Diophantine Approximation and Abelian Varieties: Introductory Lectures

Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, ZUrich F. Takens, Groningen

1566

B. Edixhoven J.-H. Evertse (Eds.)

Diophantine Approximation and Abelian Varieties Introductory Lectures

Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo Hong Kong Barcelona Budapest

Editors Bas Edixhoven Institut Mathematique Campus de Beaulieu F-35042 Rennes Cedex, France Jan-Hendrik Evertse Universiteit Leiden Wiskunde en Informatica Postbus 9512 NL-2300 RA Leiden, The Netherlands

Mathematics Subject Classification (1991): 14G05, 14G40, 11J99, 11G35

ISBN 3-540-57528-6 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-57528-6 Springer-Verlag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. 9 Springer-Verlag Berlin Heidelberg 1993 Printed in Germany 2146/3140-543210 - Printed on acid-free paper

Preface

From April 12 to April 16, 1992, the instructional conference for Ph.D-students "Diophantine approximation and abelian varieties" was held in Soesterberg, The Netherlands. The intention of the conference was to give Ph.D-students in number theory and algebraic geometry (but anyone else interested was welcome) some acquaintance with each other's fields. In this conference a proof was presented of Theorem I of G. Faltings's paper "Diophantine approximation on abelian varieties", Ann. Math. 133 (1991), 549-576, together with some background from diophantine approximation and algebraic geometry. These lecture notes consist of modified versions of the lectures given at the conference. We would like to thank F. Oort and R. Tijdeman for organizing the conference, the speakers for enabling us to publish these notes, C. Faber and W. van der Kallen for help with the typesetting and last but not least the participants for making the conference a successful event.

Contents Preface List of C o n t r i b u t o r s Introduction Terminology and Prerequisites Diophantine Equations and Approximation, by F. BEUKERS 1 Heights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The Subspace Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Weil Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Vojta's Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II

III

IV

V

VI

v ix x ooo xnl

1 1 2 5 5 9

D i o p h a n t i n e A p p r o x i m a t i o n a n d its Applications, by R. TIJDEMAN 1 Upper Bounds for Approximations . . . . . . . . . . . . . . . . . . . . . 2 Lower Bounds for Approximations . . . . . . . . . . . . . . . . . . . . . 3 Applications to Diophantine Equations . . . . . . . . . . . . . . . . . . .

13 13 15 17

Roth's Theorem, by P~. TIJDEMAN 1 The Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Variations and Generalisations . . . . . . . . . . . . . . . . . . . . . . .

21 21 29

T h e S u b s p a c e T h e o r e m of W . M . S c h m i d t , by J.H. EVERTSE 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 About the Proof of the Subspace theorem . . . . . . . . . . . . . . . . .

31 31 33 37

Heights on A b e l i a n Varieties, by J. HUISMAN 1 Height on Projective Space . . . . . . . . . . . . . . . . . . . . . . . . . 2 Heights on Projective Varieties . . . . . . . . . . . . . . . . . . . . . . . 3 Heights on Abelian Varieties . . . . . . . . . . . . . . . . . . . . . . . . 4 Metrized Line Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . .

51 51 52 55 58

D. M u m f o r d ' s " A R e m a r k o n M o r d e l l ' s C o n j e c t u r e " , by J. T o P 1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63 63

~176176 vnl

2 3 4 5 6 VII

Mumford's Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interpretation, Consequences and an Example . . . . . . . . . . . . . . The Proof assuming some Properties of Divisor Classes . . . . . . . . . Proof of the Divisorial Properties . . . . . . . . . . . . . . . . . . . . . . Effectiveness and Generalizations . . . . . . . . . . . . . . . . . . . . . .

Ample Line Bundles and Intersection Theory, by A . J . DE JONG 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Coherent Sheaves, etc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Ample and Very Ample Line Bundles . . . . . . . . . . . . . . . . . . . 4 Intersection Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Numerical Equivalence and Ample Line Bundles . . . . . . . . . . . . . 6 Lemmas to be used in the Proof of Thm. I of Faltings . . . . . . . . . .

VIII The Product Theorem, by M. VAN DER PUT 1 Differential Operators and Index . . . . . . . . . . . . . . . . . . . . . . 2 The Product Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 From the Product Theorem to Roth's Lemma . . . . . . . . . . . . . . . IX

X

XI

64 64 65 66 67

69 69 69 71 71 74 75

77 77 79 81

G e o m e t r i c P a r t of F a l t i n g s ' s P r o o f , by C. FABER

83

F a l t i n g s ' s V e r s i o n of Siegel's L e m m a , by R . J . KOOMAN

93

A r i t h m e t i c P a r t of F a l t i n g s ' s P r o o f , by B. EDIXHOVEN 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Construction of Proper Models . . . . . . . . . . . . . . . . . . . . . . . 3 Applying Faltings's version of Siegel's Lemma . . . . . . . . . . . . . . . 4 Leading Terms and Differential Operators . . . . . . . . . . . . . . . . . 5 Proof of the Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . .

97 97 97 98 103 105

X I I P o i n t s of D e g r e e d o n C u r v e s over N u m b e r F i e l d s , by G. VAN DER GEER

111

X I I I " T h e " G e n e r a l C a s e of S. L a n g ' s C o n j e c t u r e ( a f t e r F a l t i n g s ) , by F. OORT 1 The Special Subset of a Variety . . . . . . . . . . . . . . . . . . . . . . . 2 The Special Subset of a Subvariety of an Abelian Variety . . . . . . . . 3 The Arithmetic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Related Conjectures and Results . . . . . . . . . . . . . . . . . . . . . .

117 117 119 121 121

Bibliography

123

Contributors F. B e u k e r s Universiteit Utrecht, Mathematisch Instituut, Postbus 80.010, 3508 TA Utrecht, The Netherlands. B. E d i x h o v e n Universit~ de Rennes, Institut Math~matique, Campus de Beaulieu, 35042 Rennes, France. J.H. Evertse Universiteit Leiden, Wiskunde en Informatica, Postbus 9512, 2300 RA Leiden, The Netherlands. C. F a b e r Universiteit van Amsterdam, Wiskunde en Informatica, Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands. G. van der G e e r Universiteit van Amsterdam, Wiskunde en Informatica, Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands. J. H u i s m a n Universiteit Utrecht, Mathematisch Instituut, Postbus 80.010, 3508 TA Utrecht, The Netherlands. A.J. de J o n g Universiteit Utrecht, Mathematisch Instituut, Postbus 80.010, 3508 TA Utrecht, The Netherlands. R.J. Kooman Lijtweg 607, 2341 Oegstgeest, The Netherlands. F. O o r t Universiteit Utrecht, Mathematisch Instituut, Postbus 80.010, 3508 TA Utrecht, The Netherlands. M. van der P u t Universiteit Groningen, Wiskunde en Informatica, Postbus 800, 9700 AV Groningen, The Netherlands. R. T i j d e m a n Universiteit Leiden, Wiskunde en Informatica, Postbus 9512, 2300 RA Leiden, The Netherlands. J. Top Universiteit Groningen, Wiskunde en Informatica, Postbus 800, 9700 AV Groningen, The Netherlands.

Introduction

Although diophantine approximation and algebraic geometry have different roots, today there is a close interaction between these fields. Originally, diophantine approximation was the branch in number theory in which one deals with problems such as approximation of irrational numbers by rational numbers, transcendence problems such as the transcendence of e or 7r, etc. There are some very powerful theorems in diophantine approximation with many applications, among others to certain classes of diophantine equations. It turned out that several results from diophantine approximation could be improved or generalized by techniques from algebraic geometry. The results from diophantine approximation which we discuss in detail in these lecture notes are Roth's theorem, which states that for every algebraic number a and for every 8 > 0 there are only finitely many p,q E Z with la - p/q[ < Iq1-2-a, and a powerful higher dimensional generalization of this, the so-called Subspace theorem of W.M. Schmidt. Here, we would like to mention the following consequence of the Subspace theorem, conjectured by S. Lang and proved by M. Laurent: let F be the algebraic group (~*)n, endowed with coordinatewise multiplication, V a subvariety of F, not containing a translate of a positive dimensional algebraic subgroup of F, and G a finitely generated subgroup of F; then V f3 F is finite. We give a brief overview of the proof of Roth's theorem. Suppose that the equation Ic~ - P/ql < q-2-6 has infinitely many solutions p, q E Z with q > 0. First one shows that for sufficiently large m there is a polynomial P(X1,...,X,~) in Z[X1,..., X,,] with "small" coefficients and vanishing with high order at ( a , . . . , a). Then one shows that P cannot vanish with high order at a given rational point x = (Pl/ql,... ,P,~/q,~) satisfying certain conditions. This non-vanishing result, called Roth's Lemma, is the most difficult part of the proof. From the fact that la - P/ql < q-2-~ has infinitely many solutions it follows that one can choose x such that la -Pn/q,~l < q~2-~ for n in { 1 , . . . , m}. Then for some small order partial derivative P~ of P we have P~(z) # O. But Pi(x) is a rational number with denominator dividing a := qdl "" "q,n dr, , where dj = degxr Hence IPi(x)l > 1/a. On the other hand, Pi is divisible by a high power of Xj - a and IPJ/qJ - a l is small for all j in {1 . . . . ,m}. Hence Pi(x) must be small. One shows that in fact IP~(x)] < 1/a and thus one arrives at a contradiction. Algebraic geometry enables one to study the geometry of the set of solutions (e.g., over an algebraically closed field) of a set of algebraic equations. The geometry often predicts the structure of the set of arithmetic solutions (e.g., over a number field) of these algebraic equations. As an example one can mention Mordell's conjecture, which was proved by G. Faltings in 1983 [21]. Several results of this type have been proved by combining techniques from algebraic geometry with techniques similar to those used in the proof of Roth's theorem. Typical examples are the Siegel-Mahler finiteness theorem for integral points on algebraic curves and P. Vojta's recent proof of Mordell's conjecture. In these lecture notes, we study the proof of the following theorem of G. Faltings ([22], Thm. I), which is the analogue for abelian varieties of the result for (Q-*)"

xi mentioned above, and which was conjectured by S. Lang and by A. Weil:

Let A be an abelian variety over a number field k and let X be a subvariety of A which, over some algebraic closure of k, does not contain any positive dimensional abelian variety. Then the set of rational points of X is finite. (Note that this theorem is a generalization of Mordell's conjecture.) The proof of Faltings is a higher dimensional generalization of Vojta's proof of Mordell's conjecture and has some similarities with the proof of Roth's theorem. Basically it goes as follows. Assume that X ( k ) is infinite. First of all one fixes a very ample symmetric line bundle s on A, and norms on s at the archimedian places of k. Let m be a sufficiently large integer. There exists x = ( x l , . . . , Zm) in Xm(k) satisfying certain conditions (e.g., the angles between the xi with respect to the NSron-Tate height associated to s should be small, the quotient of the height of xi+l by the height of xl should be big for I < i < m and the height of xl should be big). Instead of a polynomial one then constructs a S l , . . . , Sm) d on a certain model of X m global section f of a certain line bundle s over the ring of integers R of k. This line bundle is a tensor product of pullbacks of s along maps A m ~ A depending on a - e , the si and on d; in particular, it comes with norms at the archimedian places. By construction, f has small order of vanishing at z and has suitably bounded norms at the archimedian places of k. Then one considers s~,...,sm) a on Spec(R); the Arakelov degree of the metrized line bundle x * s the conditions satisfied by the xl give an upper bound, whereas the bound on the norm of f at the archimedian places gives a lower bound. It turns out that one can choose the parameters e, ~, the si and d in such a way that the upper bound is smaller than the lower bound. We mention that the construction of f is quite involved. Intersection theory is used to show that under suitable hypotheses, the line bundles s S l , . . . , sin) d are ample on X m. A new, basic tool here is the so-called Product theorem, a strong generalization by Faltings of Roth's Lemma. On the other hand, Faltings's proof of Thm. I above is quite elementary when compared to his original proof of Mordell's conjecture. For example, no moduli spaces and no l-adic representations are needed. Also, the proof of Thin. I does not use Arakelov intersection theory. Faltings's proof of Thin. I in [22] seems to use some of it, but that is easily avoided. The Arakelov intersection theory in [22] plays an essential role in the proof of Thm. II of [22], where one needs the notion of height not only for points but for subvarieties; we do not give details of that proof. The only intersection theory that we need concerns intersection numbers obtained by intersecting closed subvarieties of projective varieties with Cartier divisors, so one does not need the construction of Chow rings. The deepest result in intersection theory needed in these notes is Kleiman's theorem stating that the ample cone is the interior of the pseudoample cone. Unfortunately, we will have to use the existence and quasi-projectivity of the N6ron model over Spec(R) of A in the proof of Lemma 3.1 of Chapter XI; a proof of that lemma avoiding the use of NSron models would significantly simplify the proof of Thm. I. We believe that for someone with a basic knowledge of algebraic geometry, say Chapters II and III of [27], everything in these notes except for the use of N6ron models is not hard to understand. In the case where X is a curve, i.e., Mordell's conjecture, the proof of Thin. I can be considerably simplified; this was done by E. Bombieri in [9].

xii Let us now describe the contents of the various chapters. Chapter I gives an overview of several results and conjectures in diophantine approximation and arithmetic geometry. After that, the lecture notes can be divided in three parts. The first of these parts consists of Chapters II-IV; some of the most important resuits from diophantine approximation are discussed and proofs are sketched of Roth's theorem and of the Subspace theorem. The second part, which consists of Chapters V-XI, deals with the proof of Thm. I above. Chapters V and VII provide the results needed of the theory of height functions and of intersection theory, respectively. Chapter VIII contains a proof of the Product theorem. This theorem is then used in Chapter IX in order to prove the ampleness of certain s sl . . . . , sin) d. Chapter X gives a proof of Faltings's version of Siegel's Lemma. Chapter XI finally completes the proof of Thm. I. Chapter VI gives some historical background on how D. Mumford's result on the "widely spacedness" of rational points of a curve of genus at least two over a number field lead to Vojta's proof of Mordell's conjecture. The third part consists of Chapters XII and XIII. Chapter XII gives an application of Thin. I to the study of points of degree d on curves over number fields. Chapter XIII discusses a generalization by Faltings of Thm. I, which was also conjectured by Lang.

Terminology and Prerequisites

In these notes it will be assumed that the reader is familiar with the basic objects of elementary algebraic number theory, such as the ring of integers of a number field, its loealizations and completions at its maximal ideals, and the various embeddings in the field of complex numbers. The same goes more or less for algebraic geometry. To understand the proof of Faltings's Thm. I the reader should be familiar with schemes, morphisms between schemes and cohomology of quasi-coherent sheaves of modules on schemes. In order to encourage the reader, we want to mention that Hartshorne's book [27], especially Chapters II, w167 and III, w167 and w167 contains almost all we need. The two most important exceptions are Kleiman's theorem on the ample and the pseudo-ample cones (see Chapter VII), for which one is referred to [28], and the existence and quasi-projectivity of NSron models of abelian varieties (used in Chapter XI), for which [11] is an excellent reference. At a few places the "GAGA principle" (see [27], Appendix B) and some algebraic topology of complex analytic varieties are used. A less important exception is the theorem of Mordell-Weil, a proof of which can for example be found in Manin's [52], Appendix II, or in [70]; Chapter V of these notes contains the required results on heights on abelian varieties. Almost no knowledge concerning abelian varieties will be assumed. By definition an abehan variety over a field k will be a commutative projective connected algebraic group over k. We will use that the associated complex analytic variety of an abelian variety over C is a complex torus. Since these notes are written by various authors, the terminologies used in the various chapters are not completely the same. For example, Chapter I uses a normalization of the absolute values on a number field which is different from the normalization used by the other contributors; the reason for this normalization in Chapter I is clear, since one no longer has to divide by the degree of the number field in question to define the absolute height, but it has the disadvantage that the absolute value no longer just depends on the completion of the number field with respect to the absolute value. Another example is the notion of variety. If k is a field, then by a (algebraic) variety (defined) over k one can mean an integral, separated k-scheme of finite type; but one can also mean the following: an (absolutely irreducible) aff/ne variety (defined) over k is an irreducible Zariski closed subset in some affine space K ~ (K a fixed algebraically closed field containing k) defined by polynomials with coefficients in k, and a (absolutely irreducible) variety (defined) over k is an object obtained by glueing afl:ine varieties over k with respect to glueing data given again by polynomials with coefficients in k. As these two notions are (supposed to be) equivalent, no (serious) confusion should arise.

Chapter

I

Diophantine Equations and Approximation by Frits Beukers

1

Heights

Let F be an algebraic number field. The set of valuations on F is denoted by MR. Let I.l~, or v in shorthand, be a valuation of F. Denote by F. the completion of F with respect to v. If F~ is R or C we assume that v coincides with the usual absolute value on these fields. When v is a finite valuation we assume it normalised by IPI~ = 1/p where p is the unique rational prime such that IPl, < 1. The normalised valuation II.ll~ is defined by

I1~11,., = I~1~~'~Q'J'rF~ with the convention that p = oo when v is archimedean and Q ~ = •. non-zero x E F we have the product formula

(1.1)

rl

For any

li~ll,, = 1.

v

Let L be any finite extension of F. Then any valuation w of L restricted to F is a valuation v of F. We have for any x E F and v E MF,

(1.2)

[Ix II,., = II Ilxll.,,, wlv

where the product is over all valuations w C ML whose restriction to F is v. The absolute multiplicative height of x is defined by

H(x) = I'Imax(1, Ilxll~). v

It is a consequence of (1.2) that H(x) is independent of the field F which contains x. The absolute logarithmic height is defined by

h(x) = log H(x). Let P" be the n-dimensional projective space and let P E P~(F) be an F-rational point with homogeneous coordinates (x0, x l , . . . , z ~ ) . We define the projective (absolute) height by h(P) = ~--: log max(llzoll~, , Ilzlll,,,..., II~,,ll.).

2

F. BEUKERS

Again, h(P) is independent of the field F containing P. Therefore the projective height can be considered as a function on P"(F). Notice that the height h(x) of a number coincides with the projective height of the point (1 : x) C p1. The projective height has the fundamental property that, given ho, there are only finitely many P e Pn(F) such that h(P) < ho. Let V be a non-singular projective variety defined over F. Let r : V '--* pN be a projective embedding also defined over F. On V(F), the F-rational points of V, we take the restriction of the projective height as a height function and denote it by he. In general the construction of heights on V runs as follows. First, let D be a very ample divisor. That is, letting fo, f l , . . . ,f,, be a basis of the space of all rational functions f defined over F with (f) > - D , the map r : V ~ P" given by P ~ ( f o ( P ) , f , ( P ) , . . . ,f,~(P)) is a projective embedding. The height hD is then simply defined as he. If D1, D2 are two linearly equivalent very ample divisors, then hD, -- hD~ is known to be a bounded function on V(F). Now let D be any divisor. On a non-singular projective variety one can always find two very ample divisors X , Y such that D + Y = X. Define hD = hx - hr. Again, up to a bounded fuction, hD is independent of the choice of X and Y. We summarize this height construction as follows. 1.3 T h e o r e m . There exists a unique homomorphism linear divisor classes

--*

real valued functions on V(F) modulo bounded functions

denoted by c H hc + 0(1) such that: if c contains a very ample divisor, then hc is equivalent to the height associated with a projective embedding obtained from the linear system of that divisor. We also recall the following theorem. 1.4 T h e o r e m . Let c be a linear divisor class which contains a positive divisor Z. Then h~(P) > 0(1) for all P e

v(-F), P r supp(Z)

For the proof of the two above theorems we refer to Lang's book [36], Chapter 4. Finally, following Lang, we introduce the notion of pseudo ample divisor, not to be confused with the pseudo ample cone. A divisor D on a variety V is said to be pseudo ample if some multiple of D generates an embedding from some non-empty Zariski open part of V into a locally closed part of projective space. One easily sees that there exists a proper closed subvariety W of V such that, given h0, the inequality hD(P) < ho has only finitely many solutions in V ( F ) - W. 2

The

Subspace

Theorem

For the sake of later comparisons we shall first state the so-called Liouville inequality.

I. DIOPtIANTINE EQUATIONS AND APPROXIMATION

3

2.1 T h e o r e m ( L i o u v i l l e ) . Let F be an algebraic number field and L a finite extension. Let S be a finite set of valuations and extend each v E S to L. Then,/'or every a E L, a ~ O we have 1

IX II~ll~ > - -

~es

- H ( a ) [L:~"

P r o o f . Let us assume that [[a[] ~ < 1 for every v 9 S. If not, we simply reduce the set S. Let SL be the finite set of valuations on L which are chosen as extension of v on F . Using the product formula we find that

1-I I1~11~ = ws

1-I I1~11~ 1 we.St

->

I ~ m a x ( l , II~ll~.) -1 we.St

>

H

max(1,1lall~) -1 =

1

H(a)"

wEML

The proof is finished by noticing that

II~llv = II,HI~ :FI/[L=:F~I > II~ll~ :~ [] Liouville applied more primitive forms of this inequality to obtain lower bounds for the approximation of fixed algebraic numbers by rationals. In our more general setting, let a be a fixed algebraic number of degree d over F . Then it is a direct consequence of the previous theorem that c(a) 1-I (ll x - ~11~) > H(x)-----~

(2.2)

yES \.

for every x E F with x # 0. Here c(a) is a constant which can be taken to be ( 2 H ( a ) ) -d. Using such an inequality Liouville was the first to prove the existence of transcendental numbers by constructing numbers which could be approximated by rationals much faster than algebraic numbers. In 1909 A. Thue provided the first nontrivial improvement over (2.2) which was subsequently improved by C.L. Siegel (1921), F. Dyson (1948) and which finally culminated in Roth's theorem, proved around 1955. The theorem we state here is a version by S. t a n g which includes non-archimedean valuations, first observed by Ridout, and a product over different valuations. 2.3 T h e o r e m ( R o t h ) . Let F be an algebraic number field and S a finite set of valuations of F. Let e > O. Let a C Q and ext-~nd each v to F ( a ) . Then 1

l - I ( l l x - ~11~) < H(x)=+-----:

yES

has only finitely many solutions x E F. A proof of Roth's original theorem can be found in Chapter III of these notes. Around 1970 W.M. Schmidt extended Roth's techniques in a profound way to obtain a simultaneous approximation result. Again the version we state here is a later version which follows from work of It.P. Schlickewei.

F . BEUKERS

4

2.4 T h e o r e m ( S e h m i d t ' s S u b s p a c e t h e o r e m ) . Let F be an Mgebraic number t~eld and S a finite set of valuations of F. Let H1, H2,... ,H,~ be m hyperplanes o f P n in general position. For each i let Hi be given by Li(xo,..., xn) = 0 where Li is a linear form with coe~cients in Q. Extend each v E S to the field generated by the coefficients of the Li. Let e > O. Then the points P = (Xo,... ,x~) E Pn(F) which satisfy m 1-I 1-[min

~,~Si=l ~

Li(xo,..., xj

~Cn)

v

<

1 H(P) n+l+"

lie in a finite union of proper hyperplanes of P'~. For a rough sketch of the proof of Schmidt's original theorem we refer to Chapter IV. Let us rewrite these theorems, starting with Roth's theorem. Take logarithms. Then we find that log [ix - a l l , < - ( 2 + e)h(x) yES

has finitely many solutions. The function on the left can be considered as a distance function, measuring the S-adic distance between x and a. Minus this function can be considered as a proximity function which becomes larger as x gets closer to a. Roth's theorem can now be reformulated as follows. For all but finitely many x E F we have log Ilx -

-- ~

~11~ <

(2 +

,)h(x).

yES

We can reformulate the Subspace theorem similarly. There is a finite union Z of hyperplanes in P~ such that for all P E P n ( F ) , P ~/Z, we have logm~x - 3

vs

i=l fi Li(xo~.

~xi. , xn) , < (n + 1 + e)h(P).

The left hand side of this inequality can be considered as a proximity function measuring the S-adic closeness of the point P to the union of hyperplanes uim=lHi. For comparison, under the same assumptions a trivial application of the Liouville inequality would give that for all P 6 P " ( F ) , P r Hi (i = 1 , . . . , m ) ,

Zlogm~tx i=1 fi ii(xo~xJ

v6S

3

,xn) v

<[L:F]mh(P)+O(1)

where L is the field generated over F by the coefficients of the Li. As a comment we would like to add that the Subspace theorem, in the way we have formulated it here, has two crucially distinct ranges of applicability. The first which we call the arithmetic one is when m < n + 1. The Subspace theorem gives a non-trivial result as long as [L : F]m > n + 1. However if we allow points P in some finite extension of F it may happen that [L : F] decreases and the statement of the theorem becomes trivial. The second range is when m >__n + 2, which we like to call the geometric realm. Here it does not matter if we extend the field F , the theorem remains non-trivial.

I. DIOPtIANTINE EQUATIONS AND APPROXIMATION 3

Well

5

Functions

On general varieties we can also define proximity functions. In that case they are known as Well functions. The general definition is too cumbersome to state here, and we refer to [36], Chapter 10. Instead we give a short recipe for the construction of Well functions on projective varieties. First construct sets of positive divisors Xi, (i = 1 , . . . , n ) and Yj, (j = 1 , . . . , m ) such that D + Xi ~ Yj for every i,j and such that the Xi have no point in common and the Y/have no point in common. Let flj E F ( V ) be such that (fi~) = Yj - X~ - D for each pair i,j. Extend the valuation to all of F . For each P E V ( F ) we define ~D,,(P) = mjax m in log Ilfij(P)ll,. Of course ID,v depends upon the choice of the fij, but it is known that two Well functions associated to the same divisor D differ only by a bounded function on V(F). For more details on Well functions we refer to Lang's book [36], Chapter 10. Consider for example the elliptic curve in 1P2 with affine equation

y2 = x 3 + Ax + B,

A , B C F.

Let D be three times the point at infinity. Choose f l l = x, f12 = Y and f13 = 1. Notice that (x) = {oc, (0,-Fx/~)} - D and (y) = {2-torsion r c~} - D. Notice that when B -~ 0 the zero divisors of x and y are disjoint. The corresponding Well function is

x~,. = log max(l, IIxllv, IlYllv)As another example take for D the union of hyperplanes U,%lHi in P~ from the Subspace theorem. Let L be the product of all Li. For the functions fij we choose m

flj-

xj L(xo,...,x,~)

j = O,...,n.

The pole divisor of each f i j is precisely D and the zero divisor is m times the hyperplane xj = 0. Our v a d i c proximity function (Well function) reads

.kD,,(P) = log max 9

x~ v" L(xo:::: ,x,~)

The Subspace theorem can again be reformulated. There is a finite union of hyperplanes Z in F'~ such that for all P C P~(F), P r Z we have

~_, .~D,,,(P) < (n + 1 + c)h(P). yES

4

Vojta's

Conjecture

In the beginning of the 1980's P. Vojta [80] discovered an uncanny similarity between concepts from diophantine approximation and from value distribution theory for complex analytic functions. Theorems in the latter area, translated via Vojta's dictionary to diophantine approximation, yielded statements which can be considered

6

F. BEUKERS

as very striking conjectures. Although we are far from being able to prove these conjectures, they look like a fascinating guide line in further development of diophantine approximation and diophantine equations. Here we state only Vojta's "Main Conjecture" without going into the analogy with value distribution theory. A normal crossing divisor D of a non-singular variety V is a divisor which has a local equation of the form zlz2 - ' - zm = 0 near every point of D for suitably chosen local coordinates Zl, Z2,---, Zdim(V) on V. 4.1 C o n j e c t u r e ( V o j t a ) . Let V be a non-singular projective variety defined over the algebraic number field F and let A be a pseudo ample divisor. Let D be a normal crossing divisor defined over a finite extension o f F . Let K be the canonical divisor of V. Let S be a finite set of valuations on F and for each v E S let AD,, be a proximity function for D. Let e > O. Then there exists a Zariski closed subvariety Z of V such that for all P E V ( F ) , P ~ Z we have,

AD,,,(P) + hK(P) < eha(P) + 0(1). yES

The subvariety Z will be referred to as an exceptional subvariety. Actually, an exceptional subvariety is nothing but a Zariski closed subvariety, but it sounds more suggestive for this occasion. One could argue whether or not the condition "nonsingular" is really necessary. The "normal crossings" condition is vital however. In the example of the Subspace theorem this condition comes down to the condition that the hyperplanes lie in general position. As an example consider V = P~. To determine the canonical divisor consider the differential form ~ = d x l d x 2 . . , dx,~ in the affine coordinates (1 : Xl : x2 : . . . : x~) on U0 = {x E ?'nix0 -~ 0}. There are no zeros or poles on U0. But if we rewrite fl with respect to (x0 : . . . : 1: . . . x ~ ) on Ui = {x C P~Ix~ # 0 } w e f i n d

-

1

,,+1 dxo.. 9dS:~ ... dx,~. Xo

Hence s has a pole of order n + 1 along x0 = 0 and K = - ( n + 1)H where H is the hyperplane at infinity. So by linearity of heights we find that hi,- = - ( n + 1)h where h is the ordinary projective height. Thus Vojta's conjecture for V = P= reads as follows. For all P E P ( F ) , P r Z we have

AD,v(P) < (n + 1 + e)h(P) + O(1). yES

So in particular, when D is a union of hyperplanes in general position we recover the Subspace theorem again, except that Z is now known to be a union of hyperplanes. If we take for D any hypersurface and, to fix ideas, F = Q, we obtain the following interesting consequence. 4.2 C o n j e c t u r e . Let D be a hypersurface in P~ defined over Q of degree d >_ n + 2 with at most normally crossing singularities. Suppose it is given by the homogeneous equation Q ( x o , . . . , x ~ ) = 0 where Q has coefficients in Z. Let S be a finite set (possibly empty) of rational primes. Then the set of points ( X o , . . . , x ~ ) E Z TM, g c d ( x o , . . . , x~) = 1 such that Q ( X o , . . . , x~) only contains primes from S lies in a Zariski dosed subset of P~.

I. DIOPItANTINE EQUATIONS AND APPROXIMATION Proof.

7

We apply Vojta's conjecture with the proximity functions

)~D,v

= log max i

_L Q(xo,...,x~)

v

and the set of valuations S O c~. The conjecture implies that for any e > 0 the set of projective n + 1-tuples ( x 0 , . . . , x=) E Z ~+1 with g c d ( x 0 , . . . , xn) = 1 which satisfy

~esuoo

log max

i

x_~

Q(xo,...,

x,,)

~ > (n + 1 + e)h(xo,.., x,,)

lies in an exceptional subvariety. The inequality can be restated as

log

IIQ(xo,...,=,,)ll,, < -(', + 1 + e)h(xo,... ,xn) + Y~

v ESucr

logm~x(llx~ll,,)

v ESUoo

The sum on the right is precisely dh(xo,..., x,~) since the sum includes the infinite valuation and the max~ is 1 for all finite v. So the set of solutions to

~_, logllQ(xo,...,x,,)ll, < ( d - n - X - e ) h ( x o , . . . , x , ) vESuoo

lies in an exceptional subvariety. Suppose Q(xo,..., x,) is composed of primes only from S. Then by the product formula the left hand side of the latter inequality is zero and we have a solution of the inequality because d > n + 1. This proves our corollary. [] Application of this corollary to the case n = 1 yields the Thue-Mahler equation F ( x , y ) = p~l...p~, where F is a binary form of degree at least 3 and distinct zeros. Application of the corollary to Q = XoXl " 9x=(xo + xl + . ' . + x,,) gives us the S-unit equation in n + 2 variables. Application of the corollary in the case n = 2 already poses us with problems that no one knows how to solve. We obtain a so-called ternary form equation

(4.3)

Q(x,y,z)

where Q is a ternary form of degree d and P l , . . . , P r are given primes. A number which is composed of only the primes P l , . . . ,Pr will be called an S-unit. We assume that the curve Q(x, y, z) = 0 has at most simple singularities. Let us distinguish three cases, the case d = 1 being considered trivial. d = 2 Here it is very easy to construct examples having a Zariski dense set of solutions. Suppose that Q is indefinite and that there are integers x0, Y0, Z0 such that Q(xo, y0, z0) = 1. We indicate how this implies the existence of a Zariski dense (in P~) set of solutions to Q(x, y, z) = 1, x, y, z E Z. Take any triple of integers Zl, Yl, Zx E Z and consider the binary form Q(Ax0 + # x l , Ay0 + #yl, Az0 + #zl) which has the shape A2 + AA# + B # 2. Because Q is indefinite we can choose Xl, yl, zl in infinitely many ways such t h a t A s - 4 B is positive and not a square. By writing down an infinite set of solutions A, # of the Pellian equation A2 + AA/~ + B # 2 = 1 for each such xl,YhZl we arrive at our dense set of solutions.

8 d=3

F. BEUKERS This case seems to be very interesting. When Q = 0 has a singular point it is often possible to construct a dense set of solutions to (4.3) by considering straight lines through the singularity in a way similar to the previous case. When Q = 0 is a non-singular curve, things seem to be more difficult, but we can still give examples of dense solution sets. The equation x s + y3 + z 3 = 1, for example, has a Zariski dense set of solutions x, y, z E Z. This can be inferred from a construction by D.H. Lehmer [42].

d>_4 The above corollary implies that the solutions are contained in an exceptional subset of p2. Of course the arguments given in cases d = 2, 3 are very ad hoc. It would be very interesting to have a more systematic treatment which the author is presently trying to work out. For the proof of the following Corollary we refer to [80], Chapter 4.4. 4.4 C o n j e c t u r e ( H a l l ) . For any c > 0 there exists c(e) > 0 such that ix3 _ y21 > c(~)x~ -~

for any x, y C Z with y2 ~ x 3, It is known that 1/2 is the best possible exponent. The totally new and very remarkable ingredient in Vojta's conjecture is the occurrence of the term hK(P ). It makes the following kind of statement possible. 4.5 C o n j e c t u r e ( B o m b i e r l ) . Let V be a projective variety over F and suppose that it is of general type. Then V ( F ) is contained in a Zariski dosed subset of V. P r o o f . Apply Vojta's conjecture to the case D = 0, A = K and e = 1/2. Because V is of general type, K is pseudo ample. We find that for any P E V ( F ) outside a certain exceptional subvariety Z the height hK(P) is bounded. Because K is pseudo ample we conclude that V ( F ) - Z is finite. [] Thus we see that in case D = 0 Vojta's conjecture still makes highly non-trivial predictions thanks to the occurrence of hK(P). Roughly speaking the case D = 0 can be seen as an example of diophantine approximation without actually approximating anything! More particularly, when V is an algebraic curve of genus >_ 2 defined over F, the number of F-rational points should be finite (Mordell's conjecture). This was proved by Faltings [21] in 1983, and later by Vojta in 1989 following an admirable adaptation of Siegel's diophantine approximation method. So it was Vojta himself who vindicated his insight that one can do diophantine approximation without approximants. Finally there are some consequences of Vojta's conjecture which have recently been proved by Faltings as Theorems 5.5 and 5.6. We shall discuss them in the next section.

I. DIOPHANTINE EQUATIONS AND APPROXIMATION

5

9

Results

In order to be able to appreciate the results we first establish a trivial result which is the analogue of Liouville's inequality. With the same notations as in Vojta's conjecture it reads as follows. 5.1 T h e o r e m (Liouville). Suppose that the divisor D is ample and defined over a finite extension L o f F . Then we have for all P E V ( F ) , P q[ D, ~ , A D , , ( P ) < [L: F ] h D ( P ) + O(1). v

P r o o f . Choose m such that m D is very ample. Let f0, f l , . . . , f N be an L-basis of the linear system corresponding to m D . A good proximity function is given by A~,,mD(P) = log m~x I[f~(P) llv. Since the A~,mD are functions bounded from below for all v and bounded from below by 0 for almost all v we find that A~,mo(P) < ~ l o g m f x yES

IIf,(P)ll, + 0(1)

v

where for each v we have chosen an extension to L. The term on the right can now be bounded by [L : F]hmD(P) q- O(1). Division by m on both sides yields our desired result. [] In the previous section we have already considered the Subspace theorem relation with Vojta's conjecture. In 1929 C.L. Siegel applied his theorem on proximation of algebraic numbers by algebraic numbers from a fixed number the situation of algebraic curves. His result, as extended by Mahler, can be preted as follows.

and its the apfield to reinter-

5.2 T h e o r e m ( S i e g e l - M a h l e r ) . Let C be a non-singular projective curve o[ genus >_ 1 and defined over an algebraic number field F. Let S be a finite set of valuations on F. Let h be a height function on C(-F) and let D be a positive divisor on C defined over F. Then, for any e > O, we have AD,,,(P)

<

eh(P)

yES

for almost a11 P E C ( F ) (i.e. finitely many exceptions). P r o o f . (sketch) First we embed C into its Jacobian variety J and let h be a height on J. It is known that if we prove the theorem for h restricted to C we have proved it for all h. Although Siegel had to work with a weaker result we can nowadays profit from Roth's theorem. It is a fairly direct consequence of Roth's theorem that AD,.(P) < 3#h(P) yES

10

F. BEUKERS

for almost all P E C(F), and where # is the maximum multiplicity of the components of D. Actually we can have 2 + e instead of the factor 3, but the latter will do. Let e > 0 and suppose that there exists an infinite subset E C C(F) such that ~ AD,.(P) > eh(P)

(5.3)

yES

for all P E E. Choose m E N such that m2c > 4#. It is known via the weak MordellWeil theorem that J ( F ) / m J ( F ) is finite. Let a l , . . . , a ~ be a set of representatives of J ( F ) / m J ( F ) . There exists a representative, al say, such that mQ + el E E for infinitely many Q E J(F). We denote this set of Q by E'. The covering w : J ~ J given by wu = mu + al provides an unramified cover of C by an algebraic curve U. It follows from (5.3) that (5.4)

Z "~D,v(mQ + a,)

> eh(mQ+al)

yES

:

m~A(Q) + 0(1)

> 4#h(Q) + 0(1) for all Q E E'. Notice that AD,,(mQ + a l ) is a proximity function for Q E U(F) with respect to the divisor w*D on U. Because w is unramified the maximum multiplicity of the components of w*D is also #. Direct application of Roth's theorem to the curve U shows that )~D,~(mQ + al) > 3#h(Q) yES

has only finitely many solutions Q E U(F). This is in contradiction with (5.5). Hence (5.3) has only finitely many solutions, as asserted. [] It is well known that from Siegel's theorem we can derive the finiteness of the set of integral points on affine plane curves. This is done by taking for D the divisor at infinity. Consider for example the elliptic curve

y~ = z 3 + Ax + B,

A,B E F

considered previously. In the section on Weil functions we saw that the function log max(i, IIx ]iv, Ilyll~) is a good proximity function to the point R at infinity, counted with multiplicity 3. Suppose we are only interested in points (x, y) on E with x, y in OF, the ring of integers of F. Take for S the set of infinite valuations. Then Siegel's theorem implies that log max(I, Ilxllv, IlYllv) > e h ( l :

x : Y)

v]oo

has finitely many solutions for any e. If x, y E OF we see that the left hand side of the inequality equals h(1 : x : y). Hence, if we take e = 1/2 say, we have a solution of the inequality. Siegel's theorem tells us that there are only finitely many such solutions. In 1989 Vojta proved the Main Conjecture for the case of algebraic curves. The method used is an adaptation of a method of Siegel to the geometric case. Originally Vojta used some highly advanced results from the theory of arithmetic threefolds. However, Bombieri gave in 1990 a version which uses only fairly elementary notions of algebraic geometry. It seems that Vojta's ideas had broken a dam because Faltings very soon, in 1990 and 1991, produced two papers where the following two fascinating theorems are proved.

