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The Geometry and Cohomology of Some Simple Shimura Varieties. (AM-151), Volume 151 : Annals of Mathematics Studies - Michael Harris

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

Annals of Mathematics Studies

Paperback Published: 4th November 2001
ISBN: 9780691090924
Number Of Pages: 288

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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.

"...clearly and carefully written. In sum, it represents an awe-inspiring achievement and is a model of good exposition."--Bulletin of the American Mathematical Society, Volume 40, Number 2

Introductionp. 1
Acknowledgementsp. 15
Preliminariesp. 17
General notationp. 17
Generalities on representationsp. 21
Admissible representations of GL[subscript g]p. 28
Base changep. 37
Vanishing cycles and formal schemesp. 40
Involutions and unitary groupsp. 45
Notation and running assumptionsp. 51
Barsotti-Tate groupsp. 59
Barsotti-Tate groupsp. 59
Drinfeld level structuresp. 73
Some simple Shimura varietiesp. 89
Characteristic zero theoryp. 89
Cohomologyp. 94
The trace formulap. 105
Integral modelsp. 108
Igusa varietiesp. 121
Igusa varieties of the first kindp. 121
Igusa varieties of the second kindp. 133
Counting Pointsp. 149
An application of Fujiwara's trace formulap. 149
Honda-Tate theoryp. 157
Polarisationsp. 163
Polarisations IIp. 168
Some local harmonic analysisp. 182
The main theoremp. 191
Automorphic formsp. 195
The Jacquet-Langlands correspondencep. 195
Clozel's base changep. 198
Applicationsp. 217
Galois representationsp. 217
The local Langlands conjecturep. 233
A result on vanishing cyclesp. 257
Bibliographyp. 261
Indexp. 269
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780691090924
ISBN-10: 0691090920
Series: Annals of Mathematics Studies
Audience: Tertiary; University or College
Format: Paperback
Language: English
Number Of Pages: 288
Published: 4th November 2001
Country of Publication: US
Dimensions (cm): 22.86 x 15.24  x 1.91
Weight (kg): 0.41

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