Jean Lannes

Jean Lannes
École Polytechnique · Centre de Mathématiques Laurent Schwartz

About

71
Publications
1,205
Reads
How we measure 'reads'
A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text. Learn more
1,125
Citations
Introduction
Skills and Expertise

Publications

Publications (71)
Preprint
Full-text available
Let $V$ be an elementary abelian $2$-group and $X$ be a finite $V$-CW-complex. In this memoir we study two cochain complexes of modules over the mod2 Steenrod algebra $\mathrm{A}$, equipped with an action of $\mathrm{H}^{*}V$, the mod2 cohomology of $V$, both associated with $X$. The first, which we call the "topological complex", is defined using...
Chapter
This chapter starts with a general discussion of reductive groups and their root data. Our aim is to review the Satake isomorphism for the Hecke ring of a reductive group scheme over the p-adic integers \(\mathbb {Z}_p\), as well as the Harish-Chandra isomorphism for the center of the universal enveloping algebra of a complex reductive Lie algebra....
Chapter
In this chapter, we explain Arthur’s description of the discrete automorphic representations of classical groups in terms of selfdual cuspidal automorphic representations of GLn. In agreement with the general philosophy of this book, we restrict our exposition to the level 1 automorphic representations, but provide a concrete form of the famous Art...
Chapter
In this chapter, we give many examples of specific cases of the Arthur-Langlands conjecture concerning automorphic forms for SOn or Siegel modular forms. They rely on concrete constructions of automorphic forms which are either classical (using theta series and results of Rallis and Böcherer), more recent (Ikeda liftings), or new (applications of t...
Chapter
First we deal with the notion of d-neighbors (d positive integer, very often prime) for even unimodular lattices, introduced by M. Kneser, and with the associated Hecke operators; numerous examples are given. Then we analyse in depth the d-neighborhoods between a Niemeier lattice with roots and the Leech lattice; this sheds some light on the “holy...
Chapter
In a first part we explain how Theorem E of the introduction leads to a solution of the p-neighbor problem which involves in particular the four integers τj,k(p) introduced in Chap. 9, which are “genus 2 analogs” of the Ramanujan τ(p). Using the analysis made in Chap. 3 of neighborhoods of the Leech lattice, we determine τj,k(p) for p ≤ 113 (Theore...
Chapter
We first introduce the Hecke ring of a \(\mathbb {Z}\)-group G and discuss it basic properties (local-global structure, compatibility with isogenies, criterion for commutativity…). An elementary description of the Hecke rings of classical groups is given. Then, we recall the notion of a square integrable automorphic form for G, and that of a discre...
Chapter
Most of this chapter may be read independently. We first recall known properties of the Siegel theta series of even unimodular lattices in rank 16 (Witt, Igusa, Kneser) and 24 (Erokhin, Borcherds, Nebe-Venkov…). Then we give two proofs of Theorem A of the introduction (the p-neighbor problem in dimension 16): a short one relying on a construction o...
Chapter
This chapter essentially recalls classical material. First we introduce the definitions and results from the theory of symmetric bilinear forms, quadratic forms, alternating forms, and their associated (classical) groups, that are used in this book. Next we explain the relation between root systems and even unimodular lattices. We recall in particu...
Chapter
This long chapter is the technical heart of the book. We first apply Arthur’s theory to PGSp4 ≃SO3,2 to study the standard parameter of the 4 vector-valued genus 2 Siegel modular forms of interest for Niemeier lattices. Then, we apply Arthur’s theory to give two short, but conditional, proofs of Theorem E of the introduction, as well as a full proo...
Book
This book includes a self-contained approach of the general theory of quadratic forms and integral Euclidean lattices, as well as a presentation of the theory of automorphic forms and Langlands' conjectures, ranging from the first definitions to the recent and deep classification results due to James Arthur. Its connecting thread is a question abou...
Article
Full-text available