L DIOPHANTINE EQUATIONS AND APPROXIMATION

11

5.5 T h e o r e m (Faltings). Let A be an abelian variety over F and E a subvariety, also defined over F. Let h be a height on A and v a valuation on F. Let e > O. Then We h a v e <

for almost every point P E A ( F ) - E.

This theorem settles a conjecture of S. Lang about integral points (with respect to E) on abelian varieties. In [80], Chapter 4.2 it is shown that Vojta's conjecture implies this theorem. 5.6 T h e o r e m (Faltings). Let A be an abelian variety defined over a number field F. Let X be a subvariety o[ A, also defined over F. Then the set X ( F ) is contained in a finite union of translated abelian subvariety of X . We observe that Theorem 5.6 is again a consequence of Vojta's conjecture. For this we remark that a subvariety of an abelian variety is either of general type or a translated abelian subvariety. Then we apply the following argument. If X is of general type, the exceptional subvariety is a lower dimensional subvariety. If a prime component X ' of this exceptional subvariety is again of general type we repeat our argument for X'. When we have finally reached dimension zero, we are left with an exceptional subvariety which is a finite union of translated subtori. The only technical detail we have to be aware of is that X and other components of exceptional varieties may be singular. In that case we must resolve singularities to be able to apply Vojta's conjecture. It is nice to notice that Theorem 5.6 partly solves two conjectures proposed by S. Lang. The first one is that an algebraic variety which is (Kobayashi) hyperbolic has only finite many F-rational points for a given number field F. In 1978 it was proved by M. Green that a subvariety X of an abelian variety which does not contain any translate of a positive dimensional abelian subvariety is indeed hyperbolic. Thus Theorem 5.6 settles Lang's conjecture for this particular type of hyperbolic variety. Theorem 5.6 is also interesting in another respect. Notice that X ( F ) = A ( F ) N X . We know that A ( F ) is a finitely generated group. Thus the question of the nature of X ( F ) can be considered as an example of problems where we intersect finitely generated subgroups of algebraic groups with their subvarieties. This is precisely the subject of Lang's second conjecture we alluded to. The conjecture, which has recently turned out to be a theorem, reads as follows. 5.7 T h e o r e m . Let G be either an abelian variety over C or a power of the multiplicative group C*. Let F be a subgroup of G of finite Q-rank. Let V be a subvariety of G. Then V N F is contained in a finite union of translates o[ algebraic subgroups of G. In the case when G is a power of the multiplicative group this was proved in 1984 by M. Laurent [41], using the Subspace theorem and a fair amount of Kummer theory. As for the case when G is an abelian variety defined over a number field it was shown in 1988 by M. Hindry [29] that the truth of the above theorem for the special case of finitely generated F implies the truth of the full theorem. The proof of this result is heavily based on work of M. Raynaud. Once that is done we only need to invoke Theorem 5.6 to prove the above theorem in case G is defined over a number field. The general case follows by a specialisation argument.

Chapter II

Diophantine Approximation and its Applications by Rob Tijdeman

1

Upper

Bounds

for Approximations

Litt. [85], [581, [261 and [651. As with many other areas, it is difficult to say when the development of the theory of diophantine approximation started. Diophantine equations have been solved long before Diophantos of Alexandria (perhaps A.D. 250) wrote his books on Arithmetics. Diophantos devised elegant methods for constructing one solution to an explicitly given equation, but he does not use inequalities. Archimedes's inequalities 3 ~ < ~r < 3~ and Tsu Ch'ung-Chih's (A.D. 430-501) estimate ~ = 3.1415929... for zr = 3.1415926... are without'any doubt early diophantine approximation results, but the theory of continued fractions does not have its roots in the construction methods for finding good rational approximations to 7r, but rather in the algorithm developed by Brahmagupta (A.D. 628) and others for finding iteratively the solutions of the Pell equation x 2 - dy 2 = 1. Euler proved in 1737 that the continued fraction expansion of any quadratic irrational number is periodic. The converse was proved by Lagrange in 1770. Lagrange deduced various inequalities on the convergents of irrational real numbers. In particular, he showed that every irrational real a admits infinitely many rationals p/q such that (1.1)

a -p


Alternative proofs of this inequality are based on Farey sequences and on the Box principle. In 1842 Dirichlet used the latter principle to derive the following generalisation: Suppose that aij (1 < i < n, 1 < j <_ m) are n m real numbers and that Q > 1 is an integer. Then there exist integers q i , . . . ,qm, P l , . . . ,P~ with

1 _< max([ql[,..., ]qm[) < ~n/m and [ c ~ i l q x + . . . + a ~ m q , , - p , l < _ Q -1

(1
14

R. TIJDEMAN

(By taking m = n = 1 we find 1 < q < Q, whence [a - P / q l < 1/q 2") A further generalisation resulted from the development of the geometry of numbers. Minkowski's Linear Forms Theorem (1896) reads as follows. Suppose that (flij), 1 < i < n, 1 <_j <_ n, is a real matrix with determinant -4-1. Let A 1 , . . . , A n be positive reals with A1A2 " " A n = 1. Then there exists an integer point x = (Xl,... ,x=) # 0 such that ]flilXl "4-''' -~ flinXn[ < Ai

(1 < i < n - 1)

and [flnlXl

-[- " " " -[-

/~nnXn[ ~ An.

(Dirichlet's Theorem corresponds with n + m inequalities in n + m variables.) The proof of Minkowski's Linear Forms Theorem is based on his Convex Body Theorem which says: any convex compact set K in I~n, symmetric about the origin, with the origin as interior point and with volume V ( K ) > 2 ~, contains a point in

zn\{0} For j = 1 , 2 , . . . , n , let Aj = Aj(K) be the infimum of all A _> 0 such that AK contains j linearly independent integer points. The numbers A1, A2,..., A,, are called the successive minima of K and satisfy 0 < A1 < A2 _< ... _< An < oo. Minkowski's Convex Body Theorem implies that A~V(K) _< 2n. In 1907 Minkowski published the following refinement (Minkowski's Second Convex Body Theorem): 2n - - < A1A2--'AnV(K) < 2 n. n!

--

Both bounds can be attained. The continued fraction algorithm provides a quick way to actually find rational approximations, but the theorems of Dirichlet and Minkowski don't. In [43] a polynomial-time basis reduction algorithm is introduced which is practical and theoretically still rather strong. It implies, as an effective version of Dirichlet's cited theorem: There exists a polynomial-time algorithm that, given rational numbers otij (1 < i < n, 1 < j < m) and Q satisfying Q > 1, finds integers q l , . . . ,q,~, pl, . . 9 pn for which 1 < max(lq~l,..., Iq,,I) < 2('n+n)~/4'nQn/'n and Ioqlql+...+aimqm-pil
-1

(1 < i < n ) .

The basis reduction algorithm has been used in the theory of factorisation of polynomials into irreducible factors, lla optimisation theory and in several other areas. See [31]. In Section 3 some applications to diophantine equations will be mentioned.

II. DIOPIIANTINE APPROXIMATION AND ITS APPLICATIONS

2

15

Lower Bounds for Approximations

Litt. [8] and [65]. In 1844, Liouville proved that some number is transcendental. He derived this result from the following approximation theorem.

Suppose a is a real Mgebraic number of degree d. Then there is a constant c(a) > 0 such that c(c~)q -d for every rational number p/q distinct from a. (We tacitly assume q > 0). Liouville deduced that numbers like z~=l ~oo 2-~., are transcendental. Liouville's Theorem implies that the inequality (2.1)

a-

p < q-~

has only finitely many rational solutions p/q if # > d. Thue showed in 1909 that (2.1) has only finitely many solutions if # > d/2 + 1. Then Siegel (1921) in his thesis showed that this is already true if # > 2x/~. A slight improvement to # > ~ was made by Dyson in 1947. Finally Roth proved in 1955 that (2.1) has only finitely many solutions if # > 2. If d :> 2, then comparison with (1.1) shows that the number 2 is the best possible. If d = 2, then Liouville's Theorem is stronger than Roth's one. Chapter III of these notes contains a proof of Roth's Theorem. In a series of papers published between 1965 and 1972, W.M. Schmidt proceeded an important step forward. One of his results is the following extension of Roth's Theorem.

Suppose a is a real algebraic number. Let k >_ 1 and 5 > O. Then there are only finitely many algebraic numbers fl of degree <_ k with

I~

-

fll

< H(fl) -(~+'+6)

where H(fl) denotes the classical absolute height. This result follows from the so-called Subspace Theorem which, in its simplest form, reads as follows.

Suppose Ll(X),... ,Ln(x) are linearly independent linear forms in x = ( X l , . . . , x , ) with algebraic coett~cients. Given 6 > O, there are finitely many proper linear subspaces T1,... ,T,~ of R ~ such that every integer point x ~ 0 with I L l ( X ) " . L.(x)[ < Ixl -~ lies in one of these subspaces. ([x[ denotes the Euclidean length of x.) In Chapter III it will be shown that Roth's Theorem is equivalent to the case n = 2 of the above stated Subspace Theorem.

16

R. TIJDEMAN

Apart from Liouville's Theorem, all theorems stated in this section up to now are ineffective, that is, the method of proof does not enable us to determine the finitely many exceptions. However, the method makes it possible to derive upper bounds for the number of exceptions. I shall mention two results giving upper bounds for w in the Subspace Theorem. The first result is due to Schmidt [66].

Let L 1 , . . . , L ~ be linearly independent linear forms with coemcients in some algebraic number field of degree d. Consider the inequality (2.2) ILl(X).--L,(x)[ < Idet(L1,... , L . ) [ Ixl -~

where 0 < 6 < a

The set of solutions of (2.2) with x C Z ~,

Ix I > max((n!)S/6,H(nl),... ,H(L~))

is contained in the union of at most [(2d) 226"~-2] proper linear subspaces of R ~. The second result is due to Vojta [82]. Essentially it says that, apart from finitely many exceptions, which may depend on 6, the solutions of (2.2) are in the union of finitely many, at least in principle effectively computable, proper linear subspaces of R ~ which are independent of 6. Mahler derived in 1933 a p-adic version of the Thue-Siegel Theorem. Ridout and Schneider did so for Roth's Theorem. The p-adic versions of Schmidt's theorems have been proved by Schlickewei. We shall see that they have important applications. Vojta proved the p-adic assertion of the above mentioned result himself. For more information on the Subspace Theorem see Chapter IV of these notes. Liouville's Theorem is also the starting point of a development of effective approximation methods. Hermite, in 1873, and Lindemann, in 1882, established the transcendence of the numbers e and 7r, respectively. The Theorem of Lindemann-Weierstrass (1885) says that file ~1 + . . . + fl~e~" ~ 0 for any distinct algebraic numbers a l , . . . , a~ and any nonzero algebraic numbers 131,..., fl~. A new development started in 1929 when Gelfond showed the transcendence of 2 v'~. In 1934 Gelfond and Schneider, independently of each other, proved the transcendence of c~z for a, ~ algebraic, a ~ 0, 1 and ~ irrational. Alternatively, this result says that for any nonzero a]gebraic numbers al, a2, ill, ~32 with log al,log a2 linearly independent over the rationals, we have Da log al + ~32log a2 5L 0. In 1966 Baker proved the transcendence of e~~ 1 . . . a ~ " for any algebraic numbers a l , . . . , a ~ , other than 0 or 1, and ~ 1 , - - - , ~ , provided that either fl0 :~ 0 or 1,/31,... , ~ are linearly independent over the rationals. This follows from the theorem that if a l , . . . , a ~ are nonzero algebraic numbers such that their logarithms log al, 9 9 log cr~ are linearly independent over the field of M1 rational numbers, then 1, log a l , . 9 log a~ are linearly independent over the field of M1 algebraic numbers.

IX. DIOPIIANTINE APPROXIMATION AND ITS APPLICATIONS

17

The above mentioned results have p-adic analogues and also analogues in the theory of elliptic functions. For example, Masser [46] showed that if a Weierstrass elliptic function p(z) with algebraic invariants g2 and gz and fundamental pair of periods ~l,w2 has complex multiplication, then any numbers U l , . . . , u , for which p(ul) is algebraic for i = 1 , . . . , n are either linearly dependent over Q(wl/~2) or linearly independent over the field of algebraic numbers. The effective character of the results can be expressed in the form of transcendence measures. In 1972-77 Baker [7] derived the following important estimate. Let o~1,... , a n be nonzero algebraic numbers with degrees at m o s t d and (classical absolute) heights at m o s t A1, ..., A~ (all > 2), respectively. Let bx,...,bn be rational integers of absolute values at most B (> 2). Put A = bl log ol1 + - ' - + b n log a=. Then either A = 0 or

loglA[ > -(16nd) 2~176 f l l o g A j

log

logA~

logB.

j=l

The constants have been improved later on and the log(It log) factor has been removed. The best bounds known at present, both in the complex and in the p-adic case, are due to Waldschmidt and his colleagues. They use a method going back to Schneider, whereas Baker's proof can be considered as an extension of Gelfond's proof of the transcendence of an. These linear form estimates have important applications to diophantine equations. Similar, but weaker, estimates have been obtained for linear forms in Ul,... , u n such that u~ is a pole of go(z) or gg(u~) is an algebraic number, for i = 1,...,n.

3

Applications to Diophantine Equations.

Litt. [49], [72] and [67]. Results on diophantine approximations have been applied in various areas. I may refcr to applications in algebraic number theory (class number problem, factorisation of polynomials), numerical mathematics (uniform distribution, numerical integration) and optimisation theory (when applying basis reduction algorithms, geometry of numbers). I shall deal here with applications to diophantine equations. This is very appropriate, since the last decade has also shown striking applications of arithmetic algebraic geometry to diophantine equations. It is quite likely that a merging of the theories of arithmetic algebraic geometry and diophantine approximations, as strived after in these Proceedings, would provide a new and solid basis for the theory of diophantine equations. This is a challenge for the young generation. An immediate consequence of Thue's approximation result is as follows. The equation

(3.1)

f ( x , y ) = aox ~ + a l x ~ - l y + . . . + a~y ~ = k # O,

where n > 3, and f ( x , y) E Q[x, y] is irreducible, has only a finite n u m b e r of solutions (in rational integers x, y).

18

R. TIJDEMAN

The corresponding consequence of Roth's approximation result is that the equation (3.2)

f ( x , y) = P ( x , y),

where f is as above and P ( x , y) E Q[x, y] is any polynomial of degree m < n - 2, has only finitely many solutions.

Schinzel used a suggestion of Davenport and Lewis to show that in the latter theorem the condition m < n - 2 can be replaced by m < n. We note that, by a completely different method, Runge obtained a general result in case f is reducible, but f - P is irreducible. Runge showed in 1887 that under these conditions there are only finitely many solutions provided that f is not a constant multiple of a power of an irreducible polynomial. Schinzel's result incorporates Runge's theorem and a result from Siegel's famous 1929-paper. It states that if f - P is irreducible, f is homogeneous of degree n and P is any polynomial of degree < n such that (3.2) has intlnitely many integer solutions, then f is a constant multiple of a power of a linear form or an irreducible quadratic form.

The p-adic analogues of approximation theorems led to important applications to Sunit equations. For simplicity I state results for rational integers, but corresponding results hold for integers from some algebraic number field or even finitely generated integral domains, see [20]. Let P l , . - - , P s be given prime numbers and denote by So all rational integers composed of p l , - - - , p s . In 1933 Mahler showed that the equation

(3.3)

x + y= z

in

x , y , z C So

with

gcd(x,y)=l

has only a finite number of solutions. He applied his p-adic version of the Thue-Siegel method. A general and in some respect best possible result was proved by Evertse in 1984. He used Schlickewei's p-adic version of Schmidt's Subspace Theorem to show that for any reals c, d with c > O, 0 <_ d < 1 and any positive integer n, there are only finitely m a n y ( x l , . . . , x,~) E Z ~ such that (i) x 1 -[- ... -[- x n = O, (ii) xil + " " + xi, ~ 0 for each proper non-empty subset { i l , . . - , i t } of { 1 , . . . , n}, (iii) g c d ( x , , . . . , x~) -- 1 and (iv) n

(3.4)

I-[(Ixk[ ]'I [Xk[p) < c k=l

pES

--

max l
I~klu.

For the many diverse applications of this and related results I refer the reader to the survey paper [20]. The proofs of the above mentioned results are ineffective. So it is impossible to derive upper bounds for the size of the solutions by following the proofs. However, it is possible to give upper bounds for the numbers of solutions. A remarkable feature is that these bounds depend on very few parameters. Lewis and Mahler derived in 1961

H. DIOPHANTINE APPROXIMATION AND ITS APPLICATIONS

19

an explicit upper bound for the number of solutions of (3.3) depending on p a , - . - , p , . In 1984, Evertse improved this upper bound to 3 • 72~+3 which depends only on s. Schlickewei [63] gave an upper bound for the number of solutions of (3.4) with c = 1, d = 0 in Evertse's theorem. This bound depends only on n and s. Bombieri and Schmidt [10] proved that the number of primitive solutions of the Thue equation (3.1) is bounded by cln s+a where cl is an explicitly given absolute constant and s denotes the number of distinct prime factors of k. Recently, Schmidt derived good upper bounds for the number of integer points on elliptic curves. Effective methods make it possible to compute upper bounds for the solutions themselves. In 1960, Cassels obtained an effective result on certain special cases of equation (3.3) by applying Gelfond's result on the transcendence of a a. Baker's estimates in 1967 on linear forms of logarithms in algebraic numbers caused a breakthrough. Baker himself gave upper bounds for the solutions of the Thue equation (3.1) and the super-elliptic equation (3.5)

ym = P(x)

where m _> 2 is a fixed given positive integer and P(x) E Z[x] a polynomial with at least three simple roots if m = 2, and at least two simple roots if ra _> 3. (Later, the conditions were weakened.) Baker and Coates derived an upper bound for the number of integer points on some curve of genus 1. (This bound has been sharpened by Schmidt.) Coates used the p-adic estimates to obtain bounds for the Thue-Mahler equation (3.1) with m unknown and in S, equation (3.3), and the equations y: = x3+k with k unknown and in 5'. There are many later generalisations and improvements of the bounds. Baker's sharpening made it possible to deal with diophantine equations which cannot be treated by the mentioned ineffective method. E.g. Schinzel and Tijdeman showed that equation (3.5) with m, x, y variables and P(x) E Z[x] a given polynomial with at least two distinct roots implies that ra is bounded. Tijdeman also showed that the Catalan equation x m - y~ = 1 in integers ra, n, x, y all > 1 implies that x m is bounded by some effectively computable number. The bounds obtained in equations involving a power with both base and exponent variable are, however, so large that it is not yet possible to solve such equations in practice. The best bounds for linear forms known at present make it, however, possible to solve Mahler's equation (3.3) and Thue and Thue-Mahler equations (3.1) completely. Additional algorithms are needed to achieve this. To give a typical example, Tzanakis and de Weger [77] considered the Weierstrass equation y: = x 3 - 4 x + 1. They reduced it to some Thue equations, of which f ( x , y) = x 4 - 12x2y 2 - S x y 3 + 4 y 4 = 1 is a typical example. This leads to some equations x

--

~y

~

c~' 5 2~ 5 3~3,

where 51, 52, 53 is a fixed fimdamental set of units of Q(O) and ~ is a zero of f ( x , 1). A suitable linear form estimate yields ,naN(lax I, ]a2], [a3]) < 104x. Subsequently the basis reduction algorithm of Lenstra, Lenstra and Lovs is applied. The first time yields an upper bound 72, the second time an upper bound 10. Checking the remaining values yields four solutions, (0, 1), ( 1 , - 1 ) , (3, 1) and (-1,3). They correspond to the solutions (x, zky) = (2, 1), (10,31), (1274,45473), (114, 1217), respectively, of the equation y2 = x3 _ 4x + 1. In this way Tzanakis and de Weger determined the 22

20

R. TIJDEMAN

integral solutions of the equation. For an introduction to the available additional techniques, I refer to [86]. The following results illustrate the power of the described method. (a) [86] The Mahler equation z + y = z subject to gcd(x,y) = 1, z < y, z y z composed of the primes 2, 3, 5, 7, 11 and 13, has exactly 545 solutions of which all large ones are explicitly stated. (b) [86] The equation x + y = z 2 in integers x > y, z > 0 such that both x and y are composed of primes 2, 3, 5 and 7 and that gcd(x, y) is squarefree, has exactly 388 solutions. The largest one is (x, y, z) = (199290375,-686, 14117). (c) [78] The Thue-Mahler equation x 3 _ 2 3 x 2 y + 5 x y 2 + 2 4 y 3 ~-

-4-2z13~5z37z~

in integers x, y, zl, z2, z3 and z4 with x > 0 has exactly 72 solutions. The largest is given by (z, y) = (48632,-3729).

C h a p t e r III

Roth's Theorem by Rob Tijdeman

This chapter is based on Schmidt [65] and as to the proof of Lemma 1.6 on Wirsing

[s9].

1

The

Proof

1.1 T h e o r e m ( R o t h , [61]). Suppose a is real and algebraic of degree d > 2. Then for each ~ > O, the inequality a - p- < q-(2+~) q

(1.2)

has only finitely m a n y solutions in rationals p/q. 1.3 R e m a r k s . (1) By Dirichlet's Theorem the number 2 in (1.2) is best possible. (2) If a is of degree 2, then Liouville's Theorem implies the stronger inequality (1.4)

a-

p > c(a)q -2 > 0

for all rationals p/q. For no single a of degree > 3 do we know whether (1.4) holds. It is likely that (1.4) is false for every such a (that is, that every such a has unbounded partial quotients). (3) Lang conjectured in 1965 that for a of degree > 3 a - p < q-2(log q)-K has only finitely many solutions if K > 1, or at least if K > Ko(a).

[]

The first lemma is a straightforward application of the box principle. For a matrix A = (ajk) with rational integer coefficients, put [A[ = max lajkI. For an integer vector Z = ( z l , . . . , Z y ) , put tz[ = max([zl[,... ,[zN[).

22

R . TIJDEMAN

1.5 L e m m a (Siegel). Let A be an M x N matrix with rational integer coeflTcients, not MI zero, and suppose that N > M. Then there is a z E Z N with Az = 0,

z

:~ 0,

Izl _< (NIAI) MIeN-M)

Proof. Put Z = [(NIAI)M/~N-M)]and Lj(z) = EN=lajkZk for j -----1 , . . . , m and z = ( z x , . . . , z u ) . For z E Z N with 0 < zk < Z (1 < k < N) there are at most NIAIZ + 1 possible values for Lj(z). Hence Az takes at most (NIAIZ + 1) M values. Since (NIAIZ + 1) M < (NIAI)M(z + 1) M < (Z + 1) N, there are z (1) :fi z0) in the considered set with Az (1) = Az (2). Then z := z0) - z0) satisfies the conditions of the lemma. []

The second lemma states that most of the values il/dl + " " + im/d,~ with 0 < ih < dh (h = 1 , . . . ,rn) are close to m/2. 1.6 L e m m a ( C o m b i n a t o r i a l L e m m a ) . Suppose d l , . . . ,din E Z>_I and 0 < e < 1. Then the number of tuples ( i l , . . . , i m ) C Z m with 0
(h=l,...,m)

and

h= ~-~[~ldh ih 7922 -> s

is at most ( d x + l ) . . . ( d m + l ) / ( 4 m e 2 ) . P r o o f . We may consider il,. 9 i,, as independent stochastic variables such that ih is uniformly distributed on { 0 , . . . , dh}. Define the stochastic variable X = ~"~=1 ih/dh. Then X has expectation # X = m / 2 and variance a 2 = Var(i,/d,) + . . . + Var(i,,,/d,,,). We have dh ( i ~ 1) 2 1 2dh+l 1<1 Var(ih/dh) = Z -dh +-----~-- 6d------~ 4 - -4"

ih=O

Hence a 2 < m/4. By Kolmogorov's generalisation of Chebyshev's inequality, we have Prob(lX - ~'l >- c) < a2/c~. Thus 1

Prob([X - m / 2 [ ~_ era) ~_ 4me----~. [] For a polynomial P ( X ) = P ( X 1 , . . . , X , , ) E Z[X1,..., Xm] and i = (i1,..., ira) E Z~o, put PI(X)

-

=

1 0q il!...im! OX~'

Z'"Z jl=O

i1""

0 i" 0--~~ P ( X )

'--m

j,~=0

It follows that Pi(X) has integer coefficients. Let aa . . . . ,ar, E C and dl,...,d,,, E Z_>I. The index i(P) of P with respect to ~ = ( a l , . . . , a m ) and (da,..., d~) is the

IV. ROTrI'S THEOREM

23

least value a for which there is a tuple i = (il,.-. ,i,,) with il/d~ + ... + i,,,/d,~ = a and Pi(a_) -r 0. (If P = 0, then the index is defined to be oo.) Note that i ( P Q ) = i(P) + i(Q) and i ( P + Q) >_min(i(P), i(Q)). The third lemma provides the construction of a polynomial with high index at some given point. For P E Z[X1,..., X,,] we denote the maximum of the absolute values of the coefficients of P by IP I1.7 L e m m a ( I n d e x T h e o r e m ) . Suppose a is an algebraic integer of degree d > 2. Let e > O, and let m E Z with m > d/2~ 2. Let d l , . . . , d m E N. Then there is a polynomial P in Z[X1,...,Xm], P r O, such that (i) P has degree < dh in Xh, (ii) P has index > m(1 - e)/2 with respect tO ( a , . . . , a) and ( d l , . . . , d,,),

(iii) IPl < C~'++a". P r o o f . Write P(X1, 9 . . , X m ) = E j , =d~ 0 - - ' E j m = 0 z ( j ~an ,...,Jm)X~' ... X m j m , where the z ( j l , . . . , jm) are integers which have to be determined such that (ii) holds, i.e., Pi(_a) = 0 for il/dl + ' " + i,,/dm < m(1 - e)/2. By taking all these expressions together, we obtain A0z + aAlZ + - " + o~dl+'"+dmAdl+...+dm g = 0 where the A i are M • N integer matrices with IAil _< ga~+...+d,~,where N = (dl + 1 ) . . . (d,, + 1) and M is the number of tuples i with il/dl -Jr.'.-1- im/dm < m(1 - e)/2. Using that a is an algebraic number of degree d, we get B0z + ~ B l Z + . . . + o ~ d - l B d - l Z = 0

where the Bi are integer M • N matrices with [Bi[ < C d'+'''+dm. Since 1,ct,... , a d-1 are Z-linearly independent, we have B0z = 0, Blz = 0 , . . . , Ba-lZ = 0. Hence Bz = 0 where B is a d M • N integer matrix with IBI < C~ ' + + a ' . By the Combinatorial Lemma, we have M< (dl+l).-.(dm+l)_ N < N 4me2 4me-------2_ -~. Now Siegel's lemma implies that there is a non-zero integer vector z such that Bz = 0,

]z I _< (NIBI) dM/(N-dM) <_ N]B I <_ Ca3'+'''+d'.

Note that the constants C1, C 2 and C3 depend only on a.

[]

The fourth lemma gives a sufficient condition for a polynomial to have a small index with respect to the approximation vector (pl/q~,... ,p,,/qm) and ( d : , . . . , d,,). 1.8 L e m m a ( R o t h ) . Let m E Z, m > 1, and ~ > 0. There exists a number C4 = C4(m,e) > 1 with the following property: Let d l , . . . ,d,~ be positive integers with dh >_ C4dh+~ for h = 1 , . . . , m - 1 . Let (p~, q:), . . . , (p~, qm) be pairs ot'coprime integers with _ qd~ for h = l , . . . , m. (1.9) qhda > and qh > 22mc~ Let P ( X I , . . . , X , ~ ) E Z[X~,...,Xm] be a polynomial of degree < dh in Xh for h = 1 , . . . , m and with (1.10) [p[C~ < qd,, P 7~ O. Then the index of P with respect to (P~/ql,... ,P,~/q,~) and ( d l , . . . ,d,,,) is < e.

24

R,. TIJDEMAN

P r o o f . We use induction on m. The case m = 1. Write P ( X ) = ( q l X - p l ) I R ( X ) where 1/dl is the index of P with respect to Pl/ql and dl. The polynomial R ( X ) has integer coefficients by Gauss's Lemma. Since q~ divides the leading coefficient of P, we obtain q{ < Iel _< by (1.10). By choosing C4(1,e) = e-1 we get 1/dl <_ e. The induction step. Let m _> 2. Suppose Roth's Lemma is true for m - 1, but not for m, and C4(m - 1,f) has been defined for 0 < f < 1. We shall apply the case m - 1 of Roth's Lemma to some Wronskian which is constructed as follows. Consider a decomposition

q~,/c,

k

(1.11)

P ( X 1 , . . . , Xm) = ~ C j ( X l , . . . , Xm-1)r 5=1

where r Ck and C a , . . . , r are polynomials with rational coefficients and k is minimal. Since the choice r = X ~-1 (j = 1 , . . . , d m + 1) is possible, we have (1.12)

k _< d m + 1.

The minimality of k implies that both r Ck and ~bl,... , ~bk are linearly independent over the reals. We consider differential operators oil+...+im

A=

ox{1.

. . ox

m

and call il + "-" + im its order. A (generalised) Wronskian of r minant of the form det(AiCj) (1 < i < k, 1 _< j _< k)

r

is any deter-

where A a , . . . , A k are operators as above, with Ai of order _< i - 1 for i = 1 , . . . , k . We shall choose the Ai in such a way that det(AiCj) does not vanish. Such a choice is possible in view of the following lemma. 1.13 L e m m a . Suppose that r Ck are rational functions in X1, . . . , Xm with real coefficients, and linearly independent over the reals. Then at least one Wronskian of r Ck is not identically zero. P r o o f . We use induction on k. If k = 1, then det(AiCj) = r 7~ 0. Suppose now that r Ck are k _> 2 rational functions satisfying the hypotheses of Lemma 1.13. Then 1, r162 ,r162 are also linearly independent over the reals. It follows that

0 r

0 r

OXj r

OX~r

are linearly independent over the reals for some j with 1 < j _< m. By the induction hypothesis there exists a Wronskian of these functions which is not identically zero. This induces a Wronskian of 1, r162 ,r162 which is not identically zero. It follows from a simple induction argument that for any rational function r a Wronskian of r162 r 1 6 2 r162 can be written as a linear combination of Wronskians of r Ck with coefficients which are rational functions involving only the partial derivatives of r By taking r = 1/r we can conclude that there exists a (generalised) Wronskian of Ca, r ,r which is not identically zero. []

IV.

ROTH'STHEOREM

25

We can now define the Wronskian to which Roth's Lemma for m - 1 will be applied. By Lemma 1.13 there exist operators 1

0il "t''"+srn--I

A~ = i1! " " i,,_~! OX~ 1 . . . v ~ X im-:.~_l

with orders il + " " + ira-1 _< i - 1 _< k - 1 for i = 1 , . . . , k , such that U ( X ~ , . . . , X m - ~ ) := det(A'iCjh_
By Lemma 1.13 V(Xm) := det ( ~

1-1), 0i-1

OX~_x CAX,~ )

) l
# O,

since all the other generalised Wronskians of the prescribed type vanish. Put W ( X ~ , . . . , Xm) = det

1

(j - 1)[ OX~-

A~ip

] l
Then, by (1 [11),

W ( X l , . . . , X m ) = U(Xl,...,Xm-1)V(Xm) r O. The entries in the determinant defining W are of the type Pil..-i~-~5-a, whence these entries have rational integer coefficients. Thus W is a polynomial with rational integer coefficients. The determinant W is a polynomial in P and its partial derivatives. Hence a lower bound for the index t9 of P with respect to ( P l / q l , ' ' ' , P , , J q m ) and ( d l , . . . , din) implies a lower bound for the index | of W with respect to the same values. We shall compute such a lower bound. On using the conditions of Lemma 1.8 we obtain, for il+'"+im-l
i2

im-a j - 1 din-1 dm >t~ dm j-1

dl d2

>0

il+-..+im-1 dm-1

j-1 dm

dm-1

>9_C~_ 1_j-1

dm

dm

Note that this index is also non-negative. By expanding W we get using the formulas for the index of the sum and of the product of functions, (1.14)

(1.15)

0

>

k-1 ~-~max(0 - C ; 1 5=0

>_ -kez' +

~

[~kl

j)

,0) > -kC41 -JvZ(O - dm

5=0

+ 1) _> -kc; I +

Next we shall use the induction hypothesis to show that 19 cannot be large. Let 0 < ~ < ~ < 1. The factorisation W = U V induces the factorisation W ( X l , . . . , X , , ) = U*(X1, . . . , X , , _ l ) V * ( X , , )

26

R. TIJDEMAN

where U* and V* have rational integer coefficients. Note that IPi~...i,,_lj_ll < 2dl+"+d-.Ip t _< 2~+'"+a,,q~''/c~, by (1.10). Since the number of summands in the determinant expansion of W does not exceed k! < k k-1 _< k d~ _< 2 kum, it follows that

IW I < 2ka~(2dl+...+dmqd11/O,)k < (22,~q~/C~)d~k< q2~lk/C~ in view of (1.9). This yields the estimates 2dl k/C4

[U*[ < q~

,

2dl k/C4

[V*] ~_~ 111

~2dmk/C4 ~-~ '~m ,

by (1.9).

We will now apply the induction hypothesis to U*(X1,...,Xm_I) with respect to (Pl/qx,...,Pm-1/q,,~-l) and (kdl,...,kdm-1). If we choose C4(m,e) in such a way that C4(m, ~) >_2C4(m - 1,5), then IU'I <_ q21d~k/c'(m'O <

qf~k/c,(~-x,O.

We conclude therefore that the index of U" with respect to (px/q~,... ,P,,~-x/q,,~-~) and (dlk,...,dm_lk) is at most 5. Similarly we can conclude by applying the induction hypothesis for m = 1 to V(Xm) that the index of V* with respect to Pm/q,, and d~k is at most 5. Since W = U'V*, the index of W with respect to (Pl/qx,...,Pm/qm) and (dlk,... ,d~k) is at most 2L Hence (1.16)

0 < 25k,

where, as before, @ is the index of W with respect to (Pa/q~,...,P,~/q,~) and ( d l , . . . , din). A suitable choice of 6 yields a contradiction to our assumption v~ > e. Namely, choose C4(ra, e) so large that Ca(m, e) > 4 c ~. Then, by (1.14),

e > -kC~ 1 + ~k/2 > ~2k/4. On the other hand, choosing 5 < e2/8, we obtain from (1.16) that

0 < ~2k/4. [] Suppose cr is an algebraic number of degree d _> 2 such that, for some 5 with 0<5<1, (1.17)

a - qP- < q-~-6

has infinitely many rational solutions p/q. We shall use the Index Theorem to construct a polynomial P with very high index with respect to ( a , . . . , a) and arbitrary (dl,..., d,~). We shall show below that this implies that for a suitable choice of solutions Pl/q~ ..... p,,Jq,,~ of (1.17) the polynomial P has still high index with respect to (pl/ql,... ,p,,/qm) and ( d l , . . . ,d,~). On the other hand, we shall choose d l , . . . ,din in such a way that P has to have low index with respect to (Pl/q~,... ,P,,,/q,~) and ( d l , . . . , d,~) by Roth's Lemma. This contradiction will complete the proof.

IV. ROTII'S THEOREM

27

P r o o f . (of Roth's Theorem) Let cr be an algebraic number with denominator a0 satisfying (1.2). Then

( aoa )

~-

ao < q~

1 < q2~-612

for q > qo(a). Hence we conclude that it suffices to prove Roth's Theorem for algebraic integers a. We assume that a is an algebraic integer of degree d > 2 with Ja I < 1 and ~ some number with 0 < ~ < 1/2 such that (1.17) has infinitely many rational solutions p/q. Let P be the polynomial constructed in the Index Theorem with respect to a, e = 5/12, m > d/2e 2 and arbitrary d l , . . . ,din. Then P has index > m(1 - e)/2 with respect to ( a , . . . , a ) and ( d l , . . . ,din). We first choose solutions Pl/ql,... ,P,,,/q,,, of (1.17) and integers d l , . . . ,d,~ as follows: (a) choose (Pl,ql) with (1.18)

ql > max((6C1) 1/~, C~', 22"~C~)

(b) choose solutions (p2, q2),..., (pro, qm) such that (1.19)

log qh+l > (1 + e)C4 log qh

(1 < h < m - 1)

[This step makes the proof ineffective.] (c) choose dl so large that (1.20)

edl log qa >-- log qm

(d) for h = 2 , . . . , m , choose dh such that dl(l+c) 9 qdl < qdh < Ul

(1.21)

(This is possible since q~dl > qm > qh.) The conditions of Roth's Lemma are satisfied, since (1.18) and (1.21) imply

dh = dhlogqh logqh+l > 1 . ( 1 + e ) C 4 = C 4 > 1 , dh+l dh+~log qh+l log qh - 1 + e whence d, > d~ > ... > d,~, and further (1.21) and (1.18) imply

q~h > q~, > 22rnC~dl > 22mC.dh. Hence the polynomial P has index < e with respect to (p~/q~,...,pm/qm)

and to

(dl,..., d~,). We finally show, in order to obtain a contradiction, that P has index > e with respect to (pl/q~,... ,Pm/qm) and (d~,... ,din). So we have to prove that for i with (1.22)

il i,~ d-~ + " " + ~ < e

we have Pi(P~/q~,... ,P~/q,,) = 0. Note that a(o)

....

28

R. TIJDEMAN

whence, using ]a] < 1,

where the m a x i m u m extends over all j with jh <-- dh for h = 1 , . . . , m. Expand Pi(X) in a Taylor series around _~ = ( a , . . . , a ) ,

P,(X) = EPJ(-a).

"'"

....

im

3

According to the Index Theorem, Pj(~) = 0, i f j x / d x + . - . +j,,Jd,,~ <_ m(1 - e)/2, so certainly if (ji - il)/dl + ' " + (jm - im)/d,~ <_ m(1 - 3e)/2. Furthermore, by (1.23),

J

J

Hence, for T ( X ) := Pi(X) = ~ T(j)(X1 -- a) ~ . " " (Xm - a) jm J we have T(j) = 0

Jl

if

and

jm

m

~+...+~_<

5-(1-

3e),

IT(J)[ _< (6C1) rod'. J

It follows that, on denoting by * the j with j l / d l + ' "

IT(~I"'"

,

P")l _~ (6Cl)mdl max Pl _ qm J ql

~

+ j m / d m > m(1 - 3e)/2,

Jl...

im P_.m _ c~

< (6c1)md,m~x ((q~l)J,/d,...(q~,)~,/~,)-2-'

(by (1.17))

J

< (6Cl)rod1
-~-')

<

J q~'~d~(qdl) -'~(1-300+6/2)

<

(qa~...qa~)(~-(1-3~}(l+S/2)}/O+O

<

(qla. . . . qa,,,)-i

(by (1.21))

(by (1.18)) (by (1.21))

(since 6 = 12e < 1/2.)

