Memoires de la SMF 97 (2004), vi+110 pages
Resume :
Dans cet article, nous abordons quelques questions d'analyse
harmonique sur les groupes reductifs p-adiques. Plus precisement,
nous nous interessons a la transformation de Satake des
distributions unipotentes stables dans le cas des groupes
deployes. Ce probleme est motive, d'une part par les travaux de M.
Assem sur le calcul des integrales orbitales unipotentes, et
d'autre part par ceux de J.-L. Waldspurger sur la determination
de l'espace des distributions unipotentes stables. Cette question
est facile pour les groupes lineaires mais inconnue en general.
Dans ce travail, nous traitons le cas des groupes . Pour n =2,
nous demontrons que ces transformees de Satake s'expriment comme
des fonctions regulieres sur le tore reel unitaire de dimension 2.
Nous montrons ensuite que ces fonctions peuvent egalement etre
retrouvees par la transformation de Satake des distributions de
toute autre nature: les traces tordues compactes d'une famille
explicite de representations de . Ce phenomene peut s'expliquer
par l'endoscopie tordue entre et comme l'a remarque Arthur. Pour
n >2, on demontre dans un certain nombre de cas que les
transformees de Satake de telles traces sont effectivement des
fonctions regulieres, d'une forme commune, sur le tore reel
unitaire de rang n. On l'a en particulier verifie pour . On
s'attend a ce que ceci reste vrai pour n quelconque. Grace a ces
calculs, on propose alors une conjecture assez precise qui decrit
les transformees de Satake des distributions unipotentes stables
sur .
Mots clefs : Analyse harmonique, Groupes reductifs p-adiques,
Algebres de Hecke, Transformation de Satake, Integrales orbitales
unipotentes, Stabilite, Representations, Traces, Endoscopie
tordue
Abstract:
Stable unipotent orbital integrals and their Satake transforms
In this article, we are concerned with some questions arising
from harmonic analysis on p-adic groups. More precisely, we are
interested in Satake transforms of stable unipotent distributions
in the case of split groups. This problem is motivated, on one
hand, by M. Assem's work on the computation of unipotent orbital
integrals, and on the other hand, by J.-L. Waldspurgers' on the
determination of the space of stable unipotent distributions.
This question is easy for general linear groups but unkown in
general. In this work, we deal with the groups . For n=2, we show
that these Satake transforms are regular functions over the rank-2
unitary real torus. We then show that these functions can be
recovered by the Satake transform of some distributions of a
totally different kind: the twisted compact traces of an explicit
familly of representations of . This phenomenon may be explained
by twisted endoscopy between and as remarked by Arthur. For n
> 2, we show, in some cases, that the Satake transforms of
these traces are actually regular functions, of a common form,
over the rank-n unitary real torus. In particular, we have
verified it when . We expect that it is true in general. Thanks
to these computations, we then propose a quite precise
conjecture, that describes the Satake transforms of stable
unipotent distributions on .
Key words: Harmonic analysis, p-adic reductive groups, Hecke
algebras, Satake transform, Unipotent orbital Integrals,
Stability, Representations, Traces, Twisted endoscopy
Asterisque 295 (2004), xiv+300 pages
p-adic Hodge theory and values of zeta functions of modular forms
Kazuya Kato
Asterisque 295 (2004), 117-290
ISBN : 2-85629-157-0
Resume :
Theorie de Hodge p-adique et fonctions zeta de formes modulaires
Si f est une forme modulaire, nous construisons un systeme
d'Euler attache a f, ce qui nous permet d'obtenir des bornes pour
les groupes de Selmer de f. Une loi de reciprocite explicite
permet de relier ce systeme d'Euler a la fonction zeta p-adique
de f, ce qui nous permet d'obtenir un resultat de divisibilite en
direction de la conjecture principale pour f ainsi que des
minorations pour l'ordre d'annulation de cette fonction zeta p-adique.
Dans le cas particulier ou f est attachee a une courbe elliptique
E definie sur , nous prouvons que la fonction zeta p-adique de E
a un zero en s=1 d'ordre superieur ou egal au rang du groupe des
points rationnels de E.
Mots clefs : Forme modulaire, systeme d'Euler, groupe de Selmer,
loi de reciprocite, fonction zeta p-adique, courbe elliptique
Abstract:
If f is a modular form, we construct an Euler system attached to
f from which we deduce bounds for the Selmer groups of f. An
explicit reciprocity law links this Euler system to the p-adic
zeta function of f which allows us to prove a divisibility
statement towards Iwasawa's main conjecture for f and to obtain
lower bounds for the order of vanishing of this p-adic zeta
function. In particular, if f is associated to an elliptic curve
E defined over , we prove that the p-adic zeta function of f has
a zero at s=1 of order at least the rank of the group of rational
points on E.
Key words: Modular form, Euler system, Selmer group, reciprocity
law, p-adic zeta function, elliptic curve
ISBN: 1-57146-103-5
Binding: Hardcover
Page Number: 125
Year Published: 2004
These are the proceedings of the joint seminar by M.I.T. and
Harvard on the current developments in mathematics for the year
2003. Established in 1995, this seminar has been continued on the
third weekend of November every year. The organizing committee
for the seminar consisted of distinguished mathematicians from
the mathematics departments of both institutions: Barry Mazur,
Wilfried Schmid, and S.T. Yau from Harvard, and David Jerison,
Tom Mrowka, and Richard Stanley from M.I.T. This year, the
seminar was dedicated to Prof. Wilfried Schmid and Prof. George
Lusztig.
The 2003 speakers included Sergey Fomin, Henryk Iwaniec, Claire
Voison, and Andrei Zelevinsky,
We would like to thank each of the contributors: without their
participations, the seminar would not have been possible. We
trust that these proceedings will be of interest to many
mathematicians. And we hope that many of you will be able to join
us for future seminars.
Contents
Cluster algebras: Notes for the CDM 2003 Conference - Sergei
Fomin and Andrei Zelevinsky .1
AutomorphicNumber Theory - Henryk Iwaniec .35
On some problems of Kobayashi and Lang: Algebraic approaches -
Claire Voison 83
ISBN: 0-471-22554-1
Paperback
192 pages
August 2004
Author Information
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Hardcover
286 pages
October 2004
Stock market investors have very different reactions to downside
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ISBN: 0-471-66984-9
Hardcover
442 pages
March 2005
Description
This pioneering resource tells the full story of Fourier analysis
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Table of Contents
Preface.
Introduction.
1. Fourier Coefficients and Fourier Series.
2. Fourier Series and Boundary Value Problems.
3. L2 Spaces: Optimal Contexts for Fourier Series.
4. Sturm-Liouville Problems.
5. A Splat and a Spike.
6. Fourier Transforms and Fourier Integrals.
7. Special Topics and Applications.
8. Local Frequency Analysis and Wavelets.
Appendix: Complex Numbers.
Exercises.
Exercises.
References.
Index.