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165

p-Adic Monodromy and the Birch and

Swinnerton-Dyer Conjecture A Workshop

on p -Adic Monodromy and the Birch and Swinnerton-Dyer Conjecture

August 12-16, 1991 Boston University, Boston, Massachusetts

Barry Mazur Glenn Stevens

Editors

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p-Adic Monodromy and the Birch and

Swinnerton-Dyer Conjecture

CoNTEMPORARY MATHEMATICS

165

p-Adic Monodromy and the Birch and

Swinnerton-Dyer Conjecture A Workshop on p -Adic Monodromy and the Birch and Swinnerton-Dyer Conjecture

August 12-16, 1991

Barry Mazur Glenn Stevens

Editors

American Mathematical Society Providence. Rhode Island

EDITORIAL BOARD Craig Huneke, managing editor

Clark Robinson J. T. Stafford Linda Preiss Rothschild Peter M. Winkler

The conference on p-Adic Monodromy and the Birch and Swinnerton-Dyer Con-jecture was held at Boston University, Boston, Massachusetts, August 12-16, 1991, with support from the National Science Foundation, Grant DMS-9109048, and the Mathematics Trust of Harvard University.

1991 Mathematics Subject Classification. Primary 11G40; Secondary 14F30.

Library of Congress Cataloging-in-Publication Data Conference on p-adic Monodromy and the Birch and Swinnerton-Dyer Conjecture (1991 : Boston University)

p-adic monodromy and the Birch and Swinnerton-Dyer conjecture/ Conference on p-Adic Mon-odromy and the Birch and Swinnerton-Dyer Conjecture, August 12-16, 1991, Boston University; Barry Mazur, Glenn Stevens, editors.

p. em. -(Contemporary mathematics; v. 165) Includes bibliographical references. ISBN 0-8218-5180-2 (acid-free) 1. p-adic analysis-Congresses. 2. Homology theory-Congresses. 3. Birch-Swinnerton-Dyer

conjecture-Congresses. I. Mazur, Barry. II. Stevens, Glenn, 1953- . III. Title. IV. Series: Contemporary mathematics (American Mathematical Society); v. 165. QA241.C6883 1991 512'.74-dc20

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Contents

Preface ix

List of Talks xiii

On monodromy invariants occurring in global arithmetic, and Fontaine's theory 1

by BARRY MAZUR

A p-adic Shimura isomorphism and p-adic periods of modular forms 21 by ROBERT F. COLEMAN

Numerical solution of the p-adic hypergeometric equation 53 by ROBERT COLEMAN and JEREMY TEITELBAUM

Iwasawa £-functions and the mysterious £-invariant 63 by JOHN W. JONES

p-adic variants of the Birch and Swinnerton-Dyer conjecture for elliptic 71 curves with complex multiplication

by KARL RUBIN

On standard p-adic £-functions of families of elliptic cusp forms 81 by KOJI KITAGAWA

p-adic pairings 111 by KI-SENG TAN

Variation of the canonical height in algebraic families 123 by JOSEPH H. SILVERMAN

Kronecker's polynomial, supersingular elliptic curves, and p-adic periods 135 of modular curves

by EHUD DE SHALIT

Trivial zeros of p-adic £-functions by RALPH GREENBERG

149

vii

viii CONTENTS

Higher weight modular forms and Galois representations 175 by BRUCE W. JORDAN

On the conjecture of Mazur, Tate, and Teitelbaum 183 by RALPH GREENBERG and GLENN STEVENS

Formes modulaires et representations Galoisiennes a valeurs dans un 213 anneau local complet

by HENRI CARAYOL

A p-adic conjecture about derivatives of £-series attached to modular 239 forms

by NAOMI JOCHNOWITZ

Euler systems and refined conjectures of Birch Swinnerton-Dyer type 265 by HENRI DARMON

On p-adic £-functions of Mumford curves 277 by CHRISTOPH KLINGENBERG

Preface

In recent years we have witnessed significant breakthroughs in the theory of p-adic Galois representations and p-adic periods of algebraic varieties. G. Faltings' proof of the p-adic Hodge-Tate conjecture and the recent work of Fontaine, Hyodo and Kato on the monodromy conjecture for varieties with semistable reduction provide fundamental tools for future work in arithmetic geometry. Progress in rigid analysis promises a useable theory of p-adic integration. Hida's theory of A-adic modular forms is proving to be a powerful tool for studying p-adic Galois representations attached to automorphic forms. Recent work in each of these areas suggests mysterious connections to special values of associated £-functions.

