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.

• 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

"...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

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.

• 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

"...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

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.

• 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

"...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