The Grothendieck Festschrift, Volume II: A Collection of Articles Written in Honor of the 60th Birthday of Alexander Grothendieck

563 Pages·1990·20.71 MB·English

by  Pierre Cartier

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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
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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.
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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)'<^-"""'*