In this memoir, we study the even unimodular lattices of rank at most 24, as well as a related collection of automorphic forms of the orthogonal and symplectic groups of small rank. Our guide is the question of determining the number of p-neighborhoods, in the sense of M. Kneser, between two isometry classes of such lattices. We prove a formula for...
Book
La théorie classique des suites de Sturm fournit un algorithme pour déterminer le nombre de racines d’un polynôme à coefficients réels contenues dans un intervalle donné. L’objet principal de ce mémoire est de montrer qu’une généralisation adéquate de la théorie des suites de Sturm fournit entre autres choses: • une notion d’indice de Maslov pour...
Chapter
Full-text available
Soient R un anneau commutatif et L un R-module libre de dimension finie. La notation L* désigne le R-module dual de L: L* = HomR (L,R).
Chapter
Full-text available
L’énoncé du théorème en question nécessite quelques préparatifs.
Chapter
Le coeur de ce chapitre est le paragraphe 6.2 où l’on décrit sept analogues du théorème 4.1.9. La concaténation de ces huit résultats fournit une version algébrique du théorème de périodicité de Bott réelle. En 6.1 on présente une version algébrique du théorème de périodicité de Bott complexe; ce paragraphe est là, principalement pour servir d’écha...
Chapter
On montre dans ce chapitre comment utiliser certaines idées des chapitres précédents pour obtenir des variantes des résultats de Sharpe [Sh]. Dans ce chapitre, l’anneau (commutatif) R est a priori arbitraire.
Chapter
Les mots «composante connexe» ci-dessus font référence à la condition (v) de la proposition 3.1.1 ci-après (la «proposition clé»). Afin que l’énoncé de cette condition soit irréprochable, il nous faut clarifier au préalable quelques notations et conventions.
Article
Full-text available
Nous montrons dans cet article comment les connaissances acquises sur les espaces fonctionnels de source le classifiant du groupeZ/p ([La2], [DS]) et l’utilisation de MU-résolutions instables permettent d’obtenir des résultats sur les espaces fonctionnels de source le classifiant d’unp-groupe abélien fini ou d’un tore si l’on impose au but d’avoir...
Chapter
Soient V un 2-groupe abélien élémentaire (en d’autres termes, un \({\Bbb F}\) 2-espace vectoriel de dimension finie) et A l’algèbre de Steenrod modulo 2; H*V désigne la cohomologie modulo 2 de V. Un H*V-A-module instable est un A-module instable M muni d’une structure de H*V-module définie par une application H*V ⊗ M → M qui est A-linéaire (l’exemp...
Article
Soient p un nombre premier et V un p-groupe abelien elementaire. Nous developpons une theorie de Smith pour les H * V-A-modules instables (la cohomologie modulo p equivariante d'un espace muni d'une action de V est le type meme d'un tel objet). Celle-ci est un analogue algebrique de la Theorie de Smith classique qui concerne la cohomologie modulo p...
Article
Nous dcrivons une nouvelle mthode de calcul de la cohomologie de MacLane des corps finis. Cette thorie est intimement relie aux extensions du groupe additif dj tudies par L. Breen et l'homologie de Hochschild topologique de M. Bkstedt (et donc la K-thorie stable). Notre approche utilise de manire cruciale l'annulation de la cohomologie de MacLane d...
Article
Here we give a very simple construction of Brown-Gitler spectra, using only the existence of certain projective bicommutative Hopf algebras as proved by Schoeller and the analysis of the homology of fibration sequences of infinite loop spaces given by Moore and Smith.
Article
The main object of this note is to prove the following generalisation of a theorem of Serre. A simply connected space of finite type whose mod. 2 cohomology is nilpotent (and non-trivial) has infinitely many homotopy groups which are not of odd torsion. Incidentally we show that for every fibrationF( → ί )E ( → p )B, satisfying certain mild condi...
Article
We first give a new proof of a conjecture of J.-P. Serre on the homotopy of finite complexes, which was recently proved by C. McGibbon and J. Neisendorfer. The emphasis is on a property of the mod. 2 homology of certain spaces: their quasi-boundedness as right modules over the Steenrod algebra. This property is preserved when one goes from a simply...
Article
Full-text available

Network

Cited By