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Progress in Mathematics Volume 235 Series Editors Hyman Bass Joseph Oesterle´ Alan Weinstein Geometric Methods in Algebra and Number Theory Fedor Bogomolov Yuri Tschinkel Editors Birkhauser¨ Boston • Basel • Berlin Fedor Bogomolov Yuri Tschinkel New York University Princeton University Department of Mathematics Department of Mathematics Courant Institute of Mathematical Sciences Princeton, NJ 08544 New York, NY 10012 U.S.A. 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. c 2005 Birkhauser¨ Boston All rights reserved. This work may not be translated or copied in whole or in part without the writ- ten permission of the publisher (Springer Science+Business Media Inc., Rights and Permissions, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter de- veloped is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed in the United States of America. (HP) 987654321 SPIN 10987025 www.birkhauser.com Preface 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 conics 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.Adif- 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, Fedor Bogomolov August 2004 Yuri Tschinkel Contents Beauville surfaces without real structures Ingrid Bauer, Fabrizio Catanese, Fritz Grunewald .................... 1 Couniformization of curves over number fields Fedor Bogomolov, Yuri Tschinkel .................................. 43 On the V -filtration of D-modules Nero Budur ..................................................... 59 Hecke orbits on Siegel modular varieties Ching-Li Chai ................................................... 71 Ax–Kochen–Erˇsov Theorems for p-adic integrals and motivic integration Raf Cluckers, Fran¸cois 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 Tam´as Hausel ...................................................193 viii Contents Mahler measure for dynamical systems on P1 and intersection theory on a singular arithmetic surface Jorge Pineiro, Lucien Szpiro, Thomas J. Tucker .....................219 A Combination of the Conjectures of Mordell–Lang and Andr´e–Oort Richard Pink ....................................................251 Motivic approach to limit sheaves Markus Spitzweck ................................................283 Counting points on cubic surfaces, II Sir Peter Swinnerton-Dyer ........................................303 Quantum cohomology of isotropic Grassmannians Harry Tamvakis .................................................311 Endomorphism algebras of superelliptic jacobians Yuri G. Zarhin ..................................................339 Beauville surfaces without real structures Ingrid Bauer1, Fabrizio Catanese1, and Fritz Grunewald2 1 Department of Mathematics, University of Bayreuth, D-95440 Bayreuth, Germany [email protected] [email protected] 2 Mathematisches Institut, Universit¨atsstrasse 1, D-40225, Germany [email protected] Summary. Inspired by a construction by Arnaud Beauville of a surface of general 2 type with K =8,pg = 0, the second author defined Beauville surfaces as the surfaces which are rigid, i.e., without nontrivial deformations, and which admit an unramified covering which is isomorphic to a product of curves of genus at least 2. In this case the moduli space of surfaces homeomorphic to the given surface consists either of a unique real point, or of a pair of complex conjugate points corresponding to complex conjugate surfaces. It may also happen that a Beauville surface is biholomorphic to its complex conjugate surface, while failing to admit a real structure. The first aim of this note is to provide series of concrete examples of the second situation, respectively of the third. The second aim is to introduce a wider audience, in particular group theorists, to the problem of classification of such surfaces, especially with regard to the problem of existence of real structures on them. 1 Introduction In [2] (see p. 159) A. Beauville constructed a new surface of general type with 2 K =8,pg = 0 as a quotient of the product of two Fermat curves of degree 5 by the action of the group (Z/5Z)2. Inspired by this construction, in the article [4], dedicated to the geometrical properties of varieties which admit an unramified covering biholomorphic to a product of curves, the following
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