Table Of ContentArithmetic: Geometry
Cortmm 1.994
Edited by
Fabrizio Catanese
Uzzi‘I/‘ersity ofPisa
‘99? CAMBRIDGE
UNIVERSITYPRESS
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©lstituto Nazionale di Alta Matematica Francesco Severi 1997
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no reproduction of any part may take place without
the written permission of Cambridge University Press.
First published 1997
Printed in the United Kingdom at the University Press, Cambridge
A catalogue recordfor this book is available from the British Library
ISBN 0 521 59133 3 hardback
Contents
Introduction
List of talks
Geometric theory of G~functions
Yves Andre Kr Francesco Baldassarri
Growth of Mordell—W’eil groups in anticyclotomic towers
Massimo Bertolini
Algebraic cycles on degenerate fibers 45
5', Bloch, Hr Gillet ((7, C Soulé
The equivariant Thue—Siegel method 70
E. Bombieri
Differential algebraic geometry and diophantine
geometry: an overview
Alexandru Buinm
Extensions of motives associated to symmetric powers of elliptic
curves and to llecke Characters ofimaginary quadratic fields 99
Christopher Dening‘er
Motivic vunctions and regularized determinants ll 138
Christopher Deninger
The determinant of cohomology in etale topology
Gerd Faltings
How to construct curves over finite fields with many points 169
Gerard van (ler Geer (Qt hilarcel van (ler Vlug‘t
Heights of subvarieties over illvfields 190
Walter G11hler
Canonical liftings and dense sets of (TM—points
Frans Oort
viii Contents
Finiteness results for Chow groups ofvarieties over number fields 235
Claudio Pedrini
Diophantine geometry in characteristic 1): a survey 260
José Felipe Voloch
Computations 011 commutative group varieties 279
Gisbert i/Viisthofz
Introduction
The I.Nrd.A.M. Symposium ’Arithmetic Geometry" took place on the days
October 16 ,, October ‘2'] in the "Palazzone7 of the Scuola Normale located
in Cortona (Arezzo, Italy). The Symposium is one of a series of Symposia
organized for many years under the patronage of the Istituto Nazionale di
Alta lVIatematica7 which granted financial support to the main speakers; we
also acknowledge a. financial contribution from the G.N.S.A.G.A. (Group of
Algebra and Geometry of the ONE. (Italian National Council of Research)
which was used to support other participants. Nowadays geometric methods
are becoming almost ubiquitous in the treatment of questions stemming from
arithmetic and number theory. This trend. originating in the work of A. “’eil.
led to a series of programmes and conjectures through the work of several
mathematiciansr Grothendieck and Lang to name a few: it culminated in the
last dozen ofyears to remarkable successes, including Faltings’ solution of the
Mordell conjecture and \V'iles’ proofof Fermat’s last theorem. In Italy7 during
the blossoming of the classical school of algebraic geometry? mathematicians
like Bianchi, Beppo Levi, Scorza and others realized the importance of the
geometric methods in the study of diophantine equations This classical tra—
dition has been revived nowadays in Italy, and it looks like an excellent idea
to try to bring together many ofthe. leading mathematicians actively working
in this area where very fruitful interaction is advancing between arithmetic
and geometric methods. The success of the Symposium relics in our opinion
not only on the very lively discussions which lead to refinements and improve—
ments of several conjectures presented in the talks. but also on the broad
spectrum of topics and methods covered; the variety of approaches and topics
touched upon is also reflected in the contributions appearing in these Pro—
ceedings. The papers published here are, as we originally wanted7 of mixed
nature. in that we have some surveys of the current status of the field7 but
also many significant new research results. Afirst theme which was considered
ranged from the theory of elliptic curves over number fields and p—adic fields.
to the arithmetic theory of algebraic groups. including the classical subject of
Abelian varieties and their moduli. incorporating also the study of Shimura
varieties and their rational points through differential geometric and cohomo—
logical methods, A central role was occupied by the study of integral and
rational points, from several points of view: the study of equations over finite
fields. effective versions of the Thue—Siegel method. the approaches through
differential algebra or through p—adic differential equations. extensions of the
x Introduction
classical theory of heights and the problems arising from an attempt to prov
duce a unified treatment of the global case in any characteristic. One of the
main themes was centered arouud the cohomological approach. encompassing
the standard conjectures about L—functions, the study of Chow groups over
number fields, the role of the determinant of cohomology. the extension to
Chow homology groups of the theory of degenerations of Hodge structures.
and finally results and open problems concerning Chow motives and the eX—
istence of mixed motives related to algebraic K—groups. There is no need to
dwell on afurther description, thelist ofthepapers and the articles themselves
being available in the following pages; we want now only to express our deep
gratitude to the anonymous referees for the precious help they gave us. \We
finally thank the participants who attended the Symposium (we apologize for
any involuntary omission): F. Amoroso. F. Baldassarri, L. Barbieri Viale . M.
Bertoliui. J.B. Bost, A. Buium. G. Canuto. B. (Thiarellotto. C. liliberto, C.
De (71011Ci11i. 1. Del Corso. (7'. Derringer, R. Dvornicich7 G. Faltings7 D. Franco,
G. van der Geer, H. Gubler. G. Harder. K. Kiinnemann. M. McQuillan7 M.
Manaresi. M. Manetti. J.L. lVlerel, M. Mendes Lopes. F. Cort. C. Pedrini, A.
Perelli. D. Portelli, G. Puglisi, M. Salvetti, N. Schappacher. C. Soulé, C. Viola.
J.F. \"oloch. U. Zannier.