On the other hand, T ( p l / q l , . . . ,Pm/qm) is a rational number with denominator dividing qal 9-. q,,am . Thus Pi

Pl

Pm ""

\ qi

if (1.22) holds. []

IV. ROTII'S THEOREM

2

Variations

29

and Generalisations

Roth's Theorem can be restated in the following symmetric form. 2.1 T h e o r e m . Let L ~ ( X , Y ) = a X + ~ Y and L : ( X , Y ) = "~X + ~Y be linearly independent linear forms with algebraic coefficients. Then, for every 5 > O, the inequalities

0 < IL~(x,y)L2(z,y)l < max(Ix l, lYl) -~

(2.2)

have only finitely many solutions in integers x, y.

Proof.

Theorem 2.1 implies Roth's Theorem, since [ a - x/y[ < [y[-2-6 implies

ly(~ - - y ) t < lyl -~ << (ma~(l~l, lYl))-~ On the other hand, Roth's Theorem implies Theorem 2.1 by the following argument. Assume IL2(~,Y)I _> ILx(x,y)[. We have ILl(X,y)l >> lYI'I~/Y +/~/~1 and

IL2(~,Y)I >> I~L~(~,y)- ~L~(x,y)l = I~'~-/~I'IYl > IYl. Hence

so there can be only finitely many solutions x / y .

[]

Ridout (1955) extended Roth's Theorem to p-adic numbers and LeVeque did so for approximations by elements from some fixed number field. In 1960 Lang formulated the following common generalisation. 2.3 T h e o r e m . Let K be an algebraic number field, S a finite set of places of K containing all infinite places. Let L be a finite extension of K. For each v E S choose a fixed extension of[ ]~ (normalised valuation) to L. For v E S, let a, E L. Then the inequality I I rain(l, [~ - a.[.) < H~c(~)-2-~ in ( E K yES

has only finitely many solutions. See S. Lang, [36], p. 160. Of course, there is also a symmetric form of Theorem 2.3. It says that for (a~: ft.) in •I(L) there are only finitely many ( ( : r/) in PI(K) such that

I~o~ +/3~1~

1-I max(f([., [~[.) < H K ( ( :

~)-2-~.

yES

This implies immediately that the S-unit equation (2.4)

x+y+z=0

inx, y a n d z i n C 9 ~

has only finitely many solutions. Indeed, for v E S choose (a~ :/~.) = (1 : 0), (0: 1) or ( - 1 : -1) according as [x[~, [Y]vor Iz[~ is the smallest of [x[~, [y[~ and [z[,. Then

.~s m~--s

~.) << ~ max(l~l~, IYl.) ~ -- HK(~,yp'

30

R. TIJDEMAN

since x , y , z E 0~, whence I~1. = lylv = jzlv = 1 for v ~ S. Thus there are only finitely many solutions (x : y : z) of (2.4). It is essential that in Theorem 2.3 the unknown ~ belongs to some fixed number field K . The 2 in the exponent is the best possible, as we see from the following result. 2.5 T h e o r e m . Let K be a reM algebraic number field. Then there exists a constant cK such that for every real a not in K there are infinitely many ~ E K having

[a - ~l < ck m a x ( l ,

I~I2)HK(~)-=.

See Schmidt, [65], p. 253. If only the degree of ~ is fixed, the exponent will depend on the degree. 2.6 T h e o r e m . Suppose a is a reM algebraic number. Let k > 1, 5 > O. Then there are only finitely many algebraic numbers ~ of degree < k with

[a -~1 < H(~) -k-l-a, where H denotes the classical absolute height. See Schmidt, [65], p. 278. The number k + 1 in the exponent is the best possible. Theorem 2.6 is a consequence of the Subspace Theorem and it therefore belongs to Chapter IV. A c k n o w l e d g e m e n t . I am indebted to Dr. J.H. Evertse for making a draft for part of this text and for making some critical remarks.

Chapter IV

T h e S u b s p a c e T h e o r e m of W.M. Schmidt by Jan-Hendrik

1

Evertse

Introduction

In Chapter III about Roth's theorem, the following equivalent formulation of this theorem was proved: 1.1 T h e o r e m . Let la(X, Y ) = a X +/3Y, 12(X, Y) = "~X +6Y be linearly independent linear forms with (real or complex) algebraic coefficients. Then for every e > O, the number of solutions of

(1.2)

l/~(z,y)f~(x,y)l <max(lzl, lyl) -~

in (x, y) E g 2 with gcd(z, y) = 1

is finite. A natural generalisation of (1.2) is the inequality (1.3)

Ill(X).-./,(x)l < Ix[-~

in x E Z",

where Ixl = max(IZll,...,Iz,[) for x = ( x a , . . . , z , ) i n Z" and where l l , . . . , l , are linearly independent linear forms with algebraic coefficients. Inequalities of type (1.3) might have infinitely many solutions (Xl,...,xn) with g c d ( X l , . . . , xn) -- 1. Consider for instance

(1.4)

(Xl + X'/2X2 "t- M/3X3)(Xl "1- %/~X2 -- V/3X3)(Xl -- ~/'2X2 -- ~/'3X3) • IX[ -r

in (Xl,X2,Xa) E Z a, where 0 < e < 1. It is easy to check that every triple (xl,x2,x3) E Z a w i t h x 3 = 0, xa > 0, x: < 0 a n d x ~ - 2 x 2 2 = 1 satisfies (1.4). Hence (1.4) has infinitely many solutions with gcd(xl, x2, x3) = 1 in the subspace xa = 0. Similarly, (1.4) has infinitely many such solutions in the subspaces Xl = 0 and x2 = 0. It appears that the set of solutions of (1.4) with xlx2xa r 0 is contained in finitely many proper linear subspaces of Q3. Namely, one has 1.5 T h e o r e m . (Subspace Theorem, W.M. Schmidt, 1972 [69]). For every e > 0 there are a finite number of proper linear subspaces T1,... ,Th of Q~ such that the set of

solutions of (1.3) is contained in T1 tO ... tOTh.

32

J.H. EVERTSE

Theorem 1.5 can be reformulated in a more geometrical way as follows. The Subspace topology (notion introduced by Schmidt) on Qn is the topology of which the closed sets are the finite unions of linear subspaces of Q". Then Theorem 1.5 states that for every e > 0, the set of solutions of (1.3) is not dense in Q" w.r.t, the Subspace topology. The Subspace Theorem has a generalisation over number fields, involving nonarchimedean absolute values. The theorem below is equivalent to a result proved by Schlickewei [64]. The following notation is used: K is an algebraic number field; {11" II~ : v E MK} is a maximal set of pairwise inequivalent absolute values on K , normalised such that the Product Formula 1-I. Ilxllv = 1 for x E K* holds (of. [14], p. 151); each II'llv h~s been extended in some way to Q; HK(x) = 1-I. Ilxll~, where llxll~ = max(llzlll~,...,llz=llv) for x e IP=-I(K); ~' is a finite subset of MK; {llv,... , l ~ } (v E S) are linearly independent sets of linear forms in Q[X1,..., Xn]. 1.6 T h e o r e m . For every e > 0 there are a/~nite number T1,... ,Th of proper hyperplanes of P=-I(K) such that the set of solutions of (1.7)

II I~I IIZ~v(x)ll~< i=l Ilxll~

HK(x) . . . .

in x e P " - I ( I ( )

~S

is contained in T1 U -.- U Th. Note that (1.3) implies that II'?=l(lll~(x)lL~/llxll~ ) < gQ(x) . . . . for x e Z n, where I1"11~ is the usual absolute value. Hence Theorem 1.6 implies Theorem 1.5. The following slight generalisation of Theorem 1.6, due to Vojta [80], is more convenient for certain applications. A set of linear forms {l~ . . . . ,lm } in Q [ X 1 , . . . , X,] is said to be in general position if each subset of cardinality _< n of {ll,...,l,~} is linearly independent. For v E S, let { l l , , . . . , l . . . . } be a set of linear forms in Q [ X 1 , . . . , X,] in general position. 1.8 T h e o r e m . For every e > 0 there are a tinite number T1,... ,Tn of proper hyperplanes of P ~ - I ( K ) such that the set of solutions of my

(1.9)

1-I II Ill~v(x)ll______~ < HK(x) . . . . ~si=, Ilxll~

in x e F ' ~ - I ( K )

is contained in T1 U ... U Th. P r o o f . (Theorem 1.6 implies Theorem 1.8.) If m~ < n then we may assume that { l l ~ , . . . , Im~, Xm~+~,..., X= } is linearly independent. Put l~ = l~ for i <_m~, l~ = X~ for i > m~. Thus, IIZ~(x)I[~ _< Ilxllv for x 9 I( ~, i > m~. Now suppose that m~ _> n. Fix x 9 K ". There is a permutation (j~,... ,jm~) of ( 1 , . . . ,my) such that IIb,(x)ll~ <_ Ilb=(x)ll~ _< ". <_ Illjmo(x)ll~ 9 Put l'i, = lj,,~ for i = 1 , . . . ,n. For i > n, the set {lj,,...,lj,_,,lj~} is linearly independent. Hence for i > n, k = 1 , . . . , n we have Xk = aikllj~ A- " " -4- aik,n_llj,~_~ "4- C~iknlji for suitable coefficients crikh. It follows that for i > n, Ilxll~ << max(lllj~(x)ll~,..., Illj,(x)ll~) = Illj,(x)ll~.

V. T]tE SUBSPACE THEOREM OF W . M . SCHMIDT

33

Hence every solution x of (1.9) satisfies one of a finite number of inequalities

II fI IIZ~(x)t[-------z~<< H g ( x ) . . . .

.

By applying Theorem 1.6 to each of these inequalities we obtain Theorem 1.8. 2

[]

Applications

In 1842, Dirichlet proved the following result:

Let { a l , . . . ,am} be a Q-linearly independent set of real numbers. Then for some C > O, the inequality [cexX, + - - - + a=x=[ < Clx[ - " + '

in x E Z ~

has infinitely many solutions. In 1970, Schmidt ([68], see also [65], p. 152) proved that the exponent - n + 1 cannot be replaced by a smaller number if a l , . . . , aN are algebraic. 2.1 T h e o r e m . Let a i , . . . , a n be algebraic numbers. inequality (2.2) 0 < [o/ix 1 Ac---Ai- cenXn[ • Ix[ - n + l - e

Then for every e > 0, the in x E Zn

has only finitely many solutions. P r o o f . Induction on n. For n = 1, the assertion is trivial. Let n > 1. We may assume that aX ~ 0. Then the set of linear forms {alX1 + "" + a~Xn,)(2 . . . . , Xn} is linearly independent. For every solution x of (2.2) we have (2.3)

I(axxl + . . . + c e , x , ) x 2 . . . x , I < Ixi -~.

By Theorem 1.5, the set of solutions of (2.3), and hence (2.2), is contained in finitely many proper linear subspaces of Q=. Let T be one of these subspaces. Without loss of generality we may assume that celX1 + ' " + ce, Xn = j31X1 + ' - - + ~ , - i X,_ 1 identically on T, where ~ , . . . , ~,_~ are algebraic numbers. Hence for every solution x of (2.3) with x E T we have (2.4)

0 < I]~lXl "-[-""" "]-/~n_lXn_l[ < Ixl -~+~-~ < max(Izll,..., ]x,_al) -'+~-~.

By the induction hypothesis, (2.4) has only finitely many solutions in ( X l , . . . , X,_l) E Z "-1. Hence (2.2) has only finitely many solutions with x E T. Since there are only finitely many possibilities for T this completes the induction step. [] From Theorem 2.1, one can derive the following generalisation of Roth's theorem. Here, the height H(~) of ( is the maximum of the absolute values of the coefficients of the minimal polynomial of (. 2.5 T h e o r e m . For every algebraic number a C C, integer d _> 1, and e > 0, there are only finitely many algebraic numbers ~ E C o[ degree d such that (2.6)

Ice - ~1 < n(~) -d-l-e"

34

J . H . EVERTSE

P r o o f . Let ~ be an algebraic number of degree d satisfying (2.6). We may assume that ~ is not a conjugate of a. Denote by f ( X ) = xd+lX d + x d X d-~ + " . + xx E Z[X] the minimal polynomial of ~. Then H(~) = Ixl and f ( a ) # O. From, e.g., the mean value theorem it follows that tf( )l/l - r <
-- If(~)l <<~ ts < H(~) -d-~ = Ixl -d-~.

0 < Ix~ + x2~ + . . . + Xd+l~al

- ~IH(~) <

By Theorem 2.1, there are only finitely many x E Z d+l satisfying (2.7). This implies that there are only finitely many ~ of degree d with Is - ~1 < H(~) -d-~-'[] We consider the equation (2.8)

alxl+...+a~xn=l

inxl,...,x,~EG,

where G is a finitely generated multiplicative subgroup of Q" and a l , . . . ,an E ~ ' . A solution ( X h . . . , x n ) of (2.8) is called non-degenerate if E i e t aixi ~ 0 for each non-empty subset I of { 1 , . . . , n }. 2.9 T h e o r e m . ([18], [59]). Equation (2.8) has only finitely many non-degenerate solutions. 2.10 L e m m a . There is a finite set U such that for every solution (either or not non-degenerate) of (2.8) there is an i E {1,... ,n} with xi E U. P r o o f . Induction on n. For n = 1, the assertion is trivial. Let n > 1. There are an algebraic number field K and a finite set of places S of K, containing all archimedean places, such that G is contained in the group of S-units {x E K : [[x[[~ = 1 for v • S}. Define the linear form Xo := alX1 + "" + a,~X,~. Then {Xo, X 1 , . . . ,Xn} is in general position. Let x = (x~,...,xn) be a solution of (2.8) and put x0 := 1. Then, from the fact that x 0 , . . . , x n are S-units and from the Product Formula, it follows that rives [[x0xl-.. x~ll. = 1. Further, HK(X) = lives HXHv9Hence

H II~0-"Hx]l~ ~z~*=ll~ = H~.(x)-.-, yeS

Now Theorem 1.8 implies that x E TIU-..UTh where T I , . . . , Th are proper hyperplanes of Pn-~(K) independent of x. Now let x be a solution of (2.8) lying in a proper projective subspace T of P ~ - I ( K ) defined by an equation b,X, + ... + b, Xn = 0. By combining this with (2.8) we obtain E,e3 c~xi = 1, where ci (i E J) are elements of K* depending only on a l , . . . ,an, T , and J is a proper subset of {1,... ,n}. Now the induction hypothesis implies that xi E UT for some i E J, where UT is a finite set depending on a l , . . . , an and T. Then clearly, for every solution ( x l , . . . , xn) of (2.8) there is an i with xi E U := UT, U *.. U UTh. [] P r o o f . (of Theorem 2.9.) Induction on n. For n = 1, the assertion is again trivial. Let n > 1. In view of Lemma 2.10, it suffices to show that (2.8) has only finitely many non-degenerate solutions with x , = c, where c E U is fixed. A solution with x~ = aX1 is degenerate so we may assume that a,c r 1. Then clearly, (xx,... ,xn-1) is a non-degenerate solution of (1 - a=c)-lalxl + ... + (1 - anc)-la~-lxn-1 = 1. By the induction hypothesis, there are only finitely many possibilities for x ~ , . . . , X , _ l . []

V. TIIE SUBSPACE TIIEOREM OF W.M. SCIIMIDT

35

We now prove a generalisation of Theorem 2.9 which was part of a conjecture of Lang (cf. [36], p. 220), and proved by Laurent [41]. The product of x = (xl . . . . , xn) and y = ( y l , . . . , y~) in (Q*)~ is defined by x . y = ( x l y i , . . . , x~y~). We may view (Q*)~ with this coordinatewise multiplication as an algebraic group (a so-called linear torus). For every n-tuple i = (il . . . . ,in) in (Z>0) ~ we write X i = X~ ~ . . . X ~ ". An algebraic subgroup of (Q*)~ is a set H = {x E (Q*)~ : fl(x) = 0 . . . . ,ft(x) = 0}

(f~,...,f, EQ[X~,...,X,,])

which is closed under coordinatewise multiplication and inversion. It is not difficult to prove that every algebraic subgroup is of the form {x E (~*)n : x i = x.i

for (i,j) E I},

where I is a finite set Of pairs from (Z_>0)~. We shall not need this fact. 2.11 T h e o r e m . ([41D Let V be an algebraic subvarlety of (Q*)" and G a t~nitely generated subgroup of (-~*)n. Then V N G is a finite union of "families" u . (H M G), where H is an algebraic subgroup of (~')~ and u E G is such that u . H C V. 2.12 R e m a r k . Faltings [23] proved the other part of Lang's conjecture on p. 220 of [36], which is the analogue of Theorem 2.11 for abelian varieties, see Chapter XIII. In this analogue, one has to take instead of (Q*)~ an abelian variety A over an algebraic number field K and instead of G the group of K-rational points A ( K ) of K. [] 2.13 R e m a r k . Laurent [41] proved in fact a more general result than Theorem 2.11, in which V is a subvariety of (C*) ~ and G a subgroup of (C*) ~ of finite rank, i.e., G [] has a finitely generated subgroup Go such that G/Go is a torsion group. P r o o f . (of Theorem 2.11.) Theorem 2.11 can be derived from Theorem 2.9 by using combinatorics. We have v = {x E ( ~ ' ) ~ : / , ( x )

= 0 . . . . , f , ( x ) = 0},

where fk(X) = ~

a(i, k)X i,

iECk

with Ck being a finite subset of (Z>0)" and a(i, k) in Q*, for k = 1 , . . . , t, i in Ck. Let P denote the collection of subsets of cardinality 2 of C1 U. 99U Ct. To every x E V we associate a subcollection $x of T', which consists of the sets {p, q} with the following property: there are k E {1 . . . . ,t} and a subset C of Ck such that p, q E C, a(i, k ) x i = 0, iEC a(i, klx i r 0 iEC'

for each proper, non-empty subset C' of C.

36

J . H . EVERTSE

For each subcollection g of P , put

V(E) = {x E V : & = E} and define the algebraic subgroup of (Q*)~: H(E) = {x E (Q*)~ : x i = x j 2.14 L e m m a .

for each {i,j} in s

Let ~ C__79 and u E V($). Then u - H ( E ) C_ V.

P r o o f . There are pairwise disjoint subsets E 1 , . . . , E 8 of C1 such that C1 = E1 U 9-. U Es and such that for l = 1 , . . . , s , a(i, 1)u i = 0, iEEl

~ a(i, 1)u i -r 0 lEE'

for each nonempty E' C El.

Since u E V(E) we have Eu = g. Hence if i and j belong to the same set El, then {i,j} E g which implies that x i = xJ for every x E H(E). Therefore, for every x E H(E) we have ~--~ a(i, 1)(u- x)i = x i~ }-~ a(i, 1)ui = 0 iEE~ iEE~

for / = 1 , . . . , s ,

where il is a fixed tuple from Eu By taking the sum over all l we get f l ( u 9x) = 0 ft(u" x) = 0 for all x E H(E). Hence for all x E H(E). Similarly, f2(u" x) . . . . . u.

H(E) c_ V.

[]

2.15 L e m m a . For each subcollection s of 7~ there is a finite subset W(s such that

o f V ( s MG

V(E) na c_ U u.(H(g) na). uEW(s P r o o f . Theorem 2.9 can be reformulated as follows: Let a0, . . . , a~ E Q* and let Go be a finitely generated subgroup of Q*. Then there is a finite set U, such that xl/zj E U for every non-degenerate solution ( x 0 , . . . , x=) in G~ +1 of aoxo + "" + a~xn = 0 and for a l l i , j E { 0 , . . . , n } . Now let x E V(e) f i G a n d take { p , q } E g. Choose a s e t C with p , q E C and C C_ Ck for some k E { 1 , . . . , t } as in the definition of gx. By applying the above reformulation of Theorem 2.9 to ~ i e c a(i, k)x i = 0, we infer that x P / x q E U, where U is some finite set, depending only on f l , . . . , ft, G. We can choose the same set U for each {p, q} E C. Thus, xP/x q E U

for each {p, q} E g.

This implies that there is a finite subset W ( g ) of V(g) fi G such that for every x E V(E) M G there is an u E W ( g ) with x P / x q = u P / u q for each {p,q} E g, in other words, ( x / u ) p = ( x / u ) q for each {p,q} E C or x E u . ( H ( g ) f i G ) . This implies Lemma 2.15. [] Lemma 2.15 implies that

YnCC-U

U

,.,.(H(E)nC),

e uEW(E) where the union is taken over all subcollections g" of 7'. By Lemma 2.14, these two sets are equal. This completes the proof of Theorem 2.11. []

V. TItE SUBSPACE TItEOREM OF W . M . SCtIMIDT

3

37

A b o u t t h e P r o o f of t h e S u b s p a c e t h e o r e m

We give an overview of the proof of Theorem 1.5. For certain details, we refer to [65], Chaps. V, VI. We remark that it suffices to prove Theorem 1.5 for the case that the linear forms in the inequality (3.1)

Ih(x) . . . l ~ ( x ) l < Ixl -r

in x C Z ~

have real algebraic integer coefficients. Namely, if for instance some of the coefficients of l~ are complex, then write l= = I~1 + V/-~-l~, where l~l, l=2 are linear forms with real algebraic coefficients. Choose l'~ from {l=l,/,2} such that { l l , . . . , l~-l, l'~} is linearly independent. Then II',(x)l _ It~(x)l for x e z ~. Hence (3.1) remains valid if we replace l~ by l~. Similarly, we can replace l l , . . . , l~_1 by linear forms with real algebraic coefficients. Thus, we can reduce (3.1) to a similar inequality with linearly independent linear forms l~,..., l'~ with real algebraic coefficients. Choose a positive rational integer a such that the linear forms l~' := al I (i = 1 , . . . , n) have real algebraic integer coefficients. Then for [xl sufficiently large, we get [/~'(x).-./~(x)l < Ix[-~/2. 3.2

R e d u c t i o n to a S t a t e m e n t a b o u t P a r a l l e l e p i p e d s

In what follows, we assume that /1,..., In are linearly independent linear forms in n variables with real algebraic integer coefficients. For each tuple A = ( A 1 , . . . , A~) of positive reals, we consider the parallelepiped II(A) = {x r R*: II,(x)[ _< A, for i = 1 , . . . , n } . Put Q(A) := m a x ( A , , . . . , An, A~-I,..., A;1). Theorem 1.5 follows from the following result about the parallelepipeds II(A): 3.2.1 T h e o r e m . For every e > O, there are t~niteIy m a n y proper linear subspaces T 1 , . . . , Th of Q~ such that for all tuples A = (A1,..., A~) of positive reals with (3.2.2)

A 1 A 2 " " A~ < Q(A) -r

and Q(A) suftlciently large, the set II(A) A Z ~ is contained in one of the spaces TI, . . . , Th.

In the sequel, constants implied by tile Vinogradov symbols << and >> depend only on ll,...,l~,e. By a <<>> b we mean a <~ b and a >> b. P r o o f . (Theorem 3.2.1 implies Theorem 1.5) Let x he a solution of (3.1) with h ( x ) . . . / = ( x ) # 0. Put Ai = II~(x)l

for i = 1,... ,n.

Then x r n ( A ) n Z '~,

A~... A,~ < ]x] -r

38

J.H. EVERTSE

We estimate Q(A) from above. Suppose that the number field K generated by the coefficients of I i , . . . , l ~ has degree D. On the one hand, we have I/i(x)l << Ixl . On the other hand,

INK/Q(Idx))l

II,(x)l = ii}~)(x)l... Ig}D)(x) I >> Ixl -D, where l~2)(x),...,/}O)(x) are the conjugates of/,(x) over Q. Hence for Ix I sufficiently large we have Q(A) < Ixl 2D. Therefore, AI "" " A,~ < Q ( A ) -~/2D.

Moreover,

(3.2.3) hence Q(A) is large if

Ixl << max(lll(x)l,..., Ig,(xl) _< Q(A), Ixl is large. Now Theorem 3.2.1 implies

that x E I I ( A ) n Z" c

T1 12 . . . U Th for certain proper linear subspaces T 1 , . . . , Th of Q~ independent of x.

[] 3.2.4 R e m a r k . If (3.2.2) holds and Q(A) is sufficiently large, then rank(lI(A) n Z n) _< n - 1, where rank(II(A) N Z ") is the maximal number of linearly independent vectors in H(A) N Z ". Namely, take Xa,..., xn in H(A) N Z n. Then Idet(xl""'x~)l

I det(/i(xJ))l

=

Idet(q,...,l,)l << m~x Ih(xoo))..-t,(x,(~))l

<

A a ' " A m < Q(A)-~,

where the maximum is taken over all permutations cr of {1,... ,n}. Since Q(A) is sufficiently large, this implies that I d e t ( x i , . . . , x,~)] < 1. But this number is a rational integer, hence must be 0. Therefore, {xx, : . . , x~} is linearly dependent. [] Analogous to Roth's theorem we could try to prove Theorem 3.2.1 as follows. For m sufficiently large, we can construct a polynomial P ( X 1 , . . . ,Xm) in m blocks of n variables, with rational integral coefficients, which is divisible by high powers of li(Xh), for i = 1,... , n and h = 1,... ,m. Suppose that Theorem 3.2.1 is not true. Then for every m we can choose A 1 , . . . , Am such that the sets II(Ah))NZ ~ (h = 1 , . . . , m) are in some kind of general position. Prove that ]Pi(Xl,--. ,xm)l < 1 for all xh E II(Ah) NZ ~ (h = 1 , . . . , m ) and all partial derivatives Pi of P of small order. Then for these X h . . . , xm and partial derivatives we have P i ( x l , . . . , xm) = 0. Suppose we were able to prove that on the other hand, Pi(X~,...,Xr~) =/ 0 for some Xh E H(Ah) N Z n (h = 1,... ,m) and some small order partial derivative. Thus, we would arrive at a contradiction. Unfortunately, as yet such a non-vanishing result has been proved only for the special case that rank(H(Ah) n Z n) = n - 1 for h = 1 , . . . , m. Therefore, we proceed as follows. In Step 1 of the proof of Theorem 3.2.1 we show that it suffices to prove Theorem 3.2.1 for parallelepipeds H(A) with rank(H(A)NZ ~) = n - l , by constructing from each H(A) with rank(H(A) n Z ~) < n - 1 a parallelepiped H'(B) in Z N with rank(II'(B) N Z N) = N - 1, where in general N > n. For this, we need geometry of numbers. Then, in Step 2 we prove Theorem 3.2.1 for parallelepipeds H(A) with rank(II(A) n Z ~) = n - 1 in the way described above.

V. THE SUBSPACE THEOREM OF W.M. SCHMIDT 3.3

39

S t e p 1 o f t h e P r o o f o f T h e o r e m 3.2.1

If r a n k ( I I ( A ) N Z n) = r < n - 1, then II'(B) will be contained in the exterior product A"-k(R'~) ~ R(~), for some k with r < k < n - 1. We need to recall some facts about exterior products. P u t N := (~). Let a , , . . . ,aN denote the subsets of cardinality n - k of { 1 , . . . ,n}, ordered lexicographically: thus, al = { 1 , . . . , n - k}, a2 = { 1 , . . . , n - k - 1, n - k + 1 } , . . . , a r c _ x = { k , k + 2 , . . . , n } , aN = { k + l , k + 2 , . . . , n } . Let {e, = ( 1 , 0 , . . . , 0 ) , e 2 , . . . , e , } be the standard basis of R n and { E , , . . . ,EN} the standard basis of Rrc. The multilinear mapping from R ~ • • ~ ( n - k times) to R N sending ( X l , . . . , x,_k) to Xl A 9.. A xn-k is defined by 9 ei~ A . . . A ei._, = 0 if ix,.,. ,Q_, are not all distinct; 9 eix A . - . A ei._, = + E i where { i l , . . . , i~-k } = ai, the sign is + if the permutation needed to rearrange i x , . . . , i n - k into increasing order is even, and the sign is if this permutation is odd. It is easy to verify that if { a l , . . . ,an} and { b x , . . . ,b~} are two bases of R" related by bi = ~ j % , ~ i j a j for i = 1 , . . . , n , then { A 1 , . . . , A N } , and { B 1 , . . . , B N } with A i = ai~ h ' " . Aai,_k and Bi = bi~ A . - . Abe._ k where al = {il < "'" < i~-k}, are two bases of R N related by:

N Bi = ~ E i j A j

for i = 1 , . . . , N , with =ij = det(~pq)pe~,,qe~:.

j=,

(We write ai = {i, < - . - < the following fact: 3.3.1 L e m m a .

i , - k } if ai = { i ~ , . . . , i , - k } and i~ < - . . < i , - k . ) We need

Let k E { 1 , . . . , n}. If { a , , . . . , a~} and { b x , . . . , b , } are two bases of

R n, then {ax,..., ak) and { b l , . . . , bk} generate the same vector space if and only if { A x , . . . , A r c - , } and { B , , . . . , B N - , ) generate the same vector space. P r o o f . Use that { a ~ , . . . ,ak} and { h i , . . . ,bk} generate the same space r EIN = 0 for i = 1 , . . . , N - 1 r { A , , . . . , A N _ , } and { B , , . . . , B N _ x } generate the same space. [] Lemma 3.3.1 implies that there is an injective map fk from the collection of kdimensional subspaces of N n to the collection of ( N - 1)-dimensional subspaces of W v such that if V has basis {ax,... ,ak} and ak+x,... ,a= are any vectors such that { a x , . . . ,a=} is a basis of R ~, then { A 1 , . . . , A N - x } is a basis of fk(g). Now let l , , . . . ,l= be linearly independent linear forms in n variables with real coefficients (at this point, the coefficients need not be algebraic). Write h(X) = (c~, X) (usual scalar product in Rn), put Ci = % A . . . A ci._~ for i = 1 , . . . , N where i, < .-- < i , - k and { i x , . . . , i = - k } = ai, and define the linear forms on Rrc: LI(X) = ( C i , X )

(i = 1 , . . . , N ) .

Note that Lx,...,Lrc are linearly independent. ( B x , . . . , Brc) of positive reals, define

For every N =

IIk(B)

=

{x E Rrc: ILi(xll _< B~ for i = 1 , . . . , N } ,

Q(B)

=

m a x { B 1 , . . . , B N , B~X,..., B~ x}.

(~)-tuple B =

40

J.H. EVERTSE

3.3.2 P r o p o s i t i o n . For every e > 0 and for every tuple A = (A1,... ,An) of positive reals with Q ( A ) sufficiently large and (3.3.3)

A I ' " Am < Q ( A ) -~,

1 _< rank(II(A) f3 Z ~) _< n - 1

there are k with rank(H(A) fq Z '~) < k < n - 1 and a tuple of positive reals B = with N := (:) such that

(B1,...,~N)

(i) B~ . . . BN < Q(B)-r (ii) rank(ilk(B) N Z N) = N - 1; (iii) II(A) A Z '~ is contained in a k-dimensional linear suhspace V of R n such that f k ( V ) is the R-vector space generated by Ilk(B) fq Z N. Now assume that Theorem 3.2.1 has been proved for tuples A with rank(II(A) fq Z ~) = n - 1. Apply this to IIk(B) where k and B are determined according to Proposition 3.3.9_. It follows that there are finitely many ( N - 1 ) - d i m e n s i o n a l subspaces Slk),..., S~ff) of R W such that for every B satisfying (i), (ii), the set IIk(B) N Z N is contained in one of the spaces SJ k) (j = 1 , . . . , tk). From (iii) and the fact that fk is injective, we infer that for every tuple A with (3.3.3), the set H(A) A Z ~ is contained in one of the spaces Tff ) (k -- 1,...,n - 1, j = 1,..., tk), where Tff ) is the unique k-dimensional subspace of ~,~ with fk(Tff )) = SJ k). This implies Theorem 3.2.1 in full generality. Proposition 3.3.2 can be proved by combining arguments from [65], Chap. IV, w167 3, 6, 7, Chap. VI, w167 15. Our approach is somewhat different. We need two lemmas. 3.3.4 L e m m a . (Minkowski's Second Theorem). Let C be a convex body in R ~ which is symmetric about O. For A > O, put AC := {Ax : x ~ C}. Deh'ne the n successive minima A I , . . . , A n of C by A~ = inf{A > 0 : rank(AC N Z n) = i}. Th en

2n --

<

/~1

"'"

"~n

n! -

Proof.

"

vol(C) <

2n

E.g. [11], Lecture IV.

[]

The next lemma is to replace Davenport's lemma and Mahler's results on compound convex bodies used by Schmidt ([65], Chap. IV, w1677). 3.3.5 L e m m a . Let W be an R-vector space with basis { b l , . . . , b~}, and let l l , . . . , l~ be linearly independent linear functions from W to R . Suppose that

I/,(bj)l _< #j

for i = 1 , . . . , n, j = 1 , . . . , n,

where #1 ~- "'" ~- #,~. Then there are a permutation ~ of {1,... ,n} and vectors V1

----

bl

v2

=

b2 -t- ~21bl

Vr~

=

b,~ q- ~nlbl q- "'" -1- ~...... abn-1

with ~j E Z for 1 < j < i <_n, such that

(3.3.6)

II~(o(vAI

_

<-

9 i+j

#min(i,j)

~ri=

1,...,n,j

= 1,...,n.

V. THE SUBSPACE THEOREM OF W.M. SCHMIDT

41

Proof. We proceed by induction on n. For n = i, Lemma 3.3.5 is trivial. Suppose that Lemma 3.3.5 holds true for n - 1 instead of n, where n >_ 2. Let W ~denote the space generated by bx,... ,b~-1. There are a l , . . . , a , E R, not all zero, such that

al/l(X) + " "

+ a./n(x) = 0

Choose g(n) e { 1 , . . . , n } such that -aj/a,~(,q, we have (3.3.7)

/~(,)(x)

= ~

fljlj(x)

I~(.)1

for x E W'.

= m a x ( l a l l , . - . , [a-I). Thus, with flj =

for x e W', where I~J[ < 1 for j :~ a(n).

J#~(~) By the induction hypothesis, there are a permutation ,r { 1 , . . . , n}\{,~(n)}, and vectors Vl = b , , . . . , v~_~ = b ~ _ l + ~ - l . a b , + ' with~ij E Z f o r l _ < j < i _ < n - l , suchthat

It~(0(vj) I _< 2~+J#mm(,,j)

- 1) of the set "+~-1,~-2b~-~

for i = 1,... ,n - 1, j = 1 , . . . ,n - 1.

By (3.3.7) we have I/~(~)(vj)l _< Z~_--( I/~(i)(vj)l -< ( 2~+j + 22+j + " "

+ 2~-x+J)#J <

2'~+J#j for j = 1 , . . . , n - 1. Hence (3.3.8)

2i+J#mln(iS)

I/~(i)(vj)l <

for i = 1 , . . . , n, j = 1 , . . . , n -- 1.

It suffices to show that there are ~a,... , ~ - 1 C Z such that v~ := b , + ~1Vl + " " + satisfies I/~(o(v.)l _< 2~+"#~ for i -- 1,... ,n. (3.3.9)

~n_lVn_l

Define the vectors aj

=

(/~0)(vj),...,l~(,)(vj))

(j = 1 , . . . , n -

b

=

(/n(i)(bn),...,/~(~)(bn)).

1),

Since {/1,... ,l~) is linearly independent and {Vl,... ,V~-l} is a basis of W ' , and by n--1 (3.3.7), the set { a l , . . . , a~-i } is a basis of the space xn = ~j=l 13,,(j)xj. Hence there are t l , . . . ,tn--1 E • such that b

.

t~a~ .+

+. tnlan-1 . + ( 0. , . . . , 0, a),

Choose ( 1 , . . . , ~ - 1 E Z such that I~J + t i l b~ + ~1vl + " " + (n-lV~-l. Thus, (/a(l)(Vn),...,

l~(n)(Vn))

with a

/~(n)(b~) -

~ - ' ~(j)l~(j)(b.). Ej_-,

-< 89for j = 1 , . . . , n -

= (~lal -~- 9- 9-'[- ( ~ n _ l a n _ l + ( 0 , . . . ,

1. P u t v~ :=

0, Or)

with I~/1_< ~1 f o r j = l , . . . , n - 1 . Putai=0fori=l,...,n-landa,,=a. For i = 1 , . . . ,n we have, by assumption, I/~(,)(b~)l _< #,. Further, I/3il _< 1 for i ~ x(n). Hence lal _< n#~. Therefore, la~] _< i#~ for i = 1 , . . . ,n. Together with (3.3.8) this implies for i = 1 , . . . , n,

II~(,)(v.)l

~n-ll.(i)(Vn-1) Jv Oli[ _< (ll~(,)(vi)l + ' " + II~(,)(v,,-a)[)/2 + I~,1

=

I~,t.(,)(v,) +---+

_< {(2 i+x + 2 i+2 + " " + 2i+"-x)/2 + i}#i -< 2'+"#i 9 Hence (3.3.9) is satisfied. This completes the proof of Lemma 3.3.5.

[]

42

J.H. EVERTSE

P r o o f . (of Proposition 3.3.2) Let A = (A1,..., A,) be a tuple satisfying (3.3.3). The parallelepiped H(A) is convex and symmetric about 0. Let ) i l , . . . , A, be its successive minima. The volume of H(A) is I d e t ( h , . . . , l,,)l-~A~ ... A,~. Hence by Lemma 3.3.4, (3.3.10) Put ~

)i1""-)inA1-" .A~ <<>> 1.

:= rank(H(A) n Z~). Then )i~ < 1 < )i,+1. Further, by (3.3.10) and (3.3.3),

An --~ ()~1 ' ' " )in) 1In >> (A, ... An) - V " > Q(A) ~/~. The integer k in Proposition 3.3.2 is chosen from {r,r -4- 1,... ,n - 1} such that the quotient ) i k / ) i k + l is minimal. Thus, - )ik -

<

)ik+l --

)ir+l

Hence (3.3.11)

)it+2 - )ik -

)in ]

--

~n

<< Q ( A ) - ~ / " ( ~ - 1 ) .

)ik+l Let g l , - . . , g~ be linearly independent vectors such that gj C )IjH(A) A Z ~ For j =

for j = 1 , . . . , n .

1 , . . . , N , put ii.j = Ai, ---Ain_~,

Aj = )ii, '")ii,_k,

Gj = gil A . - . Agi,_~,

where aj = {il < ... < i~-k}. Let V be the vector space generated by g l , . - . ,gk. We know that )iT _< 1 < )ir+l. Hence the space generated by H(A) N Z ~ is the same as that generated by g l , . - . ,gr. Further, r < k. Hence H ( A ) A Z = C V. Denote by W the space generated by G 1 , . . . , GN-1. Thus, (3.3.12)

H(A) N Z ~ C V,

W = fk(V).

Laplace's rule states that for any vectors a l , . . . ,a=-k, b l , . . . ,b=-k E R n, (ai A . - . A a~_k, bl A - . - A b~ k) = det((ai, bj)l
la(r)l --< Ai~)i~'(j~)'"Ai,,_k)i':o,,_k) = ftviq. Hence ILp(Gq)l <_ (n

-

]~)!Apiq

for p ---- 1 , . . . , N, q -- 1 , . . . , N.

