Arithmetic Algebraic Geometry

449 Pages·1991·9.494 MB·English

by  Gerard van der Geer, Frans Oort, Jozef Steenbrink (auth.), G. van der Geer, F. Oort, J. Steenbrink (eds.)

Progress in Mathematics
Volume 89
Series Editors
J. Oesterle
A. Weinstein
G. van der Geer F. Oort J. Steenbrink
Editors
Arithmetic Algebraic
Geometry
Springer Science+B usiness Media, LLC
G. van dcr Geer F.Oort
Mathematisch lnstituut Mathematisch Instituut
Universitcit van Amsterdam Rijksuniversiteit Utrecht
Plantage Muidcrgracht 24 Budapestlaan 6
10 18 TV Amsterdam 3508 TA Utrccht
Thc Ncthcrlands The Netherlands
J. Steenbrink
Mathematisch 1n stituut
Katholieke Universiteit Nijmegen
Toemooiveld
6525 ED Nijmegen
The Netherlands
Printed on acid-free paper.
© 1991 Springer Science+Business Media New York
OriginalIy published by Birkhauser Boston, Inc in 1991
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987654321
ISBN 978-1-4612-6769-0 ISBN 978-1-4612-0457-2 (eBook)
DOI 10.1007/978-1-4612-0457-2
Contents
Participants VII
Contri bu tors IX
Introduction
Well-Adjusted Models for Curves over Dedekind Rings 3
T. Chinburg and R. Rumely
On the Manin Constants of Modular Elliptic Curves 25
B. Edixhoven
The Action of Monodromy on Torsion Points of Jacobians 41
T. Ekedahl
An Exceptional Isomorphism between Modular Varieties 51
T. Ekedahl and B. van Geemen
Chern Functors 75
1. Franke
Curves of Genus 2 Covering Elliptic Curves and an Arithmetical
Application 153
G. Frey and E. Kani
Jacobians with Complex Multiplication 177
1. de long and R. Noot
a
Families de Courbes Hyperelliptiques Multiplications Reelles 193
1.-F. Mestre
Series de Kronecker et Fonctions L des Puissances Symetriques de
Courbes Elliptiques sur Q 209
l.-F. Mestre and N. Schappacher
Hyperelliptic Supersingular Curves 247
F.Oort
v
Letter to Don Zagier 285
A.N. Parshin
The Old Subvariety of l,,(pq) 293
K. Ribet
Kolyvagin's System of Gauss Sums 309
K. Rubin
The Exponents of the Groups of Points on the Reductions of an Elliptic
Curve 325
R. Scho(){
The Generalized De Rham-Witt Complex and Congruence Differential
Equations 337
1. Stienstra
Arithmetic Discriminants and Quadratic Points on Curves 359
P. Vojta
The Birch-Swinnerton-Dyer Conjecture from a Naive Point of View 377
D. Zagier
Polylogarithms, Dedekind Zeta Functions, and the Algebraic K-Theory
of Fields 391
D. Zagier
Finiteness Theorems for Dimensions of Irreducible r..-adic
Representations 431
Y.C. Zarhin
vi
Participants
The following is the list of participants of the conference Arithmetic Algebraic
Geometry. Texel . 89. which was held on Texel Island during the last week of
April 1989.
J. van Beele (Leiden), F. Beukers (Utrecht), S. Bloch (Chicago), J. Brinkhuis
(Rotterdam), H. Carayol (Strasbourg), T. Chinburg (Philadelphia), B. Edixho
ven (Utrecht), T. Ekedahl (Stockholm), J. Franke (Berlin), G. Frey (Saar
briicken), J. van Geel (Gent), B. van Geemen (Utrecht), G. van der Geer (Am
sterdam), R. de Jeu (Chicago), A.J. de Jong (Nijmegen), W. van der Kallen
(Utrecht), T. Katsura (Tokyo), J. van der Lingen (Amsterdam), R. Livne (Tel
Aviv), J.-F. Mestre (Paris), J.P. Murre (Leiden), R. Noot (Utrecht), F. Oort
(Utrecht), M. van der Put (Groningen), K. Ribet (Berkeley), K. Rubin (New
York), N. Schappacher (Bonn), e.G. Schmidt (Groningen), R. Schoof
(Utrecht), J.-P. Serre (Paris), B. de Smit (Amsterdam), J. Steenbrink (Nij
megen), P. Stevenhagen (Amsterdam), J. Stienstra (Utrecht), J. Top (Utrecht),
R. Versseput (Amsterdam), P. Vojta (Berkeley), J. Wildeshaus (Cambridge),
D. Zagier (Maryland, Bonn), Y.G. Zarhin (Moscow).
vii
Contributors
T. Chinburg Department of Mathematics, University of Pennsylvania, Phila
delphia, PA 19104, USA
1. de long Mathematisch Instituut, Katholieke Universiteit Nijmegen, Toer
nooiveld, 6525 ED Nijmegen, The Netherlands
B. Edixhoven Department of Mathematics, University of California, Berkeley,
CA 94720, USA
T. Ekedahl Matematiska Institutionen, Stockholms Univcrsitet, Box 6071, S-
11385 Stockholm, Sweden
1. Franke Karl-Weierstrass-Institut fUr Mathematik, Mohrenstrasse 39, DDR-
