TABLE OF CONTENTS: Introduction 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
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