We apply Lemma 3.3.5 to the forms .4:-1L~,..., ft~v~LN and the vectors G ~ , . . . , GN. It follows that there are a permutation ~c of {1,... , N } and a basis {V~,... ,VN-1} of the space W generated by {G1,... , GN_ 1}, with V 1 , . . . , VN-1 E Z N, such that /~-(lp) l/~(p)(Vq) I _< 2~N(n _ k)!imi,,(~,,q)

for p = 1 , . . . , N, q = 1 , . . . , N - 1.

V. THE SUBSPACE THEOREM OF W.M. SCHMIDT

43

Choose B = (B1,..., BN) such that (3.3.13)

B,,(v ) =

22N(n -- k)!Av,4,~(v )

B,,(N)

22N(n -- k)!iN-1A~(N).

=

(p = 1 , . . . , g

- 1),

Thus, ]Lv(Vq) [ ~ B v for p = 1 , . . . Y , q = 1 , . . . , N - 1, i.e.,

Vl,... ,VN-1 e Hk(B) [q Z N.

(3.3.14)

We show that B satisfies (iN), (iii) and (i). Proof. (of (iN)) Let the succcessive minima of Hk(B) be v i , . . . , VN. The volume of Hk(B) is [det(L1,..., LN)[-1BI ... BN. Together with Lemma 3.3.4 this implies that V l " ' V N >> ( B ~ . . . BN) -~. Equation (3.3.14) implies that VN-1 _< 1. Hence (3.3.15)

~'N )>)>(BI " " B N ) - ' .

Note that -41""-4N -- ( A I " ' A n ) (":1), that AI""AN ---- (Ax'-'A~) ("~1), and that { k + l , k + 2 , . . . , n } . Hence

aN-1 = { k , k + 2 , . . . , n } , a N =

~N-1

/~k'~k+2 " " " "~n

~g

-- /~k+lhk+2 " ' ' / ~ n

*~k ----- /~k+l"

Together with (3.3.13), (3.3.10) and (3.3.11), this gives (3.3.16)

B,-"BN

<<

i1""~NAl"'ANiN_l/iN =

=

(h,...hnA,...An)("-~[1)hk/hk+l

<~
Together with (3.3.15) this implies that ~'N > 1 if Q(A) is sufficiently large. Hence rank(Hk(B) N Z N) = N - 1. [] Proof. (of (iii)) From (3.3.14), the fact that {V1,...,VN-1} is a basis of W and from rank(Hk(B) N Z N) = N - 1, it follows that the space generated by Hk(B) N Z N is equal to W. Together With (3.3.12) this proves (iii). [] Proof. (of (i)) We estimate Q(B) from above in terms of Q(A) and insert this into (3.3.16). Note that gi E hill(A) CIZ n and g~ -~ 0. Hence 1 _< [gl[ (< max([/l(gl)[,...,

[/n(gl)[) --~ AIQ(A).

Therefore, hi >> Q(A) -1. Since each Aj is the product of n - k hi's, and each hi is bounded below by hi, this implies that ij >> Q(A) -(~-k)

for j = 1 , . . . , N .

On the other hand, by (3.3.10), ~j _< (hi " " An)hi ~ << (AI"'" A~)-IQ(A) k _< Q(A) ~+k. By combining this with (3.3.13), and using that each Aj is the product of n - k Ai's we get Q(B) << max(A1,...,AN,i~-~,...,i~)max(.J.i,...,AN, A.i-1,...,A~,~) << Q(A)('+k) +('-k) = Q(A) 2~. Together with (3.3.16) and Q(A) sufficiently large this implies that B I ' " BN < Q(B) -r

as required.

1-I []

44 3.4

J.H. EVERTSE Step 2 of the P r o o f of T h e o r e m 3.2.1

For this subsection we shall refer to [65], Chapters V, VI for details; l l , . . . , l~ will be linearly independent linear forms in n variables with real algebraic integer coefficients, and 0 < e < 1. A g e n e r a l i s a t i o n of R o t h ' s l e m m a . Let n > 2, m > 2. For h = 1 , . . . , m , denote by Xh the block of n variables (Xh~,...,Xh,~). For a tuple d = ( d l , . . . , d m ) of positive integers, denote by T~(d) the set of polynomials in m n variables P ( X l , . . . ,Xm) E Z[X1,... ,Xm] such that P is homogeneous of degree dh in the block of variables Xh. We shall consider polynomials P E T/(d) as functions on the m-fold cartesian product ~ • ... • Q". For a tuple i = (i11,..., im.~) E (Z>0) "~, put 1 f 0 '~qa (___0__0_'~i~" P i ( X l , . . . ,Xm) -- i l l ! ' - iron! \~-~11] ""\OXmn] P" Note that for P E 7~(d), the coefficients of Pi are integers. The factorials have been included to keep the coefficients of Pi small: if H ( P ) denotes the height (maximum of absolute values of coefficients) of P, then H(Pi) < 2d'+"+dmH(P). For a tuple i as above, put ihl + " " " + ihn

(i/d) h=l ~

dh

3.4.1 Definition. Let x = (Xl,. 9,xm) E ~ x ... x Q'~ (m times). The index of x w.r.t. P, denoted by i(x, P), is the largest number a such that Pi(x) = 0 for all i with (i/d) < a. Clearly, a E Q. [] Every linear subspace V of Q~ of dimension n - 1 can be given by an equation alX1 + "'" + a~Xn = 0 with a l , . . . , a ~ E Z, gcd(aa,. .. ,an) -- 1 which is uniquely determined up to sign. The height of V is defined by H(V)

:= m a x ( l a l l , . . . , [a,I).

3.4.2 P r o p o s i t i o n . (Generalisation of Roth 's /emma). Let 0 < a < 1, 0 < 7 < n - l , m a positive integer, dl,. 99 dm positive integers, V 1 , . . . , V,~ (n - 1)-dimensional linear subspaces o f Q ~, and P a polynomial with

dh

forh=l,...,m-1; > C1 dh+l H(Vh) d~ >_ H(V~y 1" for h = e , . . . , m ,

H(Vh) P H(P)

>- "~2c~-1("-1)2 E

for h = 1 , . . . , m ,

7~(d),

<_ H(V1) c3~(~-1) 2d',

where C1, C~ and C3 are positive numbers depending only on m and a. Then there is a point x E V1 • . . . x Vm with i(x, P) < ma.

Y. TIIE SUBSPACE THEOREM OF W . M . SCHMIDT

45

P r o o f . ([65], pp. 191-194.) The idea is as follows. For n = 2, Proposition 3.4.2 is a homogeneous version of Roth's lemma, except for the additional parameter 7- But with this parameter, the proof of Roth's lemma does not essentially change, cf. [65], pp. 137-148. The general case is reduced to n = 2 as follows. We may assume by permuting the variables in each block Xh if necessary, that Vh is given by an equation ahlX1 + " " + ahnX,, = 0 with ahl,...,ah= 9 Z, g c d ( a h l , . . . , a h , ) = 1, ahl = max(lahxl,..., lah, I), and g := gcd(ahl, ah2) < a(h~-2)/(~-~). Put P* = P ( X ~ , . . . ,X~,), where X~, = ( X h l , X h 2 , 0 , . . . ,0) and Vh* = {ah~X~ + ah2X2 = 0}. We have H(V~) = ahx/g and therefore H(Vh) ~/(~-~) < H(Vr, ) <_ H(V~). Now the conditions of Proposition 3.4.2 are satisfied with n = 2, and P*, VI*,... , V,], 3'* := "7/(n - 1) replacing P, V1,...,Vm, % Hence / ( x ' , P * ) < ma, with x* = ( x ; , . . . , x ~ , ) , x ~ = (ah2,--ahl) for h = 1 , . . . , m . This implies that i ( x , P ) < m a for x = (xa,...,Xr~) with Xh = (ah2,--ahl, 0 , . . . , O) 9 Vh for h = 1 , . . . , m. []

C o n s t r u c t i o n of t h e a u x i l i a r y p o l y n o m i a l . Introduce new variables

Uhj = l j ( X h l , . . . , X h = )

(h = 1 , . . . , m , j = 1 , . . . , n ) .

Since the linear forms 11,..., l~ are linearly independent, we can express every polynomial P C T~(d) as P = Y~ dR (5) UiTl1UiT~2 " ' ' UJ"~" , J where j = ( j i l , . - . ,jm~) runs through all tuples of non-negative integers with jhl + 9"" + Jh,~ = clh for h = 1 , . : . , m. Similarly, we can express Pi as pl =

uL-:. J

We need the following generalisation of Roth's Index theorem, which states that there is a P C 7~(d) of small height such that if (i/d) is small then dp, i(j) = 0 unless j l j d l + ' " + j,~i/dm ~ m / n for i = 1 , . . . , n . 3.4.3 P r o p o s i t i o n . (Polynomial theorem). Let a > 0 and d = ( d l , . . . , a m ) be any tuple of positive integers with m > C4(n, l l , . . . , l~, a). Then there is a polynomial P E T~(d) with the following properties: (i) H ( P ) < Casl+'''+d~, and [dp,i(j)[ _< Csa~+'''+d"~ for all pairs i, j, where Cs = Ch(n, 11,..., ln, a); (ii) if (i/d) < 2ma, then dp, i(j) = 0 unless ] ~ = ~ j h i / d h - - m i n i <_ 3nma for i = 1,...,n. P r o o f . This is Theorem 7A of [65], p. 180. Its complete proof is on pp. 176-184 of [65]. The idea is as follows. First one shows the existence of a polynomial P E T/(d) such that

H(P) (3.4.4) dp(j)

< C6(n,l~,...,l,~,a) d~++d'~ = 0 for all tuplesj with ~h=~ "~ Jhi/"d h < ( 1 / n - - a ) m

for at least o n e i 9 { 1 , . . . , n }

46

J . H . EVERTSE

(This is the Index Theorem on p. 176 of [65]). The proof is similar to that of the Index Theorem of Roth: one counts the number of tuples j in (3.4.4) by using a combinatorial argument (cf. [65], p. 122, Lemma 4C) and then applies Siegel's Lemma to the equations dR(j) = 0. It is straightforward to show that P satisfies (i). Further, from the definition of index it follows that if (i,d) < 2ma then dp, i(j) = 0 unless ~h~lJh,/dh > (1In -o')m -- 2ma = m / n -- 3ma for i = 1 , . . . , n. But for such i, j we also have

E 3 h l = ~ - ' ~ J h k -dh k = l h = l

jhk~ --

h=l

This implies (ii).

[]

P r o o f . (of Theorem 3.2.1) By Cz, C s , . . . , we denote positive numbers depending only on n, lx,..., ln, e. For a given tuple A = ( A 1 , . . . , A,) with rank(II(A) VI Z ") = n - 1, let VA denote the Q-vector space generated by II(A) N Z =. We must compare Q(A) with H(VA). 3.4.5 L e m m a . Let A = (A1,... ,An) be a tuple of positive reals with (3.4.6) A I ' - " A~ < Q(A) - ' ,

rank(II(A) Cl Z ") = n - 1,

H(VA) > Cr,

where Cr is sumciently large. Then Q(A) c8 < H(VA) < Q(A) c9. P r o o f . This is essentially Lemma 11A of [65], p. 195. Schmidt used reciprocal parallelepipeds to prove this; we give a more straightforward proof. Suppose that li(X) = (ci,X) (i = 1 , . . . , n ) , and that the coordinates of c l , . . . , c , are contained in an algebraic number field of degree D. Let g l , . - - , g n - 1 be linearly independent points in II(A) N Z ~. For k = 1,... ,n, put Ak := d e t ( ( c , gj))a_<~_
IA~I << A~.. "Ak-,Ak+l" .A,, << Q(A)-~A~-1.

We have gl A .. 9A g~-i = $a, with )t E Z and a being a vector with coprime integer coordinates. Further, VA has equation ( a , X ) = 0. Therefore, H(VA) = ]a]. Put c~ := c~ A... Ack_~ ACk+l A.-. Ac,. Thus, by Laplace's rule, Ak = (c~, g~ A.-. Ag=-l). Therefore, (3.4.8) IAkl = I~1. I(e~,a)] > I(e~,a)l for k = 1 , . . . , n . Since {c~,... ,c~,} is linearly independent we have by (3.4.8) and (3.4.7)

H(VA) (3.4.9)

= lal << max(IA, I,...,IA~l) << Q(A) -r max(A~-',..., A~ x) << Q(A) 1-~

Let E = {i: 1 < i < n, A~ > Q(A)-#2}. First suppose that E :~ 0 and Ak :~ 0 for some k E" E. For this k we have by (3.4.8) that (c~, a) ~ 0. By an argument similar to

V. THE SUBSPACE TtIEOREM OF W . M . SCHMIDT

47

that in the derivation of Theorem 1.5 from Theorem 3.2.1, we have I(c~, a)I >> l a r D. Together with (3.4.8), (3.4.7) this implies that H(VA) = lal >> Q ( A ) #2D. This proves Lemma 3.4.5, provided that E # O, and Ak # 0 for some k E E. Assume that E -=- 0 or that Ak = 0 for all k E E. If E = 0 then choose arbitrary linearly independent h l , . . . , h , , - 1 from Z ~. If E =/ 0 and Ak = 0 for all k E ~ , then there are linearly independent integer vectors h i , . . . , h , - 1 such that det((ci, h/))~<~<~, ~#k, 1<j<~-1 = 0 for each k E E. We can choose h i , . . . , h~_~ from a finite set independent of A, since there are only finitely many possibilities for E. Now it follows from H(VA) _> C7, that there is an x E II(A) M Z ~ outside the space generated by h ~ , . . . , h~-l, that is, with ] det(x, h i , . . . , h~_~)]-~ 0. Since this number is a rational integer we have I det(x, h ~ , . . . , h~_~) I :> 1. On the other hand, we have det(x, h l , . . . , h ~ _ l )

=

(Oct(l,,

..,l,))

-1

k--1 ---- (det(/x,..., in)) -1 E "}-Olklk(x)'

Further, lakl << 1, Ilk(x)l < mk for k ~ E. Hence by

where ~k := det((c,,hj))i#k. (3.4.0),

1 _< Idet(x, h l , . . . ,h~-l)] << Y~ Ak << Q(A) -~/2 << H(VA) -#2. But this contradicts that H(VA) > Cr. Therefore, E # O and Ak # 0 for some k E E. [] We need another simple non-vanishing result for polynomials. 3.4.10 L e m m a . Let f ( X l , . . . , X r ) E C [ X 1 , . . . , X r ] be a non-zero polynomial of degree _< Sh in Xh for h = 1,... ,r and let B 1 , . . . , B ~ be positive reals. Then there are rational integers xm,..., xr, il, . . . , iT with IxhI <_ Bh, ox

]

0 < ih <_ sh/B~

for h = 1,... ,r,

"

P r o o f . This is a special case of Lemma 8A on p. 184 of [65], which is sufficient for our purposes. For r = 1, Lemma 3.4.10 follows from the fact that f cannot be divisible by the polynomial { X ( X - 1)(X + 1)(X - 2)(X + 2 ) - - . ( X - a ) ( X + a)} b, where a = [B~], b = [sl/B1] + 1, since the latter polynomial is of degree > &. It is straightforward to complete the proof of Lemma 3.4.10 by induction on r. [] We assume that Theorem 3.2.1 is false. Then, by Proposition 3.3.2, there is an n _> 2 such that if A runs through the tuples A = (A1,... ,As) of positive reals with (3.4.11)

rank(H(A) A Z n ) = n - 1 ,

AI'"Am
-~,

H(VA)_>CT,

48

J.H. EVERTSE

then VA runs through infinitely many (n - 1)-dimensional linear subspaces of Q~. In view of Lemma 3.4.5, this implies that the set of numbers Q(A) for which A satisfies (3.4.11 ) is unbounded. Note that Ai = Q(A) ~ for i = 1 , . . . , n , where q01,...,q0. are reals depending on A with [q0i[ _< i for i = 1 , . . . , n , and c21 + ' " + ~ < -e. The set of real points (tpt,... , T~) satisfying these inequalities can be covered by finitely many cubes {(~t . . . . ,~9n) : Ci -- e/2n < ~i <_ci for i = 1,... ,n}, where (3.4.12)

[cil _< 1

for i = 1 , . . . , n ,

c, + . . . + c~ < - e / 2 .

Hence there is a tuple ( q , . . . ,c,) with (3.4.12), such that the set of numbers Q(A) for tuples A with rank(II(A) C/Z")

=

n-

At...A,~

< <

Q(A) -~, Q(A) c'

(3.4.13)

Ai

1, for i--- 1 , . . . , n ,

H(VA) > c~ is unbounded. We choose a sufficiently small, depending only on n and e, with 0 < a < 1In. Let m be the smallest integer exceeding the number C4(n, It,..., 1,~,o) in Proposition 3.4.3. Choose At . . . . . Am with (3.4.13) such that for sufficiently large Ct0, C11, (3.4.14)

Q(At) ~ el0,

Q(Ah+I) _> Q(Ah) C1~

for h = 2 , . . . ,m.

Put Qh := Q(Ah), Vh := VAh for h = 1,... ,m. Similarly as in the end of the proof of Roth's theorem, we choose a tuple d = ( d l , . . . , d,~) of positive integers such that (3.4.15) and (3.4.16)

dtalogQ1 > log Q,~, d 1 log Qt ~ dh log Qh -< (1 + a)dl log Qt

for h = 2 , . . . , m.

The latter is possible since adl log Q1 >_ log Qh for h = 2 . . . . , m. Let P C 7~(d) be the polynomial from Proposition 3.4.3. Put 3' := Cs/Cg, where Cs, C9 are the constants from Lemma 3.4.5. Let C1, C2 and C3 be the constants from Proposition 3.4.2 and Cs the constant from Proposition 3.4.3. We verify that a, m,7, dl,... , din, Vb..., Vm and P satisfy the conditions of Proposition 3.4.2 for suitable Cm, Cn: namely, for sufficiently large Ct0 and Cat we have by (3.4.16), (3.4.14), H(Vh) > C7, Lemma 3.4.5 and the upper bound for H(P) in Proposition 3.4.3 that

dh dh+l

dh log Qh log Qh+l - _> (1 + o ) - l C l l > e I for h = 1 , . . . , rn - I, dh+l log Qh+l log Qh n(Wh)dh ~ QCsdh:> O1Csdl = Q1C9dl-), ~ [i(Wt)Ch.v for h = 1,... ,m - 1, H(Vh) > . QC, . > . QlC,. > C~o, > ~z'~w-'(~-'?

for h = 1, ... ,m

and

o H (Vl)C3"y(n-1)-2d, --> "~If)C3C87(n-l)-2dl>- CO3CS~(n-l)-2d' > C~ d' > C~'+''+din > H(P).

V. THE

SUBSPACE

TIIEOREM

OF W.M.

SCHMIDT

49

Now Proposition 3.4.2 implies t h a t there is a point x C V1 • • Vm with i(x, P ) < ma. W h a t we actually want is a point x = ( X l , . . . ,xm) with xh E H(Ah) C/Z ~ for h = 1 , . . . , m and small index w . r . t . P . There is such a point in a slightly larger set. Namely, choose a tuple i with ( i / d ) < a such t h a t Pi does not vanish everywhere on V1 • " " • Vm. Choose linearly independent vectors g h l , . . . ,gh,n-1 from H(Ah) N Z ~ for h = 1 , . . . , m and define the polynomial n-1

n-1

)

T h e n f is not zero. By L e m m a 3.4.10, there are integers Y n , . . . , Ym,,~-l, and integers k l l , . . . , k . . . . 1 with

[Yh~l <_na-1, I*(Y) :=

O <_khi
for h = l , . . . , m , i = l , . . . , n - 1 ,

\OYm,n-1]

~

f(Y11,-.-,Y . . . . 1) :# 0.

P u t xh : = E~-~ Yh~gh~ (h = 1 , . . . , m ) , x = ( x ~ , . . . , x m ) , AA := ( A A I , . . . , A A ~ ) for A = ( A 1 , . . . , A~). Note t h a t f * ( y ) is a linear combination of terms Pi+j(x), where J = (j11,. . 9,jm~) with El=x3hl ~-1 k h l for h = 1 , . . . , m. Hence n 9 = ~"~/=1 (3.4.17)

Xh C II(ncr-~Ah) N Z n for h -- 1 , . . . m, m k h l + ' " + k h ' n - 1 < 2ma. i(x, P ) < ma + Eh=l dh --

Hence there is a tuple i with ( i / d ) < 2ma, Pi(x) # 0. Note t h a t Pi(x) E Z. Below we shall show that [Pi(x)[ < 1. Thus, our initial assumption t h a t T h e o r e m 3.2.1 is false leads to a contradiction. P u t uhi = li(xh) for h = 1 , . . . , m , i = 1 , . . . ,n. By Proposition 3.4.3 we have (3.4.18)

~j~[ <__ t.J12 r,d~+...+d~ ~.o.A ~*~. it~11 IoA, . ..,~m, I [Pi(x)[ < - - ~ l d p , i(j)[[u~ ...urn ~ m n I, j J

where the sum, m a x i m u m , respectively, are taken over all tuples j = (j11,... ,jm~) n. Fix such a tuple j. By (3.4.17) with [~h~=l jh~/dh -- mini <_3nma for i = 1,. and (3.4.13) we have

[Uhi] _
for h = 1 , . . , m, i = 1,. . . , n.

Hence (3.4.19) log lu~l".umn I <

i=1 h=l

c,

\dh]

dh log Qh + (dl+"" +din)log

.

Further, by (3.4.16) and a < 1 we have m

~~.(jh,/dn)dhlogQt,

rn

<_ dllogQ1. (1 + a). Y'~.(jh~/dn)

h=l

h=l

_< dl log Q1- (1 + a)(rn/n + 3nma) < mn-ldl log Q1 " (1 + 7n2a), rn

E(jh,/ah)dh log Qh > 61 log Q1 h=l

h=l

_> dl log Ql(m/n - 3nine) >_mn-ldl log Q1. (1 - 7n2a).

50

J.H. EVERTSE

By inserting this into (3.4.19), and using (3.4.12), we get log 1u:1'1'...uJ,7,~,l< (mn-ld, log Q,). ~ + (d, + . . - + din)log(ha-'), with

r~

n

= ~'~c, + 7n:a~_, I~,1 -< -~/2 + 7n~o < -~/3, i=l

i:1

if we choose a < e/42n 3. Together with (3.4.18) this yields, for sufficiently large C10, IPi(x)l -< ~laPa~+'"+d"'~-md'r

_< (C~3Q-~r

md`< 1,

which is the contradiction we wanted. This completes the proof of Theorem 3.2.1.

[]

Chapter V Heights

on

Abelian

Varieties

by Johan Huisman

1

Height on Projective Space

We will follow to a great extent Chapter 6 of [14]. If K is a number field and v a finite place of K, that is, v corresponds to a prime ideal p of the ring of integers of If, then we define a norm II ' II. on K by

rlxll:

(1

~ o~dp(~)

,

where Np is the absolute norm of p. If v is an infinite place, that is, v corresponds to an embedding dr of K in ~ or v corresponds to a conjugate pair {a,~} of embeddings of K in C, then we define a norm II 9Ilv on K by

Ilxllv =

{ I~(x)[, i f v i s r e a l , I~(x)lL if v is complex.

Clearly, for any place v of K, the homothety y ~-~ xy transforms a Haar measure # on the completion Kv of K at v into [l~lIv .~. Let MK be the set of places of K, M ~ the set of infinite places and M~. the set of finite places. Then we have the product

formula

l-[ lixll,, ]-I Io-(x)l = y[ IIxIIv--1,

vEl~Ilh,

a:K~C

vEMK

for every x E K*. This can be easily seen as follows (cf. [87], Ch. IV, w Thin. 5). Let A be the ring of addles of K. The product formula will follow if we prove that any Haar measure on A is invariant under the homothety A,: y ~ xy of A, for any x E K*. Since A / K is a compact topological group and ),, induces an isomorphism A, of A / K , any Haar measure on A / K is invariant under ),,. Moreover, since K is discrete and the restriction ~ of A, to K is an isomorphism of K as a topological group, any Haar measure on K is invariant under A,. Therefore, any Haar measure on A is invariant under A,. If P = (x0:'-" :x~) is in P~(K) then we define the height of P relative to K by

HK(P) = 1-[ max{llx0II,,...,llx, ll~}. vEhCK

Observe that, by the product formula, this is well defined.

52

J. HUISMAN

1.1 E x a m p l e . If K = Q and P 9 F'~(Q) then we may assume P = ( x 0 : " ' : x~) with x~ C Z and g c d ( x o , . . . , x~) = 1. Then

HQ(P) = m a x { l x o l , . . . , {x~[}. [] It is clear that Hs,'(P) >>_1, for every P 9 IP~(K). If L is a finite extension of K and P 9 F'~(K) then

HL(P) = HK(P) [L:K]. Hence, we can define the absolute height H on P~(K) by 1

H ( P ) = HL(P) rc~, where L is some number field containing the coordinates of P. It will be convenient to define the (logarithmic) height h on P~(7-() by

h(P) = log H ( P ) .

(1.1.1)

1.2 T h e o r e m

Let n, d and C be integers. Then

(Northcott). {P 9

[H(P)
and

[K(P):K]


is a finite set. For a proof the reader is referred to [14], Chapter 6 or [70], w 2

Heights

on

Projective

Varieties

We will define height functions on a projective algebraic variety V over a number field K , using morphisms from V into projective space. Suppose

f: V ---~ ~ is a morphism of algebraic varieties over K. Then one defines the (logarithmic) height on V relative to f by

hi: V ( K ) P

~ ~

h(f(P)).

Let us call real-valued functions h and h' on the set V(I-() equivalent, denoted by h -,, h', if [h - h'[ is bounded on V ( K ) . It turns out that the height hi depends only, up to equivalence, on the invertible sheaf s = f*O•-(1). 2.1 T h e o r e m . Let V be a projective algebraic variety over K. g: V --~ ~'~ are morphisms over K such that f ' O p ~ ( 1 ) ~ g*O?m(1)

then h I and hg are equivalent.

If f: V --+ r

and

VL

HEIGHTS ON ABELIAN VARIETIES

53

P r o o f . Recall that a morphism r V --+ IPk is uniquely determined by (the isomorphism class of) the invertible sheaf g = r and the global sections s~ = r E F(V,s Therefore, it suffices to prove the theorem in the case that m > n and f = zr o g, where ;r is the rational map ~r:W~ . . . - - - ~

(x0:--.:xm)

~-~

y~

(x0:-..:x.).

Clearly, hg - h i > 0. To prove that hg - h I is bounded from above, observe that g(V) n ~ ( X 0 , . . . , X~) is empty. Since g(V) is closed, let I C K [ X o , . . . , X m ] be its defining homogeneous ideal. Then < I + ( X o , . . . , X , j = ( X o , . . . ,Xm) in K [ X o , . . . , Xm]. Therefore, there exist a positive integer q and F~j E K [ X 0 , . . . , Xm] such that r ~

Xn+ - E ]gijXj c l,

i = O, . . . , m - n.

j=0

We may assume Fij to be homogeneous of degree q - 1. Denote the coefficients of F# by aijk. If L C_ K is a finite extension of K and w is a place of L then we define 0 if w is finite, 1 ifwisreal, 2 if w is complex

e~=

and put c~ = ( n + l ) ~

(

q-l+m

.max

lla~j~H~.

m

Choose P E g(V)(L), say P = ( x o : ' " : xm) with xl E L. It is easy to see that ][z~+i[[E < e~ . m a x [Ixjll~-I. max [IxjI[~, 3~_m

- -

fori=0,...,m-n.

j<_n

Put

1

c~ = max { 1, c$ }, then

max IIx~ll~ < c~. max IIx/ll~. --

i<_m

j<_n

In particular,

H L ( X o : ' " : xm)

=

II ?<~x IIx, ll~ wEML

-

J. HUISMAN

54 where d = [L: K]. Therefore h ( x 0 : " ' : xm) < h ( x 0 : ' " : x,) + c,

where c = ~ ~-]veMKClv which neither depends on P nor on L. Hence hg - h f is bounded from above. [] As a consequence, we can define, up to equivalence, a height function hc for every invertible sheaf s on V which is basepoint-free (i.e., generated by global sections). For, choose a morphism f over K from V into pn such that (1)

/: "" f l o p

(Such a morphism exists since/: is basepoint-free.) Then, by Theorem 2.1, hc = hj

depends only, up to equivalence, o n / : . More precisely, one defines hz as the equivalence class of h 1. That is, if T I ( V ( K ) ) is the group of equivalence classes of real-valued functions on V(7(), we have hL ~ 7-/(V(I()). However, often we will treat hL as a real-valued function, keeping in mind that hL is only defined up to equivalence. It is easy to prove that, for any basepolnt-free invertible sheaves/: and .A4 on V, (2.2)

hc~

~

hc + h ~ .

As a consequence, for any invertible sheaf/: on V, we can define, up to equivalence, a height function h~ by hL = hLl - hL~, where s

and 1:2 are basepoint-free invertible sheaves on V such that : ~ 1. = /:1 ~ / : 2

(Such sheaves always'exist; see [27], p. 121.) By (2.2), this does not depend on s s Hence the following result due to A. Weil. 2.3 T h e o r e m . Let V be a projective algebraic variety over K . There exists a unique homomorphism h: PicV , ~(V(7()) such that (i) if V = Pn then hop.O) is the usual height h on projective space, (ii) if W is a projective algebraic variety over K and f: V --* W is a morphism over K then hi.c = hco f , for a n y / : E PicW.

It is then easy to prove, using Theorem 1.2, the following finiteness theorem. 2.4 T h e o r e m . Let V be a projective algebraic variety over K . invertible sheaf on V then, for all reM numbers C and d, the set {Pc is a ~nite set.

V ( K ) I hL(P) < C

and

[K(P):K]
I f / : is an ample

VI. HEIGHTS ON ABELIAN VARIETIES

55

Observe that it makes sense to call an element of 7-/(V) bounded from below (or above). 2.5 T h e o r e m . Let V be a projective algebraic variety over K. If s is an invertible sheaf on ,V and s is a global section then hL is bounded from below on the set

{P e v(TO l s(P ) ~ 0}. Proof.

Choose basepoint-free invertible sheaves s

and s

on V such that

s ~ s | s Let So,...,s~ be global sections of s s n + b . . . ,sin of s such that

that generate s

Choose global sections

3 @80,...~S @Sn,$n+l,...,8 m

generate s

Then, whenever P e V()-() with s(P) 7~ O,

hz,(P)

= >_ = =

h(s@so(P),...,s| h(s @ s o ( P ) , . . . , s | h(so(P),...,s,(P)) hz2(P ).

Therefore, h~ is bounded from below on the set of P E V(I-0 such that s(P) -fi 0.

3

[]

Heights on Abelian Varieties

We will need the Theorem of the Cube. 3.1 T h e o r e m ( T h e o r e m of t h e C u b e ) . Let X1, X2, X3 be complete algebraic varieties over the field K and let P~ ~ Xi( K). Then, an invertible sheaf s on X1 xX2 xXa is trivial whenever its restrictions to {P1 } • X2 • X1 x {P2 } x X3 and X1 x X2 x {P3} are trivial. Proof. (After [52], Chapter II, w Let us give a proof when char(K) = 0, since this is the case we are interested in. Then it suffices to prove the theorem for K -- C. (Here one uses that the Xi, Pi and /2 are defined over a subfield Ko of L which is of finite type over Q and that for K ---* L any field extension, X any quasi-compact separated K-scheme and s any invertible sheaf on X, s is trivial if and only if its pullback to XL is trivial.) Before we continue the proof let us recall the following definition. A contravariant functor F from the category of complete complex algebraic varieties with basepoints into the category of abelian groups is called of order < n if for all complete complex algebraic varieties X o , . . . , X~ with basepoints, the natural mapping

F(II x,) i=o

II F(II x,)

j=o

i#j

56

J. HUISMAN

is an isomorphism. As an example, the Theorem of the Cube states that the functor Pic is of order _< 2. To finish the proof we switch to an analytic point of view. Let OX,h denote the sheaf of analytic functions on X. According to the GAGA-principle, PicX ~ H~(X, O*x,h), for any complete complex algebraic variety X. The long exact sequence associated to

0 ~

g

' Ox,h 2!50*x.h ~

0

implies the existence of an exact sequence

H i ( X , OX,h) ~

H ' ( X , Ok,h) ~

H2(X, Z).

Since both Hi( ., OX,h) and H2( ., Z) are functors of order < 2, the functor H i ( ., O];,h ) is of order < 2. This proves the theorem. [] 3.2 C o r o l l a r y . Let X be an abelian variety over the fidd K and let Pi: X 3 ~ X be the projection on the ith factor. Let pq = Pi + Pj and Pijk = Pi -t- pj + Pk. Then, for any invertible sheaf s on X , the invertible sheaf

on X a is trivial. P r o o f . Taking restrictions of this sheaf t o O x X x X , X x O x X a n d X x X x O yields a trivial sheaf. The conclusion follows from the Theorem of the Cube.

[]

Since this corollary expresses a relation between sheaves on X 3, we have immediately, by Theorem 2.3, the following fact about heights on abelian varieties. 3.3 T h e o r e m . If X is an abelian variety over a number field K then, for any invertible sheaf s on X ,

hc( P + Q + R) - hL( P + Q) - hL( P + R) - hL( Q + R) + hz( P) + hL( Q ) + hL( R) ,~ O, as functions on X ( K ) 3. Let us denote for an abelian variety X over K the multiplication-by-n mapping from X into itself by [n], for any integer n. Recall that an invertible sheaf s is called symmetric (resp. antisymmetric) whenever [-1]*/2 ~ s (resp. [-11"s ~ s As a consequence of Corollary 3.2, one can prove the following. 8.4 C o r o l l a r y . If X is an abelian variety over the field K and s is an invertible sheaf on X then

| for any integer n. In particular,

~ s

t s

if s is symmetric, if s is antisymmetric.

VI. HEIGItTS ON ABELIAN VARIETIES

57"

Again, this translates into properties of heights on abelian varieties. 3.5 T h e o r e m . /t' X is an abelian variety over the number field K and s is an invertible sheaf on X then

for any integer n. In particular, hL o [n] ,.,

n2hc, nhz,

if s is symmetric, if s is antisymmetric.

The property of hc in Theorem 3.3 will imply the existence of a canonical height function, the Ndron-Tate height relative to s /~c: X (2-()

, R,

as stated in the following theorem. 3.6 T h e o r e m . If X is an abelian variety over the number field K and s is an invertible sheaf on X then there exist a unique symmetric bilinear mapping bc: x(-i-0 x X(-[() --* R and a unique linear mapping IL: X(-[0 --~ R, such that

~:x(~)

, ~t,

defined by ]zc(P) = 89

P) + IL(P),

is equivalent to hL. Moreover, if s is symmetric then Iz = 0 and if s is ample then bc is positive definite on X(-[() @ R. P r o o f . The existence and uniqueness of bc and IL follow from the lemma below, whose proof can be found in [52], Appendix II, w If L: is symmetric then in virtue of Theorem 3.5

89

P) + nl~(P) = hL(nP) = n2]zz:(P) = 89

P) + n2ls

for any integer n. Hence lc = 0. If s is ample then [-1]*/: is ample too. Hence, M = / : @ [ - 1 ] * s is ample. Moreover by uniqueness b~ = ~bL 1 + ~b[-l].~ 1 1 = ~b~a. Therefore it suffices to prove that bc is positive definite on X ( K ) @ R for any symmetric ample invertible sheaf s Since then ]~c = ~bc, 1 it follows from Theorem 2.4 that for any finitely generated subgroup A of X(I-() and for any C E R the cardinality of the set

{P e A l b c ( P , P ) < C} is finite. It is not difficult to prove that this implies that bc is positive definite on

x ( K ) | R.

[]

3.7 L e m m a . Let G be an abelian group and h: G ---* R a function such that

h(P + Q + R) - h(P + Q) - h(P + R) - h(Q + R) + h(P) + h(Q) + h(R) ~ O, as functions on G 3. Then there exists a unique symmetric bilinear mapping b: G x G R and a unique homomorphism l: G ---* R such that h .~ h, where

h(P) = lb(p, P) + l(P).

58 4

J. HUISMAN Metrized

Line

Bundles

The main interest of this section is to show that metrized line bundles on projective R-schemes give, in a natural way, a height function (i.e., not just a class of functions modulo bounded functions, but a specific element of such a class). For example, the N6ron-Tate height can be obtained really nicely in this way, see for example [50], Chapter III. For the moment let X be a complex analytic variety (i.e., a complex analytic space as in [27], App. B, which is reduced and Hausdorff), g a coherent O x - m o d u l e (e.g., a locally free Ox-module of finite rank). We denote by COX the sheaf of continuous complex valued functions on X. If U C X is an open subset and f E C}(U) then f denotes the complex conjugate of f (i.e., tof , where t : C ~ C is the complex conjugation). We define a (continuous) hermitean form on ~" to be a morphism of sheaves (., .): ~" @ ~' --. C~ such that: (i) (.,.) is bi-additive, (ii) (fsl,S2) = f(sx,s2) for all U C X, f E Ox(U), sl,s2 E g(U), (iii) (s2,sl) = (Sl,S2) for all U C X , si,s2 E E(U). A hermitean form is called a hermitean metric if moreover it is positive definite: (s,s}(z) > 0 for all U C X, s E s and z E U such that s(x) # O. The norm associated to a hermitean metric on s is the morphism of sheaves I1 II: E --, cox, s H (S,S) 1/2. Of course a hermitean metric is determined by its norm (use some polarization identity). If s is an invertible Ox-module then the norms of hermitean metrics on it can be easily characterized: they are the maps of sheaves II" I1:Z; ---* COX such that Ilfs[I = Ifl II~ll and Ilsll(x) > 0 if s(x) r 0. In this case we will also call the norm associated to a hermitean metric a hermitean metric. One can define differentiable, C ~ or real analytic hermitean forms on E just by replacing the sheaf C~ by the sheaf of that kind of functions. Let s := COx | g; then g' is a coherent COx-module. For any C~c-module s we define c--7:= COx| L" and for any open U C X, s E E'(U) we let ~ := 1 | E C--7(U). The maps s ~ ~ give an anti-linear isomorphism E' ~ ~ (i.e., one has ~ = 1 | f s = f| = f.~). With these definitions, giving a hermitean form (., .) on a coherent Oxmodule s is the same as giving a morphism of C~-modules r s | g--7__. COx, such that r N~) = r | Using this description it is immediate that for Y ~ X a morphism of complex analytic varieties one has the notion of pullback for hermitean forms. In particular, for all z E X we get a hermitean form (., .)(x) on the C-vector space g(z) (the fibre of L" at z, not the stalk). These (-,.)(z) vary continuously in the sense that if one expresses everything in coordinates then the coexTacients of the matrix obtained that way are continuous. Of course, to give a continuous (-, .) is the same as to give a continuous family of (., .)(x)'s. 4.1 E x a m p l e . The formula ( f , g ) = f(-j defines a hermitean metric on Ox.

