Progress in Mathematics Volume 224 Series Editors H. Bass J. Oesterle A. Weinstein Modular Curves and Abelian Varieties John Cremona Joan-Carles Lario Jordi Quer Kenneth Ribet Editors Springer Basel AG Editors: J. E. Cremona K. A. Ribet University ofNottingham Department of Mathematics School of Mathematical Sciences MIC 3840 University Park University of California Nottingham NG7 2RD Berkeley, CA 94720-3840 UK USA e-mail: [email protected] e-mail: [email protected] J.-C. Lario and J. Quer Facultat de Matematiques i Estadistica Universitat Politecnica de Catalunya Pau Gargallo, 5 08028 Barcelona Spain e-mail: [email protected] [email protected] 2000 Mathematics Subject Classification llGxx, 14Gxx, 14Kxx A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at <http://dnb.ddb.de>. This work is subject to copyright. All rights are reserved, whether the whole or part ofthe material is concerned, specifically the rights oftranslation, reprinting, re-use ofillustra tions, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use whatsoever, permission from the copyright owner must be obtained. © 2004 Springer Basel AG Originally published by Birkhituser Verlag in 2004 Softcover reprint of the hardcover 1s t edition 2004 Printed on acid-free paper produced of chlorine-free pulp. TCF 00 ISBN 978-3-0348-9621-4 ISBN 978-3-0348-7919-4 (eBook) DOI 10.1007/978-3-0348-7919-4 987654321 www.birkhauser-science.com Table of Contents Preface ................................................................... vii IRENE 1. Bouw AND STEFAN WEWERS Stable reduction of modular curves 1 KEVIN BUZZARD On p-adic families of automorphic forms ............................. 23 GABRIEL CARDONA Q-curves and abelian varieties of GL2-type from dihedral genus 2 curves ......................................... 45 JANOS A. CSIRIK The old subvariety of Jo(N M) ....................................... 53 LUIS V. DIEULEFAIT Irreducibility of Galois actions on level 1 Siegel cusp forms 75 NOAM D. ELKIES On elliptic k-curves .................................................. 81 JORDAN S. ELLENBERG Q-curves and Galois representations 93 EKNATH GHATE On the local behaviour of ordinary modular Galois representations ...................................... 105 JOSEP GONZALEZ, JOAN-CARLES LARIO AND JORDI QUER Arithmetic of Q-curves .............................................. 125 JAYANTA MANOHARMAYUM Serre's conjecture for mod 7 Galois representations 141 BARRY MAZUR AND KARL RUBIN Pairings in the arithmetic of elliptic curves 151 KEN McMuRDY Explicit parametrizations of ordinary and supersingular regions of Xo(pn) ................................. 165 TETSUO NAKAMURA Elliptic Q-curves with complex multiplication ........................ 181 ELISABETH E. PYLE Abelian varieties over Q with large endomorphism algebras and their simple components over ij ................................. 189 VI Table of Contents KENNETH A. RIBET Abelian varieties over Q and modular forms ......................... 241 VICTOR ROTGER Shimura curves embedded in Igusa's threefold ..................... . .. 263 WILLIAM A. STEIN Shafarevich-Tate groups of nonsquare order .......................... 277 Preface It would be difficult to overestimate the influence and importance of modular forms, modular curves, and modular abelian varieties in the development of num ber theory and arithmetic geometry during the last fifty years. These subjects lie at the heart of many past achievements and future challenges. For example, the theory of complex multiplication, the classification of rational torsion on el liptic curves, the proof of Fermat's Last Theorem, and many results towards the Birch and Swinnerton-Dyer conjecture all make crucial use of modular forms and modular curves. A conference was held from July 15 to 18, 2002, at the Centre de Recerca Matematica (Bellaterra, Barcelona) under the title "Modular Curves and Abelian Varieties". Our conference presented some of the latest achievements in the theory to a diverse audience that included both specialists and young researchers. We emphasized especially the conjectural generalization of the Shimura-Taniyama conjecture to elliptic curves over number fields other than the field of rational numbers (elliptic Q-curves) and abelian varieties of dimension larger than one (abelian varieties of GL2-type). This book evolved as a byproduct of that