Modern Birkhauser Classics Many of the original research and survey monographs in pure and applied mathematics published by Birkhauser in recent decades have been groundbreaking and have come to be regarded as foun dational to the subject. Through the MBC Series, a select number of these modern classics, entirely uncorrected, are being re-released in paperback (and as eBooks) to ensure that these treasures remain ac cessible to new generations of students, scholars, and researchers. The Grothendieck Festschrift Volume II A Collection of Articles Written in Honor of the 60th Birthday of Alexander Grothendieck P. Cartier L. lUusie N.M. Katz G. Laumon Yu.I. Manin K.A. Ribet Editors Reprint of the 1990 Edition Birkhauser Boston • Basel • Berlin Luc lUusie Pierre Cartier Universite de Paris-Sud Institut des Hautes Etudes Scientifiques Departement de Mathematiques F-91440 Bures-sur-Yvette F-91405 Orsay France France Gerard Laumon Nicholas M. Katz Universite de Paris-Sud Princeton University Departement de Math6matiques Department of Mathematics F-91405 Orsay Princeton, NJ 08544 France U.S.A. Yuri I. Manin Kenneth A. Ribet Max-Planck Institut ftir Mathematik University of California D-53111 Bonn Department of Mathematics Germany Berkeley, CA 94720 U.S.A. Originally published as Volume 87 in the series Progress in Mathematics Cover design by Alex Gerasev. Mathematics Subject Classification (2000): 00B15, 00B30, 01A60, 01A75 (primary); 11F37, 11F72, 11G05,11G40,11M41,11R29,11R34,11S23,11S31,11S37,12H05,13K05,13N10,14A99,14B15, 14C17, 14D05, 14E20, 14F05, 14F20, 14F30, 14F32, 14F35, 14F40, 14F99, 14G05, 14G10, 14H25, 14H40,14L05,14L15,14M10,14M15,17B10,17B67,18A99,18B40,18E30,18F20,20F34,20F36, 20L05, 22E35, 22E50, 32C15, 32C38, 32S50, 32S60, 57M07, 58G25. 58G26, 58J50, 58J52 (secondary) Library of Congress Control Number: 2006936966 ISBN-10: 0-8176-4567-5 e-ISBN-10: 0-8176-4575-6 ISBN-13: 978-0-8176-4567-0 e-ISBN-13: 978-0-8176-4575-5 Printed on acid-free paper. ©2001 Birkhauser Boston BirkhdUSCr All rights reserved. This work may not be translated or copied in whole or in part without the writ ten permission of the pubUsher (Birkhauser Boston, c/o Springer Science-f Business Media LLC, 233 Spnng Street, New York, NY 10013, USA), except for brief excerpts m 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. 9 8 7 6 5 4 3 21 www. birkhauser. com (IBT) p. Cartier L. Illusie N.M. Katz G. Laumon Y. Manin K.A. Ribet Editors The Grothendieck Festschrift A Collection of Articles Written in Honor of the 60th Birthday of Alexander Grothendieck Volume II 1990 Birkhauser Boston • Basel • Berlin Pierre Cartier Luc Illusie Institut des Hautes Departement de Mathematiques Etudes Scientifiques Universite de Paris-Sud 91440 Bures-sur-Yvette Centre d'Orsay France 91405 Orsay Cedex France Nicholas M. Katz Gerard Laumon Department of Mathematics Departement de Mathematiques Princeton University Universite de Paris-Sud Princeton, NJ 08544 Centre d'Orsay USA 91405 Orsay Cedex France Yuri Manin Kenneth A. Ribet Steklov Institute Department of Mathematics Vavilova 42 University of California Moscow, 117966-GSPl Berkeley, CA 94720 USSR USA Printed on acid-free paper. © Birkhauser Boston, 1990. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or other wise, without prior permission of the copyright owner. Permission to photocopy for internal or personal use, or the internal or personal use of specific clients, is granted by Birkhauser Boston, for libraries and other users registered with the Copyright Clearance Center (CCC), provided that the base fee of $0.00 per copy, plus $0 20 per page is paid directly to CCC, 21 Congress Street, Salem, MA 01970, U.S.A. Special requests should be ad dressed directly to Birkhauser Boston, 675 Massachusetts Avenue, Cambridge, MA 02139, U.S.A. 3428-2/90 $0.00 + .20 ISBN 0-8176-3428-2 ISBN 0-8176-3429-0 Three-volume set ISBN 3-7643-3428-2 ISBN 3-7643-3429-0 Three-volume set Printed and bound by Edwards Brothers, Inc., Ann Arbor, Michigan. Pnnted in the U.S.A. 9 8 7 6 5 4 3 21 CONTENTS VOLUME II Une nouvelle interpretation de la formule des traces