. ,

fe

Assume that n = 2p+ 2p + 1. Assume that dimG n = dimG dim n _i. Then Assume that Assume that Gn = - G(0,0, c 3 ,..., 0, c n | 0,0,63,0,65,...,0,0) 0,0, e 3 ,0, e5 , • • •, 0,0) (by (byCorollary Corollary9.4) 9.4)

Monodromy of of Iterated Iterated Integrals Monodromy

277 277

and n-l

= {aX, {aX, aY+ Tn(C) = aY +

£

k+1 rrk+l((YX)Yk_l)+ ((YX)Yk-1) + ka

k=2,k-even

n-l k 1 n 1 ((1 ((YX)Yk_l) + /3n — kek)cka ((l-e )cka ++Ek/3k) ekpk)((YX)Y - )+p n((YX)Y - )

]T

with with

k=3,k-odd

aEC*; Vfc,fteC; CKGC*; Vk,/3k EC;

p{a,ez-faeb-fa

.. .,&) = 0}

forsome polynomial p(x, y 3 ,2/5,..., for Q[x, ±, 2/3, y5,. y 5 ,..., y5,. . yn) e€ Q[x, . yn). . ,

. ,

Assume that that for some some a$ a0 there there are flu. Then there is Assume are two different different j3 is gg in in n. Then with Gn(C) such such that g(X) g(X) = = X, g(Y) = = Y+ Y + . . . + bn((YX)Yn-1) with bn ^ 0. 0. from the the proof proof of of Lemma Lemma 9.5 9.5that that Gn = 0, . . ,0,0 Then itit follows follows from — G(0, G(0,0,0,c3, C3,0,..., 0, 0 | 0,0,63,0,... 0,0, £3,0,...,,0,1) 0,1) and and dimG n = dim dimG n _i + 11.. Hence for any a E G C* C* there there is aa unique unique /3n corresponding corresponding totothat Observe that /?„ is an an algebraic algebraic that a. a. Observe function of ofa,a,/33, /?3, /?5,... ,, which which depends depends only only on a. a. Hence Hence /3n = = p(a) p(a) for for some p(x) G somep(x) e Q[rr, ^] • Choose any /f EG G n (C). Then Choose Then f(X) ==tX tX and and .

.

.

I

n-l

f(Y) + f(Y) == tY tY+

Yl

((1 ((!—"Ek)Ck(t ^)C*C —"tk+1) + Ekbk) ((YX)Yk_l)

k=3,k-odd

+

—

. 6 n _ 2 EGC.C.Acting any e C*,b3,b5,. for some t G 63,65,..., Actingbybyf /onon anyelement elementofof Tn(C), we get get .

n-l

((at)X, {{crt)X,(at)Y+ (at)Y+

Yl

rk(at)k+l((yX)ykl) rk(at)k^1((YX)Yk-1)

k=2, fc-even k-even fc=2, n-l

+

J2

1 tk+1) (a(i («(1 —- Ek)Ck(t ek)ck(t —- t^ ) ++aekbk aekbk + + (1 (1—

k

k=3,k-odd

+ + ek(3ktk+1)((YX)Yk-1) + + ( a C n— ^ - r + ^ ++ ^ Therefore we get Therefore we get

p(at) = p(at) = cna(t - tn+1) + p(a)tn+1. p(x) = ax n + 1 + c n x. The equality ((n)(—2iri) This implies implies p(x) C(n)(-27ri) = a(2ni)n+1 + implies p(x) p(x) = = cnx and cn(27ri) implies and cn = = —((ii). — C(^). Therefore Thereforewe weget get dimG n = dim dimG n _i

if and only if C(n) eG Q. Q.