[]

Suppose now that K is a number field and that f : X --* Spec(K) is a variety over K. Then for every a: K --~ C we have the variety X~ over C obtained from f : X ~ Spec(K) by pullback via Spec(a): Spec(C) ~ Spec(I(). Taking C-valued points then gives for all ~r a complex analytic variety X~(C) = {P: Spec(C) --* X I foP = Spec(a)}. If we denote g = toe then we have maps P ~ T := PoSpec(t):

59

VI. HEIGHTS ON ABELIAN VARIETIES

X~(C) ~ X~-(C); these maps are anti-holomorphic isomorphisms. For C a coherent O x - m o d u l e let s denote the coherent Ox~(c)-module induced on X~(C); for U C X and s E $(U) let s~ E $~(U~(C)) denote the section induced by s. 4.2 D e f i n i t i o n . A hermitean metric on $ is a set of hermitean metrics (-, .), on the s a: K ---* C, such that for all open U C X, s, t E $(U), a: K ---+C and P E U~(C) one has (s~-, t~-)~-(T) = {s,, G)~(P) [3 Another way to phrase the last condition is as follows: if we denote by E' the C~(c)-module on X(C) := {P:Spec(C) --+ X } = LI~ X~(C) induced by s and by

Foo:X(C) ~ X(C) the map P ~ P, then one has F~o{S,t) = (s,t) for all U C X , s, t E $(U). For our purposes we do not need this condition, but we impose it anyway in order to be compatible with the existing literature (e.g., [76]). W i t h our conventions concerning hermitean metrics on invertible sheaves we get the following definition. 4.3 D e f i n i t i o n . A hermitean metric on an invertible O x - m o d u l e / ~ is a set of hermitean metrics 11" H~: s ~ C~c~(c), a: g ~ C, such that Ils~-ll~-(T) = IIs~ll~(P) for all U C X , s E / : ( U ) , a and P E Uo(C). [] 4.4 E x a m p l e . Let X := P ~ , Z: := Ox(m) and denote by x 0 , x l , . . . ,x~ the homogeneous coordinates on X. By definition, a local section s of s can be written as F/G, where F and G in K[xo, xl,..., x,~] are homogeneous and d e g ( F ) - d e g ( G ) = m. The following formula defines a hermitean metric on s

o ( F ) ~ (ao: a ~ : - - - : a=) = ](aF)(ao,a,,...,a~)l a(G)

I(o-G')(ao,a,,...

1

~--~" (1~ol ~ + ta, I~ +--. + la,,l~)", '~

where a: K ---* C and (a0, a l , . . . , a , ) E C n+l. This metric will be our standard metric on Ol,,,(m). It is characterized, up to scalar multiple, by the property that at every a it is invariant under the natural action of the unitary group U ( n + l ) . [] As before, let R be the ring of integers in K . Let X be an integral projective R-scheme and let Z: be an invertible Ox-module. By a hermitean metric o n / : we will mean a hermitean metric on the pullback s of L: via XK := X • ---+X . By a metrized line bundle on X we will mean an invertible COx-module L: with a given hermitean metric II" II" In particular, these definitions apply to Spec(R) itself. Let (s I]" ]l) be a metrized line bundle on Spec(R), let M := P(Spec(R), s and let s E M be non-zero. Then the degree of (L:, I]" II) is the real number: (4.4.1)

deg(E) = log#(M/Rs)-

~

]ls~{I,~

a:K~C

(one easily checks that the right hand side is independent of s). Let us now explain how a metrized line bundle (s ]I" ]I) on an integral projective R-scheme X gives a height function on the set X ( 7 ( ) of 7g-rational points of the

60

J. HUISMAN

variety X K over K. Let P 6 X()-(). Then there is a finite extension K ' of K such that P is defined over K', i.e., P comes from a unique point P 6 X ( K ' ) . Let R' be the ring of integers of K t Then P can be uniquely extended to an R'-valued point (still denoted P) P: Spec(R') ~ X by the valuative criterion of properness (see [27], Thm. 4.7). By pullback via P we get a hermitean line bundle P*/~ on Spec(R'). The absolute height of P with respect to (s I1" II) is then: (4.4.2)

h(c,ll.ll)(P) := [ K ' : Q]-a deg(P*s

(one easily verifies that this is independent of K'). The following theorem states that h(c,lt-II) is in the class of height functions associated to s by Thm. 2.3. 4.5 T h e o r e m . ff (Z;, [I 9 [I) is a metrized line bundle on X then h(c,H) and hc are equivalent as functions on X ( K). P r o o f . It suffices to prove the theorem for s very ample, so we assume that X is a closed subscheme of ~'~ and that 1; is the restriction of O(1) to X. In virtue of Lemma 4.6 below we may choose a convenient hermitean metric on s Let us then choose the metric given by the formula: I(c~F)(ao, a , , . . . ,a,)l 1 an) = I(aG)(a~ 9'a=)l'maxi<- n lail

a(F) l or(G) a (a~ al : " ' :

(compare with Example 4.4). We claim that for this metric h(c,ll.ll) is just the standard absolute height on P=(7-() of 1.1.1. In order to verify this for P 6 X ( K ' ) , where K ' is a finite extension of K, we may base change to the ring of integers R' of K'. Hence it is sufficient to check our claim for all K and all P 6 X ( K ) . We will compute deg P*(.9(1), where P is an R-rational point of X C_ F~. We may assume that P*xo is a nonzero section of F(Spec R, P*O(1)). Then, P* o( 1 )IRP * x o =~ ( ~ Rxi (P ))IRXo(P) "~ (~R~o (P ))/R Hence, Xi

# P * O ( 1 ) / R P * xo =

I~ m~x I[--(P)IIo

vC.M~ t_n

=

XO

a

v~M~ i
~ IIz~176

Therefore,

degP*O(1)

=

logm
~ v~M~

-

~

-

log

vEM~.,

=

vEM~

Ilxo(P)ll~ m a x , < ~ IIx,(P)II~

[K: Q]hc(Pg) []

VI. HEIGIITS ON ABELIAN VARIETIES

61

4.6 L e m m a . Let s be a line bundle on X . If [[. I] and [[-[[~ are hermitean metrics on s then there exist real numbers q,c2 > 0 such that clllsll(P) _< Ilslr(P) _< c2llsll(P ) for any local section s of s and any P E X(C) where s is defined.

P r o o f . Since X is a projective R-scheme, X(C) is a compact analytic variety. Therefore the continuous functions I[. [['/[[. 66and [6. [[/[[- [[' on it are bounded. []

Chapter VI

D . M u m f o r d ' s "A R e m a r k o n Mordell's Conjecture" by Jaap Top

Throughout this exposition K will denote a number field and C / K will be an absolutely irreducible (smooth and complete) curve of genus g _> 1. In fact more generally o n e can take for K any t~eld equipped with a product formula as defined in [70, p. 7]. We make the standing assumption that over K, a divisor class of degree 1 on C exists (this can always be achieved after replacing K by a finite extension). Our main reference is the paper [53] referred to in the t i t l e above plus the descriptions given in [70], [9] of Mumford's result.

1

Definitions

Fix once and for all a divisor class a on C of degree 1. The jacobian J = J(C) of C is an abelian variety of dimension g defined over K which represents divisor classes of degree 0 on C (see [14, p. 168] for a formal definition). In particular, the class a gives rise to an injective morphism

j=j~

: C'--*J : P ~ ( P ) - a

defined over K. A. Weil proved in 1928 (case K a number field) that J(K) is a finitely generated abelian group. This raises the possibility to study the set C(K) of K-rational points on C as a subset of J(K). The first successful attempt along this line seems to be due to Chabauty (1941). He combined the idea of considering C(K) as a subset of a finitely generated abelian group with Skolem-type p-adic analysis and proved:

If 9 = genus(C) > rank(J(K)) then C(K) is finite. (cf. [70, w for a sketch of the proof; Coleman's paper (1985) [13] for effective bounds on the number of solutions in some cases.) All other attempts starting from this basic setup seem to involve heights. Depending on a one defines a divisor

0 = j(C) + . . . + j ( C )

g~-I

CJ

64

J. T o P

with divisor class 0 9 Pie(J). To it one associates a canonical height ho on J(K-) and a bilinear form (x,y) = ho(x + y) - ho(x) - ho(y). Since 0 is ample (this follows from the fact that 0 defines a principal polarization on J; cf. [14, p. 186, Thm. 6.6 and p. 117, Prop. 9.1]), (.,.) is positive definite on J(K)/torsion. To verify this, note that (x, x) = -~ho+[-1]*o(x), 1 now 0 is ample, hence 0 + [-1]*0 is ample and symmetric. The rest of the argument is a trivial exercise (see [70, w The existence of this non-degenerate bilinear form was discovered by N6ron and by Tate (mid-60's). Combined with Well's result it gives J ( K ) | R the structure of a euclidean space E. An easy consequence is

# {x 9 J(K); Ilxll <_ T} = (pos. const.).Trk-l-o(Trk). 2

Mumford's

Inequality

The previous section raises the possibility to study rational points on C by means of the sequence C(K) ----* J(K)/torsion C E in which the first map is finite to one (in fact, by Raynaud's proof of the ManinMumford conjecture [60] there is a bound on the number of points in the fibres independent of the field K); and the second map gives J(K)/torsion as a lattice in euclidean space E. Mumford's result on this which we want to discuss here can be seen as a first step towards Vojta's proof (simplified by Bombieri) of Mordell's conjecture (1989/90) and even to Faltings's theorem which is the topic of this book. The result is 2.1 T h e o r e m ( M u m f o r d , 1965). With the notations as above, there is a real number c > 0 such that t'or all x # y E C ( K ) the inequality Ilxll 2 + IlYll2 - 2g(x,y) > -c(1 + Ilxll + Ilyll)

holds. 3

Interpretation,

Consequences

and

an Example

Since for g < 2 the left hand side is non-negative, Mumford's inequality is at most interesting for g _> 2. In that case, for K a number field, if one considers points in C ( K ) only then the result is trivially implied by the truth of Mordell's conjecture. Needless to say, this will not be used here. One way to interpret the inequality is as follows. Suppose 0 < r < v / ~ - 1. Then for x 9 C(7-() with Ilxll >> 0, no y 9 C ( K ) is in B(gx, rll*ll) := the ball with center gx and radius rllxll. Indeed, were y 9 C ( K ) inside this ball, then of course Ilyll-< (r + g)ll~ll and

>

-c

- c(1 + ~ + 9)llxl[

+ (g= -

1 - ~=)llxll =

VII. D. MUMFORD'S " A REMARK ON MORDELL'S CONJECTURE"

65

which implies that Ilxl[ is small. Mumford's inequality implies 3.1 C o r o l l a r y . There is a real number c2 such that # { x E C(K);

Ilxll

<

T) <

c21ogT.

The same result (with of course a different c2) is true with I1 II replaced by any h = h~ on C; with 5 a divisor of positive degree. Comparing the above rcsult with the "density" of rational points in J(K) one says that the set C(K) C J ( K ) is "widely spaced" (this notion is formally introduced by Silverman [75]; also Serre uses it [70, p. 105], in fact already in the French version from 1980 (p. 2.7: "trhs espacSs")). (of Cor 3.1). Take c3 = 5 c + 1/2, then for x,y E C(K) with 2[Ix]l >

Proof.

IlYI[

>

IIxll > cz one has Ilxll 2 + Ily!l 2 _> c~llxll + c~llyll _> 541 + Ilxll + Ilyl[), hence

(~,y)

Ilxtl~+l[yll=+c(a+lt~ll+llyll)

3 (llx[l+

Ilyll~ < 3

Now take T > > 0. The x E C(K) with I]xl] < T can be subdivided in the x's with I[xH _< c3 and the ones with c32j < [[xll _< c32j+l (for 0 _< j < c41ogT). In one such "interval" different points satisfy (][x[[-~x, []y[I-ly) < 3/(2g). The maximal number of unit vectors in E satisfying this angle constraint is easily seen to be bounded in terms of dim E = rank J ( K ) , say bounded by c5. Then # {x G C(K); ][x[[ _< T} _< #torsion in J ( K ) . (#{][x][ _< cz} + cbc41ogT) <_c2 logT. [] 3.2 E x a m p l e . Take C, D curves defined over Fq such that a non-constant morphism D ~ C over Fq exists. Put K = Fq(D). a point x E C(K) corresponds to a morphism ~2, : D ~ C defined over Fq and for h(x) we can take h(x) = deg(T,). Let ~r denote the Frobenius homomorphism on K. Clearly with x also r"x C C(K), and h(r"x) = q"h(x). Hence in this example

{x E C(K); h(x) < T} >_ (pos. const.), log T. [] 4

The

Proof

assuming

some

Properties

of Divisor

Classes

Let D be a (smooth, complete, irreducible) curve of genus g over an algebraically closed field k. Write JD for its jacobian and use as above a divisor class a of degree 1 to define j and 0. Denote by p~ the projection JD • JD ~ JD onto the ith factor and let s : JD • JD ~ JD be summation (s(x, y) = x + y). Furthermore define a' E JD by a' = (29 - 2)a - KD in which KD is the canonical class on D. Lastly, 7b : JD ~ JD will denote translation over b E JD. The following properties of divisor classes are used in the proof of Mumford's result:

66

J. T o P

[-ll*0 =

1. 0' := 2.

j*O' =

r:~,O.

ga.

3. (j x j)*(s*O - p*lO - p~O) = a x D + D x a - A .

Theorem 2.1 is obtained from this as follows. We return to the situation in w167 and we will use the statements above in this context. Write ho(x) = l ( x , x) + g(x); here g is linear. Then 89 x) - g(x) = h o ( - x ) = ho,(X) = h,*o,o(x)_ = ho(x - a') = 71 (x, x) - (x, a') + 89(a', a') + g ( x - a') = l ( x , x) + g ( x ) - (x, a') (here use that ho(O) = 0). The conclusion is that .i

e(x) = 89 a').

Next, using heights on C x C, identifying x with j ( x ) (as we already did a couple of times) and ignoring the (bounded) functions arising from the fact that we choose actual heights depending on a divisor class:

h~(x, y) = ho•

~) + hc•

y) - h(j•

y) + h(j•

y)

"+h(jxj)*p;o(x, Y)

=

ho(x) + ho(y) - ho(x + :j) + ho(z) + ho(y) = lho,(z) + ~ho,(U)

_

1 {llxll ~ + iiyll = _ 2 g ( x , y ) - ( x , a ' ) -- (y,a'>}. 2g

g

g

(~,u)

Since A is effective, ha is bounded from below on {x r y}, hence the result follows.

5

Proof of the Divisorial

Proof.

Properties

(of (1)) The class 0 is by definition represented by the divisor |

[

Pi - ( g - 1 ) a ] ;

PieD

hence 0' by O' = {[(g - 1)a - EPi]}. Therefore it suffices to show that for general g-1 g-1 ~ i = 1 Pi a unique ~~i=1 Qi exists such that ~ Pi + ~ Qi is canonical. This follows from Riemann-Roch: one has g ( ~ Pi) - g(KD -- ~ Pi) = 0. Since g ( ~ Pi) r 0 (constant morphisms are in L(any effective divisor)), it follows that also the class of KD -- E Pi is represented by an effective divisor. A slightly more geometric argument runs as follows. In case D is not hyperelliptic one may assume that D is canonically embedded; then the canonical divisors are precisely the hyperplane sections, and g - 1 points determine a hyperplane and therefore a Canonical divisor. For hyperelliptie D, write [] for the hyperelliptic involution. Then ~2 Qi = ~ tPi is canonical.

Proof.

(of (2)) Take e

6

Pic~

generic and write b =

a -

c.

The embedding of

D into JD using b instead of a will be denoted jb. We will show ff, O' = ga - c which by specialization implies the desired formula. For x E D to satisfy jb(x) E O' is equivalent to x + E i g-1 = l Pi "~ ga - c (with ,',., denoting linear equivalence). Now 9a - c is a generic divisor of degree g, hence by

VII. D. MUMFORD'S "A REMARK ON MORDELL'S CONJECTURE"

67

Riemann-Roch its divisor class can be written as ~ = 1 Qi in a unique way (in fact, the 9th symmetric power of D is birational to JD). In other words, the possible choices for {x, { ~ P / } } are just the partitions of { Q i } in two sets with 1 and g - 1 elements, respectively. Hence the divisor of these x's is ~ Qi "~ g a -- c. E] P r o o f . ( o f (3)) Recall the seesaw principle [14, w p. 109-110] (which we only state in the special case D x D): I f ~ E P i c ( D x D ) is t r i v i a l r e s t r i c t e d t o all v e r t i c a l and o n e h o r i z o n t a l fibre, t h e n 6 = O.

Apply this to ~ = (j • j )*(s*O - p~O - p~O) + A - a • D - D x a. This is a symmetric divisor class, hence it suffices to show 61,• is trivial. Consider qo = T , : D -+ D x D : y ~ ( x , y ) . Clearly ~ 0 * ( A - a x D - D x a ) ,-, x - O - a = x-a. Furthermore, pxo(j x j)o~o = j , p2o(j • j)oh0 = constant and so(j • j)%o = r , _ ~ o j . Hence in Pic(D): r

Now for any b E Pic~ j*r;O

= j*T;_oO -- j*O + x -- a.

one has

=

j * r : r : , O ' (use (1)) = j*__b+i,. _(2,_a)=0' =

=

ga - (29 - 2)a - b + I(D (from the proof of (2)) = K D -- b -

(9 - 2)a.

It follows that ~7*a = ( K D -- z + a -- (g -- 2)a) -- (KD - (g - 2)a) + x -- a = 0 as required. [] 6

Effectiveness

and

Generalizations

In what follows, K is a number field. One of Paul Vojta's ideas was to replace the quadratic part [tx[[2 + [ly[[2 - 2 g ( x , y ) in Mumford's result by more generally A[[x [[2 + #[[y[[2 _ 2gu (z, y), with A, ~, u suitably chosen, depending on x, y. This leads to considering an other divisor in C x C, namely A(a • C ) + # ( C • a) - v ( j • j ) * ( s ' 0 - p'~O - p~O) ~ ( )~ - v ) a x C + (# - v ) C x a + v A .

To obtain a lower bound for this, we need firstly that this divisor class is effective. Secondly we want that a positive representative of this class does not contain (x, y). In fact, this second condition can be weakened to "does not contain (x, y) with very high multiplicity". Since one wants to choose ~X,#, v and hence the divisor class depending on (x, y), this is a problem. Vojta, Faltings and Bombieri each showed us a different way to cope with this difficulty, and the final result is 6.1 F a c t . T h e r e e •

> 1 such that forx,y

co one has

(x,y) <

E C(T 0 with

Ilxll > c' a n d IlYll/llxll >

~llxll Ilyll

Note that this easily implies Mordell's conjecture. It should be remarked that both c and c' can be chosen independent of the field K . They only depend on the curve C, on the divisor a and on chosen embeddings of C and of C • C into projective space, using fixed very ample divisors N a and M a ( a x C ) + M ~ ( C x a), respectively. This information yields a radius p, an angle a > 0 and a number n, such that for any number field L _~ K the points in C ( L ) can be described as follows: there are the ones within the ball B(0, p), plus each cone with angle a contains outside this ball at most n other points.

Chapter VII

A m p l e Line Bundles and Intersection Theory by Johan de Jong

1

Introduction

This chapter gives an overview of the results from intersection theory we need for the proof of Faltings's theorem. Meanwhile we will try to give a coherent account of (this part of) intersection theory and we will try to show what a beautiful theory it is. We would like to stress here, that it should be possible for anyone with some basic knowledge of (algebraic) geometry to prove all the results mentioned in this chapter after reading the first 40 pages of Hartshorne's Lecture Notes [28]. 2

Coherent

Sheaves,

etc.

X will always denote a projective variety over the algebraically closed field k. An affine open subvariety of X is an open U C X such that U is isomorphic as a variety to a closed subvariety Z C A N for some N. 2.1 E x a m p l e . If X is a closed subvariety of some P~ with homogeneous coordinates T o , . . . , T~ then for any non-zero homogeneous polynomial P(To,..., Tn) the set

{(to:...: t~) e x(k)I P(t0,... ,tn) # o} is an affine open subvariety of X.

[]

The affine open subsets form a basis for the topology on X and hence a sheaf on X is determined by the sets of sections over the afline open U C X. The structure sheaf Ox of X is determined by the rule

Ox(U) = I'(U, Ox) = F(U, Ou) = k[T1,..., TN]/I if U C X is isomorphic to the affine variety determined by the (prime) ideal I C

k[T1 . . . . , TN]. The other sheaves occurring in this chapter will always be sheaves of Ox-modules, that is, they will be sheaves of abelian groups ~" on X endowed with a multiplication

70

A.J. DE JONG

Ox x U ~ ~" such that for each open U C X, 3r(U) becomes a module over Ox(U). A sheaf of Ox-modules ~- is said to be generated by global sections if there exists a family of sections (si)i~z, si E F(X, 3r ) such that the map of sheaves (~iclOX

' ~,

Oi~tfl ~ Y~ fisi i

is surjective (i.e., any local section of ~- should be locally a finite linear combination of the sl). A coherent sheaf of Ox-modules is a sheaf U such that - for each open affine U G X , ~'(U) is a finitely generated Ox(U)-module, - if U C V C X are open affine then U(U) = U(V) |

Ox(U)

The fundamental theorem on sheaves/coherent sheaves is 2.2 T h e o r e m . For any sheaf .T on X, Hi(X,.T) = 0 for all i > d i m X . If Y: is coherent, then HI(X, .T) is a t~nite dimensionM k-vector space for a11 i. This theorem allows us to define the Euler-Poincard characteristic of f ' : oo

X(5r) = y ~ ( - X ) i dimk Hi(X, .T). i=0

Using the long exact cohomology sequence, we see that for an exact sequence of coherent sheaves 0 --~ .T1 ~ 5r2 ~ .T'3 ---*0 we have X(Ot-2) = X(.T1) + X(.T3). 2.3 E x a m p l e s . (a) I d e a l s h e a v e s . An ideal sheaf is a subsheM 2" C O such that for each U C X open, Z(U) is an ideal of Ox(U). It is coherent, since for each open affine U C X the ring Ox(U) is a Noetherian ring. An ideal sheaf 2" determines a closed subset Z of X Z={xEXlforallxEUCXandfE2"(V):

f(x)=0}

and it determines a sheaf of rings Oz = Ox/2". The pair (Z, Oz) is (what we will call) a closed subscheme of X. It is usually denoted by Z. A closed subvaviet v is a closed subscheme (Z, Oz) such that the sheaf of algebras Oz is without zero divisors. E x a m p l e : X = A~, F(X, Ox) = k[t] D I = (t"). In this case we get (Zn, Oz.) with Zn = {0} and F(X, Oz.) = k[t]/(t~). Hence Zn # Z~, unless n = n', and only Z1 is a closed subvariety. If 5r is a coherent sheaf on X then we can construct an ideal sheaf 2" as follows:

Z(U) = {f E Ox(U) [ multiplication by f : 5r[u ~ .T[u is the zero map} The closed subscheme Z associated to this is called the support of 5r: Z := supp(~'). E x a m p l e : The support of Oz is (Z, Oz). (b) L i n e b u n d l e s . A line bundle or invertible sheaf is a coherent sheaf • on X such that each point x E X has an affine neighborhood U C X such that s ~ Ox(U) as Ox(U)-modules, i.e., s ~ Oxlu.

VIII. AMPLE LINE BUNDLES AND INTERSECTION THEORY

71

If we have two line bundles s and s then we can form the tensor product s | s It is again a line bundle. Using the tensor product for multiplication, the set of line bundles up to isomorphism becomes an abelian group with identity Ox and inverse s = H o m o x ( s Ox). This group is denoted by Pic(X), it is the Picard group of X. For morphisms f : X ~ Y there is a pullback f*: Pic(Y) -~ Pic(X) defined as follows: if the line bundle /3 on Y is trivial on the open sets Ui which cover Y and if its transition functions are f~d E P(Ui (3 Uj, O~) then f * s is the line bundle on X which is trivial on the open sets f - l U i and has transition functions fl,jof E F(f-a(Ui NUd), Ojc ). The pullback is a homomorphism of abehan groups. [] 3

Ample

and

Very

Ample

Line

Bundles

Recall that a line bundle s on X is said to be very ample if it is generated by global sections and for a basis s 0 , . . . , s n of F ( X , / : ) the map r ....... : X ---* F'~ is a closed immersion. The line bundle s is said to be ample if there exist N E l~l such that s174 := s | | s ( N factors s is very ample. A cohomological criterion for being ample is the following. 3.1 T h e o r e m ( S e r r e - G r o t h e n d i e c k ) . equivalent:

For a line bundle s on X the following are

1. f-. is ample, 2. for every coherent sheaf ~ on X we have Hi(X, Y | s174 = 0 for all i > O, n>>O, 3. for every coherent sheaf ~ on X, the sheaf jr | s174 is generated by globM sections for n >> 0. 3.2 C o r o l l a r y . Let f: X --o y be a finite morphism of projective varieties and s a line bundle on Y. If s is ample then f * s is ample on X . If f is surjective and f * s is ample then s is ample. The proof uses the theorem above and comparison of cohomology of sheaves on X and on Y. 3.3 C o r o l l a r y . For any line bundle s on X there exist very ample line bundles 1~1 and s such that s ~- s | s P r o o f . Suppose M is very ample on X. For some n E N the line bundle/3 | ,M | is generated by global sections. It is easy to see that /:1 = /3 | M | | f14 and /:2 = M | are very ample (and/~ ~ s | s [] 4

Intersection

Numbers

Let ~" be a coherent sheaf on X and let s

s be line bundles on X.

4.1 P r o p o s i t i o n . The function ( n l , . . . , nt) ~-* X(s polynomial of total degree at most dim(supp(Jr)).

TM

|174163

|

is a numerical

72

A . J . DE JONG

P r o o f . We prove the proposition by induction on dim(suppgV). If dim(supp5v) = 0 then we have s ~ a | 1 7 4 1 6 3| | -~ 5v since each s is trivial in a neighborhood of the support of ~-. Further, Hi(X, ~-) = 0 unless i = 0 since ~- is supported in dimension zero. Hence our function is constant and equal to X(~-) = dimk H ~ For the induction step, by Cor. a.3 above, we may assume that each s is very ample. Hence we can choose a section s E F ( X , s whose zero set is "transversal" to supp(~'), i.e., such that dim((s = 0) A supp(Z)) < dim(suppgV). Thus we see that the kernel and cokernel of multiplication by s: 0 --* K: --* Y-- -:h Z | s

--+ Q --* 0

are sheaves with lower dimensional support. Hence for all i we have that x(z.F ~ |

| z.? u'+l~ |

| r-.?TM o .r) - x(~c~ ~ |

| z.?", | .r)

is a polynomial in n l , . . . ,nt of total degree at most d i m ( s u p p Z ) - 1. From this the result follows. [] 4.2 D e f i n i t i o n . If (Z, Oz) is a closed subscheme of dimension d a n d / : a , . . . ,s are line bundles then we define (s . . . . . s 9Z), the intersection number of Z with s 9 9 s to be the coefficient of nl ...ha in the polynomial X(s | "@s174 [] 4.3 R e m a r k s . (a) The intersection number is an integer (any numerical polynomial f of degree _< d in n l , . . . ,na can be written uniquely as a Z-linear combination of the functions ( ~ ) . . . . . ( ~ ) with k~ > 0 for all i and ~ i ki <_ d). (b) It is additive: (z:, | z:'l . . . . .

,c,~. z ) = (z:, . . . . .

c d . z ) + (z:', . . . . .

,c,,. z )

and symmetric: the intersection number is independent of the order o f / : 1 , . . . , s (c) Suppose that the irreducible components Zi of Z with dim Zi = dim Z have generic points Z l , . . . , Zk. Then the rings Oz,z, (= localisation of Oz at zi) are zero dimensional and of finite length. The multiplicity of Zi in Z is defined as m z , z = length of Oz, z~. One can show that k

(~, . . . . .

~cd. z) = ~ mz,.z" (C, . . . . . ~ . Z,) i=1

This one proves by showing that Oz has a filtration by coherent sheaves Oz = Zo > E

D - . . D.F~

with l = ~i=lk mz,,z, such that for all j with 0 _< j < l there exists i in {1,. .. , k} and morphisms

~1~+1

, Gj

,--

~

---,

Oz,

with kernel and cokernel supported in dimension smaller than d.

VIII. AMPLE LINE BUNDLES AND INTERSECTION THEORY (d) The proof of the proposition above also gives a method for computing /~d " Z). Suppose the section s C F(X,/~d) is such that 9s: s

|

Oz --, Oz

73 (~1 .....

(*)

is injective. In this case the quotient sheaf O z / s O z is the structure sheaf of a closed subscheme which we will denote by Z n Ha, where Ha = {z E X I s(x) = 0}, and we get ( q . . . . . q . z ) = ( q . . . . . q - 1 . (Ha n Z)) R e m a r k s . (1) Ha is a divisor representing/~a, i.e., such that/~a = O(Ha) (2) If s is very ample, then (*) is injective for general s. Repeating this, we get that (/~1 . . . . . /Za 9Z) is equal to the number of points in the intersection H1 N ... n Ha N Z counted with multiplicities, if the divisors H1,..., Ha are chosen general enough: (q ..... q.

Z) =

~

e p ( / t , , . . . , / ~ a ; Z)

PEHI n...nHano7

By the proof of the proposition above, the local multiplicity of H1 N -'" N Ha n Z at P is: ep(H1,..., Ha; Z) = dimk Ox,p/ (Zz,p + flOx,p + " " + faOx,e) here f l , ' " ,

fa are local equations defining H 1 , . . . , Ha.

(e) The number (/~a. Z) := (/~ . . . . . /Z- Z) (d factors/~) can also be computed by the rule: X(E.| | Oz) = ( f a . Z)(d!)-i . na + lower order terms If s is ample we will call (s Z) the degree of Z with respect to s and write deg c Z for it. Suppose now that X = P" and Z C pn is an irreducible subvariety of dimension d > O. By remark (d): dego0 ) Z = # ((general n-d-plane) n Z) = deg Z, the classical definition of the degree of Z. Using the fact that any hyperplane in P~ intersects Z non-trivially and induction on d we see that we always have that deg Z > 0. This more or less proves one direction of the following theorem. 4.3.1 T h e o r e m ( N a k a i c r i t e r i o n ) . A line bundle s on X is ample if and only if for every closed subvariety Z C X we have (E.d 9Z) > O. (f) The behavior of the degree under a finite morphism f : X ~ Y is the following: deg/. L Z = deg(f: Z --~ f ( Z ) ) , deg L f ( Z ) Here Z C X is a closed subvariety and/~ is an ample line bundle on Y. S k e t c h of p r o o f a n d e x p l a n a t i o n of deg(f: Z --+ f ( Z ) ) . The image f ( Z ) of Z under f is again a closed subvariety. The map f defines an inclusion f*: k ( f ( Z ) ) k(Z) of function fields and this is a finite field extension. Put n := deg(f: Z --~ f ( Z ) ) := [k(Z) : k(f(Z))]. If this field extension is separable, then generically for p E

74

A.J. DE JONG

f(Z) we have f - l ( p ) = {ql,... ,qn} C Z. Hence, if Hlf'l...NHdAY(Z) = {pa,... ,Pn} then f'H1 fq "" f) f'Ha M Z -- {qll,..., qln,..-, qna,..., qnn). The result follows, t:] (g) Suppose that k = C and that X is smooth. Then X(C) also has the structure of a complex manifold, which we will denote by X ~. A line bundle/: on X gives rise to a line bundle /:an on X ~ which is determined by an element of Hx(x an, O~n). The exponential sequence: 0 "-~ 2 r i Z --* O ~ L~ O ~ ~ 0 gives us the first Chern class Cl(if) E H2(X ~, Z). On the other hand, an irreducible subvariety Z C X will give rise to an analytic subvariety Z ~ C X ~ . After triangulizing it, we see that it gives rise to a class [Z] = [Z ~] 6 H2d(X~', Z). For a general closed subscheme Z we put [Z] := ~ mz,,z" [Z~] E H2d(X~, Z) i The connection between intersection numbers and cup product in cohomology is now: (El . . . . .

ifd" Z) = (Cl(ifl) A . . . A El(if.d) , [Z])

where ( , ): H2d(X~, Z) x H2d(Xa", Z) ~ Z is the canonical pairing. 5

Numerical

Equivalence

and

Ample

Line

[]

Bundles

We write FI(X) = {~niCi I Ci C X is a closed subvariety of dimension 1} for the free abelian group generated by the set of all curves in X. By abuse of language we call an element of Fl(X) a curve on X. A curve ~n,Ci is effective if ni >_ 0 for all i. Our intersection number gives us a bilinear pairing Pic(X) • F a ( X ) - - * Z

(E,~niCi) HY~ni(E.CO

We say that two curves C1, C 2 (resp. line bundles El, if2) are numerically equivalent if (/:. Ca) = (/:" C2) for a l l / : E Pic(X) (resp. (/~1" C) = (/~2" C) for all C E Fl(X)). N o t a t i o n : C1 ~ C2 (resp. ]~1 ~ ~2)" 5.1 Definition. AI(X) := ( P i c ( X ) / - ) | R and AI(X) := (FI(X)/-) | R. (By definition the intersection number gives a non-degenerate pairing between these two R-vector spaces.) [] 5.2 R e m a r k . If k = C and X is smooth then by Remark 4.3(g) we know that the intersection products ( E . C ) only depend on the first Chern class el(E) E H2(X ~, Z). Hence we conclude that Al(X) is a subquotient of H2(X ~, R) (it is actually a subspace!). This proves the following general theorem in this special case. [] 5.3 T h e o r e m . Aa(X) is finite dimensional. In general one reduces to the case where X is a smooth surface and then the theorem is a consequence of the Mordell-Weil theorem for abelian varieties over function fields.

VIII. AMPLE LINE BUNDLES AND INTERSECTION THEORY

75

5.4 R e m a r k . If/:~ - s then (s .Z:2..... s = (~'1 "s ..... s always. Hence, by the Nakai criterion, whether or not s is ample depends only on the numerical equivalence class of s i.e., it depends only on the point in A I ( X ) determined by/2. [3 A subset S of a R-vector space V is called a cone if for all x, y in S and for all real n u m b e r s a > 0 and b > 0, o n e h a s a x + b y E S. Let A t ( X ) C A~(X) be the cone generated by the classes of effective curves. We know that if s is ample then (s C) > 0 for all C E A t ( X ) . Hence the cone po (the ample cone) spanned by the classes of ample line bundles is contained in the pseudo-ample cone

P = {D E A I ( X ) I D . C > 0 'v'C E A+(X)} We will have to use the following basic theorem once: 5.5 T h e o r e m . po is the interior of P. (The ample cone is the interior of the pseudoample cone.) 6

Lemmas

to be used

in the

Proof

of Thm.

I of Faltings

Suppose X is an abelian variety over k and ~: is an ample line bundle on X. We denote by [HI the morphism given by multiplication by n on X: [n]: X --* X , x ~ nx. 6.1 L e m m a . [n]*s --- s P r o o f . (for k = C) We can write X =' = Cg/L and L ~ Z 2g. The induced map [n]*: H2(X ~ , Z) ~ H~(X ~', Z) is equal to multiplication by n 2, as is seen from the identification H~(X ~, Z) ~- A 2 H x ( X ~ , Z) '~ A 2 L*. [] Suppose we have a product situation X = X1 • X2. Let us denote by pl: X ~ X1 the projection onto X1. Further, we assume given line bundles s s on X1 and line bundles s s on X. Finally, Z C X is a closed subvariety of dimension d. 6.2 L e m m a . If dimp~(Z) < k then (p~(s

. . . . . P;(r.k) " s

..... s

Z) = O.

P r o o f . By linearity of the intersection product and of p~ we may assume s t:k are very ample. By our assumption we can find divisors H 1 , . . . , Ilk (divisors of sections of s s such that HI n . . . n H~ n p~(Z) = 0 (recall that dimpl(Z) < k). Hence also p~H1 gl ... VIp~H~ Cl Z = 0. This implies by Remark 4.3(d) that (P~(~I) . . . . . ~d" Z) ~-- (~k+l . . . . . ~-~d" (p;H1 N . . , n p ; H k ( 1 Z ) ) ~- (~k+l . . . . . ~d" 0) = 0

I-1 Put P := P"' • .-. • P " ' . On it we have the line bundles s := pr~(Op-,(1)).

76

A.J. DE JONG

6.3 Exercises. (a) Sections F E P(P, s ~d' | | s nomials of multidegree ( d l , . . . , din).

correspond with multihomogeneous poly-

(b) Pic(P) = Z[s O ' " @ Z[s and A'(P) = R m (same basis). The pseudoample cone is {(Xl,...,Xrrt) ~ R rn [ Vi:xi k 0}, the ample cone is {(xl,...,x,~) E R m [ Vi: xl > 0}. If Z C P is an irreducible subvariety, then we get degrees indexed b y e l , 9..,era with ~ e / = dim(Z): the numbers (s ~ "s ~ . . . . . s These numbers are > 0; use induction on dim Z and the fact that one can always find a section in s "transversal" to Z. [] 6.4 L e m m a . (Prop. 2.3 of [22].) Suppose X C P is a dosed subscheme which is an intersection of hypersurfaces of multidegree ( d l , . . . , d,~). ff the Xj are irreducible

components of X with multiplicities mj, and of the same codimension t, then

E mJ(L; 1 . . . . . c:,m- xA _< (~;1 . . . . . c:m. (L~d, |

| c~dm),, p)

P r o o f . The difficult point in this lemma is the fact that the Xj do not need to be the irreducible components of X of the maximal dimension. The proof is by induction on t (t = 0 is trivial). First we choose t polynomials F1,... ,Ft of multidegree ( d l , . . . ,d,~) in the ideal of X, such that each Xj is an irreducible component of their set of common zeros. (Having chosen F 1 , . . . , Fk one simply chooses for Fk+l a polynomial which is non-zero on all the components of V(F1) N - . . A V(Fk) which contain an Xj.) Then we might as well replace X by the closed subscheme defined by F1,..., Ft (this will only make the mj bigger) and enlarge the set of Xj to include all the components of X of codimension t. Let us denote by Y the closed subscheme defined by F 1 , . . . , F t - a and by Y~ the components of Y of codimension t - 1 containing some Xj. The multiplicity of Y~ in Y is nl. By our choice of Ft we have: the irreducible components of (Ft = 0) f3 Y~ are the Xj say with multiplicities kij. The formula mj = ~ i k~jni is a consequence of the fact that the sequence F1,..., Ft is a regular sequence in the local ring Op,, i (xj E Xj a generic point). Hence we get: E-~j(s

..... s ~.x~)

=

J

Ek'jn'(s

. . . . . Z.~~ ' . x A

i,j

=

E n,(g? ..... g~-

( ~ = 0) n Y,)

i

=

~ n,(C~ . . . . . .

c~.~ . c . v,)

i

< here s = J~l~dl ~ - . .