1080 Berlin, GDR
C. Frey Institut fur Experimentelle Mathematik, Universitat Essen, D-4300
Essen, FRG
E. Kani Department of Mathematics and Statistics, Jeffery Hall, Queen's Uni
versity, Kingston, Ontario, K7L 3N6, Canada
i.-F. Mestre Departement de Mathematiques et d'Informatique, Ecole Nor
male Superieure, Mathematiques, 45 rue d'Ulm, F-75230 Paris Cedex 05,
France
R. Noot Mathematisch Instituut, Rijksuniversiteit Utrecht, Budapestlaan 6,
3508 T A Utrecht, The Netherlands
F. Oort Mathematisch Instituut, Rijksuniversiteit Utrecht, Budapestlaan 6,
3508 T A Utrecht, The Netherlands
A.N. Parshin Steklov Mathematical Institute, UI. Vavilova 42, Moscow
117966, GSP-I, USSR
K. Ribet Mathematics Department, University of California, Berkeley, CA
94720, USA
K. Rubin Department of Mathematics, Ohio State University, Columbus, OH
43210, USA
R. Rumely Department of Mathematics, University of Georgia, Athens, GA
30602, USA
N. Schappacher Max-Planck-Institut fur Mathematik, Gottfried-Clarenstrasse
26, D-5300 Bonn 3, FRG
ix
R. Schoof Mathematisch Instituut, Rijksuniversiteit Utrecht, Budapestlaan 6,
3508 T A Utrecht, The Netherlands
J. Stienstra Mathematisch lnstituut, Rijksuniversiteit Utrecht, Budapestlaan 6,
3508 T A Utrecht, The Netherlands
B. van Geemen Mathematisch Instituut, Rijksuniversiteit Utrecht, Budapest
laan 6, 3508 T A Utecht, The Netherlands
P. Vojta Department of Mathematics, University of California, Berkeley, CA
94720, USA
D. Zagier Max-Planck-Institut fi.ir Mathematik, Gottfried-Clarenstrasse 26, D-
5300 Bonn 3, FRG
Y.G. Zarhin Research Computing Centre, USSR Academy of Sciences, Pusch
ino, Moscow Region 142292, USSR
x
Introduction
Es zeigt sich hier einmal mehr, dass die Zahlentheorie zwar
mit recht die Konigin der Mathematik genannt wird, sie aber
ihren Glanz, wie auch Koniginnen selbst, nicht so sehr aus
sich selbst als vielmehr aus den Kriiften ihrer Untertanen zieht.
G. Faltings (1984)
At the moment fascinating developments are taking place in arithmetic
algebraic geometry. Very prominent among these is that of Arakelov ge
ometry. This is a way of "completing" a variety over the ring of integers of
a number field by adding fibres over the archimedean places. In this way
the analogy between algebraic number fields and function fields of algebraic
curves is extended to a more precise analogy between arithmetic varieties
and varieties fibered over a complete curve. Thus a completely new tool
for attacking arithmetic problems has emerged from the Russian school of
arithmetic algebraic geometry. The importance of this development lies
not only in its direct results (like the proof of the Mordell Conjecture), but
also in the link it establishes between number theory and complex analytic
geometry.
Another fascinating development is the appearance of the relation be
tween arithmetic geometry and Nevanlinna theory, or more precisely be
tween diophantine approximation theory and the value distribution the
ory of holomorphic maps. Vojta has formulated conjectures generalizing
Mordell's Conjecture by using the unexplained similarity between non
degenerate holomorphic maps from en to a complex variety V and sets
of infinitely many K-rational points of a variety V defined over a number
field K which are non-degenerate (in the sense that they do not lie in a
proper Zariski-closed subset of V).
Besides these developments there is a lot of other activity in the field:
algebraic geometry is applied in many ways to solve arithmetic problems.
It was against this background that we decided to organize a conference on
arithmetic algebraic geometry on Texel Island in 1989. The present volume
appears on the occasion of this conference, though the contributions do not
always correspond to lectures given at the conference.
We would like to take the opportunity to thank the participants of
the conference and especially the speakers, who made the conference into
a success. vVe would also like to express our gratitude to the institutions