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

Publication subTitle ：The Geometry and Cohomology of Some Simple Shimura Varieties. (AM-151)

Publication series ：Annals of Mathematics Studies

Author： Harris Michael;Taylor Richard;;  

Publisher： Princeton University Press‎

Publication year： 2001

E-ISBN: 9781400837205

P-ISBN(Paperback): 9780691090900

Subject： O187 algebraic geometry

Keyword： 数学

Language： ENG

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Description

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

Chapter

Cover

Title

Copyright

Dedication

Contents

Introduction

Acknowledgements

I Preliminaries

I.1 General notation

I.2 Generalities on representations

I.3 Admissible representations of GLg

I.4 Base change

I.5 Vanishing cycles and formal schemes

I.6 Involutions and unitary groups

I.7 Notation and running assumptions

II Barsotti-Tate groups

II.1 Barsotti-Tate groups

II.2 Drinfeld level structures

III Some simple Shimura varieties

III.1 Characteristic zero theory

III.2 Cohomology

III.3 The trace formula

III.4 Integral models

IV Igusa varieties

IV.1 Igusa varieties of the first kind

IV.2 Igusa varieties of the second kind

V Counting Points

V.1 An application of Fujiwara's trace formula

V.2 Honda-Tate theory

V.3 Polarisations I

V.4 Polarisations II

V.5 Some local harmonic analysis

V.6 The main theorem

VI Automorphic forms

VI.1 The Jacquet-Langlands correspondence

VI.2 Clozel's base change

VII Applications

VII.1 Galois representations

VII.2 The local Langlands conjecture

Appendix. A result on vanishing cycles

Bibliography

Index

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