~ j~/~drn

(induction) < (s . . . . . . s

s

s [-7

Chapter VIII

The Product Theorem by Marius van der Put

1

Differential Operators and Index

In this chapter, k will be a field of characteristic zero. 1.1 D e f i n i t i o n . A vector field on a variety X over k is a k-linear derivation D of the sheaf Ox, i.e., D: Ox ---* Ox is k-linear and for all open subsets U C X and all f , g E Ox(V) one has:

D(fg) = D(f)g + fD(g) In other words, a vector field on X is a global section of the sheaf H o m o x (fl~/k, Ox) (see [271, II.S). [] 1.2 D e f i n i t i o n . Let r > 0. A differential operator on X of degree < r is a k-linear endomorphism L of the sheaf Ox such that every x E X has an open neighborhood U on which L can be written as a sum of expressions fD1 " 9D8 (~ < r), where each D, is a vector field on U and f r Ox(V). (The factor f can be omitted for r > 0.) [] 1.3 R e m a r k . This definition is not the same as the one in [25], IV, w but one can show that the two definitions are equivalent (note that k has characteristic zero). [] 1.4 E x a m p l e . Let X = P~ with homogeneous coordinates x 0 , . . . , x~. Let E be any k-derivation of k ( x 0 , . . . , x~) such that E(xi) is homogeneous linear for all i. Then E induces a vector field D on P~ where D = the restriction of E to k(xl/Xo,..., x,/xo). Indeed D(xi/xj) = E(xi/xi) = ( x j E ( x , ) - x i E ( x j ) ) / x ~ E k[xo/xj,...,xn/xj]. A calculation shows that every vector field on P~ is obtained in this way. In affine coordinates Yl = Xl/xO,..., yn = xn/xo the vector field D has the form: y]~(degree < 1) ayi + (degree < 1 ) ~ yi~u; i=l

"

/=1

ffi

[] 1.5 E x a m p l e . Let m > 1. Define 79 = p~l • . . . • p~m and let x.(i) denote the homogeneous coordinates of P~'. Let R = k [ z . ( 1 ) , . . . , z . ( m ) ] be the (multi-)graded

78

M. VAN DER PUT

ring of P . As in the previous example one can show that any vector field D on P is induced by a k-derivation E of R which respects the multigrading, i.e., E(x.(i)) is a k-linear combination of the elements x. (i) and

Ix.(i)~ xo(i)Z(x.(i)) " x.(i)E(xo(i)) D ~xo(i)] = Xo~ A calculation shows that every differential operator L on P can globally be written as a polynomial in global vector fields; it follows that L acts on R and respects the multi-grading. [] L e t m > 1, X x , . . . , X , ~ varieties over k and X := X1 x . . . x Xm. Any vector field D on X can be written uniquely as D = ~ i ~ l Di, where Di is a vector field on X in the direction of the i-th factor, i.e., the image of D~ under the projection pri#~: X ~ lIj#~ Xj is zero (note that g~/k = q ) i p r * ~ , / k ). 1.6 D e f i n i t i o n . A differential operator L on X is of multi-degree < ( r l , . . . ,rm) if, locally, it can be written as a sum of expressions fD1 " " D s , where the Di are vector fields in the directions of the factors of X and for all j the number of Di in the j t h direction is at most rj. [] Let d l , . . . , d,~ be positive integers and let f :~ 0 be a section of a line bundle s on X. In the following definition these d~ will be used to define a kind of weighted total degree for differential operators on X: the weighted degree of a vector field in the ith direction will be 1~all. 1.7 D e f i n i t i o n . A differential operator L on X is of weighted degree < r (or < r) if, locally, it is a sum of differential operators of some multi-degree < ( r l , . . . ,rm) with r,/d, + . . . + r,~/d,~ _< r (resp. < r). [] 1.8 D e f i n i t i o n . Let x E X and 8 a generator of the stalk s of s at x. Then f = gs in s for a unique g E Ox,~. We define the index i ( x , f ) of f at x, with respect to the weights d l , . . . , d,~, to be a E Q>o with cr maximal for the property:

(L(g))(x) = 0 for every differential operator L defined on some neighborhood of x and of weighted degree < a. (This property of a and g is easily seen to be independent of the choice of s.)

[]

Note that in this definition x is not necessarily a closed point of X. For x any point of X and g any element of Ox,~: we define the value g(z) E k(x) of g at x by g(x) := i*~g, where is: Spec(k(x)) -~ X is the inclusion. 1.9 E x a m p l e . Let Xi = A 1 for all i, s = Ox, x = 0 and f

=

~

a , 1-, . . . , z m. X

,,1 9 ..x~,~

( il ,...,i,,, )

Then

i ( x , f ) = min{ix/dl + . . . + i,,/dm [hi ......i,, # O} []

IX. THE PRODUCT TtIEOREM

79

1.10 Definition. In the situation of Def. 1.8, let a E R. The closed subscheme Z~ of X on which the index of f is at least cr is defined by the sheaf of ideals in Ox generated locally by the L(g), where L is a differential operator of weighted degree < a and f = gs with s a local generator of s [] 1.11 E x a m p l e . Let 79 be as above, let ex,... ,era be non-negative integers and let Z: be the line bundle O ( O , . . . , era). We have already seen that global differential operators on 79 act on F(79, O ( e a , . . . , e,~)) and that the sheaf of differential operators of multi-degree < ( j l , . . - , J m ) is generated by its global sections. It follows that Z~ is defined by the homogeneous ideal of R generated by the L(f), where L is of multidegree < ( j l , - . . ,jm) with j~/dl + ' " + j~/d,~ < a. Note that all L(f) are global sections of O ( e l , . . . , era). I'-I 2

The

Product

Theorem

2.1 T h e o r e m . Suppose that k is algebraically dosed. Let m, n l , . . . , n m be positive integers. Let 79 :-- p~l x .-. x p~m. For every e > 0 there exists r E R such that if." 1. d l , . . . ,din are positive integers satisfying dl/d2 ~_ r , . . . ,dm-1/dm ~_ r, 2. f E F(79, ( 9 ( d l , . . . , d,~)) is non-zero, and 3. for some a, Z is an irreducible component of Z~ and of Z~+~ (here the index of

f b taken with respect to the weights d l , . . . , din) then: (i) Z is a product of dosed subvarieties Zi of P m, i.e., Z -- Z1 x ... x Zm, (ii) the degrees deg(Zi) are bounded in terms ore and nl + ' " +nm only. P r o o f . Let pri: 79 --* P~ be the ith projection. Let Zi := priZ and fl := dim Zi. Then Zi is irreducible and closed in P~i and of course Z is contained in Z1 x -.- x Z,~. We have Z = Zl x -.. • Zm if and only if ~ fi = dim Z. Let s be the line bundle pr~O(1) on 79 and consider the intersection numbers s . . . . . /~'~ 9Z for ratuples ( e l , . . . , e,,) of non-negative integers with E e i = dim Z (see Chapter VII). We claim that ~] fi = d i m 2 if and only if there exists only one such mtuple (ea,... ,e,~) with s . . . . . . Z:~m 9Z > 0. Namely, if ~]fi = d i m Z and ei > fl for some i, then s . . . . . Z:~" 9Z = 0 because there exist el hyperplanes in pm whose common intersection with Zi is empty (see Chapter VII, Lemma 6.2). Suppose now that ~] fi > dim Z. There exist fl hyperplanes in P"I such that their common intersection with Z1 is a non-empty set of dimension zero. It follows that there exist e l , . . . ,e,~ with el = fl and/::~ . . . . . . Z:~,~ 9Z :f 0. For some i we must have ei < fi. But in the same way we show the existence of e~,..., e~ with e~ = fi and Z:~~ . . . . . Z:~,- 9Z -fi 0. For what follows we need a lower bound for the multiplicity mz, z~, of Z in Zo. Recall (Chapter VII, w that rnz,z, is by definition the length of the local ring Oz,,,,, where 7/is the generic point of Z. The lower bound for mz, z, we want is a consequence of the assumption that Z is an irreducible component of both Z~ and Z~+,. First some notation. Let pr>i: 79 ~ lIj>i pn3 and Pr>i: T~ ~ lIj>i F'nJ denote the projections. Let 5~ := dim(pr>~2) - dim(pr>~Z); hence ~1 + ' " + 5,~ = dim Z. Let .T/

80

M. VAN DER PUT

be the fiber of (pr_>iZ ~ pr>iZ) over pr>ir/, let r/i := pr_>ir/be the generic point of .~ and let ki := k(rli ). Then .T/is a closed subvariety of P~k~+~of dimension ~5i. 2.2 L e m m a .

With the notations as above, suppose that dl >__ "'" > din, then we

have:

rn

mz,z~ >_ (e/codim( Z) ) c~

H d~i'-~' i=1

P r o o f . We want to get ti,j (1 < i < m and 1 _< j < ni) in O7,,~, such that: 1 (i) the dtij form a basis of the free O7, o-module f~7'/k,~, (ii) the ti,j with j > 3, form a system of parameters of the regular local ring OT,,n of dimension codlin(Z), (iii) the images in fl~/k,n of the dt~j with j < ~5~form a basis of that free Oz.n-module, (iv) each tij is a pullback via pr>i , i.e., ti,j is constant in the first i ~ l directions. Such a system ti,j can be gotten inductively, starting with i = m. Let 1 < i < m and suppose that we have already chosen the t~,j with t > i. Take sj E 0~,,~, 1 < j <_ 5i, such that the dsj form a O~-,,n~-basis of f~dk,+x,n~" Let sj, 6i < J -< ni, be a system of -1 parameters of the regular local ring Op~,~+~,n~. Put t~,j := pr>is j. Let the k-derivations O,.j: O7,, -~ O~,,, be the dual basis of the basis dti,j of ~,/~,,. From the last property of the ti,~ it follows that each Oi,~ is a linear combination of vector fields in the directions of the lth factors, with l _< i (the "_<" is explained by the passage to the dual basis). It follows that Oi,j is of weighted degree _< 1/di, since we suppose that d~ _> .-. _> d,~. From now on we only consider O,j with j > 5i. The number of those is codlin(Z). Let Io C 07, and I~+~ C 07, be the ideal sheaves of Z, and Z~+~, respectively. By construction, we have LI, C Io+~ C Iz for all differential operators L of weighted degree < e. In particular, this holds for all L = YIi,j 0~,~'' with a~,j < ~di/codim(Z). Let ai be the largest integer, less than or equal to r It follows that the ideal Izo,,~ of 07, ,~ is contained in the ideal generated by the Ui,j'+1(1 < i < m , j > * ~ ) . Hence: m

length(Oz,,,) > I I ( a l + 1) '~'-*' > (e/codim(Z)) r176 i=1

II r i=1

[] Let ( e l , . . . , era) be an mtuple of non-negative integers with E i ei = dim Z = ~ 6 i . According to Prop. 2.3 of [22] (see Lemma 6.4 of Chapter VII) and the lemma above, one has (c~ ......

c~ . z) TM

1 _< - - c ~

......

c ~~ . ( d l ~ + . , . + d m ~ ) ~~

77ZZ , Z .

< (c~176

Hi(niC~

ei)! ~ . d'~'-*'

-ej).

Then rh = 0 and

rrt

_< c(e, codim(Z)) 1"[ d~'-~' i=1

with c(e,t) -= (t/e)tt!. 1 - lm = l d ~ ' - ~ ' = nm ,.=2~

For 1 < i < m let r/, := ~ j = i ( j

tdi/di / - l j vr~ 9

IX. TttE PRODUCT THEOREM Suppose now that s

81

. . . . . /:~m . Z ~ 0. Then for 1 < i < m one has:

ei + " " + e m _< dim(pr>/Z) = 6i + " " + 6,, It follows that T/i _> 0 for all i. Finally suppose that

di/di+l _> r > max(c(e, codim(Z)), 1) Let 7/:= r/2 + " " + T/re. Then we have: 1 _< s . . . . . .

s

Z _< c(e, codim(Z))r-"

So we find that r/ = 0. It follows that ei = 6i for all i, so by the argument at the beginning of the proof one has Z = Z1 x ... x Zm and dim(Z/) = 5i for all i. To prove the second part of the theorem, note that deg(Z1) . . . . . deg(Z,~) : / : ~ ' . . . . . s

Z _< c(e, codim(Z))

It follows that deg(Zi) 5 c(e, codim(Z)) for all i.

[]

2.3 R e m a r k . The Product theorem will be used in the following way. Let N be an integer greater than dim(T'). Suppose that f has index at least cr at a point x. Then there exists a chain P ~ Z1 D Z2 D ' " D ZN ~ x, with Zi an irreducible component of Zi,,/y. It follows that for some i one has Z~ = Zi+l. Then one applies the Product [] theorem with e = a / N . 2.4 R e m a r k . Suppose that (with the terminology of the Product theorem) f is defined over some subfield /Co of k. Then Z~ and Z~+r are also defined over k0. Let d be the number of conjugates of Z under the Galois group of k over /Co. Applying Prop. 2.3 of [22] (see Lemma 6.4 of Chapter VII) to these conjugates of Z one finds d < c(e, codim(Z)). It follows that Z is defined over an extension kl of k0 with [kl: k0] < e(e, codim(Z)). [] 3

From

the

Product

Theorem

to Roth's

Lemma

This section will not be used in the rest of the book, but is included to show, following Vojta in [81], w how one derives Roth's lemma from the following arithmetic version of the Product theorem (Thm. 3.3 of [22]). Let f , e, r, d l , . . . , d m and Z be as in the Product Theorem. Suppose that f and Z are defined over Q and choose an affine product A ~1 x --- x A ~'* C "P such that Z V I A ~ x ..- • A ~m ~ 0. On A ~' • ... • A ~" we represent f by a nonzero F E Z [ y . ( 1 ) , . . . , y . ( m ) ] . Let log [F[ := log(max lcoefficients of F[) denote the logarithmic height of F. Then one has:

1) Z = Z l x . . . x Z m 2) deg(Zi) < c(e, codim(Z)) 3) ~ d,h(Zi) <_ Cl(e ) log ]F[-~- c2(e)(d, + . . . + din) i=l

82

M. VAN DER PUT

In this statement h(T) denotes the logarithmic height of a subscheme T of PzN. We will give neither definitions nor proofs. We note that for p, q 6 Z with gcd(p, q) = 1 the subscheme "pxo + qzl = O" of P~ has logarithmic height log(max IPl, ]ql)3.1 L e m m a f R o t h ) . Let m >_ 1 and e > 0. There exist positive numbers r, C2 and

C1 such that if'. (i) d~,...,dm satisfy di/di+~ >_ r for all i, (ii) q b . . - , q m are positive integers such that Vi: q~' >_ q(', logqi >_ C~, (iii) p~,... ,p,~ are integers with gcd(pi,ql) = 1, (iv) F 6 Z [ y l , . . . , ym], F 7~ O, satisfies IPl c, _< q~' add has, for all i, degree at most di in Yi, then index(d,,...,a,,) (F, ( ~ , . . . , v,, q,,)) < (m + 1)e. The statement above is, except for minor notational changes, the statement used in Chapter III. P r o o f . We apply the arithmetic version with nl . . . . . n m = 1. Suppose that the index is >_ (m+l)e. Then there exists a decreasing set of irreducible components Z(a) of Z,~ (a = 1 , . . . , r e + l ) with 7' D Z(1) D Z(2) D "'" D Z ( m + 1) 3 ( P l , . . . , P m ] \ql Since dim(7,) = m one has Z = Z(a) = Z(a + 1) for some a. The arithmetic version applied to Z yields Z=ZI

X...xZ

m

For some i we must have Zi = {~x} q~ and so

di log ql _< Cl(e)log IFI + c~(e)(d, + . . . + d~,) hence

d, (log q,

- mc2(e)) < ca(e)log

IF I

since da + . . . + d~ <_ rod1 and dl log ql _< d~log q~. Choose C2(e) = 2mc2(e). Since log qt >- C2 one has 1 dl ~ log ql _< Cl(e)log IFI Choose Cl(e) = 3c1(e) and we find a contradiction.

[]

Chapter IX

G e o m e t r i c Part of Faltings's P r o o f by Carel Faber

We follow Faltings's [22], w Let k be an algebraically closed field of characteristic zero, A an abelian variety over k, and X C A an irreducible subvariety which does not contain any translate of an abelian subvariety B C A of positive dimension. 1 L e m m a . For m big enough the m a p am: X m --~ A m-I defined by a m ( z 1 , . . . , xm) = (2xl -- x2, 2x~ -- x 3 , . . . , 2xm-1 - xm) is t~nite. P r o o f . We use the following facts: 1. A projective morphism that is quasi-finite (i.e., whose fibres are set-theoretically finite) is finite. ([27], Ch. III, Exc. 11.2.) 2. Let f : X1 ~ X2 be a morphism of projective varieties. Let s be a line bundle on X1 that is ample on the fibres of f . Then the set of degrees of fibres of f , measured with respect to s (see Chapter VII, w is finite. Sketch of proof: We may assume that f is surjective. So f is flat on an open subset U of X1 containing the generic fibre. The image in X2 of the complement of U is a finite union of irreducible closed subsets of lower dimension, etc. ([27], Ch. III, Exc. 9.4 and Cur. 9.10). The essential points of the proof are: 1. The equations for the fibre over fire-1 := ( a l , . . . , a m - i ) : X2

~

x3 = Xm

:

2X 1 -- a 1

2x2 - as 2Xm-1

-- am-1

(to be solved with X l , X 2 , . . . ,x,n in X) show that for m > n the projection onto first n factors pm,,~:X m ~ X '~ induces a closed immersion on fibres a~,l(ffm_l) a~l(ff~_l). In particular, via pro,l: X m ~ X, we get closed immersions of fibres of into X. 2. So the maximum of dimensions of fibres of a,~ exists, and decreases with thus is constant, say equal to d, for m > m0. 3. We want to show that d = 0: then the morphism a , , has finite fibres, and are done by Fact 1 above.

the aT, m, we

84

C. FABER

4. So suppose that d > 0. Fix an ample line bundle s on A. By the degree of a subvariety Z of a fibre of a,~ we will mean the degree of pm,lZ in A, with For all respect to /:. (Note that this is the degree of Z with respect to p~,1s m > m0 we look at the d-dimensional irreducible components of fibres of ar,. Any such component is also a component of a fibre of am0 (by 1. above). Apply Fact 2 above with am0:X m~ ---* A m~ and line bundle p:,o,1/~. We conclude that for any m >_ m0 the maximum of degrees of d-dimensional irreducible components of fibres of am exists, and that it decreases with m. I.e., such degrees are bounded. 5. For m > m0 define Y,~ to be the subset of A m-1 above which the fibres of am have dimension d. Since for such m the maximum dimension of a fibre is d, the subsets Ym are non-empty and closed ([27], Ch. II, Exc. 3.22d). For m > n > m0, denote by qm,n: A m-1 ---* A ~-1 the projection onto the first n - 1 factors. Then q,~,,~(Y,,~) C Y,~. The subsets q,~,mo(Ym) of Ym0 are non-empty and closed, and any finite intersection of them is non-empty. Thus the intersection of all these subsets is non-empty. 6. Pick a point y in this intersection. Among the finitely many d-dimensional components in the fibre above y, there is at least one that occurs in fibres of am for infinitely many m; thus (by 1. above) it occurs in fibres of am for all m. Let Z be such a d-dimensional component, and consider Z as a subset of A. Note that for any irreducible closed S C A we have the following equivalence: S contained in a fibre of am r162 S C X and for all 0 < r < m there exists a br C A with 2~S + br C X. So for all r > 0 there exists a b, C A with 2rZ + br C X, and this again implies that 2~Z + br is contained in a fibre of am0, for all positive r. We conclude (by 4. above) that the degree of 2rZ (which equals the degree of 2rZ + br) is bounded uniformly in r.

7. Let G C A be the algebraic subgroup such that g E G if and only if g + Z = Z. Note that G is closed in A, so the connected component G ~ of the identity is an abelian subvariety of A. The product of the Z:-degree of 2*Z and the degree of the map 2~: Z --* 2~Z equals the (2~)*Z:-degree of Z (see Chapter VII, w Since (2")*/~ is numerically equivalent to Z:| (Chapter VII, w the (2*)*/:-degree of Z equals (4r) d times t h e / : - d e g r e e of Z. It follows that the degree of the map 2": Z ~ 2*Z, which equals the number of 2r-torsion points in G, grows like 4 *d. We conclude that the dimension of G equals d. By the definition of G we have for any z E Z that z + G C Z. So Z contains a translate of a d-dimensional abelian variety (in fact equality holds, and G is an abelian variety). 8. We have shown that am is finite for all m > m0. It is easy to show t h a t if X is a curve, then one can take m0 = 2. Finally we remark that the assumption is necessary: if x + B C X where B C A is an abelian subvariety of positive dimension, then for all b E B and for all m the point (b + x,2b + x,4b + x , . . . , 2 m - l b + x) is in the fibre over ( x , x , . . . ,x) E A m - l , so am has infinite fibres. [] We continue with [22], w Choose a big enough integer m such that the map am: X m ---* A "~-1 is finite, and choose a very ample and symmetric line b u n d l e / ~ on A, embedding A C Pn. In the sequel we will mainly use additive notation for the tensor product in the Picard group. E.g., on A • A we have the Poincar~ bundle

X. GEOMETRIC PART OF FALTINGS'S PROOF

85

7) = add*s - pr~s - pr;L; where add: A x A --~ A denotes the addition. We will also consider linear combinations of line bundles with rational coefficients, i.e., we identify a line bundle s with its image in P i c ( . ) | Q; we can do this since we are interested only in ampleness (see Chapter VII, Remark 5.4). For e, s l , . . . , s m positive rational numbers we define on A m the (rational) line bundle s s l , . . . , sin) as the rational linear combination

k

(i.i) Here

C(--s (.SIX i --

= --f-"

m--1

i=I

8 2. i pr~(s

+ ~ (SiXi -- Si+lXi+l)*(s

i=1

8i+lXi+l)*(~ ) means 1 n2

( r t s i g i --

T/,Si+lXi+I) (~[~)

for any non-zero integer n such that nsi and nsi+l are integers, so that sending (Zl,..., xm) in A m to nsizi -nsi+lXi+l is a morphism. (It follows from Cor. 3.4 that this is well-defined.) This line bundle s S l , . . . , sin) is a rational linear combination of s = pr~(s and T'i,j = Pr~d(~~) (use the Theorem of the Cube, Chapter V, Cor. 3.2):

m--1 (1.2)

s

=

(1 - e)s~f-q + ~ ( 2 - e)s~s +

i=2 m-1

+(1

-- s

-- B

SiSi+,~i,i+l,

i=l

are proportional to s~, and those of ~i5 to sisj.

so for fixed e the coefficients of s

2 L e m m a . SupposeY = ~ x . . ' 4 x Ym is aproduct subvarietyofA m. Then as a function of the si the intersection product s S l , . . . , s,~) aim(Y) 9Y is proportional t o yIm=l 8~'dim(Yi). P r o o f . The intersection number is a linear combination of terms Hi s I-i,#/P~,}"'Y, with coefficients proportional to I-[is~'. [Ii#j(sisj) e',J (see 1.2), and Z i ei + ~ i # j ei,j = dim(Y). We claim that such a term is zero unless for each i one has 2ei + ~j#i ei,j <_ 2 dim(Yi). To see this, we use the cohomological interpretation of intersection numbers (Chapter VII, w Remark g):

H c7'. H

y =

~A

[Yl)

(note that the order in the wedge product doesn't matter since the factors are of even degree). In terms of de R h a m cohomology this means:

H cte'i' H .~ei,,,,?.y i

= /yah A i c i ( c i ) e l A A,~,c,(P,.,) ~

i#j

where e I now denotes the first Chern class in H~R((A~) ~) (to integrate a volume form on a possibly singular variety like Y~", choose a finite projection Y ~ pdimY as in Lemma 5 below and perform the integration on (pdimY)~m). Note that for each

86

C. FABER

i, o ( s ~' = p7(c1(s is a pullback of a differential form of degree 2ei on Y~. Now think of A ~" as C dimA modulo a lattice. Then add: A = • A ~ ~ A m is induced by _it_:cdimA • ~limA ~ cdimA and c1(s is represented by a linear combination of terms dx A dy with x , y in HomR(C ~ama, R). The calculation: add*(dx A dy) - p;(dx A dy) - p;(dx A dy) =

=

d(xl + x2) A d(yl + y2) - dxl A dyl - dx2 A dy2

= dxl A dy2 + dx~ A dyl where xi = p~x and Yi = P'Y, shows that O(Pi,j) is a linear combination of terms p'w, A p;wj with w,,w~ in H~a(Aa"). The claim follows. As Ei(2e~ + ~ j r ei,j) = ~ i 2 dim(Yi), the intersection number is zero unless 2ei + ~jr ei,j 2 dim(Yi) for all i. [] =

3 C o r o l l a r y . There exists a positive eo such that for any e <_ r and for any product variety Y C X m the intersection number s s l , . . . , s , 0 aim(Y) 9Y is positive. P r o o f . Indeed, for si = 2 m-i we have that s s l , . . . ,sin) is the pull-back by the finite morphism ~m of the ample line bundle 4m-2s + 4m-3s + . . . + s hence it is ample on X " (see Chapter VII, Cor. 3.2). Hence for these si and for small e the bundle s s l , . . . , Sin) is ample, since the ample cone is open (see Chapter VII, Thm. 4.3.1). It follows that the implied constant in Lemma 2 (depending on e and Y) is positive for small e and arbitrary Y, showing that the intersection number s163 8 1 , . . . , S i n ) d i m ( Y ) 9Y is positive for small e, arbitrary Y and arbitrary (positive) Sl ~ . . . ,Sin.

[]

The main result of this section can now be formulated. 4 T h e o r e m . Let A be an abelian variety over an algebraically dosed fie/d k of characteristic zero. Let X C A be an irreducible subvariety which does not contain any translate of a positive dimensional abelian subvariety of A. Take m big enough as in L e m m a 1 and let s be a symmetric ample line bundle on A. For e, S x , . . . , s m positive rational numbers let s Sl,... , Sin) he the element of Pic(A m) | Q given by 1.1. Take eo as in Cor. 3. For any e < e0 there exists an integer s, such that s s l , . . . ,s,~) is ample on X TM if s~/s2 >_ s, N2/S 3 > S, . . . , Sm_l/S m ~ S. Before proving this result, we need some information concerning projections. Let L C P~ be a linear subvariety of dimension m. After a suitable choice of homogeneous coordinates x 0 , . . . , x~, one can assume that L = V ( x m + l , . . . , x~). Then L gives rise to a projection morphism r : ~ - L --~ P . . . . 1, sending (ao: " " :a~) to (am+l:--" :a~). Let (am+~:-..:a~) E ~'. . . . 1, then the map A m+l ---* P~ sending ( a o , . . . , a m ) to (a0: " " :a~) is an isomorphism between A m+l and ~r-l(am+l: -.- :a,); hence the fibres of ~r are affine. Suppose now that X C P~ is an irreducible subvariety of dimension d, such that X C/L = 0. Then ~r: X ~ ~rX is finite (since it is projective and its fibres are affine), and deg(X) = deg0r: X ~ r X ) . d e g 0 r X ) (see Chapter VII, w 5 L e m m a . Let X C P~ be an irreducible subvariety of dimension d. There exists a finite projection lr: X ~ pd of degree deg(X) and a global section s 7~ 0 in F ( P n, O( N ) ) for some N < ( n - d ) d e g ( X ) , such that the ideal sheaf of V(s) annihilates f~x/ed. Es--

1

pecially, r is dtale on X - ( X N V(s)). Moreover, N o r m s ( s i x ) in F ( P d, O ( N deg(X))) defines a hypersurface whose ideal sheaf annihilates f~XlP~. 1

X. GEOMETRIC PART OF FALTINGS'S PROOF

87

P r o o f . Let L0 C pn be a linear subvariety of codimension d+2, such that LoNX = 0. This gives a projection r 0 : P " - L 0 --~ pd+x and f o X C pd+l is a hypersurface of degree < deg(X). It follows that X C ~ro~r0X = V(F) with F an irreducible homogeneous polynomial of degree < deg(X). Let x 0 , . . . , x , be homogeneous coordinates on P" such that OF/Ox, ~ 0 and P := (0:...:0:1) • Y(F). Let ~r~:[r" - {P} ~ p,,-1 be the projection given by P. A computation shows that the ideal sheaf of V(OF/Ox,) 1 is the annihilator of ~v(F)/p._~. By induction we can assume that the lemma has been proved for ~rxX C p,-1; let ~r2:~rxX --* pd and 8~ e F(P"-~,O(Nx)) be as required. Then 7r := ~r2o~rl and s := (OF/Oxn)~r~s~ satisfy our requirements. For the construction, definition and properties of Norm~(s]x) see [25], II, w Note that in our case ~r is not necessarily finite locally free. [] We will also need the following lemma, the proof of which was communicated to me by Edixhoven. 6 L e m m a . Let X be a projective variety, let D be an ample effective Cartier divisor on X and let M be an invertible sheaf on X such that the restriction of M to D is ample. Then for d large enough we have hi(X, M | = 0 for all i > 2. P r o o f . We claim that for d big enough we have that for all i > 1 and all n > 0 the vanish. Assume this for a moment, and take cohomology groups Hi(D, Ad| d as in the claim. Also, take n big enough such that hi(X, Jtd| = 0 for all i > 1 (note D is ample, cf. [27], Ch. III, Prop. 5.3). We consider thickenings of D: let D , be the n-th infinitesimal neighbourhood of D in X, i.e., the closed subscheme with ideal sheaf 2-~+1, where :To is the ideal sheaf of D in X. Then the long exact cohomology sequence belonging to the standard short exact sequence

0 --* .M | --* .M|

--+ M|

~ 0

gives isomorphisms

Hi(D,~_x, M|

) ~-- Hi+I(X, M |

for all i > 1. So we want to show that for i >_ 1 we have Hi(D,~_l, .M| For this we use the filtration of coherent sheaves on Dn-a:

M|

~ M|

~ ' " ~ A4|

= O.

D0

whose successive quotients are

M|

M|

- 1)D)[D, . . . , M|

From the short exact sequences associated to the filtration we see that indeed

Hi( D,_I, Ad| for all i _> 1.

=0

88

C. FABER

It remains to prove the claim. We prove a more general statement. Let X be a projective variety, 2~4 and A/" ample line bundles on X and .T a coherent sheaf on X. Let ~-(a, b): = 9v | 2.'1| | A/"| Then the set $ 7 : = {(a,b) I a > O, b > O, h'(X,.T(a,b) ~ 0 for some i > 0} is finite. We may suppose that f14 and A; are very ample (replace A4 by f14m, A / b y A/"~, .~ by the direct sum OU(a, b) with 0 < a < m, 0 < b < n). Then we have an embedding of X in a product of two projective spaces. Now we mimick the argument for one projective space in [27], Ch. III, proof of Thm. 5.2: we may replace X by the product of the two projective spaces, and U by any coherent sheaf on that product. There is a short exact sequence 0 ~ 7~--, g --. Or ~ 0 with g a finite direct sum of sheaves O(ai, b,), and T~ coherent. All we need now is that s k) has no higher eohomology for j and k big enough. This follows from the corresponding statement for O; for this, use a Kfinneth formula as in [71], Ch. VII, Prop. 12 and Remarque, pp. 185-186 or in [25], Ch. III, Thin. 6.7.8, or, alternatively, use the Kodaira vanishing theorem. This finishes the proof of the claim and the lemma. [] We can now begin the proof of Thm. 4. Proof.

Let e < e0. By induction on r we will show:

For any integers N and r there exists an integer So, such that for any product subvariety Y = Y1 • " " • Y,~ of X " with dim(Y) = r and deg(Yi) < U for all i, t:(-e, sl,...,sm) is ample on Y if sl/s2 >_ so, - . 9 ,Sm-1/'Sm

~--- "30.

For r = 0 this is trivially true, and for r = m. d i m ( X ) and N = deg(X) we get the statement of the theorem. So assume the statement is proven for all r < r0, and let's try to prove it for r = r0. Let N be given. First of all, if we take the ratios sl/s2,... ,sm-a/sm big enough, then for all Y = I/1 x -.- x Ym C X TM with dim(Y) = r0 and deg(Y~) < N, there exists an effective sl,... ,s,~) to that divisor is ample divisor of Y such that the restriction of s ample. Namely, take D1 + ' " + Din, with Di = Hi x l-Ijr Yj, Hi C Y / a hyperplane section; apply the induction hypothesis with r = r 0 - 1 , same N. This will be used later on. We will show that, if sl/s2 .... ,Sr~-l/Sm are sufficiently big, and Y is as in the statement, s s l , . . . , sin) has non-negative degree on all irreducible curves C C Y. sa,... ,s,~) By Kleiman's theorem (cf. Chapter VII, Thin. 5.5) it follows that s is in the closure of the ample cone. Decreasing e a little bit then gives an ample line bundle (note that we could have started with (e + e0)/2 instead of e). So let Y be as in the statement, and C C Y an irreducible curve. The idea of the proof is now as follows. One shows that if Sl/S2,...,sm-1/s,~ are sufficiently big, for d sufficiently big and divisible there exists a non-trivial global section f of s sl,...,sm) | on Y. If the restriction of f to C is non-zero, then of course s sl,... ,sin) has non-negative degree on C. If the restriction of f to C is zero,

X. GEOMETRIC PART OF FALTINGS'S PROOF

89

one distinguishes two cases: f has small or large index along C. If f has large index along C, one uses the Product Theorem to show that C is contained in a product variety of dimension less than r0 (and of bounded degree), and one uses induction. If f has small index along C, then one constructs a derivative of f that does not vanish on C and has suitably bounded poles. How to define these two cases and how big sl/s2,..., s,,,_l/s,,, should be will follow from the computations below. Of course all estimates concerning the si/si+l will have to be done uniformly in Y and in G. As Faltings puts it, "set up the geometry": Choose projections ri: Yi --* pn~ = Pi (with deg(~ri) = deg(Y~)) and (not necessarily reduced) hypersurfaces Zi C P~ of degree deg(Zi) < nN 2, such that the ideal sheaf of Zi annihilates fll.,/p, (cf. Lemma 5; the n comes from the P" in which A i s embedded). Let 7r:Y ~ P = P1 x . . . • be the product of the 7ri; then deg(~r) is the product of the deg(Y~). We claim that any derivation 0 on Pi (there are many because Pi is a homogeneous space under GL(n~+I)) extends to a derivation on Y~ with at most a simple pole along zr*Zi. To see this, consider the exact sequence 0 --+ 7qfllp, l~ --, ~'~Yi/k I 1 --~ ~Yi/Pi -"r 0

of [27], Ch. II, Prop. 8.11. Applying Homor~(.,Tr*O(deg(Zi))) to it gives the exact sequence 1 0 ---* D e r k ( O , , 7ri9 O(deg(Z,))) ~ Homo n (Tri* flPdk, 7ri* O(deg(Zi)))

E x t ~ (a~,/p,, rZ O( deg( Zi) ) ) Let G~ in F(Pi, O(deg(Z~))) be an equation for Z~. Saying that the ideal sheaf of Z~ annihilates f~,~/p~ means that the map

9Gi." ~~Yi/pil ----+7r, O(deg(Z/)) (~Oyi ~~Yi/pil is zero. One checks that 7r~'(GiO) maps to zero in the Ext 1. If C is contained in the preimage of some Zi, i.e.,

c c ~.lz, • 1-[ j#i

then we are done by induction: note that ri: 7ri-1Zi --'4 Zi is finite of degree deg(Yi) _< N (since r~:Y~ ~ P~ is flat in codimension one), hence deg(r/1Z~) _< Ndeg(Z~) < nN a by Chapter VII, w Remark (f). So suppose that C is not contained in the inverse image of any of the Zi. Then 7r: C --* D = 7rC is generically &ale. For sl/s2,..., sm-1/s,= big enough, and d big enough and sufficiently divisible, we have that r(Y, c ( - e , s~,..., s~)~e) is non-trivial. This follows from: Sl,..., s,~) | is positive. 1. For such d the Euler characteristic x(Y, s 2. For such d and i > 2 the cohomology groups Hi(y~/~(-e, Sl,... ,Sin) @d) vanish. The second statement is a direct consequence of Lemma 6. To get the first state| ment, recall (Chapter VII, w that, as a function of d, x(Y,s is a polynomial of degree < dim(Y), and that almost by definition the coefficient of d dim(Y) is 1/dim(Y)l times the intersection number ( s .S1,... , Sin) dim(Y) .Y), which is positive by Cor. 3 and the choice of e0. So, if sl/s~,... ,s,~_l/s,~ are big enough (uniformly in Y and C), then for d sufficiently big and divisible, s sx,... ,sin) | has a non-trivial global section f on Y.

90

C. FABER

Next we consider line bundles on Am. We have the identity in Pic(A m) | Q (use the Theorem of the Cube and t h a t / 2 is symmetric): (8i$i -- $i+lXi+l)'( ~.) -~ (SIX i ~- 8i+lXi+l)*(~) : 282~i + 282+1~i+1 Suitable multiples of the two terms on the left are generated by their global sections (because s is). Hence we can find sections without common zeroes. Multiplying by these, we first find "injections without common zeroes" on Am: rn--1 2 d. ~ (sixi - si+,Xi+l) 9(~;) --* d. (2s~L1 + 4s]L2 + - ' - + 4s,~_,/~m_,

+ 2~:m)

i=l

then, using that the s are also generated by global sections, we find injections without common zeroes d . ~ ( - ( ~ , 8 1 , . . . ,sin)

--~ d ' ((2 - (i)Sl2~l -4- (4 - e)s~L2 + . . . + (2 - e)s~s

---* d . ( 4 s ~ s 1 6 3 Define dl = 4ds~, and choose an injection p:L(-e, sl,... ,sin) | ~ |163 which does not vanish identically on C (i.e., p is an isomorphism at the generic point of C). We come to the final part of the proof. Define the index i(C,f) of f along C to be the index i(z, f) of f at the generic point of C, with respect to the weights di (cf. Chapter VIII, Def. 1.8). Since p is an isomorphism at :r, we have i(x, f) = i(z,p(f)). Now note that s = 7r~O(1) since ~ri is a projection from P~ minus a linear subvariety to P~ and A was embedded in ~" via global sections of L. So p(f) can be viewed as a section on Y of ~r*O(dl,...,dm). Because 7r:Y --* P is finite and surjective, and Y is integral and P is integral and normal, we can take the norm of f with respect to 7r (cf. [25], II, w g := Norm~(p(f)) e F(P, O(dl,..., din) |162

= F(P, O(deg(Tr)dl,..., deg(Tr)d,~)

Let i(D,g) := i(Tr(x),9) be the index o f g at 7r(x) (i.e., along D = ~rC) with respect to the weights deg(Tr)di. We claim that i(D,g) > i(C,f)/deg(Tr). To see this, note that 7r is 6tale above 7r(x), so that over some 6tale neighbourhood U of ~r(x), rr: Y --. P is a disjoint union of deg(Tr) copies of U; then use the formula i(x, flf2) = i(x, fl)+i(x, .[2). Now we choose a sufficiently small positive number cr (the precise choice of r will be explained later), and we distinguish two cases. C a s e 1: i(D,9) > a. Then apply the Product Theorem (Chapter VIII, Thin. 2.1) as in Chapter VIII, Remark 2.3. To be precise: there exists a chain P # Z1 D Z2 D ' " D Zdim(P)+l ~ X with Zl an irreducible component of the closed subscheme of P where 9 has index > la/(dim(P)+l). Because of dimensions, there exists an I with 1 < l < dim(P), such that Zt = Zt+l. The Product,Theorem then shows that if the si/si+l are sufficiently big (in terms of ~, m, and n a , . . . ,nm), then Zl = V1 x ..- x Vm is a product variety of dimension < dim(P) (since f # 0) and the deg(Vi) ar.e bounded in terms of Or, rn, and nx,...,nm. Let Y' be an irreducible component of lr-lZl with C C Y'. Then dim(Y') < dim(Y), Y' = YI' • "'" x I/I is a product variety and deg(Yi' ) < deg(~ri) deg(Vi) < N deg(Vi) (note that Y/' is an irreducible component of ~r~-'Vi and

X. GEOMETRIC PART OF FALTINGS'S PROOF

91

use Chapter VII, Remark 4.3.f). By induction, if the si/si+l are sufficiently big in terms of a, m, n l , . . . , n , ~ and N, then L:(-C, S l , . . . ,am) is ample on Y', hence has positive degree on C. C a s e 2: i(D,g) < a. Then i ( C , f ) < a deg(Tr). For 1 < i < m let Oi,j, 1 < j < n,, be global vector fields on Pi which give a basis of the stalk of Homop~(~.,/~, Op~) at r i ( x ) . As r : Y ~ P is 6tale at x, we may view the Oi,j as derivations of Oy,,. Let f l be a generator of the stalk 13(-E,81, ,sin/| then f = hfl for a unique h E OF,,. By the definition of index (Chapter VIII, Def. 1.8, also recall what we mean by the value at x of some f e Or,~), there exist integers eid > 0 with ~,r ei,j = i(z, f), such that (H(h))(x) # O, where H = I-lid Oil)j (in some order). We want to define

(H(f))(x) as (g(h))(x).fx(x) C s

~., )~| |

=s

Sl,...,s,~)|

k(x)

so let us check that this is well-defined. Let u E O~,~ and f{ := u-l fl. Then h ~ = uh and H(h') = H(uh) = uH(h)+ terms like H~(u)H2(h) with H2 of lower differential degree than H, hence (H2(h))(x) = 0 by the definition of i(x, f). It follows that (H(f))(x) is well-defined. Since k(x) is the function field of C, ( n ( f ) ) ( x ) is a rational section on C of the line bundle s s l , . . . , sm)| we want to bound its poles. We have already remarked that each cgl,j extends to a derivation on Y/ with at most a simple pole along 7r*Zi. Working on an affine open on which ff.(-r S l , . . . , am) | has a generator f l we can write f = hfl and Oi,j = g~lOi,j where the Oi,j are regular derivations and gi is an equation for ~r*Zi. Then M = 17

=

i,j

+...

