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 Softcover reprint ofthe hardcover Ist edition 1991 Copyright is not claimed for works by U.S. Government employees. AlI rights reserved. This work may not be translated or copied in whole or in part without the written permis sion of the publisher Springer Science+Business Media, LLC, except for brief excerpts in connection with reviews or scholarly analysis. U se in connection with any form of information storage and retrieval. electronic adaptation. computer software. or by similar or dis similar methodology now known or hereafter developed is forbidden. The use of general descriptive names. trade names. trademarks, etc. in this publication. even if the former are not especialIy identified. is not to be taken as a sign that such names. as understood by the Trade Marks and Merchandise Marks Act. may accordingly be used freely by anyone. Permission to photocopy for internal or personal use. or the internal or personal use of specific clients. is granted by Springer Science+Business Media, LLC for libraries registered with the Copyright Clearance Center (CCC). provided that the base fcc of $0.00 per copy. plus $0.20 per page is paid directly to CCC. 21 Congress St .. Salem. MA 01970. USA. Special requests should be addressed directly to Springer Science+Business Media, LLC Camera-ready copy prepared by the authors. 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