Modular Curves and Abelian Varieties

290 Pages·2004·7.92 MB·English

by  Irene I. Bouw

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
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Washington D.C., USA
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© 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