The Geometry and Cohomology of Some Simple Shimura Varieties. (AM-151), Volume 151

Michael Harris; Richard Taylor

Published by Princeton University Press 2002

The Geometry and Cohomology of Some Simple Shimura Varieties. (AM-151), Volume 151

Michael Harris and Richard Taylor

Volume 151 in the series Annals of Mathematics Studies

https://doi.org/10.1515/9781400837205

Overview
Contents

About this book

This book aims first to prove the local Langlands conjecture for GLn over a p-adic field and, second, to identify the action of the decomposition group at a prime of bad reduction on the l-adic cohomology of the "simple" Shimura varieties. These two problems go hand in hand. The results represent a major advance in algebraic number theory, finally proving the conjecture first proposed in Langlands's 1969 Washington lecture as a non-abelian generalization of local class field theory.

The local Langlands conjecture for GLn(K), where K is a p-adic field, asserts the existence of a correspondence, with certain formal properties, relating n-dimensional representations of the Galois group of K with the representation theory of the locally compact group GLn(K). This book constructs a candidate for such a local Langlands correspondence on the vanishing cycles attached to the bad reduction over the integer ring of K of a certain family of Shimura varieties. And it proves that this is roughly compatible with the global Galois correspondence realized on the cohomology of the same Shimura varieties. The local Langlands conjecture is obtained as a corollary.

Certain techniques developed in this book should extend to more general Shimura varieties, providing new instances of the local Langlands conjecture. Moreover, the geometry of the special fibers is strictly analogous to that of Shimura curves and can be expected to have applications to a variety of questions in number theory.

Author / Editor information

Michael Harris is Professor of Mathematics at Université Paris 7--Denis Diderot. His research focuses on automorphic forms and representation theory, with a view to applications in number theory and arithmetical algebraic geometry. Richard Taylor is Professor of Mathematics at Harvard University. His main research interest is number theory, particularly the theory of modular forms, automorphic forms, and Galois representations.

Reviews

...clearly and carefully written. In sum, it represents an awe-inspiring achievement and is a model of good exposition.

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Frontmatter Publicly Available Download PDF	i
Contents Publicly Available Download PDF	vii
Introduction Requires Authentication Unlicensed Licensed Download PDF	1
Acknowledgements Requires Authentication Unlicensed Licensed Download PDF	15
Chapter I. Preliminaries Requires Authentication Unlicensed Licensed Download PDF	17
Chapter II. Barsotti-Tate groups Requires Authentication Unlicensed Licensed Download PDF	59
Chapter III. Some simple Shimura varieties Requires Authentication Unlicensed Licensed Download PDF	89
Chapter IV. Igusa varieties Requires Authentication Unlicensed Licensed Download PDF	121
Chapter V. Counting Points Requires Authentication Unlicensed Licensed Download PDF	149
Chapter VI. Automorphic forms Requires Authentication Unlicensed Licensed Download PDF	195
Chapter VII. Applications Requires Authentication Unlicensed Licensed Download PDF	217
Appendix. A result on vanishing cycles V. G. Berkovich Requires Authentication Unlicensed Licensed Download PDF	257
Bibliography Requires Authentication Unlicensed Licensed Download PDF	261
Index Requires Authentication Unlicensed Licensed Download PDF	269