278

Zdzislaw Wojtkowiak Zdzislaw

The final result isis the thefollowing. following. Proposition 13.5. 13.5. Let Let G G = limG n . Then G(02) G(62) = G and G = = n

G(0,0, c 3 ,0, c 5 ,0, c 7 , . . . , 0, c2jfe+i, 0 . . . | 0,0, e 3 ,0, e 5 , 0 . . . , 0, e2fc+i, 0 . . . ) ,

where: 62k+1 £2fc+i = 00 if and and only only if((2k if ((2k++1)1)EGQ, Q,and andthen thenC2k+1 c2fc+i = —c(2k — ((2k ++1). 1). 2 The torsorT(6 torsor T(92)(C) )(C) isisgiven given by by oo

{aX,aY+

J2

oo

k+l

k 1

rka ((YX)Y - )++

^

((i —Ek)cka+ ((

k=3,k-odd

k=2,k-even

+ek/3k)((YX)Ykl) k, /3k 'M I a €E Ct C , ,for for all all Jk, /3fc ee c}, where((2k) C2k+1 where ((2k) == _r2k(2lri)2k, -r2k(27ri)2k, and andc = —((2k+1), —C(2*:H-1),C2k+1 ^2fc+i == 00 if((2k+1) i/C(2fc+l) EG 2k+i = = 1 if((2k + 1) e2k+1 = 0, Q, Q.

Corollary 13.6. group G G contains the group group HH := {fc, Corollary 13.6. The T/ie ^ro^p contains the {/a | fa(X) fa(X) = aX, fa(Y) = aY aX, ctY \ a e Ct} C*} ifif and and only only ifif all allnumbers numbers (,(2k + 1) are are irrational. irrational. Proposition 13.2 that all ((2k+ Proof. ItIt follows follows from Proposition 13.2 that £(2fc + 1) l) are irrational if and only ifG = G(0,0,0,...,0,... 0,0,1,0,1,0,...0,1,0,1,0,...) only if G | 0,0,1,0,1,0,.. .0,1,0,1,0,...) (all (all c2 a = 0, el=0,alls2k=Oandalle2k+l=lfork=1,...). e\ = 0, all 62k = 0 and all £2fc+i = 1 for fe = 1,...).ThenwehaveHcG. Then we have H C G.

LetHCG=G(0,0,c3,0,c5,...,0,c2k+l,0,..l0,0,63,0,65,0,...,0,e2k+l, existsff €e G such that /(X) f(X) = aX 0,. .). Then 0,...). Then there there exists such that a l and .

f(Y) = aY + 9.5 implies impliesthat that GG = G G(0, where all X2k+1 X2A;+I / 0. 0, Corollary Corollary 9.5 ( 0 ,0. 0 . . . ,,0,... 0,... | 0,0,1,0,1,. ..0,1,0,...) (all = 0, 1 for fek == 0 , 0 , 1 , 0 , 1 , . . . 0 , 1 , 0 , . . . ) (all a 0, ex = e2fc = 0, 62k+1 e2fc+i = 1 for = E2k 1,2,...). 0 I

Corollary 13.7. 13.7. Let Le£ ^ be 6e the smallest closed closed subgroup subgroupofofAut(ir2) Aut(?r2)defined defined overQ, containing G andH. Then g over Q, containing G and H. Then Q = G(0,0...,0,... G(0,0..., 0,... |10,0,1,0,1,0...) 0,0,1,0,1,0 ...) fa// Ci = 0 , e i =e2k=0,e2k+1=lfork=1,2,...). = e2fc = 0, e2k+i = 1 for k = 1,2,...).

Proof. group Q Gcontains contains/ f such suchthat that f(X) f(X) = Proof. The group =X X and

f(Y) = Y + k=l

0. Lemma 9.5 implies the corollary. where all X2k+1 x2k+i ^ 0. corollary.

Monodromy of of Iterated Iterated Integrals Monodromy

279 279

13.8. Proof Proof of of Theorem TheoremDDininthe theintroduction. introduction.We Werecall recallfrom from §6 §6 that that the monodromy monodromy homomorphism homomorphism

0: iri(V(C), ot) —+ 112(V)(C) is given by

+

SS0o - • (-27rt)X,

i=O,j=O

(see §6, §6, formula formula(7)). (7)). It It is an observation Drinfeld that that the numbers (see observation of Drinfeld numbers satisfy satisfy the equation equation

-—1 1 ++

vtm = n>0,m>1 n>0,m>l — —

_exp(_ (

.

k=2i>1,j>1 i+jk k=2

(see [Dr]). [Dri). Therefore (see Therefore we can describe describe the themonodromy monodromy homomorphism homomorphism 00inin the following following way

(k_i)!c(k)xi

So

k=2 k=2 i+j=k ?+j=k

.