Organisers
E. Bombieri (IAS7 Princeton)
F. Catanese (Pisa)
A4. \Viistholz (ETH7 Ziirich)
List of talks
E, Bombiert The cluster principle and the S-unit equation
F. 0071 Canonical liftings and dense sets of CM—points
F, Baldassm‘ri Geometric theory of G—functions
C. Soule’ The virtual motives of varieties
A. Buium Differential algebra and diophantine geometry
G. Faltt'ngs Crystalline cohomology and integral models for Shimura vari—
eties
G. Harder The. action of the Galois group on the cohomology of Shimura
varieties
,].F. Voloch Diophantiue geometry in characteristic.p, a survey,
H, Gable)" Heights of subvarieties over M—fields
N, Schnappacher Heegner landscape
N]. Bertolmi p—adic analogues ofthe Birch and Swinnerton~Dyer conjecture
over imaginary quadratic fields
J.L. ill/[era Filtration by the rank of modular curves
J,B. Bast Heights of Abelian varieties
K, [i’fimze'nmmt On the analogues of the standard conjectures in Arakelov
geometry
AM. AlcQuillan Holomorphic maps and Diophantine approximation
C. Dminger Explicit motivic extensions of CM elliptic curves
E. Bo-mbtw-t Some applications of the equivariant Time-Siege] principle
G. mm (ler Gem” Curves over finite fields
(7. Pedrimf On the group of zero cycles of surfaces over number fields
G. W’iist/wlz Logarithms and integrals
Geometric theory of G-functions.
by
Yves André1 and Francesco BaldaSSarri
§0 Summary.
Our object of study are G—connections i.e. linear partial differential equa—
tions satisfied by G—functions in several variables [A], [DGS]; typical examples
are non—confluent generalized hypergeometric connections with rational param—
eters [GHE Chap. 12]. Our main result is the stability of this notion under
higher direct images, for any smooth morphism. As a by—product we obtain a
purely p—adic proof of the open local monodromy theorem which does not use
resolution of singularities.
§1 G-connections.
1.1. Notations.
Q1 = the field of algebraic numbers;
K : a number field offinite degree;
VK : the ring ofintegers of K3
S = a non-empty open affine subscheme of SpecVK;
Vs 2 0(3);
23 = { finite places of K having center on V5 } = { closed points of S }.
For any 2; 6 ES, 1) dividing the rational prime p 219(7)), we put:
K1) 2 the U-cornpletion of K,
M; = the ring of integers of K1,,
7n, = a uniformizing parameter of Vv, chosen in K,
16(1)) 2 the residue field of W.
We normalize the absolute value 1 iv corresponding to v by
[Kwopi
(1-1-1) Iplv 217‘ [Km
We denote by | |U also the unique extension of I I1, to the completion 017(2)) of
the algebraic closure of K“ and by TU its value group |C;(v)|v.
Partially sponsored by EC contract CHRXCT—93—O403 (DG 12 COMA).
AMS Subject Classification:11Gxx712H25.
1 The first author thanksthe University ofPadovafor hospitality andsupport
during the preparation of this paper.
2 ANDRE (Q7 BALDASSARRI: Geometric theory of Girlncfiionx
For an element A 6 CI) , let us consider the series
(1.1.2) g)‘(m)= Z A558.
3_
A—s750
An element A E Cp will be said to be non—Liouville ifthe radius ofconvergence
ofboth 9;(at) and 94133) is 1. .
For g = (a1, . . . ,an) 6 (R20)“, any n, we set IQlR = 2:7;011-.
1.2. Let X —f—) S’ be a smooth surjective morphism of finite type with
geometrically connected fibers. We say that X/S is a smooth S-model of the
functionfield.7: = 5(XK) overK. Analogously, we definethe notionofa smooth
K-model ofT. For X/S' as before, let 77 (resp. 77v, 1) 6 25) be the generic point
of X (resp. of the closed fiber Xk(v)). The choice of a smooth S-model X of
.7: privileges one extension of the absolute value | [v to f. Namely, the local
ring OX1,” is a DVR of f, with residue field 16(1)) and uniformizing parameter
7n}. Let | l denote the absolute value corresponding to OX)“, normalized so
as to extend | |v. We extend I |U to matrices of elements of F, by taking the
maximum ofthe values ofthe entries. We introduce the Kv-rigid analytic space
Xgig = the Raynaud generic fiber of the formal completion of X along XMD).
The functor X I—> X39 is extended to a functor E r—> 5519 from OXK-modules
to Oxzig—modules in a well—known way.
Similar definitions canobviously begivenwhenXis asmoothgeometrically
connected Vv—scheme; they will be understood in the sequel. In that case, for a
locally closed subvariety T0 ofXMy), ]T0[=]T0[X will denote the tube (ofradius
1) of To, that is the set of points of X1779 that specialize to points of To [Be].
For p E I}, 0 [0,1), and y 6 X179, we will denote by [y]xflp, or simply [y]p, the
closed tube ofy in X519. So, in terms ofan étale coordinate neighborhood in X
(U,Q) = (U,a:1, . . . ,md) ofthe specialization yo ofy in Xk(v), with mi(y) : 0, for
i = 1, . . . ,71,
(1'21) lylp:{t Elyoll |$i(t)i’u _<_p,z'=1,...,n} ‘
If p = 1 and X is afline, we abusively set [yh 2 X179. We introduce also
M(y)p = Frac((9([y]p)) = the field of meromorphic functions on Mp.
1.3. Let (M,V) be an f/K—differential module offinite rank 11, i.e. M E
.73” and
(1.3.1) sflMofogK
is an integrable f/K—connection. A model of (M,V) on X/S is a locally free
OX—module (ofrank n) M, with an integrable X/S-connection
(1.3.2) VzM flMeox ilk/S