=

i,j

i,j

i,j

where the dots stand for terms of lower differential degree. It follows that =

\

t,j

~,3

/

-el j

hence its poles are no worse than those of l-li,j gi ' = l-Ii g[*~, with el = ~ j e~5. Since Y can be covered by affine opens on which s | is trivial, we get a non-zero global section I-[,5 G~"J(H(f))(z) on C of m ?~ F|

~

k., i

(~ deg(Gi) ~ , - ,

WJ L..(--s

i=1

.-ql,

..

.

, Sm)|

m-1 s

=

deg(Gi)-eds~) |

i=1

~

( S i X i -- ,.qi+lXi+l).ff|

i=l

where, as before, Gi is an equation for Zi in F(Pi, O(deg(Zi))). Using ei/ds 2 = 4ei/di < i ( C , f ) < a d e g ( r ) and deg(Gi) < n N 2 one gets a non-zero global section on C of s 2 d e g ( r ) a - e, S l , . . . , sin) | so this line bundle has non-negative degree on C.

We still have to choose the right a: we take a > 0 such that e' := 4nN 2 deg(Tr)~r + r < e0 The argument above, with e replaced by d, then shows that ~ ( - r non-negative degree on C. Acknowledgement. these notes.

has []

I would like to thank Bas Edixhoven for his help in preparing

Chapter X

Faltings's Version of Siegel's Lemma by Robert-Jan Kooman

Siegel's Lemma in its original form guarantees the existence of a small non-trivial integral solution of a system of linear equations with rational integer coefficients and with more variables than equations. It reads as follows: 1 L e m m a . ( C . L . S i e g e l ) Let A = (aij) be an N x M m a t r i x with rational integer coemcients. P u t a = m a x i j la~jl. Then, if N < M , the equation A x -= 0 has a solution X E Z M, X ~ O, with

IIxll _< (Ma) N/(€ where [[ [[ denotes the max-norm: [Ix H = H(xl,... ,XM)][

:

maxl
Stated in an alternative form, the lemma finds a non-trivial lattice element of small norm which lies in the kernel of a linear map a from R g to R N with the Z-lattices Z M and Z N where M > N and a ( Z M) C Z N. Faltings [22] uses a more general variant of the lemma, firstly by taking a general Z-lattice in a normed R-vector space, and secondly, by looking not for one, but for an arbitrary number of linearly independent lattice elements that lie in the kernel of a and are not zero. The lemma is a corollary of Minkowski's Theorem, which we state after the following definitions: 2 D e f i n i t i o n s . For V a finite dimensional normed real vector space with Z-lattice M (i.e., M is a discrete subgroup of V that spans V) we define A~(V,M) as the smallest number A such that in M there exist i linearly independent vectors of norm not exceeding A. Further, by V I M we denote a set {vEV:v=AlVl+...+AbVb,

0<.~i
where V l , . . . , vb is a basis of M. Finally, if V, W are normed vector spaces with norms H'[[v, [['[[w, then B ( V ) denotes the unit ball {x E Y : [Ix[Iv _< 1} in Y and if a is a linear map from V to W, the norm [[a[[ of a is defined as the supremum of the II~(~)llw/ll~llv with x E V, x # O. [] We can endow V with a Lebesgue measure #v as follows. Take any isomorphism of R-vector spaces r : V ~ R b and let A denote the Lebesgue measure on R b. Then, for Lebesgue measurable A C R b, put #v(r

= A(A).

94

R.J. KOOMAN

Up to a constant, there is only one Lebesgue measure on V. Hence the quantity Vol(V) = Vol(V, I1"11,M) . - # v ( B ( V ) ) #v(V/M) does not depend on the choice of #v- Clearly, it is also independent of the choice of the basis of M in the definition of V/M. 3 T h e o r e m . (H. M i n k o w s k i ) Let V be a normed real vector space of finite dimension b and with Z-lattice M.Then 2b/b! < A I ( V , M ) . . . A b ( V , M ) . V o l ( V ) < 2b. P r o o f . See [48]. Note that B(V) is a convex closed body in V, symmetric with respect to the origin, and that AI(V, M ) , . . . , Ab(V,M) are precisely the successive minima of B ( V ) with respect to M. [] We now state Faltings's version of Siegel's Lemma. 4 L e m m a . (Lemma 1 of [22]) Assume that we have two formed real vector spaces V and W with Z-lattices M and N, respectively, and a linear map a : V ~ W such that a ( M ) C N. Let C > 2 b e a r e a l n u m b e r s u c h that the map a has norm at most C, that M is generated by elements of norm at most C and that every non-zero element of M and N has norm at least C -~. Then, for a = dim(ker(a)), b = dim(V), and U = ker(a) with the induced norm on ker(a),

,~i+l(U, V 1~ M) _< (C 3b. b!)1/(~-i) for O < i < a-1. Thus, for any subset Y of ker(a) of dimension not exceeding i we can find an element in ker(a) which is not in Y and whose norm is bounded by the right-hand side of the above equation. The lemma will be applied in situations where the dimensions a and b tend to infinity and such that b/(a - i) remains bounded. Then the right-hand side of the equation grows like a fixed power of C times a power of b. For the proof of the lemma we let I1"11be the norm on V, and II'llu its restriction to U. We endow a(V) with the quotient norm: for v* E a(V) we put

llv'll"

= inf{llvll : ~(v) = v*}.

The unit balls B(U) and B ( a ( V ) ) are now defined in a similar way as B ( V ) and Vol(g) = Vol(U, II'tlu, V n M), Vol(a(V)) = Vol(a(V), II'II','~(M)). We first prove the following lemma. 5 L e m m a . Vol(V) < 2 ~. Vol(U). Vol(a(V)). P r o o f . Let #v, #v be Lebesgue measures on V and U, respectively. On a(V) we have a unique Lebesgue measure ~u,~(v), defined by

.,,(E) = :.

(v)

fs(v') d,.r

X. FALTINGS'S VERSION OF SIEGEL'S LEMMA

95

for #v-measurable sets E C V where for v* = a(v) 9 a(V) we have f~(~') = ~ ( { .

9 u : = + v 9 E})

which is independent of v since/~u is translation invariant. We compute fB(v)(v*) for v* 9 a(Y). If v* r B(a(V)), then Hvl[ > 1 so v r B(V) for all v 9 a-a(v*) and fB(v)(v*) = 0. If v* 9 B(a(V)), then v 9 B(V) for some v 9 a-~(v*). For u 9 U with u + v 9 B(V) we have IluIIv _< Hu + vii + ][vii _< 2, hence fB(y)(v*) <_ #u(2B(V)) = 2 ~ # v ( B ( v ) ) and

(6)

# v ( B ( V ) ) <_2a . #v(B(U)) . I.t,~(v)(B(a(V)).

Furthermore, if u l , . . . ,u~ is a basis of U r3 M and Ul,... ,u b is a basis of M, then a(u~+,),... ,a(ub) is a basis of a ( M ) and

fv/M(v*) = #v({u 9 U: u + v 9 V/M}) = tzu(U/U N M) for v* 9 a ( V ) / a ( M ) and fV/M(V*) = 0 otherwise. Hence (7) # v ( V / M ) = f~(v) fv/M(V*) d#~,(y)(v*) = #u(U/U N M ) . #,~(v)(a(V)/a(M)). Finally, by the definition of Vol(V), we have that # v ( B ( V ) ) / # v ( V / M ) = Vol(V) and similarly for Vol(U) and Vol(a(Y)). Combination of (6) and (7) now yields the statement of the lemma. []

Proof.

(of Lemma 4) We have )h(a(V),M*) > C -2, since for 0 # v* 9 a(M) C

a(V) we have v ~ = a(m) with m E M and IIv*ll* = inf lira + ~'11 > II'~(m)llw ,,~u II~

> C-~, -

where H IIw denotes the norm on W. By Minkowski's Theorem we obtain A~(a(V), M*)S-~.Vol(a(V)) < 2 b-a, so that Vol(a(V)) < (2C2)b-% Further, Ab(V, M) < C, and applying Minkowski's Theorem once more, we find that Ab(V, M) b -Vol(V) > 2b/b!, whence VoI(V) > 2 b. C-b/b!. Finally we have A~(U,U A M)) > C -~ and, by Lemma 5, Vol(V) < 2~.Vol(U).Vol(a(V)), so that Aa(U, UnM);.),i+I(U, Ur3M) ~-~. Vol(U) _< 2% Hence Ai+,(U, U Cl M)

< (2~176 _<

for0
_ '/C~176 2~

<_

gol(a(V)).

Vol(V)

C,),/(=-,)

'''~ []

We give an example of how one obtains, in a general manner, a normed R-vector space with a Z-lattice. 8 C o n s t r u c t i o n . Let K be a number field, OK its ring of integers. Let X be a proper 01c-scheme and let E be a metrized line bundle on X (i.e., E is an invertible Oxmodule equipped with a hermitean metric [['1]~o n / ~ for every embedding a: K ~ C,

96

R . J . KOOMAN

see Chapter V, w for details). Then H~ is a finitely generated OK-module by [27], Ch. III, Thm. 8.8 and Rem. 8.8.1; it is torsion free if X is flat over OK. We let V = H~ s | R. Then V is a R-vector space and the image of the (canonical) map H~ s --* H~ | R is by construction a Z-lattice in it. A norm on V can be obtained as follows. Since K NQ R = G,K~ (the sum being taken over all infinite places). For v an infinite place of K let V~ := H~ where XK~ denotes the pullback of X to Spec(IQ) and /2K. denotes the line bundle on XK~ induced b y / ; . We have V = H~

s

|

OK |

R = @,,H~

s

|

K,, = O,,V,,

(for the last equality, use [27], Ch. III, Prop. 9.3). If ~: K --* C is a complex embedding giving the infinite place v, then on Vv we have the sup-norm over X(K~) = Xo(C). Note that by Def. 4.3, ~ gives the same norm on Vv. On V = (~Vv we take the max-norm associated to the norms on the V~. In other words, this norm on V is just the sup-norm over X ( C ) = H~ X~(C). []

Chapter XI

A r i t h m e t i c Part of Faltings's Proof by Bas Edixhoven

1

Introduction

In this chapter we will follow w of [22] quite closely. We start in the following situation: k is a number field, Ak is an abelian variety over k, Xk C Ak is a subvariety such that X k does not contain any translate of a positive dimensional abelian subvariety of A~, m is a sufficiently large integer as in Chapter IX, Lemma 1 Let R be the ring of integers in k. Since we want to apply Faltings's version of Siegel's Lemma (see Ch. X, Lemma 4) we need lattices in things like F(X~, line bundle). We obtain such lattices as: r(proper model of X~" over R, extension of line bundle).

2

Construction of Proper Models

First we extend Ak to a group scheme over Spec(R). Among all such extensions there is a canonical "best" one, the so-called N6ron model A ---* Spec(R) of Ak. It is characterized by the property that .A --~ Spec(R) is smooth and that for every smooth morphism of schemes T -~ Spec(R) the induced map .A(T) ~ Ak(Tk) is bijective. For more information and a proof of existence we refer to [11]; there is an "explicit" construction of .A/R when Ak is a product of elliptic curves. We will use the following properties of A ---* Spec(R): (i) it is a group scheme, (ii) every x E A~(k) extends uniquely to an x E .A(Spec(R)), (iii) .A ~ Spec(R)is quasi-projective ([11] w Thm. 1) and smooth. We choose an embedding i: A ~ ~,M. Let A be the closure of A '-~ ~,M x n pM by P ~ (iP, i ( - P ) ) , and let A ~ A be its normalization. Since A ~ Spec(R) is projective and A ~ ~ is finite (by Nagata's theorem [471, Thm. 31.H), A ~ Spec(R) is projective. By construction A contains A as an open subset, A Xspec(n)Spec(k) = Ak,

98

B. EDIXHOVEN

and the automorphism [-1] of Ak extends to A. Let s be a very ample line bundle on A. Then s := s | [-1]*s is a very ample line bundle on A whose restriction to Ak is symmetric. Finally let s = s |163 -1, where p: A ~ Spec(R). Then s is very ample on A and we have given isomorphisms s --~ [ - 1 ] ' s and Ospec(R) --=*0"s For 1 < i , j < m we have the line bundles /:i := pr*s and Pi,j := (Pri + prj)*s | pr*s -1 | pr~s -1 on A~. We choose positive rational numbers e < 1 and s as in Chapter IX, Thm. 4: if S l , . . . , s m are positive rational numbers with sl/s2 >_ S , . . . , S m - 1 / S m > S and e, < e then the restriction to X ~ of the Q-line bundle

.~C(--C% 81,...

8 m) := ,

,~1 -

-(2-~')~ @ s ,')Sl2 | ~)1 ]'~i --m

|

m--1

-~,~i+~

~[~i,i+ 1 i=1

i=2

on A~ is ample. Let A '~ denote the m-fold fibred product of A ~ Spec(R) and let s := pr~s on Am. We extend the 7)i,j to the open subset A TM of A m by the same formula as above: 7'~,j := (pr~ + pri)*s | pr~s -~ | pr~s -1. Using the construction described below it is easy to find a proper modification B ---* A m such that 1) B is normal, 2) B ---* A m is an isomorphism over .4m and 3) the 7~i,j on A TM can be extended to line bundles on B. 2.1 C o n s t r u c t i o n . Let X be an integral scheme, of finite type over Z, U C X open and normal, D C U a closed subscheme whose sheaf of ideals ID is an invertible Oct-module. Let /) be the scheme theoretic closure of D in X, let r: )( -~ X be the blow up in ID, let X ' be the normalization of the closure of r - l U in X and let /)' be the closure of D' := r - I D in X'. Then X ~ is normal, X ' --~ X is proper and an [] isomorphism over U, and If), is an invertible Ox,-module. We fix a choice of B and extensions of the Pi,j from A m to B. The pullback of the s to B are still denoted s We let Y be the closure of X ~ in B and we define yo := y N Am. Note that every x E X ~ ( k ) extends uniquely to an z E Y ~ We extend all line bundles s S l , . . . , sin) d to B by their defining formula.

3

Applying Faltings's version of Siegel's Lemma

We fix norms on s at the infinite places. This gives norms on the 1:i, the Pi,j and the s ,sin) d. In order to get some control over I'(Y, s a into using Faltings's version of Siegel's Lemma, we will embed s 1 6 2 @~ | s (a independent of the si, ~' and d) just as in the proof of Chapter IX, Thm. 4. However, this time we have to keep track of norms and denominators. We fix a finite set of generators fo, a C I, of the R-module F(A, s We choose an isomorphism r on A3: (pr 1 + pr 2 + pr3)*L: - - ~

(~

(pri + prj)*s |

l_
(~ pr~s 1
whose existence is guaranteed by the Theorem of the cube. Note that if r is another such isomorphism then r = ur for a unique u C k*. From r we can construct isomorphisms r on A~ for a, b E Z and 1 <_ i , j < m: r

(apri + bprj)*s --=--*s

|

| ~a,~

X/. ARITHMETIC PART OF FALTINGS'S PROOF

99

as follows. Let "P := (pr I -4- pr~)*L | p r ; s -1 | pr~L -1 on A~. We have (on A~' and on A~): r (pr i + prj, prt)*T' --:--* (pri, prt)*'P | (prj, prt)*T' and (~-1: [a --~ 1]*s

, [a]'C: | [a -- 1]*L-1 | s

And by definition: (apri + bprj)*/: -:+ apr,/: | bpr;/: | (apr,, bpr/)*7 ~ on A~. 3.1 L e m m a . (Lemma 5.1 of [22]) For any a, b, i, j let ~'~,b,~,j be the subsheaf of

OB-modules of the OAr-module (f_.~2 | C~2 | P@],.a/| generated by the Ca,b,ij(apri qbprj)* fi,, a E I. There exists Cl E ~( such that for all a, b, i, j there exists r E Z, 0 < r < exp(cl(1 + a 2 + b2)2) with: 1. r . ~,b,~,j C Z~ ~ | --3 r.b ~~ "P~b " t,3 (on B) 2. on Am: s | L~ | P;~,~ C r - 1 ' .~a,b,i,j P r o o f . Let us first check that the lemma holds for our choice of data if and only if it holds for any choice. If r = u - l r with u E k" then Cta,b,i, j ~- u a ( a - 1 ) / 2 T b ( b - 1 ) / 2 - a - b + 2

If us

C L~ C

and uTV,,j C 7~'.j C

u-ls

9

|

| r':i c z.', |

u-l~'~i,j

s

for all i, j then

| .p,~b

_,

r

,

c

u-(a~+b~+~b)L~~ .

b~ |

|

If 7r: B' --* B is another model satisfying the same conditions as B, then for any line bundle M on B we have: F(B',~r*M) = F(B, Tr.~r*M) = F ( B , M ) ; this shows that cl works for B' if and only if Cl works for B (if we take P[,j = 7r*Pi,j). Let us now prove statements (1) and (2) on Am. By [50] Lemme II, 1.2.1, there exists do > 0 such that s [ak can be extended to a s on A for which some isomorphism r on A 3 extends to an isomorphism over ,,4 3. This means that the lemma is true for s on A '~, hence for s and hence for s itself (on Am). To finish the proof we have to consider pole orders of the G,b,~,j(apr~ + bprj)*f~, along the (finitely many) divisors of B contained in B - Am; these divisors are irreducible components of closed fibres of B ~ Spec(R). Let V be the local ring on B at the generic point of such a divisor. Then V is a discrete valuation ring and prl and prj define V-valued points of A. The lemma is then proved by base changing from R to V, replacing A v by the N$ron model of A over V, and applying the arguments above. Note that those arguments give upper bounds on pole and zero orders on the whole model; that upper bounds on the pole orders on the whole model give upper bounds on the pole orders on V-valued points but that the same is not true for the order of zeros since the divisors of zeros of the G,b,id(apri + bprj)'f~ are not necessarily supported on closed fibres. [] Let s l , . . . , s m be positive rational numbers, let 0 < r ~ r and let d E Z, d > 0, be a sufficiently divisible square. Let di := 4ds~. Let ai (1 < i < m - l ) and fli (1 < i < m) be arbitrary elements of tile set I of labels of the f~. Multiplication by the product of the r~ "es~v~,,,+l,/~,~,i+l (si v~pr~ + s~+l v~pq+l)*f~, (with the r~ as given . ~E'ds~

by Lemma 3.1), the pr iJ~,

. r2ds~

, prlj &

~

. r2ds 2

ana prmJ~m

m

is a map from s

s l , . . . , s,,) d

100

B. EDIXfIOVEN

to @i~1 C/a'. Since the f~ (a E I) are generators of F(A, s give an embedding on B:

all such maps together

j = l i=1

where a = # I 2m-1. By construction, this embedding remains an embedding after restriction to Y; however, only on Bk = A~ we know that it is an embedding without zeros (that is, locally split). 3.2 C o n s t r u c t i o n s . 1. Let A be a commutative ring, a l , . . . , a ~ , r E A such that r E ~ i Aal. Then the homology of the (start of the Koszul) complex: 0 ~ A ~ A" --* A ~ ,

(zl), ~

x ~ (aix)i,

(ajxi

--

aixj)i,j

is annihilated by r. Proof: choose bi such that E aibi = r. Then use the "homotopy":

(x~)i ~ E~ b~x~, (x~,j),,j ~ (Ej ~x~,j),. 2. Let X be a scheme, s and Ad invertible Ox-modules, r E O x ( X ) , pj: s ~ 3`1 (1 < j < a) a finite set of morphisms such that r3`1 C ~ j p j ( s Twisting by s As in (1) we get a complex 0 --+ O x --~ | | s gives pj: O x ~ .Ad | s Gj=1(3`1 | Twisting by s gives: 0 -----}s ----+(~;_13`1 -----}~gi:~a2 -j = l " A-42 ""

|

s

The homology of this complex is annihilated by r.

[]

S l , . . . , sin) d ---* |163 In We apply these constructions to the embeddings pj: s this case we can even embed s s~,..., s,~) -d into @~=~N~=l s by multiplication . (4--e')ds?

by the ri" Csiv/~,_si+~./~,i,i+l(Siv/dprl - Si+lv~pri+l)*f~ i and by the p r l f ~ together we get a complex:

o

-~ s

Sl,..., s~) ~ - .

s

-~

@ |

'. All

s

j = l i=1

j=l i=l

of sheaves on Y whose homology on y o is annihilated by some r E Z with 0 < r < exp(q E i : l di). 3.3 P r o p o s i t i o n . (Proposition 5.2 of [22]) There exists c2 E R such that for ali .. . , S m , g positive rational numbers with 0 < e t _< e, d E Z square and sufficiently divisible there exists an exact sequence:

Sl,

a

0 ---r I~(X~ n, s

m

.$1, . . . , Srn)d) ___+F(X~n, O @ s j=l i=1

a3

m

__~ I~(X~n, O @ j = l i=1

s

such that: 1. the norms of the maps in the exact sequence and the difference between the norm on r ( x r , s 1 6 2 Sl . . . . , s,~) 4) and that induced on it from r ( x r , I ~ : 1 @~n:l s are bounded by exp(c2 ~ i dl), 2. if a section f of s s ~ , . . . , sin) d over X ~ maps to r ( Y o, @~ | s the~ there exists r E Z with 0 < r < exp(c2 E i d,) such that r f E r ( Y ~ s ~1,..., s~)e).

XL ARITIIMETIC PART OF FALTINGS'S PROOF

101

P r o o f . The exact sequence is obtained by applying F ( X ~ , - ) to the complex above. It remains to prove the statements concerning norms. To make sense of these statements we must say which norms we are talking about. The norms on the line bundles are obtained from those on/2. On k.-vector spaces such as F ( X ~ , metrized vector bundle) |

k~

we take the supremum norm over the complex valued points. On finite direct sums of normed kv-vector spaces we take the maximum norm. Finally, if V is a k-vector space with given norms on the V | kv then we take the maximum norm on

VQQR

= VQkk|

= VQk(|

= |174

~-j)) | R can be viewed as the supreNote that the norm we have put on (| m u m norm on I'(Xp Xsper Spec(R),f~jSvj), the supremum being taken over the P E X ~ ( C ) . The proof concerning the statements about norms is analogous to the arguments to bound poles and zeros at the finite places. The analog of Lemma 3.1 is independent of the choice of metrics on /2, and for a particular choice (metrics with translation invariant curvature forms) r is an isometry. [] We want to apply the following lemma (for a proof see Ch. X, Lemma 4). 8.4 L e m m a . (Prop. 2.18 of [22]) Let p: V --+ W be a linear map between normed finite dimensional R-vector spaces with lattices M and N such that p(M) C N. Let U := ker(p) and L := M N U; then L C U is a lattice. Assume that for some C >_ 2 we have: 1. p has norm < C, 2. M is generated by elements of norm <_ C, 3. any nontrivial element of M or of N has norm >_ C -a. Then t'or any proper subspace Uo C U R there exists an element f C L, f ~_ Uo such that II/ll -< ( 6,3 d i m V) dirn(V)/c~176 Now l e t U : = F ( X ~ , / 2 ( - d , 1,1/":= F ( X k~,

S l , . . . ,sm)e)|

I~j=laa @i=lm/23d~)NQe"

TM ma @~=~s174 V : = F It X k,vW=~

The lattices M and N are F(y,(Dj=l@ma i=1 /2/~') and

F(Y,~j=I=a @~i=l L;aa'~i J. Condition 1 is satisfied with C = exp(c2E;di). Condition 2 is satisfied with C of the form exp(ca E , d~) (with c3 independent of the s~) because S := e<,...,e~_>0r(Y, | c~') is a finitely generated R-algebra by the following argument. The morphism f : Y --+ X TM is proper and each s is of the form f*/2}, hence S = | ......e=_>0p ( x ,n , ( f . o v ) | 1 7 4 m ,d, ) by the projection formula ([27], Ch. 2, Exc. 5.1). Let t : X m '--+ P := ( p ~ ) m denote a closed immersion induced by /2 on each factor. Then S = @a......d=>0F(P,.7-(dl,...,d,~)), with ) r the coherent OR-module t . f , Or. One then shows that S is a finitely generated module over the multi-homogeneous coordinate ring of P. The proof of condition 3 that we give is copied fl'om [81], Lemma 13.9. 3.5 L e m m a . f 9 F(Y,|

There exists c4 C R such that /'or all d i > 0 and for all non-zero we have Ilfll >- e x p ( - c 4 E l di).

P r o o f . Let [k' : k] < oo, let R' be the ring of integers in k', P = ( P ~ , . . . , P m ) X~. (k') = Y ( R ' ) such that P * ( f ) ~ O. Then: rr~

h~cy,(P ) = [k': Q ] - ' degw(P* |

= y~ditzr.(Pi) i=1

9

102

B. EDIXHOVEN

On the other hand:

h| '

) = [k': Q]-I ( l o g # ((P* | f-.~') /R' . P*(f)) - y~ e(v')logllf(P)[[r I / ~ vqoo

where e(v') = 1 if v' is real and e(v') = 2 if v' is complex. Hence: e(v')log IIf(P)ll~' > -~-~dlhc(Pi)

[k': Q1-1 ~ vqoo

i=1

Let 7ri: Xk --~ paimXk be associated to s r*O(1) ~ / ~ , and let h be the naive height function on paimX~(k). Then there exists c4 such that [hc(Pi) - h(Tr,(Pi))[ < c4 for all k' and all Pi. Now we take the Pi such that all coordinates of the 7ri(Pi) are roots of unity; then h(TriPi) = 0. Note that these points are Zariski dense. So we have: [U: Q]-I ~

e(v')log

[[f(P)IIr

vqc~

> -e4Edi i=1

It follows that there exists an infinite place v of k with Ilfl[ .... p >_ exp(-c4 E~ d~).

[]

To get a suitable subspace U0 of U we put e' = e - c~ with 0 < a < e, let x = (Xl,... ,zr,) C X'~(k) be a smooth point and let Uo := {f E P(X~,/~(a - - e , S l , . . . ,sm)d)[index(x,f) >_ a} For this to be useful we need a lower bound for codim(Uo). Note that: m

~(Cr -- ~, S l , . . .

, .Sin) d = ~_~(--~, S l , . . .

, Sin) d | ( ~ ~_~a$2 i=1

Because /~(-r S l , . . . , sin) is ample on X ~ , /~(-r S l , . . . , sin) d has a global section that does not vanish at x. Because/Z is very ample on Xk there are global sections of/~d~,~ on Xk with prescribed Taylor expansion at xi up to order dcrs~ = di~r/4. Namely: "/~ very ample on Xk" is equivalent to "F(Xk,/~) separates points and tangent vectors" which implies "F(X~,s --* Oxk,,,/m~ is surjective for all i" which Irnl+~ad4,~ is surjective'. It follows that codimUo > implies "F(Xk, s --+ ~'xk,,~/r'~ c~( am I-[i di) aimxk, with c5 > 0 depending only on m and dimXk. In Chapter VII it was proved that d i m V _< c~(I-[i di) dimxk with c~ independent of the dl (and of a of course). It follows that (dim V)/(codim U0) _< c7~ -'~a*mx* with c7 independent of the dl and of a. Prop. 3.3 and Lemma 3.4 give us f ~t 0 in F(Y, s - r s l , . . . , sin) a) with index < (r at x and log [Ifl[ -< cs a-maamx* ~ = 1 dl (where cs is independent of the di and of a). We have proved: 3.6 T h e o r e m . (Theorem 5.3 of [22]) Let a be a rational number with 0 < a < ~. There exists a real number c(a), depending only on a (recall that ~ is fixed),

with the following property. For any smooth k-rational point x in X'~, any sequence Sl . . . . ,sin of positive rational numbers with sl/s2 > s , . . . , S m _ l / S , , > s, and any square integer d which is big enough and sumciently divisible there exists f ~ p (yo, s - r sx,... ,sin) a) with index less than cr at x and norm at the infinite places bounded by exp(c(a) Eim=l di), where di = 4ds~. []

XI. ARITHMETIC PART OF FALTINGS'S PROOF

4

Leading

Terms

and

Differential

103

Operators

This section more or less follows part of a manuscript [33] on [22] written by L. Lafforgue in 1991 (not published), see also [34] and [35]. We will need some properties of leading terms in Taylor expansions in the following situation. Let X be an integral separated R-scheme and x: Spec(R) ---+X an R-valued point of X. Let 01,... ,0,~ be R-derivations of Ox, let s be an invertible Ox-module and f a section of s vanishing up to order e along x, that is, f E F(X, I~s where I C O x is the ideal sheaf of x. Then for integers e l , . . . , e,,, > 0 with ~ , e, = e we can define (0~ ~... O~"f)(x) in F(Spec(R), z'Z:) as follows: locally on X write f = g fl with fl a generator of s and put:

(0~' ... O~'f)(x)

:= (cO;'... cO="g)(x)f,(x)

(one easily checks that the right hand side does not depend on the choice of fl, so the local construction "glues"). Also, it is important to note that (0f' . . - 0 ~ " f ) ( x ) does not depend on the order in which the derivations are applied, so that we can write (I-[i O~.'.f)(x) for it, and that for derivations 0~,..., 0',, locally of the form ~ = 0,+ terms like g.O with g E F(X, I) and 0: Ox ---+ Ox an R-derivation, one has

(l-[, i~i~' .f)(x) = (l-I, 0~' .f)(x). 4.1 L e m m a . In this situation, (l-Ii(O~'/ei!)'f)(x), which is a priori only a section of x*s over Spec(k), is actually a section o f x * s over Spec(R). P r o o f . It suffices to verify this locally, so we may assume that X is affine, that s is trivialised by fa and that f = g fl. Since f vanishes up to order e at x, g can be written as a sum of products gl " " g , with the gi in F(X, I). It suffices to verify the claim for each term, so we may assume that g is one such term, say g = gl "" "gr Now consider ((l-[~=10~')gl"" .g~)(x). This expression can be expanded by applying the product rule (O(fg) = fO(g) + gO(f)) as many times as possible. Since we evaluate at x, and the gi(x) are zero, only terms in which all gi have been derived can give rt~ a non-zero contribution. As ~ i ei = e, we see that ((lIi=x O~e i ) g x ' " g , ) ( x ) equals the sum over all partitions of { 1 , 2 , . . . , e} into sets $ 1 , . . . , S,, of cardinal,ties e l , . . . , e,,,, of the expresion

,=x \

jes~

/

i=1

so the claim that (1-I,(O~'le,!)'f)(z) is integral has now been proved.

[]

However, in w we cannot apply this result directly, because we will only know that f vanishes at x up to order e on the generic fibre Xk of X. The problem is then that in general the scheme theoretic closure in X of the closed subscheme of Xk defined by (I.Oxk) ~ is strictly contained in the closed subscheme of X defined by I" (in other words: I ~ ~ Ox fq (I'Oxk)~). Yet another complication will be that we will work in a product situation with weighted degrees for differential operators. The result we need is the following. Let m 3> 1. For 1 < i < m let p,: X, ---+P~ be a morphism of integral separated R-schemes of finite type, M , an invertible Ox~-module, G, E F(Xi, ./t4i) a global 1 section annihilating f~x#Pi and O,.i, 1 < j < hi, some set of R-derivations on 191. Let

B. EDLXIIOVEN

104

X Xm, P : = r l • • (fibred products over Spec(R)), and p: X ~ P the product of the Pi. Let Y be a separated R-scheme with a R-morphism q: Y + X that is an isomorphism on the generic fibres: qk: Ykz*Xk. Let N" be an invertible Or-module and N E F(Y,N') a global section annihilating ft~./x. Let m C Y(R) and write q(x) = ( x , , . . . , x m ) with xi E Xi(R). Suppose that for all i the morphism of k-schemes Pk:Xi,k ~ Pi,k is etale at xi,k, and that Pi + Spec(R) is smooth along pi(xi). Let Z; be an invertible Or-module, f C P(Y, t:) and let ~ be the index of flrk at xk with respect to some weights d , , . . . , d,~ > 0. Finally let ei,j >_ 0 be given with ~ i , j ei,j/di : 0-. Since the Pi,k: Xi,k ---+ Pi,k are etale at xl,k, the Oi,j can be uniquely lifted to derivations 0ij on a neighborhood of xi,k in Xi,k. Since Yk = Xk = [ I i Xi,k we can view the 0i,j as derivations in the ith direction on a neighborhood of zk in YkAs usual, one can define (I]i,i Oi,; ~ij f)(xk) in x~Z; to be (l-li50:,~'g)(xk)'fl(xk), where fl is a generator of/~xk and f = g fl. X := Xl X-''

4.2 L e m m a . In this situation,

m extends to a global section over Spec(R) of x*s | x*N ~ | @i=1 xi 9M i ei , where ei = ~ j ei,j and e = ~ i ei.

P r o o f . It suffices to check the statement locally, so after localizing R we may assume that L: has generator fl, that f = gfl and that the M i and .M are trivial. Let Ji C Op~ be the ideal sheaf of p~(x~) and let J C O p be the ideal sheaf of p(q(x)). Let Mi C O x , be the ideal of xi and M C (gx the ideal of q(x). Let I C Or be the ideal of x and Ii := O r ' q # M i . The closed subscheme of Y defned by the ideal I ~+1 + Y~im=l I ei+l is finite over Spec(R), hence affine, so isomorphic to Spec(A) for some R-algebra A. The ideals of A induced by I and tile Ii will be denoted by the same symbols. For each i, let Spec(Bi) denote the closed subscheme of Xi defined by M_[I+1. Let B := @i~__,Bi (tensor product over R); then Spec(B) is the closed subscheme of X defined by ~ , M[ ~+~. Finally, let Spec(Ci) be the closed subscheme of P~ defined by j~+x and let C := | C~ (tensor product over R). Then we have a commutative diagram Y ~ X ~ P

T Spec(A)

T ---, Spec(B)

T -~

Spec(C)

The rings A, B and C are R-modules of finite type, the morphisms C ~ B + A induce isomorphisms C | k-z-~B | kZ-~A | k. Because of the smoothness of the Pi + Spee(R) at the p(xi), the rings Ci and C are torsion free R-modules. 4.3 L e m m a . Let B be the image of B in A, then A / B is annihilated by N% 1 P r o o f . We know that N annihilates fly~x, hence where M is the image of M in A; note that M C 12 + A - M = 12 + M + I M = 12 + M. It follows that N~I c I~+t + M = M, hence N~A = N+(R + I) c R

~ = I/(I 2 + AM), also x * f~r/x I. Since A = R + I , we have N I C 12 + -M. Induction shows + M = B. []

XL ARITHMETIC PART OF FALTINGS'S PROOF 4.4 L e m m a .

105

For all i, the morphism Ci ~ Bi is injective, and Bi/Ci is annihilated

by a~'. P r o o f . The morphism Ci ~ Bi is injective since Ci is torsion free and Ci,k--r-+Bi,k. The proof that G~~ annihilates Bi/Ci is the same as the proof of the previous lemma. [] 4.5 L e m m a .

The morphism C ~ B is injective, and B / C is annihilated by l-Iim__lGei.