J

i 1 ,j 1

Let us temporarily temporarily write write X" T' for the the torsor given Theorem D D i) i) and G' Let given in Theorem Gf for group. Then Then T(6) T(0) is is aa subtorsor subtorsor of of T' T' and and the for the corresponding corresponding group. the group group G(0) U(S1)) G E T'(C). G{6) is is aa closed closed subgroup of G' because because (0(S0), (0(So),0(Si)) T'(C). The Theimage image n+2 r of the group group G' G' in in Aut(n(F)/r II(V )) has has the thesame same dimension dimension as as the the subgroup Gn of A u t ( l l 2 / r n + 2 n 2 ) . The group G(O) G(6) projects onto G n , hence T' == T(0). Gf = = G(0) G{0) and andT' T(0). The point iii) iii) of of Theorem Theorem D Dand andCorollary Corollary EEfollow follow from from Corollary Corollary 13.7 13.7 and Corollary Corollary 13.6 13.6 respectively. respectively.

§14. The forpointed pointed projective projective lines lines and and for con§14. The group cDR(V) G*DR(V) for configuration spaces. spaces. In this section we shall shall translate translate the results In section we results about the themonodromy monodromy representations from from §§5-8 into into results results about about the the group group GDR(V). cDR(V). Let Let V V= = P£\{ai,...,, a n +i} and let v be a k-point Appoint of V. V. We can assume assume that that a\ a1 = = 0, a22 = = 11 and a n +i = oo. 0, a Lie(V) isisaafree free Lie Lie algebra algebra over over k x. We recall recall that Lie(V) on Xi := := (z^_za. =)*,1,2,...,n. 2 = 1,2,..., Letn.usLet setus set X=n+\ = — ^^=1 LetXi.R Let be aR be a fc-algebra. Let Letususset set k-algebra. •

Aut*(ll(V))(R) Aut* (U(V))(R) := :={f{/ GAut(H(V)R)I Aut(U(V)R) | 3a/EGR*, i?*, 1(X1) /(*i) == afXu —afXk 3,...n+1}. f(X2) « a/X 2 , f(Xk) /(Jffc) ~ a/^ib *k= = 3, ...n + l}.

Zdzislaw Wojtkowiak Wojtkowiak

280

(R) with with the zero Here « means a conjugation by by an an element element of of 11(V) U(V)(R) zero coefcoef(R). ficients ficients at X1 X\ and X2 and ~ means conjugation by by an anelement elementofof11(V) U(V)(R). x 3 (P'(C)\{O, Similarly for yn = (P (C)\{0, 1, l,oo})"~ we set Aut*(U(yn))(R)

R*, /(X := {/ ,*^-}. 1(X23) i f i ) - az1Xj,j}. {f EG Aut(n(y n )R)| 3a/ €E JT, j

(The elements elements X2,3 Xij are as in in §7.) §7.) These functors These functors are are representable representable by by affine affine pro-algebraic pro-algebraic group group schemes schemes over over Q, Q, which we denote by by Aut* Aut* (11(V)) (11(10) and and Aut* Aut*(lI(Xi)) rereover kk and over spectively. Let us set r (Aut* ([1(V)) Out* (11(V)) := image (11(V)) /Inn(H( V))) Out*(11(10) image(Aut*(n(V ))—+ -> Aut Aut(n(F))/Inn(n(F)))

and Out*(11(yn)) Aut*(11(yn))/Inn(11(yn)). Out*(n(y n)) := Aut*(n(yn))/inn(n(yn)). We shall define an Aut* (11(10 )-torsor T We T and and the the Aut*(ll(Yn))4orsor Aut*(n(^ n ))-torsor Tn as follows. Let S1,. follows. Let S i , . . . , S n + i be geometric geometric generators generators of of iri(V(C), TTI(V(C),V),v), i.e. . . ,

loops For any fc-algebra k-algebra R we loops around a1,. a i , ... . , an_|_i respectively. respectively. For we set . ,

R*, f(S T(R) := {f f(S1) E IT, T(R) {/ eHom(iri(V(C),v);H(V)(R)) €Hom(in(V{C),v);Il(V)(R)) \ 3af G 1) = f(S2) w afX2, f(Sk) ~ a/Xfc *k=3,...,n+1} = 3,..., n + 1} and and

TTh(R) = T"(i2) = {(f eHom(vi(yn{C),y);Il{yn)(R))

such that \ 3a,, £e R* such

are as in §7.) (The elements A23 A^- are

Proposition Proposition 14.2. 14.2.We Wehave have GDR(V) QDR(V)

c Out*(H(V)). C Out*(R(V)).

