Geometry and cohomology of some simple Shimura varieties

Includes index

Includes bibliographical references

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 kindIV. 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

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.