Geometric methods in algebra and number theory

Progress in Mathematics

•

Geometri c Method s

i n Algebr a an d

Numbe r Theor y

Fedor Bogomolov

Yuri Tschinkel

Editors

Birkhause r

Progress i n Mathematic s

Volume 235

Series Editors

Hyman Bass

Joseph Oesterle

Alan Weinstein

Geometri c Method s

in Algebra and

Number Theory

Fedor Bogomolov

Yur i Tschinke l

Editors

D AI HOC THA I NGUYEN

-TRUNGTAM HOC LIEU

Birkhause r

Bosto n • Base l • Berli n

Fedor Bogomolov

New York University

Department of Mathematics

Courant Institute of Mathematical

New York, NY 10012

U.S.A.

Yuri Tschinkel

Princeton University

Department of Mathematics

Princeton, NJ 08544

U.S.A.

AMS Subject Classifications: 11G18, 11G35, 11G50, 11F85, 14G05, 14G20, 14G35, 14G40, 14L30

14M15. 14M17, 20G05, 20G35

Library of Congress Cataloging-in-Publication Data

Geometric methods in algebra and number theory / Fedor Bogomolov, Yuri Tschinkel, editors.

p. cm. - (Progress in mathematics ; v. 235)

Includes bibliographical references.

ISBN 0-8176-4349-4 (acid-free paper)

1. Algebra. 2. Geometry, Algebraic. 3. Number theory. I. Bogomolov, Fedor, 1946- II.

Tschinkel, Yuri. III. Progress in mathematics (Boston, Mass.); v. 235.

QA155.G47 2004

512-dc22 2004059470

ISBN 0-8176-4349-4 Printed on acid-free paper.

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Prefac e

The transparency and power of geometric constructions has been a source of

inspiration for generations of mathematicians. Their applications to problems

in algebra and number theory go back to Diophantus, if not earlier. Naturally,

the Greek techniques of intersecting lines and conies have given way to much

more sophisticated and subtle constructions. What remains unchallenged is

the beauty and persuasion of pictures, communicated in words or drawings.

This volume contains a selection of articles exploring geometric approaches

to problems in algebra, algebraic geometry and number theory. All papers are

strongly influenced by geometric ideas and intuition. Several papers focus on

algebraic curves: the themes range from the study of unramified curve cov￾ers (Bogomolov-Tschinkel), Jacobians of curves (Zarhin), moduli spaces of

curves (Hassett) to modern problems inspired by physics (Hausel). The pa￾per by Bogomolov-Tschinkel explores certain special aspects of the geometry

of curves over number fields: there exist many more nontrivial correspon￾dences between such curves than between curves defined over larger fields.

Zarhin studies the structure of Jacobians of cyclic covers of the projective

line and provides an effective criterion for this Jacobian to be sufficiently

generic. Hassett applies the logarithmic minimal model program to moduli

spaces of curves and describes it in complete detail in genus two. Hausel stud￾ies Hodge-type polynomials for mixed Hodge structure on moduli spaces of

representations of the fundamental group of a complex projective curve into a

reductive algebraic group. Explicit formulas are obtained by counting points

over finite fields on these moduli spaces. Two contributions deal with sur￾faces: applying the structure theory of finite groups to the construction of in￾teresting surfaces (Bauer-Catanese-Grunewald), and developing a conjecture

about rational points of bounded height on cubic surfaces (Swinnerton-Dyer).

Representation-theoretic and combinatorial aspects of higher-dimensional ge￾ometry are discussed in the papers by de Concini-Procesi and Tamvakis. The

papers by Chai and Pink report on current active research exploring spe￾cial points and special loci on Shimura varieties. Budur studies invariants

vi Preface

of higher-dimensional singular varieties. Spitzweck considers families of mo￾tives and describes an analog of limit mixed Hodge structures in the motivic

setup. Cluckers-Loeser continue their foundational work on motivic integra￾tion. One of the immediate applications is the reduction of a central prob￾lem from the theory of automorphic forms (the Fundamental Lemma) from

p-adic fields to function fields of positive characteristic, for large p. A dif￾ferent reduction to function fields of positive characteristic is shown in the

paper by Ellenberg-Venkatesh: they find a geometric interpretation, via Hur￾witz schemes, of Malle's conjectures about the asymptotic of number fields

of bounded discriminant and fixed Galois group and establish several up￾per bounds in this direction. Finally, Pineiro-Szpiro-Tucker relate algebraic

dynamical systems on P1

to Arakelov theory on an arithmetic surface. They

define heights associated to such dynamical systems and formulate an equidis￾tribution conjecture in this context.

The authors have been charged with the task of making the ideas and

constructions in their papers accessible to a broad audience, by placing their

results into a wider mathematical context. The collection as a whole offers

a representative sample of modern problems in algebraic and arithmetic geo￾metry. It can serve as an intense introduction for graduate students and others

wishing to pursue research in these areas.

Most results discussed in this volume have been presented at the conference

"Geometric methods in algebra and number theory" in Miami, December

2003. We thank the Department of Mathematics at the University of Miami

for help in organizing this conference.

New York,

August 2004

Fedor Bogomolov

Yuri Tschinkel

Content s

Beauville surfaces withou t real structures

Ingrid Bauer, Fabrizio Catanese, Fritz Grunewald 1

Couniformization of curves over number fields

Fedor Bogomolov, Yuri Tschinkel 43

On the IZ-filtration of P-modules

Nero Budur 59

Hecke orbits on Siegel modular varieties

Ching-Li Chai 71

Ax-Kochen-Ersov Theorems for p-adic integrals and motivic

integration

Raf Cluckers, Francois Loeser 109

Nested sets and Jeffrey-Kirwan residues

Corrado De Concini, Claudio Procesi 139

Counting extensions of function fields with bounded

discriminant and specified Galois group

Jordan S. Ellenberg, Akshay Venkatesh 151

Classical and minimal models of the moduli space of curves

of genus two

Brendan Hassett 169

Mirror symmetry and Langlands duality in the non-Abelian

Hodge theory of a curve

Tamds Hausel iy o