conference: most of the articles here describe the research of their authors as it was presented in their talks. We have also included a few articles that were not presented at the conference but are closely allied with the conference's theme. Among these are three that deserve special mention because they contributed strongly to the current interest in el liptic Q-curves and abelian varieties of GL2-type; they were written soon after the publication of Barry Mazur's delightful article "Number Theory as a Gadfly" (American Mathematical Monthly, Vol. 98 issue 7, 593-610), which reformulated the Shimura-Taniyama conjecture as a statement over ij. The article by Noam Elkies is a revised version of his important unpublished 1993 manuscript on el liptic k-curves; in that manuscript, Elkies explained the precise relation between elliptic k-curves and k-rational points on Atkin-Lehner quotients of the modular curves Xo(N). The editors are grateful to Professor Elkies for revising his paper for this volume. The contribution by Elizabeth Pyle is her doctoral thesis, presented in UCLA in 1995 and realized under the supervision of K. Ribet; it contains a complete characterization of the abelian varieties that should occur as factors of modular abelian varieties over the algebraic closure of the rationals, if the gener alized Shimura-Taniyama conjecture holds. Shortly after finishing her thesis, Pyle left academia and did not have the opportunity to publish her thesis. The paper by Kenneth Ribet is a reprint of a paper already published in the proceedings of a conference held in Taejon (Korea) in 1993; it contains the first conjectural char acterization of Q-curves as the modular elliptic curves over ij, and gives evidence Vlll Preface for this conjecture by relating it to Serre's conjecture on 2-dimensional modular Galois representations. (At the suggestion of a number of conference participants, we included Ribet's article in this volume, since the original publication is out of print and not available in many university libraries.) We would like to thank all the participants at the conference for their contri bution to the scientific success of the event. Special thanks are due to the contrib utors to this volume. We are also indebted to the Centre de Recerca Matema,tica: to its director Prof. Manuel Castellet for his continued support since the beginning of the project, and to its staff, Ms. Maria Julia and Ms. Consol Roca, for their superb work on the administrative and organizational tasks, in connection with the local Organising Committee. Thanks are also due to all the administrations whose financial support made the conference possible: the European Community under its program High Level Scientific Conferences (HPCF-CT-2001-00386), the spanish Ministerio de Ciencia y Tecnologia (BFM-200l-4280-E), the Generalitat de Catalunya, the Universitat Autonoma de Barcelona and the Universitat Politecnica de Catalunya. The editorial committee: John Cremona Joan-Carles Lario Jordi Quer Kenneth A. Ribet Progress in Mathematics, Vol. 224, 1-22 © 2004 Birkhiiuser Verlag Basel/Switzerland Stable Reduction of Modular Curves Irene 1. Bouw and Stefan Wewers 1. Introduction Modular curves are quotients of the upper half-plane by congruence subgroups of SL2(Z). The most prominent examples are Xo(N), XI(N) and X(N), for N 2': l. Modular curves are also moduli spaces for elliptic curves endowed with a level structure. Therefore, they are defined over small number fields and have a rich arithmetic structure. Deligne and Rapoport [4J determine the reduction behavior of Xo(p) and XI(p) at the prime p. Using this result, they prove a conjecture of Shimura saying that the quotient of the Jacobian of XI(p) by the Jacobian of Xo(p) acquires good reduction over Q«(p). Both the reduction result and its corollary have been generalized by Katz and Mazur [lOJ to arbitrary level Nand various level structures. The basic method employed in [4J and [lOJ is to generalize the moduli problem defining the modular curve in question, in such a way that it makes sense in arbitrary characteristic. By general results on represent ability of moduli problems one then obtains a model of the modular curve over the ring of integers of a subfield of Q( (N ). This model has bad reduction at all primes dividing N, and it is far from being semistable, in general. In spite of