Shimura Varieties
London Mathematical Society Lecture Note Series

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.

Introduction T. J. Haines and M. Harris; 1. Lectures on Shimura varieties A. Genestier and B. C. Ngo; 2. Unitary Shimura varieties Marc-Hubert Nicole; 3. Integral models of Shimura varieties of PEL type Sandra Rozensztajn; 4. Introduction to the Langlands-Kottwitz method Yihang Zhu; 5. Integral canonical models of Shimura varieties Mark Kisin; 6. The Newton stratification Elena Mantovan; 7. On the geometry of the Newton stratification Eva Viehmann; 8. Construction of automorphic Galois representations Sug Woo Shin; 9. The local Langlands correspondence for GLn over p-adic fields, and the cohomology of compact unitary Shimura varieties Peter Scholze; 10. Une application des varietes de Hecke des groupes unitaires Gaetan Chenevier; 11. A patching lemma Claus M. Sorensen; 12. On subquotients of the etale cohomology of Shimura varieties Christian Johansson and Jack A. Thorne.

Thomas Haines is a Professor of Mathematics at the University of Maryland, College Park. He has authored over thirty research articles, several survey articles on matters related to the Langlands program, and a monograph on commutative algebra. He has been awarded a Sloan Fellowship and a Simons Research Fellowship.
Michael Harris is a Professor of Mathematics at Columbia University and the Université Paris Diderot. He is the author or co-author of nearly 90 mathematical books and articles, and he has received a number of prizes, including the Grand Prix Sophie Germain of the Académie des Sciences, and the Clay Research Award, which he shared in 2007 with Richard Taylor.