Wilfred W. J. Hulsbergen. Coniectures in Arithmetic Algebraic Geometry
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1 Wilfred W. J. Hulsbergen Coniectures in Arithmetic Algebraic Geometry
2 As~ tvbthematics Edited by Klas Diederich Vol. E1: Vol. E2: VoI.E3: Vol. E4: Vol. E5: VoI.E6: Vol. E7: Vol. E8: Vol. E 9: Vol. E 10: Vol. Ell: Vol. E 12: Vol. E 13: Vol. E 14: Vol. E 15: Vol. E 16: Vol. E 17: Vol. E18: G. Hector/U. Hirsch: Introduction to the Geometry of Foliations, Part A M. Knebusch/M. Kolster: Wittrings G. Hector/U. Hirsch: Introduction to the Geometry of Foliations, Part B M. Laska: Elliptic Curves of Number Fields with Prescribed Reduction Type lout of printl P. Stiller: Automorphic Forms and the Picard Number of an Elliptic Surface G. Faltings/G. WOstholz et 01.: Rational Points* W Stoll: Value Distribution Theory for Meromorphic Maps W. von Wahl: The Equations of Navier-Stokes and Abstract Parabolic Equations lout of printl A. Howard/P.-M. Wong IEds.l: Contributions to Several Complex Variables A. J. Tromba IEd.l: Seminar of New Results in Nonlinear Partial Differential Equations* M. Yoshida: Fuchsian Differential Equations* R. Kulkarni, U. PinkaIiIEds.l: Conformal Geometry* Y. Andre: G-Functions and Geometry* U. Cegrell: Capacities in Complex Analysis J.-P. Serre: Lectures on the Mordell-Weil Theorem K.lwasaki/H. Kimura/S. Shimomura/M. Yoshida: From Gauss to Painleve K. Diederich IEd.l: Complex Analysis W. W. J. Hulsbergen: Conjectures in Arithmetic Algebraic Geometry *A Publication of the Max-Planck-Institut for Mathematik, Bonn Volumes of the German-language subseries "Aspekte der Mathematik" are listed at the end of the book.
3 Wilfred W J. Hulsbergen Coniectures in Algebraic A Survey
4 Wilfred W. J. Hulsbergen KMA, NL-4800 RG Breda The Netherlands Die Deutsche Bibliothek - CIP-Einheitsaufnahme Hulsbergen, Wilfred W. J.: Conjectures in arithmetic algebraic geometry: a survey / Wilfred W. J. Hulsbergen. - Braunschweig; Wiesbaden: Vieweg, 1992 (Aspects of mathematics: E; Vol. 18) NE: Aspects of mathematics / E AMS subject classification: 11G, 11M, 14C, 14G, 14H, 14K, 19D, 19E, 19F. All rights reserved Friedr. Vieweg & Sohn Verlagsgesellschaft mbh, Braunschweig / Wiesbaden, 1992 Softcover reprint of the hardcover 1st edition 1992 Vieweg is a subsidiary company of the Bertelsmann Publishing Group International. No part of this publication may be reproduced, stored in a retrieval system or transmitted, mechanical, photocopying or otherwise, without prior permission of the copyright holder. Cover design: Wolfgang Nieger, Wiesbaden Printed on acid-free paper ISSN ISBN DOl / ISBN (ebook)
5 Contents Introduction 1 1 The zero-dimensional case: number fields Class Numbers Dirichlet L-Functions The Class Number Formula Abelian Number Fields Non-abelian Number Fields and Artin L-Functions 17 2 The one-dimensional case: elliptic curves General Features of Elliptic Curves Varieties over Finite Fields L-Functions of Elliptic Curves Complex Multiplication and Modular Elliptic Curves Arithmetic of Elliptic Curves The Tate-Shafarevich Group Curves of Higher Genus Appendix B & S-D for Abelian Varieties Bloch's Version of B & S-D I-Motives, Mixed Motives and B & S-D The general formalism of L-functions, Deligne cohomology and Poincare duality theories 3.1 The Standard Conjectures Deligne-Beilinson Cohomology 3.3 Deligne Homology
6 3.4 Poincare Duality Theories Riemann-Roch, K-theory and motivic cohomology Grothendieck-Riemann-Roch Adams Operations Riemann-Roch for Singular Varieties Higher Algebraic K-Theory Adams Operations in Higher Algebraic K-Theory Chern Classes in Higher Algebraic K-Theory Gillet's Riemann-Roch Theorem Motivic Cohomology Regulators, Deligne's conjecture and Beilinson's first conjecture Borel's Regulator Beilinson's Regulator Special Cases and Zagier's Conjecture Riemann Surfaces Models over Spec(Z) Deligne's Conjecture Beilinson's First Conjecture Beilinson's second conjecture Beilinson's Second Conjecture Hilbert Modular Surfaces Arithmetic intersections and Beilinson's third conjecture The Intersection Pairing Beilinson's Third Conjecture Absolute Hodge cohomology, Hodge and Tate conjectures and Abel-Jacobi maps The Hodge Conjecture Absolute Hodge Cohomology Geometric Interpretation Abel-Jacobi Maps
7 8.5 The Tate Conjecture. 8.6 Absolute Hodge Cycles 8.7 Motives Grothendieck's Conjectures 8.9 Motives and Cohomology Mixed realizations, mixed motives and Hodge and Tate conjectures for singular varieties Tate Modules Mixed Realizations Weights Hodge and Tate Conjectures The Homological Regulator Examples and Results B & S-D revisited Deligne's Conjecture Artin and Dirichlet Motives. 213 loa Modular Curves Other Modular Examples Linear Varieties Bibliography 225 Index 233
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Under consideration for publication in Math. Proc. Camb. Phil. Soc. 1 On the computation of the Picard group for K3 surfaces By Andreas-Stephan Elsenhans Mathematisches Institut, Universität Bayreuth,