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Shimura varieties. (English) Zbl 1435.14004
London Mathematical Society Lecture Note Series 457. Cambridge: Cambridge University Press (ISBN 978-1-108-70486-1/pbk; 978-1-108-64971-1/ebook). iii, 333 p. (2020).
Publisher’s description: This is the second volume of a series of mainly expository articles on the arithmetic theory of automorphic forms. It forms a sequel to L. Clozel (ed.) et al. [Stabilization of the trace formula, Shimura varieties, and arithmetic applications. Volume 1: On the stabilization of the trace formula. Somerville, MA: International Press (2011; Zbl 1255.11027)]. 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.
Systematically develops the Langlands-Kottwitz method for Shimura varieties through the important example of unitary groups
Constructs Galois representations attached to automorphic representations of $$GL(n)$$
Includes several surveys of contemporary developments as well as original research

The articles of this volume will be reviewed individually.
Indexed articles:
Haines, T. J.; Harris, M., Introduction to Volume II, 1-21 [Zbl 1436.11004]
Genestier, A.; Ngô, B. C., Lectures on Shimura varieties, 22-71 [Zbl 1440.11001]
Nicole, Marc-Hubert, Unitary Shimura varieties, 72-95 [Zbl 1440.14138]
Rozensztajn, Sandra, Integral models of Shimura varieties of PEL type, 96-114 [Zbl 1440.14139]
Zhu, Yihang, Introduction to the Langlands-Kottwitz method, 115-150 [Zbl 1440.14141]
Kisin, Mark, Integral canonical models of Shimura varieties: an update, 151-165 [Zbl 1440.14136]
Mantovan, Elena, The Newton stratification, 166-191 [Zbl 1440.14115]
Viehmann, Eva, On the geometry of the Newton stratification, 192-208 [Zbl 1440.14140]
Shin, Sug Woo, Construction of automorphic Galois representations: the self-dual case, 209-250 [Zbl 1455.11079]
Scholze, Peter, The local Langlands correspondence for GL$$_n$$ over $$p$$-adic fields, and the cohomology of compact unitary Shimura varieties, 251-265 [Zbl 1440.11218]
Chenevier, Gaëtan, An application of Hecke varieties from unitary groups, 266-296 [Zbl 07219416]
Sorensen, Claus M., A patching lemma, 297-305 [Zbl 1440.11219]
Johansson, Christian; Thorne, Jack A., On subquotients of the étale cohomology of Shimura varieties, 306-333 [Zbl 07219418]
##### MSC:
 14-06 Proceedings, conferences, collections, etc. pertaining to algebraic geometry 11-06 Proceedings, conferences, collections, etc. pertaining to number theory 14G35 Modular and Shimura varieties 11G18 Arithmetic aspects of modular and Shimura varieties 00B15 Collections of articles of miscellaneous specific interest
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