de Selberg 1 PIERRE CARTIER et ANDRE VOROS Jacobiennes generalisees globales relatives 69 C. CONTOU-CARRERE Categories tannakiennes Ill P. DELIGNE On The Adic Formalism 197 TORSTEN EKEDAHL F-Isocrystals on Open Varieties: Results and Conjectures 219 GERD FALTINGS Representations p-adiques des corps locaux 249 JEAN-MARC FONTAINE Rectified Homotopical Depth and Grothendieck Conjectures 311 HELMUT A. HAMM and LE DUNG TRANG Automorphisms of Pure Sphere Braid Groups and Galois Representations 353 YASUTAKA IHARA Ordinarite des intersections completes generales 375 Luc ILLUSIE V vi CONTENTS Kazhdan-Lusztig Conjecture for a Symmetrizable Kac-Moody Lie Algebra 407 MASAKI KASHIWARA Euler Systems 435 V.A. KOLYVAGIN Descent for Transfer Factors 485 R. LANGLANDS and D. SHELSTAD VOLUME I Bibliographie D'Alexander Grothendieck De L'Analyse Fonctionnelle aux Fondements de la Geometrie Algebrique JEAN DIEUDONNE The presentation functor and the compactified Jacobian ALLEN B. ALTMAN and STEVEN L. KLEIMAN Some Algebras Associated to Automorphisms of Elliptic Curves M. ARTIN, J. TATE, and M. VAN DEN BERGH Cohomology of a Moduli Space of Vector Bundles V. BALAJI and C.S. SESHADRI Sur les hypersurfaces dont les sections hyperplanes sont a module constant ARNAUD BEAUVILLE Aomoto Dilogarithms, Mixed Hodge Structures, and Motivic Cohomology of Pairs of Triangles on the Plane A.A. BEILINSON, A.B. GONCHAROV, V.V. SCHECHTMAN, and A.N. VARCHENKO Theorie de Dieudonne cristalline IIL theoremes d'equivalence et de pleine fidelite PIERRE BERTHELOT and WILLIAM MESSING Complex Immersions and Arakelov Geometry JEAN-MICHEL BISMUT, HENRI GILLET, and CHRISTOPHE SOULE CONTENTS vii L-Functions and Tamagawa Numbers of Motives SPENCER BLOCK and KAZUYA KATO Bitorseurs et Cohomologie Non Abelienne LAWRENCE BREEN Non-commutative Ruelle-SuUivan type currents JEAN-LUC BRYLINSKI VOLUME III Anneau de Grothendieck de la variete de drapeaux ALAIN LASCOUX New Results on Weight-Two Motivic Cohomology S. LICHTENBAUM Symmetric Spaces Over a Finite Field GEORGE LUSZTIG Le theoreme de positivite de Tirregularite pour les 2);j^-modules ZOGHMAN MEBKHOUT The Convergent Topos in Characteristic p ARTHUR OGUS Finiteness Theorems and Hyperbolic Manifolds A.N. PARSHIN p-groupes et reduction semi-stable des courbes MICHEL RAYNAUD Drawing Curves Over Number Fields G.B. SHABAT and V.A. VOEVODSKY Sur les proprietes numeriques du dualisant relatif d'une surface arithmetique L. SZPIRO Higher Algebraic K-Theory of Schemes and of Derived Categories R.W. THOMASON and THOMAS TROBAUGH viii CONTENTS Solitons elliptiques A. TREIBICH and J.-L. VERDIER, with an Appendix by J. OESTERLE Linear Simple Lie Algebras and Ranks of Operators Yu G. ZARHIN line nouvelle interpretation de la formule des traces de Selberg PIERRE CARTIER ET ANDRE VOROS A Alexander Grothendieck^ pour son 60e anniversaire SOMMAIRE Introduction 1. La formule sommatoire de Poisson 2. Enonce de la formule des traces de Selberg 3. Determinants d'operateurs 4. Une extension de la formule des traces de Selberg 5. Une nouvelle interpretation de la formule des traces 6. Fonction zeta de deux variables. Bibliographie Introduction La formule des traces de Selberg, pour une surface de Riemann com- pacte, a courbure constante negative, est presentee habituellement comme une relation portant sur la transformee de Fourier des distributions liees aux longueurs des geodesiques periodiques, et aux valeurs propres du lapla- cien. De maniere plus precise, soit X une telle surface de genre ^f > 2, mu- nie d'une metrique riemannienne, avec une courbure constante normalisee par K = —1. On note V Tensemble des geodesiques periodiques orientees primitives sur X, et r(p) la longueur d'une telle geodesique p. On note aussi Ax I'operateur de Laplace-Beltrami sur X, et la suite des valeurs propres de —Ax est representee sous la forme 0 = Ao < Ai < A2 < • • • < A„ < An+i < •. •. Pour tout n > 0, on choisit une des racines carrees pn de A^^ — ;|. Sous forme symbolique, on peut ecrire comme suit la formule de Selberg cosh r/2 (1) ^cosr/?„ = -|(5f-l)- sinh^ r/2 n=0 T{P) \5S*™MI".MP)/2)'<^-"""'*