Ergebnisse der Mathematik und ihrer Grenzgebiete

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1 Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Foige. Band 16 A Series of Modern Surveys in Mathematics Editorial Board E. Bombieri, Princeton S. Feferman, Stanford N. H. Kuiper, Bures-sur-Yvette P. Lax, New York H. W. Lenstra, Jf., Berkeley R. Remmert (Managing Editor), Munster W. Schmid, Cambridge, Mass. J-P. Serre, Paris J. Tits, Paris K. K. Uhlenbeck, Austin

2 Gerard van der Geer Hilbert Modular Surfaces With 39 Figures Springer-Verlag Berlin Heidelberg New York London Paris Tokyo

3 Gerard van der Geer Mathematisch Instituut Universiteit van Amsterdam Roetersstraat WB Amsterdam The Netherlands Mathematics Subject Classification (1980): llf41, llg18, ISBN-13: DOl: lo.lo07/ e-isbn-13: Library of Congress Cataloging-in-Publication Data Geer, Gerard van der. Hilbert modular surfaces. (Ergebnisse der Mathematik und ihrer Grenzgebiete ; 3. Folge, Bd. 16) Bibliography: p. Includes index. I. Hilbert modular surfaces. I. TItle. II. Series. QA573.G ' This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use ofillustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. Springer-Verlag Berlin Heidelberg 1988 Softcover reprint of the hardcover I st edition 1988 Typesetting: Asco Trade Typesetting Ltd., Hong Kong Printing and bookbinding: Konrad Triltsch, Wiirzburg 2141/

4 Preface Work on this book started in 1981 following a suggestion off. Hirzebruch. I would like to thank him heartily for this suggestion and for his constant support, in particular for the invitations to visit the Max-Planck-Institut fur Mathematik in Bonn. I profited a lot from the stimulating atmosphere at this institute. The manuscript was completed during a stay at the Mathematical Sciences Research Institute in Berkeley. I acknowledge with gratitude the support of both research institutes. During the preparation of the manuscript suggestions and corrections were made by C. Faber, J. Koehl, S. Maurmann and R. Schoof. J. Koehl kindly provided the table of numerical invariants at the end of this book. I thank them for their help. I also would like to thank B. Gross and B. Mazur for their comments on the last two chapters, and J. Lagerberg for drawing the figures. Thanks are also due to the publishers for their cooperation. Finally, I would like to express my gratitude to D. Zagier. Over the years I learned a lot from him in the course of numerous discussions we had on the subject of this book and on other parts of mathematics. Moreover, he read a large part of the manuscript and his comments were most helpful. For all this and the fact that he contributed a section, VII.5 to be precise, I thank him very much. Cambridge, Mass., June 1987 Gerard van der Geer

5 Table of Contents Introduction Notations and Conventions Concerning Quadratic Number Fields I. Hilbert's Modular Group The Action of the Hilbert Modular Group The Distance to the Cusps A Fundamental Domain The Hurwitz-Maass Extension... '"... " Elliptic Fixed Points Hilbert Modular Forms The Adelic Version II. Resolution of the Cusp Singularities The Local Ring at Infinity Glueing Dividing by the Units Digression: the Elliptic r-gon Continued Fractions Resolution of Cyclic Quotient Singularities The Baily-Borel Compactification III. Local Invariants Local Chern Classes Meyer's Theorem Extension of Differential Forms IV. Global Invariants The Volume of rvy Chern Numbers of Yr.... " Inequalities for X and CI Dimensions of Spaces of Cusp Forms Representations on Spaces of Cusp Forms

6 VIII Table of Contents 6. The Vanishing of the Fundamental Group Rigidity V. Modular Curves on Modular Surfaces The Curves FN and TN Intersections with the Cusp Resolutions The Components of FN The Geometry of SO(2, 2) The Volume of the Modular Curves The Intersection Points of the Modular Curves Classification of Elliptic Fixed Points The Intersection Number of Tl and TN The Fixed Points of the Galois Involution Appendix: Modular Forms on ro(d) VI. The Cohomology of Hilbert Modular Surfaces Cohomology and Hilbert Modular Forms The Dual of TN The Generating Series of the Modular Curves The Doi-Naganuma Lifting The Intersection Number of TM and TN' The Action of the Hecke Algebra on the Cohomology The Periods of Eigenforms The Contribution of an Eigenform to the Picard Number VII. The Classification of Hilbert Modular Surfaces The Rough Classification of Algebraic Surfaces Configurations of Curves on Surfaces Classification Theorems Exceptional Curves on Hilbert Modular Surfaces Estimates for the Numerical Invariants Proof of the Classification Canonical Divisors VIII. Examples of Hilbert Modular Surfaces Preliminaries The Examples IX. Humbert Surfaces Modular Embeddings Humbert Surfaces Examples lacobians with Real Multiplication

7 Table of Contents IX X. Moduli of Abelian Schemes with Real Multiplication Abelian Schemes with Real Multiplication Modular Stacks Hilbert Modular Forms The Galois Action on the Set of Components XI. The Tate Conjectures for Hilbert Modular Surfaces Hodge and Tate Cycles Decomposition of the Cohomology and L-Series Splitting up the Galois Representation The Tate Conjectures Table 1. Elliptic Fixed Points Table 2. Numerical Invariants Bibliography List of Notations Index... "