P r o o f . The morphism C --~ B is injective since C is torsion free and Ck-:-+Bj,. Let us consider the filtration B= B 1 DB 2 D "-DB

m =C

where B j is the image in B = B1 | ' " | B,~ of @i<j Ci | @i>j Bi. The successive quotient BJ/B j+~ of this filtration is a quotient of @~<jC~| | Bj, hence by the previous lemma it is annihilated by G~~. The lemma follows. [] rn m We have now proved that h := (N e I]~=1 Gie i )g, modulo ie+l + ~i=1 i~+a, is in C. Note that (I-{,,j O~,~'g')(x) is zero for all g' C I ~+~+ ~'~=, I~'+1. Let us consider h as an element of C. Since the index at xk of flYk is a, it follows that the image of h in Ck is in the ideal Ck'Jf . . . . . . J ~ . Since the ring C/(C.J; ~..... J,~') is torsion free (this follows again from the smoothness of P --+ Spec(R) at p(q(x))), h is actually an element of C.J~ . . . . . . J ~ . The method of the proof of Lemma 4.1, plus the fact that the 0i,j are derivations on P in the direction of Pi, show that ((l-Ii,j O~J/ei,fl)h)(p(q(x))) is integral. []

5

Proof of th~ Main Theorem

5.1 T h e o r e m . (Theorem I of [22]) Let k be a number field, Ak an abe/ian variety over k and Xk C Ak a closed subvariety such that X~ does not contain any translate of a positive dimensional abelian subvariety of A~. Then Xk(k) is finite. Proof. The proof is by contradiction: we suppose that Xk(k) is not finite. We may also suppose that Xk(k) is Zariski-dense in Xk, because we can replace Xk by a positive dimensional irreducible component of the Zariski-closure of Xk(k). The situation is now as in w167 A, s m, B, Y, the Pid, c and s are fixed. We embed Xk into some projective space over k by choosing a basis of r ( X k , s Composing this embedding with a suitable projection as in Chapter IX, Lemma 5 we get a finite morphism pr:Xk ---+P~, where n = dimXk. Note that s = pr*O(1). Let X denote the closure of Xk in A and let X --~ X be a proper modification such that Xk = Xk and that pr: Xk ---+Pk extends to pr: X ~ PR (e.g., one can take X to be the closure in X Xsp~(R) P~ of the graph of pr: Xk ---+~ ) . We choose an integer g and a non-zero G in F(P~t , O(g)) which annihilates fi2/p~l (Lemma 5 of Chapter IX shows the existence of such a G annihilating f~ck/P~; multiplying such a G by a suitable integer gives the G we want). Let ~" ~ Y be a proper modification such that ~'k = Yk and that the projections pri: Yk = X'Z ---+Xk on the ith factor extend to pri: Y ~ X'. In particular, Y" is a model of X ~ dominating both Y and X ---~ (infold fibred product

106

B. EDIXItOVEN

over Spec(R)). Since Y --+ X "-'~ is an isomorphism on the generic fibres, we can choose an integer N > 0 such that N annihilates f~k _ For 1 < i < m let Pi := P~ and y/x m" Pi,R := P~. Let pi: Y -* Pi,R be the composition of pri: Y ~ X" and pr: X" ~ P ~ . Let P := lli~l Pi and PR :-~ 1-Ii~l PI,R (fibred products over k and R) and let p: Y --~ PR be ( P l , . - . ,Pro). For 1 < i < m and 1 < j _< n let 0i0 be the derivation xibc3/c3xld on PR, where the xl,j, 1 <_j <_ n, are the coordinates of the standard affine open A~ in Pi,u. Let Gi be the pullback of G to X TM via the projection on the ith factor. Finally, we choose a non-empty open subset U C Xk such that G is invertible on U and the Oi,j generate the tangent space at every point of U (this last condition just means that U is disjoint from the n + l coordinate hyperplanes). In particular, this implies that pr: Xk ---* P~ is etale on U. Everything up to here will be fixed during the rest of the proof. The following two variables a and b will be crucial; a is a rational number with 0 < a < ~, and b is any rational number. In the end we will first have to take a small enough, then b large enough. For any a and b we can find a point x = (Xx,... ,Xm) in X'~(k) such that:

1. h(xl) >_ b (here h denotes the height function on Xk(/c) associated to /3; in particular we have h(xl) = [k: Q]-I deg(x~L)) 2. h(x2)/h(Xl) ~ 282,..., h(xm)/h(Xm_l) > 2s 2 3. ( x . ~ + x )

Ak(ic) and

___ (1 - e/a)ll~llllz~+~ll (here (x,y) denotes the N&ron-Tate pairing in I1~11= = ( ~ , ~ ) )

4. xl E U(k) (here one uses that, by the theorem of Mordell-Weil, the unit ball in A(k) | R is compact). Let hi := h(zi). We choose s i e Q close to h,1/2: [s~hi - 1[ < b-~ will do. Let d be a square integer which is big enough and sufficiently divisible, and put dl := 4ds~. Suppose that b > 3. Then si/si+l > s for 1 < i < m, so we can choose a s e c t i o n f in P ( Y ~ 1 6 2 . . . . ,am) d) as in Thin. 3.6: the index o f f at x is less than a and the norm of f at the infinite places is bounded by exp(c(cr)~i dl). Since the index of f at x is less than a there exist non-negative integers e~5 such that

((YIi,jOi:~j)

f ) ( x ) ~ 0 and, if we write ei = ~jei,j, Y~iei/di = (index of f at x) < a. By Lemma 4.2 we have a non-zero integral section

f

t

!I k e,/

of the metrized line bundle M=

X*

(T __ E , $ 1 , .

. . ,Sm

d

ig

i=1

/

on Spec(R). We will compute an upper bound on the degree of M using the properties of x and of s - r .... ,8,~), and a lower bound using f ' . These bounds then give a contradiction.

XL ARITHMETIC PART OF FALTINGS'S PROOF

107

5.2 L e m m a . There exists c2 E R, not depending on b, a and the xl, such that [k: Q]-I deg x*C(~ - ~ , S l , . . . , s ~ ) ~ _< md(~ - ~/2) + c:db -a

Proof.

Let h denote the N6ron-Tate height on Ak(~:) associated to s Then there [ ( y , z ) - [k: Q]-I d e g ( v , z ) . p l < cl for

exists c~ ~ R such that Ih(y) - h(y)l < c~ and all y, z in Ak(k). Then we compute:

[k: Q]-I d e g x * s

r

d= m,--1

=

(1 + a - r

+ ~ (2 + a - e)ds~hi + (1 + a - r i=2

m-1

-

dsisi+x[k: Q]-X deg(xi, xi+,)*P

~ i=1

m-1

<

(1 + a - r

+ b-~) + ~ (2 + a - e)d(1 + b-~) + (1 + a - c)d(1 + b-~) i=2

ra-1

--

E

dsiSi+l ((xi, Xi+l) -- C1)

i=1 rr~-I

<

+ 2(m-1)d)(1--}-b -1) - E ds,s,+~(1-U4)llz, ll.llx,+~ It + 2cldmb-'

(m(a-r

i=l m--1

-- 1 ) d ) ( l + b -1) __ d ( 1 - r

<_ ( m ( o ' - r

~1/2v~hi+ ^1/21 siSi+lv/2hi i=l

+ 2 0 d m b -1 <

1)d)(1 + b-1)

(m(a-r rrL--1

-d(1 - ~/4)2 E sisi+l(hi - cl)1/2(hi+1 - Cl) 1/2 + 2Cldmb -1 i=1 rr~-I

<

+ 2(m - 1)d) (1 + b-1) - 2d(1 - ~/4) ~

(m(a - r

sin. 1i/ 2

11/2[1

Si+a.i+lkX

--

clb_l)

i=1

+20drab -1 <_ ( m ( a - r

+ 2(m-1)d) (l+b -1) - 2 d ( 1 - c / 4 ) ( 1 - c ~ b - ' ) ( m - 1 ) ( 1 - b - ~ ) 2

+2Cldmb -1 <_ ( m ( a - r

+ 2 ( m - 1 ) d ) (l+b -~) - 2 d ( 1 - r

- ( 2 + o ) b -~) + 2c~dmb -~

< m d ( a - ~/2) + b-l(2md + 2din(2 + Ca) + 20din) = rod(,, e/e) + (6 + 4c,)mdb -1 -

This means that the lemma holds with c2 = (6 + 4cl)m. 5.3 L e m m a .

There exists Ca E

R,

[k: Q]-I deg M Proof.

not depending on < md(c3a - r

b, a

[] and the xi, such that

+ c2db -1

We have: m

[k: Q] -1 deg.A4

_< rod(or - r

+ c2db-1 -~- ~ ( e i / d i ) g d i h i i=1

108

B. EDIXHOVEN m

= rnd(a

-

e/2) + c~db-' + ~_,(eJdi)g4ds~hi i=l m

~

~ d ( ~ - ~/2) + c~db-~ + ~ ( ~ , / d , ) g 4 d ( 1 + ~-')

<

~d((1 + S ~ / ~ ) o - s/2) + e~db-~

i=1

[]

This means that the lemma holds with c3 = 1 + 89/m.

5.4 L e m m a . There exists c4(a), not depending on b and the x~, such that for each infinite p/ace v of k we have log [[f'[[. _< e4(a)db -~. P r o o f . Let v be an infinite place of k. To simplify notation, we denote by ]].][ a norm at v. We have:

,=

0

Note that el < adi < di as ~i ei/di < a, and that ~ i di < 8mdb -1. The first factor of the right hand side of the formula above is easy to bound, but when it is small we will need that to bound the whole right hand side. There exists a j such that pj:E(a-r a ---* | has norm > exp(-cs~idi) at x (this es is the c2 from Prop. 3.3). Let h := pj(f); then h E F(X~,p*O(dl,...,dm)). The metric on @i/:d~ we fixed a long time ago and the pullback of the product of the standard metrics on the O(dl) differ by a factor bounded by c ~ ~a,. Hence it suffices to show that there exists c~(a), such that we have:

where the metrics are now the standard metrics. Recall that the standard metric on O(n) on ~ m ( C ) i s given by: [[FII(P0 : . - - : Pro) = IF(Po,...,Pm)I/(~-~. IP~[2)n/2 i

Let Yi = Pri(xi); then y~ E Pn(C). Let Z 0 , . . . , Z~ in F(P~, O(1)) be the homogeneous coordinates of P~, and for 1 _< j _< n let D+(Zj) C P'~ be the affine open subscheme given by Zj -fi 0; then D+(Zj) TM A~ and the Zt/Zj (l ~ j) are coordinates on D+(Zj). For r in R let D+(Zj), denote the standard polydisc of radius r in D+(Zj)(C) given by IZdZjl <_r (1 ~ j). For each i in { 1 , . . . , m } we choose j~ in { 0 , . . . , n } such that Yi E D+(Zj)~, i.e., the coordinates Yi,j of yi in D+(Zj,) satisfy [Yi,jl -< 1. We can bound the derivative, on all standard polydiscs D+(Zj)2 of radius 2, of G in some trivialization of O(g). The conclusion is that there exists c7 > 0, cr < 1/I[G][~p2"(c), such that G has no zeros in the polydiscs Ai C D+(Zj,)2 of radius ri := cr]lG]l(yi) and center y~. It follows that pr~:Xk --* P~ is etale over Ai. Let A = 1-I~=lA~. Then A is simply connected, hence p - l A C X'~(C) is a finite disjoint union of copies of A; let A ~ denote the copy that contains x.

X L ARITHMETIC PART OF FALTINGS'S PROOF

109

We trivialize O(di) on Ai using its section Z~~. In other words, on A' we write

h = ha l']i ZS,'. The standard norm of Z~,' on Ai satisfies: (n + 1) -d'/2 <_ IlZJ' II < i on A,. By definition we have:

so it remains to estimate the first factor of the right hand side. On A' we write hi = ~ a(.)z("), with (n) = (n,,j) (here the z,,j are the standard coordinates on A': zi5 = pr~(ZffZj, - yi,j)). Then we have:

(O/Oz'")'~"J ni,j !

(hl)(X) = a(,~,,,) -

1 (2~rx/,~]-)m,~

9 ,,il:., hx 11 ia zid

"

I I dzld

i,j

Hence for (n) with 2~ n,,~ = e, for all i we have la(.,,j)[ < [[ha [[s.p,~' l-I, r~-~'. Now note that [[hl IIs.p,~x' is bounded above by some exp(c~(a)db -1) by the arguments above plus the fact that the norm of f is bounded by exp(e(a) ~ i di), hence we have:

(rv (OlOz,,,)",,, ~9 (hi )(x) ni,j!

]

< e~p(4(~)~b-') II~,

--

-~'

i

A standard computation shows that on Ai one can write Oi,j = El bi,j,z(O/O~,.,) with Ibi,j,,(y,)] <_ 1 (in fact, bi,j,t(y,) 9 {0, +Yi,t})- Using this, we can write:

%' )(:) \\i,j

:

,o'/

=

( (ala:"J)"' "j '~(h,)(:) ~

(.,,,)

~

C(.,,,)(x).a(.,.,)

("~.i) where the C(~id ) are certain polynomials in the bi,j,l. Note that C(,i.j)(x ) = 0 unless for all i one has ~ j ni5 = el. From Ibid,t(x)[ < 1, and some combinatorics, it follows that:

c(,.,,.)(.,)l-< IIi

::!

e i , l ! ' ' ' e i , n!

Note that the right hand side does not depend on (ni,j). Using this one gets:

Z IC(,,,,,)(~)l < ~ 9 ei,,!:--ein! e,! (r,,.,)

(e, + ,~ - ~)! -< ri(2.,+,,_,no,) ~--]~e~. ,. <-2'~("-l)(2n)F-"~'d'

This means that finally we have:

which finishes the proof.

[]

110

B. EDIXHOVEN

5.5 L e m m a . There exists cs(a), not depending on b and the xi, such that [k: Q]-I deg.A~ > -c8(o)db -1 Proof.

By definition we have: deg 34 = log #

~

- ~ e(v)log ItI'll~ >- - [ k : Q]c4(~)db -1 vleo

[] We can now finish the proof of Thm. 5.1. Take a < r (ca as in Lemma 5.3). Then for b big enough, Lemma 5.3 and Lemma 5.5 give contradicting estimates for [k : Q]-I deg 34. []

Chapter XII Points of Degree d on Curves over Number Fields by Gerard van der Geer

Arithmetic questions on the number of points of degree d on a smooth (irreducible) algebraic curve over a number field lead to geometric questions about the curve by using Faltings's big theorem. We discuss here some questions and conjectures initiated by Abramovich and Harris in [4]. We refer to the end of this paper for a discussion of the recent literature. Faltings's theorem (previously Mordell's conjecture) says that an irreducible curve C of geometric genus g _> 2 defined over a number field K possesses only finitely many K - r a t i o n a l points. But since over an algebraic closure K of K the number of points has the same cardinality as I-( itself one is led to the question how the number of points grows with the degree of the extension L/K over which ,we consider our curve. More precisely, what can we say about

Fe(C,K) = {p E C ( K ) : [ K ( p ) : K] _ 1) we find infinitely many points of degree _< d. This raises a multitude of questions. To begin with, the question arises whether # F a ( C , K ) is finite when C does not admit a dominant K - m o r p h i s m C ~ F'1 or C --* E of degree < d; or phrased differently, if # F d ( C , L) = oo for some finite extension L/K, does this imply that C admits a dominant morphism of degree _< d to F'1 or to an elliptic curve ? As we shall see, the answer is : no! There are curves with infinitely many "interesting" points of degree d (i.e., not obtained from a morphism of degree d to ~t:~l or to an elliptic curve). To put the questions in the framework of this volume we introduce the algebraic

112

G . VAN DER GEER

variety (for d E Z>I) C (a) = Sym(d)(C) = Cd/Sd, the d-fold symmetric product of our curve. We have morphisms d

ed : C (d) "-'+Pic(d)(c)

given on geometric points by {pl,... ,Pd} ~ E P i . i=1

Here Pic(d)(c) is the variety of divisor classes of degree d. The geometric) fibres of this morphism are the linear systems:

IDI = {(P,,...,Pd): ZP' ~ D}, where ,-, denotes linear equivalence. Some remarks are in order here. (i) By Riemann-Roch we have h~ _> 1 for d > g, hence Cd as surjective. (ii) If C does not admit a dominant morphism C --+ ~,1 of degree _< d then Cd is injective. (iii) Any curve of genus g admits a morphism of degree < [(g + 3)/2] onto l?1. We need more notation. Define W~ = {O E Pic(d)(C) : h~

>_ r + 1}

and set Wd = W2 = Cd(Cr These (functors) are (represented by) algebraic varieties defined over K. Examples of these are: W~ = O C Pic(g-a)(c), the famous theta divisor of effective divisor classes of degree g - 1 on C and

w)_, = Sing(O), the singular locus of the theta divisor. We know that dim Sing(O) > g - 4. Let us now assume that C does not admit a dominant morphism to •1 of degree d. Then ed is injective. A point of C of degree d (over K) determines a K-rational point of C (d) (note that the natural local coordinates near {Pl,-.., Pd } are the elementary symmetric functions in local parameters ti near Pi) and in turn this determines a K-rational point of Wd. So if we know that ~Vd contains only finitely many K-rational points then necessarily Fd(C, K ) is finite. But here Faltings's Theorem on rational points on subvarieties of abelian varieties comes in and weaves the present theme into the texture of this volume: if Hid does not contain the translate of a positive dimensionM abelian variety then Wd(K) is finite, and then Fd(C, K ) is finite too. All this leads us to the question: when does a jacobian variety contain an abelian subvariety (always assumed to be of positive dimension) in its Wd ? One way for this

XII. POINTS OF DEGREE d ON CURVES OVER NUMBER FIELDS

113

to happen is when C is a covering of a (complete irreducible smooth) curve D. If 7r: C --* D is a morphism of degree n with g(D) h we have an induced morphism 7r*: Pic(h)(D) --+ Pic("h)(C) whose image lands in W,~h (since ed(D(h)) = Pic(h)(D)) and thus we see that for nh < d we find a translate of an abelian variety in Wd(C). This could also happen if our C is the image of a curve C' which is a d-fold covering of a curve D. Again we are led to speculate about the converse: 1 Q u e s t i o n . If C is a curve of genus g and Wd(C) for some d < g contains a maxima/ abelian subvariety of dimension h then does it follow that C is the image of a curve C ~ which admits a dominant morphism C ~ -+ C" of degree < d/h with g(C") = h ? [] Let us call this question A(d, h;g), i.e., does it hold for all C when d, h,g are fixed. A related question is: 2 Q u e s t i o n . If C admits a dominant morphism C ~ -+ C and C' admits a dominant morphism C' ~ C" of degree _< d/h and g(C") = h, then does C itself admit a dominant morphism C --+ C"' of degree < d/h with g(C") < h ? [] Call this question S(d, h;g), i.e., does it hold for all C of genus 9 and the given values of d and h ? For points of degree d we have the relevant question: 3 Q u e s t i o n . Is it true for all irreducible smooth curves of genus g over a number field K that #Fd(C, L) = cr for some finite extension L/K if and only if C admits a [] map of degree _< d to I~ 1 o r to an elliptic curve ? Call this question F(d, g). Note that positive answers to A(d, h; g) and S(d, h; g) for all h with 1 < h < d imply F(d,g) using Faltings's theorem. Indeed, we may deduce from it that C admits a morphism of degree _< d/h to a curve C ~of genus < h. Either C' is of genus _< 1 or _> 2. In the latter case the curve C' admits a morphism of degree _< h to p1 since h > [(h + 3)/2]. In any case we find a morphism of degree _< d to P1 or to an elliptic curve. We do not need consider all h in the range 1 , . . . ,g in view of the following remark. 4 R e m a r k . An affirmative answer to A(d, h; g) implies the following statement:

for d < g the variety Wd cannot contain an abelian variety of dimension > d/2. This last statement was proved to be true by Debarre and Fahlaoui [16].

[]

There are some scattered results concerning question F(d,g). Abramovich and Harris proved it in [4] for d = 2 and 3 (all g) and for d = 4 provided that in the latter case g @ 7. Using the same arguments one can also show it for other d provided g does not lie in a certain interval, e.g. d = 6 and g - 10 or g > 17. One could also consider a sharpened version of the question : suppose that #Fd(C, L) = oe for some finite extension L, but #F~(C, M) < e~ for all e < d and all finite extensions M/K, then does it follow that C admits a non-constant morphism of degree d to p1 or to

114

G. VAN DER GEER

an elliptic curve ? It seems that this sharpened question admits a positive answer for d = p, a prime and g < 2 p - 2 or g > (~) + 2. None of the questions A(d,h;g) and S(d, h; g) has a positive answer for all triples for which the question makes sense. Abramovich and Harris conjectured in [4] that F(d, g) is true for all tuples (d, g) with d > 1 and g > 0. However, this was disproved shortly afterwards by Debarre and Fahlaoui [16]. The questions A(d, h; g) and S(d, h; g) do not admit a positive answer for many triples (d, h,g). To point out some positive statements, one can see that A(d, h;g) has an affirmative answer for h = 1. Similarly, the question S(d, h; g) has an affirmative answer for g = 0. Coppens [15] proved that S(d, h,g) holds for d a prime and g large using Castelnuovo theory. He also shows that counterexamples to S(d, h,g) produce counterexamples to S(md, h, g) for large enough g. In the following we give a few principles from [16] which give rise to some affirmative answers to the questions A and S and we give an easy counterexample to A(2h, h, 2h + 1). For more results we refer to the literature, although the terrain is largely unexplored and the interested reader might find rewarding challenges there. 5 L e m m a . Assume that | C Pic(g-1)(C) contains a subvariety Z stable under translation by an abdian subvariety A C_Jac(C). Then

dim(Z) + dim(A) < g - 1. P r o o f . We may assume that Z is irreducible and meets 6),~g (otherwise, replace Z by Z + Wd(C) - We(C), where d + 1 = multiplicity of 6) in the generic point of Z). Consider the Gauss map g: 6)~g --,. P(T0(Jac(C))*),

x ~ Te,~.

If x E Z C/6)reg, then To,, contains the fixed space x + T~,0, hence we find a map g: Z n 6)reg ~

~((TJac,olTA,o)*).

But for a Jacobian the Gauss map has finite fibres (see e.g. [5]), hence dim(Z) <_ g - dim(A) - 1.

[] Similarly, if W~ contains Z which is stable by A and if d _< g - 1 + r then dim(Z) + dim(A) < d - 2r. (Apply the Lemma to Z + Wg-l-d+~ -- Wr C Wg-1 = 6).) When do we have equality ? 6 L e m m a . Suppose that C is such that Wd(C) contains Z stable under translation by an abelian variety A r (0) in Jac(C). Assume that dim(Z) + dim(A) = d and dim(Z) + d _< g - 1. Then there exists a curve B of genus h = dim(A) and a morphism r:C ---4B of degree 2 such that A TM Jac(B) and Z -- 7r*(Pic(h)(B)) A- Wa-2a(C).

XII. POINTS OF DEGREE d ON CURVES OVER NUMBER FIELDS

115

There is a variation of this Lemma in which one reads W j instead of Hid and assumes d i m ( Z ) + dim(A) = d - 2r and in which one then finds that Z = ~r*Pic(h+~)(B) + Wd-:r-2h(C). Taking Z = A one obtains the corollary: 7 Corollary. Let C be such that W~(C) contains a translate of an abelian variety A of dimension h > O. Assume that d <_ g - l + r . Then dim(A) _~ d / 2 - r . If d <_ ~(g - 1 + r) we have equality if and only if d is even and if there exists a curve B of genus d/2 - r and a morphism ~r: C ~ B of degree 2 such that A = ~r*(Pice/2(B)). Note that this implies the validity of A(2h, h;g) for h < g/3. It is not difficult to construct counterexamples to A(2h, h, 2h + 1) for h _> 4. One uses P r y m varieties of double covers as follows. Let C ---* D be a double etale cover of a smooth irreducible curve of genus g(D) = h + 1. Then the genus g(C) equals 2h + 1 and we can consider the P r y m variety P = P r y m ( C --+ D) = ker{Nm : Jac(C) ~ J a c ( D ) ) ~ Here Nm is the norm map. This connected component of this kernel has dimension h and can (up to translation) be identified with {d 9 Wg-l=2h(C): ~r.(d) = KD, h~

- 0(mod2)},

where KD is the canonical divisor class of D. The claim is then that for general D and h >_ 4 there does not exist a double covering p : C ' --~ B with g(B) = h and a non-constant morphism q: C' --* C. Indeed, suppose that it does exist. Then we find q.(p*(Jac(B))) inside Jac(C). For D general it is known that the jacobian Jac(D) is simple and that the P r y m P is not isogenous to a jacobian for h >__4. We see that Jac(C) is isogenous to a product P x Jac(D), both factors of which are simple and not isogenous to each other. It follows that q.(p*(Jac(B))) is a point. But this is impossible because then for all D E Jac(B) we have deg(q)p*(D) = q*q.p*(D) = 0 which contradicts the fact that the kernel of p* is finite.

8

Additional Remarks

If we change the point of view somewhat (fi'om the jacobian to the abelian variety in its Wd(C)) we might pose the question differently: which curves do lie on a given abelian variety ? Answers to this type of questions (very interesting in their own right) can also be helpful in that they put restrictions on the possible abelian varieties. Instead of curves over a number field one might consider curves over function fields and ask these questions there. Questions A and S make sense for curves over any field.

9

Discussion of the Literature

Many of the questions treated here were first stated explicitly in a paper of Abramovich and IIarris [4] (the title of which is not what you would think). The merits of this interesting paper lie more in the questions it poses than in the answers it provides. The questions "A", "S" and "F" were raised there (though in a slightly different form and question "F" was stated as a conjecture there). The thesis of Abramovich [1]

116

G . VAN DER GEER

contains additional material. The first author wrote a corrigendum [3] to [4] (the title of which is not what he thinks) pointing out a number of gaps and misprints in [4], e.g., the proof of Lemma 6. In Lemma 7 one should read : rk+l - rk >_ rk -- rk-1. Besides that there are more lapses. In particular, Lemma 8 (p. 223-224) of [4] is false. Abramovich tells me that he manages to salvage Theorem 2 with a lot of effort, but so far he did not publish how he did that. The main conjecture of [4] was disproved by Debarre and Fahlaoui [16]. They give partial answers to the questions A(d, h; g) and S( d, h; g). Debarre and Klassen [17] classify all points of degree d on a smooth plane curve of degree d. In the paper [6] of Alzati and Pirola the reader will find some related results. Finally, in the paper by Vojta [84] the reader will find a different approach to points of given degree over a number field using arithmetic surfaces.

Chapter XIII

"The" General Case of S. Lang's Conjecture (after Faltings) by Frans Oort

In this talk we discuss (see Thm. 3.2 below) the main result of [23]. It proves the conjecture stated in [37], page 321, lines 12-15, and it generalizes the main result of [22], which is the theme of this conference. Also see [81], Theorem 0.3, and w We assume (for simplicity) that Q c K C k = k, i.e., K is a field of characteristic zero contained in an algebraically closed field k. 1

The

Special

Subset

of a Variety

1.1 C o n s t r u c t i o n . Let X be a variety over an algebraically closed field k. Consider all abelian varieties C, and all non-constant rational maps

f :C...-+X. Let Sp(X) C X be the Zariski closure of all the images of these maps. This is called the special s u b s e t of X. (Note that a rational map does not have an "image" in general, but f is defined on a non-empty open set, and the closure of that image is well-defined, etc.) If Y is a (reducible) algebraic set, Y = UiXi, then we write Sp(Y) := U, Sp(X~). [] 1.2 Q u e s t i o n . Do we really need to take the Zariski closure, or is all images already closed?

just the union of []

This seems to be unknown in general, but in the case considered in the next section this is true. 1.3 E x a m p l e . In the situation of [22] we have Sp(X) = 0, and Thin. 3.1 and Thm. 3.2 below generalize the main result of [22]. [] Note that an elliptic curve can be mapped onto ~,l, hence we see that Sp(X) contains all rational curves contained in X. In fact, for every group variety G and for every non-constant rational map f from G into X the image of f is contained in Sp(X), and Sp(X) could be defined using all such maps.

118 1.4

F. OORT T h e Kodaira D i m e n s i o n

For a variety X over k one can define the Kodaira dimension, denoted by Jc(X) (see [79], w We briefly indicate the idea: take a complete normal model of X, let K x be its canonical divisor, let n be a rational integer. If every section in the sheaf f~(nKx) is zero (for every positive rational integer n) we write x(X) = -(x~. In all other cases we write ~(X) for the maximum (for all n) of the dimensions of the image of the (multicanonical) rational map defined by n K x . One can show that this is a birational invariant. We say that X is of g e n e r a l t y p e (or: of hyperbolic type) if dim(X) = x(X). Some examples: ( d = l ) For algebraic curves we have:

a(c)=0

r

g(E)=l

r

g(C) >_2 r

a(E)=O g(C) = 1 r

C is of general type

We have: x(pd) = --co for d > 0, a(F ~ = 0 and a(abelian variety) = 0. 1.5 C o n j e c t u r e .

(5;. Lang, cL [38], page 17, 3.5) ( G T ) :

Sp(X) ~ X ~

X is of general type.

1.6 R e m a r k s . If X is defined over K, then Sp(Xk) is also defined over K , i.e., there exists a K-closed subset Sp(X) C X so that Sp(X)k = Sp(Xk) (but note that the abelian varieties C and rational maps f : C . . . ---* Xk need not be defined over K). For algebraic surfaces not of general type one sees that indeed Sp(X) = X; for K3-surfaces, see [51], page 351. It seems that Conjecture 1.5 for surfaces of general type is still open. [] 1.7 C o n j e c t u r e ( L a n g ' s c o n j e c t u r e ) . that K is of finite type over Q, then

(LC): Let X be defined over K , suppose

# { x [ x E X ( K ) and x ([ Sp(Xk)} < oc

(cf. [38], page 17, 3.6). For ( G T ) and (LC) also see [39], page 191. Note that for a curve X the special subset is empty iff the genus g ( X ) > 2. Hence ( G T ) is easily proved for curves, and (LC) is "Mordell's conjecture", proved by Faltings (first for number fields, later for fields of finite type over Q, cf. [24], page 205, Theorem 3). If the conjectures ( G T ) and (LC) are true we could conclude that the following conjecture holds: 1.8 C o n j e c t u r e . (cf. [80], page 46) If X is a variety defined over K , a field of finite type over Q, and X is of general type, then the Zariski closure in X of X ( K ) is a proper subset of X . (This conjecture was mentioned by Bombieri in 1980, cf. [55], page 208). 1.9 R e m a r k . Note that the definition of Sp(X) is purely "geometric", but we shall see the importance of this notion for arithmetic questions. []

XIII. "THE" GENERAL CASE OF S. LANG'S CONJECTURE

2

119

T h e Special S u b s e t of a Subvariety of an A b e l i a n Variety

We denote by A an abelian variety over k, and by X C A a closed subvariety. Notation: Z ( X ) = {x E X [ 3B C A, an abelian variety, d i m B > 0, and x + B C X}.

Note that Z ( X ) C Sp(X). From the fact that any rational map between abelian varieties extends to a morphism of varieties (see [14], Chapter V, Thin. 3.1), and moreover that any morphism of varieties between abelian varieties is, up to translation, a homomorphism of abelian varieties (see [14], Chapter V, Cor. 3.6), it follows that Sp(X) is the Zariski closure of Z ( X ) . 2.1 T h e o r e m . (Ueno; Kawamata [32]; Abramovich [1], Thins. 1,2) Z ( X ) = Sp(X), in particular Z ( X ) is dosed in X . For every component Zi of Z ( X ) we write Bi := Stab(Zi)~ then we have dim(B/) > 0. 2.2 L e m m a . (cf. [54], Lemma 1.5; [221, proof of Lemma 4.1; [11, 1.2.2, Lemma 3) Let A be an abeIian variety over k, let W C A be a dosed subvariety (in particular W is irreducible and reduced), and let t C Z>I. Suppose that t W = W . Then W is a translate of an abelian subvariety of A. (Comment: by t W = W we mean that the map [t]: A ---* A maps the subvariety W onto itself; note that this map is finite, and hence it suffices to require that for all w in W we have t. w E W; in that case t W C W , and we conclude t W = W by remarking dim(tW) = dim W.) 2.3

S k e t c h of a P r o o f o f T h e o r e m 2.1

Using Lemma 2.2 we sketch a proof of that theorem. We choose an integer q E Z>I. Following Faltings we define for every m > 1 the map Fro: X

' A m-1

TM

by: Fra(ax,...,am) :----(qa, - a2,qa 2 - a3,...,qam_ 1 - am).

Define I I ' by the cartesian square (i.e., pull-back diagram): Xm

T y~

---.-+ A,~-I

T 9=(q--1)-A h--~ X

where g(x) := ((q - 1 ) x , . . . , (q - 1)x). Let Y,~ C A • X be the image of (projl , h): Y~' ---, d • X. Note: (a,x) EY~ a n d a l = x + b ' v = : c a i = x + q i - l b ' f o r

1
120

F. OORT

Note that

( x + t / , x ) EYm

~

( x + b ' , x + q b ' , . . . , x + q ' ~ - ' b ' ) E Y ", x+b~ E X , . . . , x + q m - l b ~ E X .

Suppose B' C X is closed, x E X, and b' E B' with x + b' E X; then

qB' = B' ~ (z + b', x) ~ Y,~; we are going to try to prove the "opposite implication". We observe: a) the maps (projl , h): Y~ --+ Yr~ C A x X make Ym+~ C Ym C A x X into a descending chain (2 _< m) of closed subschemes; clearly this is stationary, let n be chosen such that Ym = Y~ =: Y for all m > n; b) for x E X consider the fiber

Y(x) := h-l(x) C A x {x} ~ A (as a closed set of A); note that ( ( x , . . . , x ) , x ) Y = Y,~ = Ym+l (for m >_ n) implies

(~c + b,x) E Ym ~

E Is', hence x E Y(x); note that

(x + b,x) E Ym+~ ~

(x + qb, x) E Y,~;

we write B" :-- - x + Y(x), and we see that multiplication by q maps B" into itself, hence a power t = q? > 1 of it maps some highest dimensional irreducible component B' of B" onto itself; now we apply Lemma 2.2 with W -- B'; we see that B' is a translate of an abelian subvariety B of A; c) hence Z(X) = {x E X I dim(Y(z)) > 0}; the projection map Y --+ X is proper, so by general theory we conclude that Z ( X ) is a closed subset of X; the equality Z ( X ) = Sp(X) has now been proved; d) let Zi be an irreducible component of Z(X) and let x/ be its generic point; the method of step (b) (with some minor changes due to the fact that ki := k(xi) r k) applied to xi gives a non-zero abelian subvariety B~ of A | ki; by rigidity of abelian subvarieties (cf. [40], page 26, Thm. 5) B~ is defined over k, i.e., we have Bi C A with B~ = B i | ki; by construction, Bi stabilizes Zi. 2.4

The Ueno Fibration

Let X C A over an algcbraically closed field k, and let B be the connected component of the stabilizer of X: 0 B : = { a E A [ a + X CX}rea. This is an abelian subvariety of A. One can consider the quotient X / B , i.e., the image of X under the mapping A ~ A/B. The mapping X ~ X / B is called the Ueno-fibration of X (e.g. cf. [79], pp. 120/121). The Kodaira dimension of X equals dim(X/B) (cf. [79], Thin. 10.9), hence conjecture ( G T ) (see w holds for subvarieties of abelian varieties, also cf. [32], Thm 4. For connections with a conjecture made by A. Bloch in 1926, cf. [57] and [32]. For Z(X), when X is contained in a semi-abelian variety, cf. [56], Lemma 4.1. Also see [21, Thm. 1.

XIII. "TItE" GENERAL CASE OF S. LANG'S CONJECTURE

3

The

Arithmetic

121

Case

In this section we suppose that k is an algebraic closure of K. The crucial result is:

Suppose K is of finite type over Q, and A is an abelian variety over K, and X C A is a K-closed subset; then

3.1 T h e o r e m . (cf. [23], Thm. 4.1, and w

# { ~ I z e X ( K ) a.d 9 r Sp(X~)) < ~ .

(If x E X then x | k E Xk, but this point we still denote by x. This theorem and its proof are generalizations of results and methods from [22].) 3.2 T h e o r e m . (cf. [23], Thm. 4.2, notations and assumptions as in the previous theorem) (Either X ( K ) is empty, or) there exist ci E X ( K ) and abelian subvarieties

Ci C A (defined over K) such that

X(K) = U (cj + C/K)). j=l

In other words, the irreducible components of the Zariski closure X ( K ) of X ( K ) in X are translates of abelian subvarieties of A. P r o o f . Induction on dim(X). For dim(X) = 0 it is trivial. If X ( K ) is finite we are done. Let T be a positive dimensional irreducible component of X(K). Note that T is geometrically irreducible since T(K) is Zariski dense in T. From Thm. 3.1 it follows t h a t Tk = Sp(Tk). Let B := S t a b ( T ) % ; note t h a t (Stab(Tk)) ~ = ((Stab(T)~ hence Bk is a non-zero abelian subvariety of Ak by Thm. 2.1, and we see that B is an abelian subvariety of A. By induction, T / B is a translate of an abelian subvariety C / B of A/B, where C is an abelian subvariety of A containing B. It follows that T is a translate of C. [] 3.3 R e m a r k s . This theorem proves the conjecture by Lang from 1960, cf. [37], page 29, which is a special case of (LC). It does happen that the dimension of the Zariski closure of X ( K ) depends on the choice of K (and hence I do not agree with "...that the higher dimensional part of the closure of X(number field) should be geometric; i.e., independent of" that number field, cf. [81], page 22, lines 11-12). As an easy example one can take X = A = E, an elliptic curve which has only finitely many points rational over K. One can remark that the top dimensional part is not geometric, but its dimension eventually is. Note that Thm. 3.2 does not generalize directly to semi-abelian varieties (e.g., take an elliptic curve E C p2 over a number field K such that E(K) is not finite, and remove the 3 coordinate axes, obtaining U C G m • Gin; one can also take a singular rational curve or a conic instead of E). [] 4 4.1

Related

Conjectures

and

Results

Integral Points

Suppose S is a finite set of discrete valuations on a number field K, and let R := (.9s be the integers outside S. One can consider points "over R". For curves it turns out to be

122

F. OORT

essential that the curve has only finitely many automorphisms (this condition became natural in the Kodaira-Parshin construction), and now we have a better understanding of theorems of Siegel and Mahler: if C is either P~ - {0,1, co}, or E - {0} (where E is an elliptic curve over R), or C is a curve over K of genus at least two, then C ( R ) is finite (cf. Siegel [74], Mahler [44], Faltings [22]; see the discussion on page 319 of [37]; for the first case see [12]). For a generalization, see [19]. In [23] Faltings gives an example that one cannot hope for finiteness of integral points on an open set in an abelian varieties (in case the open set is not affine). See [36], page 219 for a conjecture concerning finiteness of number of integral points on an affine open set in an abelian variety. See [83], Thm. 0.2 for the statement of a result which generalizes this conjecture: take t2 as above, take a closed subscheme X' of a model ,4 over R of a semi-abelian variety A over K; the set of integral points on X" is contained in a finite number of translates of semi-abelian subvarieties (Definition: A group variety A is called a s e m i - a b e l i a n v a r i e t y if it contains a linear group L C A such that L is an algebraic torus (i.e., product of copies of Gm over the algebraic closure of the field of definition), such that A / L is an abelian variety, i.e., if A is the extension over an abelian variety by an algebraic torus). We see, cf. [83], Coroll. 0.5, that on a semi-abelian variety, in the complement of an ample divisor we have only finitely many integral points. 4.2

T o r s i o n P o i n t s on S u b v a r i e t i e s

Manin and Mumford studied a question which was settled by Raynaud: 4.3 T h e o r e m . (Manin-Mumford conjecture, proved by Raynaud, cf. [60], Thm. 1) Let A be a complex torus (i.e., C g modulo a lattice), and let S C A be the image of a Riemann surface into A. Suppose that S is not the Riemann surface of an dliptic curve (i.e., not the translate of a subtorus). Then the set of torsion points of A contained in S is finite. More generally one can consider a subgroup of finite type (i.e., finitely generated as Z-module, or take it of "finite rank", see below) inside an abelian variety and intersect with a subvariety (cf. [45], translation page 189, see [36], page 221). We see that [38], page 37, Conjecture 6.3 can now be derived from the existing literature (Liardet, Laurent [41], Hindry [29]: see the discussion on pp. 37-39 of [38], use Faltings [23]): Consider a semi-abelian variety A over C, a finitely generated subgroup Fo C A(C), and a subvariety X C A. Let

r := {a c A(C) I ~'~ e Z>o

with n.a

~

to}.

Then X contains a finite number bi + Bi of translates of semi-abelian subvarieties of A such that

x(C) n r

c U(b,+Bdr

c X.

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