The Geometry and Cohomology of Some Simple Shimura Varieties. (am-151) (ebook)

Argumento de The Geometry and Cohomology of Some Simple Shimura Varieties. (am-151) (ebook)

This book aims first to prove the local Langlands conjecture for GL_n 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 GL_n(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 GL_n(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.0Introduction 1
Acknowledgements 15
Chapter I: Preliminaries 17
I.1 General notation 17
I.2 Generalities on representations 21
I.3 Admissible representations of GL, 28
I.4 Base change 37
I.5 Vanishing cycles and formal schemes 40
I.6 Involutions and unitary groups 45
I.7 Notation and running assumptions 51
Chapter II: Barsotti-Tate groups 59
II.1 Barsotti-Tate groups 59
II.1 Drinfeld level structures 73
Chapter III: Some simple Shimura varieties 89
III.1 Characteristic zero theory 89
III.1 Cohomology 94
III.1 The trace formula 105
III.1 Integral models 108
Chapter IV: Igusa varieties 121
IV.1 Igusa varieties of the first kind 121
IV.2 Igusa varieties of the second kind 133
Chapter V: Counting Points 149
V.1 An application of Ftjiwara's trace formula 149
V.2 Honda-Tate theory 157
V.3 Polarisations I 163
V.4 Polarisations II 168
V.5 Some local harmonic analysis 182
V.6 The main theorem 191
Chapter VI: Automorphic forms 195
VI.1 The Jacquet-Langlands correspondence 195
VI.2 Clozel's base change 198
Chapter VII: Applications 217
VII1 Galois representations 217
VII.2 The local Langlands conjecture 233
Appendix. A result on vanishing cycles by V. G. Berkovich 257
Bibliography 261
Index 269