Shimura Varieties

A concise and comprehensive introduction to trace formula methods in the study of Shimura varieties and associated Galois representations.

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.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.