The workshop on p-adic Monodromy and the Birch and Swinnerton-Dyer Conjecture, held at Boston University August 12-16, 1991, brought researchers together with the aim of achieving a deeper understanding of the interdepen-dence between p-adic Hodge theory, analogues of the conjecture of Birch and Swinnerton-Dyer, p-adic uniformization theory, p-adic differential equations, and deformations of Galois representations. Much of the workshop was devoted to the relevance of special values of (p-adic and "classical") £-functions and their derivatives to arithmetic issues as envisioned in "Birch-Swinnerton-Dyer-type conjectures," "Main Conjectures" and "Beilinson-type conjectures" a la Green-berg and Coates.

Many of the workshop lectures focused on the phenomena of exceptional zeroes and exceptional sign-change. These phenomena typically occur when two £-functions which naively appear to "encode the same arithmetic informa-tion" nevertheless have different orders of vanishing at the central point. In this curious circumstance, it seems often to be true that the first nonvanishing Taylor coefficients "can be compared." For example the (rank zero case of the) "exceptional zero conjecture" of Mazur, Tate, and Teitelbaum relates a deriv-ative of a p-adic £-function to a value of a classical £-function multiplied by a certain p-adic period-the so-called .C-invariant. Another example is provided by the striking formula of Rubin which relates a derivative of the p-adic £-function attached to a Grossencharacter to a value of the p-adic £-function attached to the conjugate Grossencharacter. These phenomena are also related to the con-jectures of Naomi Jochnowitz which connect the 0-operator on modular forms

ix

X PREFACE

modulo pn to her conjectural v'B-operator in the p-adic theory of half-integral weight forms.

Several lectures were dedicated to the foundations of a theory of .C-invariants. In the case of ordinary p-adic Galois representations, Ralph Greenberg described a broad setting in which analogues of the .C-invariant can be defined and he conjectured that these .C-invariants occur as factors in formulas involving special values of derivatives of p-adic L-functions, in much the same way that they occur in the "exceptional zero conjecture."

In the weight two case Greenberg and Stevens proved the "exceptional zero conjecture" expressing the .C-invariant as a ratio of the derivative of a p-adic L-function and the special value of the classical L-function (at the central point) (p 2 5). Their proof relies on an interpretation of the .C-invariant as an ob-struction to deforming the p-adic Tate module. The proof makes essential use of Hida's theory and, specifically, of two variable p-adic L-functions attached to "Hida families" which p-adically interpolate both weight and character. Koji Kitagawa presented a construction, and a study of the special values, of just such p-adic L-functions. John Jones extended these ideas in another direction by ex-plaining his proof of an "exceptional zero formula" for the algebraic counterpart of the L-function in the lwasawa theory of abelian varieties.

In 1990, Christoph Klingenberg injected still another intriguing element into this circle of ideas by proving an analogue of the "exceptional zero conjecture" for a different p-adic L-function-Schneider's L-function, defined purely in terms of the p-adic uniformization theory. This seems to be the first known relation-ship between Schneider's p-adic L-function and the usual one. Unfortunately, Klingenberg was unable to attend the workshop. We are grateful to him for allowing us to include his manuscript with these proceedings.

In the higher weight case uncertainty remains as to how to define the "right" .C-invariant. Robert Coleman and Jeremy Teitelbaum gave us two "candidates" (one more general than the other), defined in terms _of p-adic integration in rigid analytic geometry. They conjecture that these candidates are equal when both are defined. Coleman's .C-invariant is defined using the Gauss-Manin connec-tion to construct a theory of integration for differential forms with logarithmic singularities. Teitelbaum's .C-invariant, defined only for newforms which come from quaternion algebras, is given in terms of his "p-adic Poisson integrals." For weight two, both Coleman's and Teitelbaum's .C-invariants coincide with the standard .C-invariant, but for higher weight they are not even known to coincide with one another. Teitelbaum reported on the results of some machine calcula-tions of the .C-invariant (in a single instance of a weight 4 newform) confirming that the two .C-invariants do indeed coincide (up to the accuracy of the calcula-tion) and moreover that this common value is related to the special value of the derivative of the appropriate p-adic L-function, as predicted by the "exceptional zero conjecture."

Barry Mazur described still another interpretation of the .C-invariant, which

PREFACE xi

was proposed by J.-M. Fontaine. This definition is based on the semi-stable monodromy conjecture which was recently proved by Kato (for sufficiently large p). The theory endows the De Rham cohomology of a variety over a local field with additional linear algebraic structure. When applied to the cohomology of Kuga-Shimura varieties this structure allows us to pick out an invariant attached to "split multiplicative" newforms which shows every indication of being the sought-for £-invariant.