Proof. Let us choose the tangential base base point. point. Let Let S\ S1 be be Proof. Let us choose the vector vector 0i as as aa tangential a a ioop loop around around0 0asasininthe thepicture picture ofof §4. §4. Let Let us us choose choose aa tangent tangent vector vector

at each point point ak. a&. Assume that that v2 V2 = T^. Let Let F be a family of paths such that 1]. Let such that 72 isis the the interval interval[0, [0,1]. Let sequence of geometric generators 81, . , S n +i be a Si, 82, S2,..., a sequence of geometric generators associated associated to to F. F. Let Let —+f 11(V)(C) 0_^ : iri(V(C), TTI(V(C), "ot) — II(y)(C) be the the monodromy monodromy homomorphism. homomorphism. It follows from that follows from Corollary Corollary 5.4 5.4 and and Theorem Theorem 6.1 6.1 that vk Vk

from from the the base base point point to to v2, V2,. -.. . •,, vn+i . .

E T(C). G

Monodromy of Iterated Iterated Integrals Monodromy of

281 281

that gDR(V) c Out*(H(V)) Aut*(n(F)). This implies implies that QDR{V) C Out*(n(F)) cC Aut*(H(V)).

if we identify (V,v)v) and and11(V) II(F) by by the thehomomorphism homomorphism uu from from Corollary Corollary we identify n^ irf(V, 2.4. 2.4. 0 Hence G(0^)

R

Aut*(n(3^ Hence T n /Inn(n(^ n )) is an an Observe that that TTh Tn is Observe is a Aut* (IT(Yn))-torsor. n))-torsor. Hence Out* Out*(n(^ n ))-torsor.

Proposition 14.3. 14.3. n)cC

Out*(H(yn)). Out* (n(y )). n

n

Proof. Corollary that 0y^yn EG T (C). Hence Proof. Corollary 7.3 implies implies that Hence the proposition proposition follows. follows.

Proposition 14.4. 4 4wewehave 14.4. For n > haveisomorphisms isomorphisms

induced induced

by the the projection projection Yn+i by yn+\ —> yn.

from Proposition 7.4, Lemma 7.5 and CorolProof. The The proposition proposition follows follows from lary 7.6. 7.6. 0

Observe that that Y4 {O, 1, oo}. Hence we have the following Observe 34 = PQ \ {0,1, following corollary. }. Hence Corollary 14.5. 14.5.For Fornn>>44we wehave have 1, oo}). {0,1, GDR(yn) « GDR{FQ \\ {O,

group 11(V) The group H(V) isis a normal normal subgroup subgroup of of n(3^ n+2 ). Hence n(>?n+ 2) C Aut (H(V)). Proposition 7.7 Aut(n(F)). 7.7 implies implies the the following following result.

Proposition 14.6. Proposition 14.6.We Wehave have GDR(V) gDR(V) .-

n(yn+2)/n(yn+2)«

gDR{v\i \\ {O, {o,1,1,oo}). oo}).

§15. §15. Conjectures. Conjectures. Let X X be be aa smooth smooth quasi-projective quasi-projective algebraic algebraic variety variety defined defined over a number field field kk and and let let a; x be a k-point ber fc-point or or aa tangential tangential base base k-point. Appoint. Denote Denote by by x k, the pro-I completion of the étale fundamental group of X x k.L fc, x)i pro-1 completion of etale fundamental group o f X x 7rf*(X k

k

Let &x ': Gal(k/k) Gal(k/k) —* representation Let —> Out(7rf*(Xx k, x k,x)i)bebethethenatural natural representationofof k the Galois group into the outer automorphisms of the pro-i completion of the Galois group into the outer automorphisms of the pro-1 completion of the étaie fundamental group. the etale fundamental group.

Zdzislaw Wojtkowiak Zdzislaw

282

representations of of iterated iterated integrals, integrals, i.e. i.e. the homomorThe monodromy monodromy representations homomorphisms