Le Probleme des Groupes de Congruence Pour SL 2
About: This article is published in Annals of Mathematics.The article was published on 1970-11-01. It has received 280 citations till now.
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01 Jan 1971
TL;DR: In this article, Bourbaki et al. implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/legal.php).
Abstract: © Association des collaborateurs de Nicolas Bourbaki, 1970-1971, tous droits réservés. L’accès aux archives du séminaire Bourbaki (http://www.bourbaki. ens.fr/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
501 citations
219 citations
01 Jan 1999
TL;DR: In this article, the authors survey the recent progress made on estimating positive eigenvalues of Laplacian on hyperbolic Riemann surfaces in the case of congruence subgroups in connection with the Selberg conjecture, as well as certain related ones.
Abstract: The purpose of this article is to survey the recent progress made on estimating positive eigenvalues of Laplacian on hyperbolic Riemann surfaces in the case of congruence subgroups in connection with the Selberg conjecture, as well as certain related ones. The results are obtained as consequences of establishing certain important cases of Langlands’ functoriality conjecture.
207 citations
Cites background from "Le Probleme des Groupes de Congruen..."
...[60]) is not valid for SL2(R), not every arithmetic subgroup (i....
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TL;DR: This expository article describes the constructions and various applications ofander graphs in pure and applied mathematics.
Abstract: Expander graphs are highly connected sparse finite graphs. They play an important role in computer science as basic building blocks for network constructions, error correcting codes, algorithms, and more. In recent years they have started to play an increasing role also in pure mathematics: number theory, group theory, geometry, and more. This expository article describes their constructions and various applications in pure and applied mathematics.
201 citations
TL;DR: In this paper, the authors relax the hypotheses on local behavior of p at X, i.e., the restriction of p to a decomposition group at X and to irreducibility hypotheses on 1p, and show that certain "potentially Barsotti-Tate" deformation problems are smooth.
Abstract: where is the reduction of p. These results were subject to hypotheses on the local behavior of p at X, i.e., the restriction of p to a decomposition group at X, and to irreducibility hypotheses on 1p. In this paper, we build on the methods of [461, [451 and [121 and relax the hypotheses on local behavior. In particular, we treat certain ?-adic representations which are not semistable at X, but potentially semistable. We do this using results of [61, generalizing a theorem of Ramakrishna [321 (see Fontaine-Mazur [22, ?131 for a slightly different point of view). The results in [61 show that certain "potentially Barsotti-Tate" deformation problems are smooth, allowing us to define certain universal deformations for with the necessary Galoistheoretic properties to apply Wiles' method. To carry out the proof that these deformations are indeed realized in the cohomology of modular curves (i.e., that the universal deformation rings are Hecke algebras), we need to identify the corresponding cohomology groups and prove they have the modular-theoretic properties needed to apply Wiles' method. As in [151 and [121, the identification is made by matching local behavior of automorphic representations and Galois representations via the local Langlands correspondence (together with Fontaine's theory at the prime ?). We work directly with cohomology of modular curves instead of
196 citations