Related to a "refined theory" of the £-invariant in the weight two case is Ehud de Shalit's proof of Oesterle's conjecture giving an explicit expression for the "tame part" of the Tate-Morikawa "q" attached to the Jacobian of a modular curve in terms of values of the j-function. Also related to a refined Birch-Swinnerton-Dyer conjecture is the work of Henri Darmon, which, inspired by Kolyvagin's work, deduces some higher rank consequences.

Canonical height pairings (which play a fundamental role in Birch-Swinnerton-Dyer type conjectures) were treated in a number of talks in this workshop: in Ki-Seng Tan's study of the arithmetic meaning of "degeneracy' of the p-adic height pairing; in Joe Silverman's detailed analysis of the variation of canonical heights, both classical and p-adic, in families; in John Coates' talk (not included here) where he gave an exposition of Perrin-Riou's p-adic height pairing in a context where a "Pancuskin condition" is satisfied, and the appropriate A-module(s) are A-torsion; and in Jan Nekovar's lecture (also not included) where a p-adic height pairing was constructed in a motivic context.

Bill Messing presented his work on the essential surjectivity of the Dieudonne functor (not included). Bruce Jordan gave an exposition of his joint work with Faltings applying crystalline techniques to study the structure of modular Ga-lois representations obtained from parabolic cohomology groups. Henri Carayol gave a detailed account of his deep results on the structure of the parabolic cohomology groups after completion at a maximal ideal in the Heeke algebra.

We would like to acknowledge our debt to the Mathematics Trust of Harvard University and to the National Science Foundation. Without the financial sup-port of these agencies our workshop would not have been possible. We would also like to thank the workshop participants for the friendly productive discus-sions and interesting lectures which took place during the conference. Most of all, we would like to thank Angelique Thayer whose expertise and cheerfulness in preparing the administrative details of the conference assured that the workshop was an enjoyable experience for all of us.

Barry Mazur Glenn Stevens

List of Talks

Barry Mazur: Fontaine's theory of p-adic monodromy. Robert Coleman: p-adic Picard-Lefschetz. Jeremy Teitelbaum: Numerical solution of a p-adic hypergeometric equation. John Jones: lwasawa £-functions and the Mysterious £-invariant. Karl Rubin: p-adic £-functions and rational points on elliptic curves with complex multiplication. Koji Kitagawa: On the standard p-adic £-functions of families of elliptic cusp forms. Ki-Seng Tan: On the p-adic height pairing. Joe Silverman: Variation of the canonical height in algebraic families. Ehud de Shalit: p-adic periods of X 0 (p). Ralph Greenberg: Trivial Zeroes of p-adic £-functions. John Coates: On Perrin-Riou's p-adic height pairing and lwasawa theory. Bill Messing: Essential surjectivity of the Dieudonne functor. Bruce Jordan: Crystalline cohomology and the Eisenstein ideal for higher weight. Glenn Stevens: On the conjecture of Mazur, Tate, and Teitelbaum. Henri Carayol: Modular forms and Galois representations over complete local rings. Naomi Jochnowitz: A p-adic conjecture about central critical values of deriva-tives of £-series. Jan Nekovar: On p-adic heights. Henri Darmon: Refined class number formulas for derivatives of £-series.

xiii

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(See the AMS catalog for earlier titles)

p-Adic Monodromy and the Birch and Swinnerton-Dyer Conjecture

Barry Mazur and Glenn Stevens, Editors '

Recent years have witnessed significant breakthroughs in the theory of p-adic Galois representations and p-adic periods of algebraic varieties. This book con-tains papers presented at the Workshop on p-Adic Monodromy and the Birch and Swinnerton-Dyer Conjecture, held at Boston University in August 1991 . The workshop aimed to deepen understanding of the interdependence between p-adic Hodge theory, analogues of the conjecture of Birch and Swinnerton-Dyer, p-adic uniformization theory, p-adic differential equations, and deformations of Galois representations. Much of the workshop was devoted to exploring how the special values of (p-adic and "classical") £-functions and their derivatives are relevant to arithmetic issues, as envisioned in "Birch-Swinnerton-Dyer-type conjectures", "Main Conjectures", and "Beilinson-type conjectures" a Ia Green-berg and Coates.

ISBN 0-8218-5180-2

9 780821 851807