Arithmetic Algebraic Geometry
Arithmetic algebraic geometry is in a fascinating stage of growth, providing a rich variety of applications of new tools to both old and new problems. Representative of these recent developments is the notion of Arakelov geometry, a way of "completing" a
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Series Editors J. Oesterle
A. Weinstein
G. van der Geer
F. Oort
J. Steenbrink
Editors
Arithmetic Algebraic Geometry
Springer Science+Business Media, LLC
G. van dcr Geer Mathematisch lnstituut Universitcit van Amsterdam Plantage Muidcrgracht 24 10 18 TV Amsterdam Thc Ncthcrlands
F.Oort Mathematisch Instituut Rijksuniversiteit Utrecht Budapestlaan 6 3508 TA Utrccht The Netherlands
J. Steenbrink Mathematisch 1nstituut 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 DOI 10.1007/978-1-4612-0457-2
ISBN 978-1-4612-0457-2 (eBook)
Contents
Participants
VII
Contri bu tors
IX
Introduction Well-Adjusted Models for Curves over Dedekind Rings T. Chinburg and R. Rumely
3
On the Manin Constants of Modular Elliptic Curves B. Edixhoven
25
The Action of Monodromy on Torsion Points of Jacobians T. Ekedahl
41
An Exceptional Isomorphism between Modular Varieties T. Ekedahl and B. van Geemen
51
Chern Functors 1. Franke
75
Curves of Genus 2 Covering Elliptic Curves and an Arithmetical Application G. Frey and E. Kani Jacobians with Complex Multiplication 1. de long and R. Noot Families de Courbes Hyperelliptiques 1.-F. Mestre
153
177
a Multiplications Reelles
Series de Kronecker et Fonctions L des Puissances Symetriques de Courbes Elliptiques sur Q l.-F. Mestre and N. Schappacher Hyperelliptic Supersingular Curves F.Oort
193
209
247
v
Letter to Don Zagier A.N. Parshin
285
The Old Subvariety of l,,(pq) K. Ribet
293
Kolyvagin's System of Gauss Sums K. Rubin
309
The Exponents of the Groups of Po