the very general results of [10], it remains an unsolved problem to describe the stable reduction of Xo(N) at p if p3 divides N (see [5J for the case p21IN). In this note we suggest a different approach to study the reduction of modular curves. Our starting point is the observation that the j-invariant X(N) -7 X(l) ~ WI presents the modular curve X(N) as a Galois cover of the projective line (with Galois group PSL (Z/N )) which is branched only at the three rational points 0, 2 1728 and 00. Let us call a Galois cover of the projective line branched only at three points a three point cover. In [12J Raynaud studies the stable reduction of three point covers under the assumption that the residue characteristic p strictly divides the order of the Galois group. His results have been sharpened in [18J and [17J. In this paper we determine the stable reduction of all three point covers with Galois group PSL2(p), using the results of [17J. As a corollary we obtain a new proof of the results of Deligne and Rapoport on the reduction of Xo(p) and XI(p), and the result of Edixhoven [5J on the reduction of XO(p2). Somewhat surprisingly, our 2 1.1. Bouw and S. Wewers proof does not use the modular interpretation of these curves. On the other hand, the determination of the stable reduction of some of the other PSL2(p)-covers (which are not modular curves) does use the fact that they are moduli spaces of a certain kind. What is so special about the group PSL2 (p)? From our point of view there are two main aspects. The first is rigidity. Let G be a finite group and C = (C1, C2, C3) a triple of conjugacy classes of G. Suppose that there exists a triple g = (gl, g2, g3) of generators of G with gi E Ci and g1g2g3 = 1. The triple g corresponds to a G-cover Y ----+ pI with three branch points. If G equals PSL2(p), such a triple g is (essentially) unique, up to uniform conjugation in PGL2(p). Therefore there exists (essentially) at most one three point cover with a given branch cycle description C, up to isomorphism. (In some cases, one needs a further invariant called the lifting invariant.) We use rigidity as follows. By a result of [17J we can construct three point G-covers Y ----+ pI with bad reduction at p by lifting a certain type of 'stable G-cover' Y ----+ X from characteristic p to characteristic zero. Here Y ----+ X is a finite map between semistable curves in characteristic p together with some extra structure (g, w) which we call the special deformation datum. In the case of the modular curve X(p) (where G = PSL2(p)) one can construct Y ----t X very explicitly; the special deformation datum (g, w) corresponds to a solution of the GauE hypergeometric differential equation. (A similar phenomenon occurs in Ihara's work on congruence relations [7J.) The PSL2 (p )-cover Y ----t pI resulting from the lifting process has branch cycle description (3A, 2A, pA). By rigidity, Y ----+ pI is isomorphic to X(p) ----t X(I). In particular, the stable reduction of X(p) is isomorphic to Y. The other nice thing about PSL2(p) is that p strictly divides its order. The results of [12J and [17J which describe the stable reduction of a given G-cover require that p strictly divides the order of G, whereas the Sylow p-subgroup of PSL2(Z/pn) is rather big for n > 1. This is the main obstruction for extending the method of the present paper to modular curves of higher p-power level. There are partial results of the authors generalizing some of the results of [12J and [17J to groups G with a cyclic or an elementary abelian Sylow p-subgroup (unpublished). It seems hopeless to obtain general results beyond these cases. But maybe a com bination of the methods presented in the present paper with the modular approach might shed some light on the stable reduction of modular curves of higher p-power level. The organization of this paper is as follows. In Section 1 we define special deformation data and explain how to associate a special deformation datum to a three point cover with bad reduction. We recall a result from [17J which essentially says that we can reverse this process. In Section 3 we introduce hypergeometric deformation data. These are special deformation data satisfying an addition condi tion (Definition 3.1). We classify all hypergeometric deformation data by showing that they correspond to the solution in characteristic p of some hypergeometric