Shimura Varieties (London Mathematical Society Lecture Note Series, Band 457)

This is the second volume of a series of mainly expository articles on the arithmetic theory of automorphic forms. It forms a sequel to On the Stabilization of the Trace Formula published in 2011. The books are intended primarily for two groups of readers: those interested in the structure of automorphic forms on reductive groups over number fields, and specifically in qualitative information on multiplicities of automorphic representations; and those interested in the classification of I-adic representations of Galois groups of number fields. Langlands' conjectures elaborate on the notion that these two problems overlap considerably. These volumes present convincing evidence supporting this, clearly and succinctly enough that readers can pass with minimal effort between the two points of view. Over a decade's worth of progress toward the stabilization of the Arthur-Selberg trace formula, culminating in Ngo Bau Chau's proof of the Fundamental Lemma, makes this series timely. "This is the second volume of a projected series of two or three collections of mainly expository articles on the arithmetic theory of automorphic forms. The books are intended primarily for two groups of readers. The first group is interested in the structure of automorphic forms on reductive groups over number fields, and specifically in qualitative information about the multiplicities of automorphic representations. The second group is interested in the problem of classifying I-adic representations of Galois groups of number fields. Langlands' conjectures elaborate on the notion that these two problems overlap to a considerable degree. The goal of this series of books is to gather into one place much of the evidence that this is the case, and to present it clearly and succinctly enough so that both groups of readers are not only convinced by the evidence but can pass with minimal effort between the two points of view. More than a decade's worth of progress toward the stabilization of the Arthur-Selberg trace formula, culminating in Ngo Bau Chau's proof of the Fundamental Lemma, has made this series timely"-- Provided by publisher Contents Introduction to Volume II Lectures on Shimura varieties Unitary Shimura varieties Integral models of Shimura varieties of PEL type Introduction to the Langlands–Kottwitz method Integral Canonical Models of Shimura varieties: an update The Newton stratification On the geometry of the Newton stratification Construction of automorphic Galois representations: the self-dual case The local Langlands correspondence for GLn over p-adic fields, and the cohomology of compact unitary Shimura varieties Une application des variétés de Hecke des groupes unitaires A Patching Lemma On subquotients of the étale cohomology of Shimura varieties The book is designed for graduate students and beginning researchers into the arithmetic theory of automorphic forms, and for all who want to know more about the Langlands program. It forms a sequel to On the Stabilization of the Trace Formula published in 2011.