Showing papers on "Fundamental lemma published in 2010"

On the cohomology of certain noncompact Shimura varieties

Abstract: Preface vii Chapter 1: The fixed point formula 1 Chapter 2: The groups 31 Chapter 3: Discrete series 47 Chapter 4: Orbital integrals at p 63 Chapter 5: The geometric side of the stable trace formula 79 Chapter 6: Stabilization of the fixed point formula 85 Chapter 7: Applications 99 Chapter 8: The twisted trace formula 119 Chapter 9: The twisted fundamental lemma 157 Appendix: Comparison of two versions of twisted transfer factors 189 Bibliography 207 Index 215

Arithmetic applications of the Langlands program

Abstract: This expository article is an introduction to the Langlands functoriality conjectures and their applications to the arithmetic of representations of Galois groups of number fields. Thanks to the work of a great many people, the stable trace formula is now largely established in a version adequate for proving Langlands functoriality in the setting of endoscopy. These developments are discussed in the first two sections of the article. The final section describes the compatible families of l-adic Galois representations that can be attached to automorphic forms with the help of Shimura varieties. To illustrate the relevance of Langlands functoriality to number theory, the article concludes with a description of the Sato–Tate conjecture for elliptic modular forms, recently proved in joint work of Barnet-Lamb, Geraghty, Taylor, and the author.

Le lemme fondamental pondéré. I. Constructions géométriques

Abstract: This work is the geometric part of our proof of the weighted fundamental lemma, which is an extension of Ngo Bao Châu’s proof of the Langlands–Shelstad fundamental lemma. Ngo’s approach is based on a study of the elliptic part of the Hichin fibration. The total space of this fibration is the algebraic stack of Hitchin bundles and its base space is the affine space of ‘characteristic polynomials’. Over the elliptic set, the Hitchin fibration is proper and the number of points of its fibers over a finite field can be expressed in terms of orbital integrals. In this paper, we study the Hitchin fibration over an open set larger than the elliptic set, namely the ‘generically regular semi-simple set’. The fibers are in general neither of finite type nor separated. By analogy with Arthur’s truncation, we introduce the substack of ξ-stable Hitchin bundles. We show that it is a Deligne–Mumford stack, smooth over the base field and proper over the base space of ‘characteristic polynomials’. Moreover, the number of points of the ξ-stable fibers over a finite field can be expressed as a sum of weighted orbital integrals, which appear in the Arthur–Selberg trace formula.

The Geometric Nature of the Fundamental Lemma

Abstract: The Fundamental Lemma is a somewhat obscure combinatorial identity introduced by Robert P. Langlands as an ingredient in the theory of automorphic representations. After many years of deep contributions by mathematicians working in representation theory, number theory, algebraic geometry, and algebraic topology, a proof of the Fundamental Lemma was recently completed by Ngo Bao Chau, for which he was awarded a Fields Medal. Our aim here is to touch on some of the beautiful ideas contributing to the Fundamental Lemma and its proof. We highlight the geometric nature of the problem which allows one to attack a question in p-adic analysis with the tools of algebraic geometry.

Relative trace formulae toward Bessel and Fourier-Jacobi periods of unitary groups

Abstract: We propose a relative trace formula approach and state the corresponding fundamental lemma toward the global restriction problem involving Bessel or Fourier-Jacobi periods of unitary groups $\mathrm{U}_n\times\mathrm{U}_m$, extending the work of Jacquet-Rallis for $m=n-1$ (which is a Bessel period). In particular, when $m=0$, we recover a relative trace formula proposed by Flicker concerning Kloosterman/Fourier integrals on quasi-split unitary groups. As evidence for our approach, we prove the fundamental lemma for $\mathrm{U}_n\times\mathrm{U}_n$ in positive characteristics.

Le lemme fondamental pond\'er\'e pour le groupe m\'etaplectique

Abstract: We state a variant of Arthur's weighted fundamental lemma for the metaplectic group of Weil, which will be an essential ingredient of the stable trace formula. Over a local field of large enough residual characteristic, we give a proof using the method of descent, which is conditional upon the weighted nonstandard fundamental lemma on Lie algebras. In view of the works of Chaudouard and Laumon, this condition is expected to hold.

Transfer to characteristic zero: appendix to

Abstract: This appendix shows that the Fundamental lemma of Jacquet-Rallis, proved by Zhiwei Yun in the positive charactersitic case, is also true in characteristic zero, when residue characteristic is sufficiently large. In fact, this follows immediately from the article "Transfer Principle for the Fundamental Lemma" by R. Cluckers, T.C. Hales and F.Loeser.

Invariant representations of GSp(2) under tensor product with a quadratic character

Abstract: Let $F$ be a number field or a $p$-adic field. The author introduces in Chapter 2 of this work two reductive rank one $F$-groups, $\mathbf{H_1}$, $\mathbf{H_2}$, which are twisted endoscopic groups of $\textup{GSp}(2)$ with respect to a fixed quadratic character $\varepsilon$ of the idele class group of $F$ if $F$ is global, $F^\times$ if $F$ is local. When $F$ is global, Langlands functoriality predicts that there exists a canonical lifting of the automorphic representations of $\mathbf{H_1}$, $\mathbf{H_2}$ to those of $\textup{GSp}(2)$. In Chapter 4, the author establishes this lifting in terms of the Satake parameters which parameterize the automorphic representations. By means of this lifting he provides a classification of the discrete spectrum automorphic representations of $\textup{GSp}(2)$ which are invariant under tensor product with $\varepsilon$. Table of Contents: Introduction; $\varepsilon$-endoscopy for $\textup{GSp}(2)$; The trace formula; Global lifting; The local picture; Appendix A. Summary of global lifting; Appendix B. Fundamental lemma; Bibliography; List of symbols; Index. (MEMO/204/957)

The Work of Ngo Bao Chau

Abstract: In August 2010, Ngo Bao Chau was awarded a Fields Medal for his deep work relating the Hitchin fibration to the Arthur-Selberg trace formula, and in particular for his proof of the Fundamental Lemma for Lie algebras. This article gives a brief introduction to his work for a general mathematical audience.

Transfer to characteristic zero: appendix to "Fundamental Lemma of Jacquet-Rallis in positive characteristics" by Zhiwei Yun

Abstract: This appendix shows that the Fundamental lemma of Jacquet-Rallis, proved by Zhiwei Yun in the positive charactersitic case, is also true in characteristic zero, when residue characteristic is sufficiently large. In fact, this follows immediately from the article "Transfer Principle for the Fundamental Lemma" by R. Cluckers, T.C. Hales and F.Loeser.