Cohomology of Drinfeld modular varieties. Part I. Geometry, counting of points and local harmonic analysis

Cohomology of Drinfeld Modular Varieties aims to provide an introduction to both the subject of the title and the Langlands correspondence for function fields. These varieties are the analogs for function fields of Shimura varieties over number fields. This present volume is devoted to the geometry of these varieties and to the local harmonic analysis needed to compute their cohomology. To keep the presentation as accessible as possible, the author considers the simpler case of function rather than number fields; nevertheless, many important features can still be illustrated. It will be welcomed by workers in number theory and representation theory. Cohomology of Drinfeld Modular Varieties aims to provide an introduction, in two volumes, both to this subject and to the Langlands correspondence for function fields. These varieties are the analogues for function fields of the Shimura varieties over number fields. The Langlands correspondence is a conjectured link between automorphic forms and Galois representations over a global field. By analogy with the number-theoretic case, one expects to establish the conjecture for function fields by studying the cohomology of Drinfeld modular varieties, which has been done by Drinfeld himself for the rank two case. The present volume is devoted to the geometry of these varieties, and to the local harmonic analysis needed to compute their cohomology. It is based on graduate courses taught by the author, who uses techniques which are extensions of those used to study Shimura varieties. Though the author considers only the simpler case of function rather than number fields, many important features of the number field case can be illustrated. Several appendices on background material keep the work reasonably self-contained. It is the first book on this subject and will be of much interest to all researchers in algebraic number theory and representation theory

this 1995 Book Introduces The Reader To Drinfeld Modular Varieties, And Is Pitched At Graduate Students.

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a Graduate Text Book, Introducing Not Only The Title Subject, But Also The Langlands Correspondence For Function Fields, Which Are The Analogues For Function Field Of Shimura Varieties Over Number Fields. The First Volume Is Devoted To The Geometry Of The Varieties And To The Local Harmonic Analysis Needed To Compute Their Cohomology. Little Is Revealed About The Second Volume Except That The Notion Of Very Cuspidal Function Plays A Crucial Role In It. Annotation C. Book News, Inc., Portland, Or (booknews.com)

Contents Preface 1 Construction of Drinfeld modular varieties 2 Drinfeld A-modules with finite characteristic 3 The Lefschetz numbers of Hecke operators 4 The fundamental lemma 5 Very cuspidal Euler-Poincare functions 6 The Lefschetz numbers as sums of global elliptic orbital integrals 7 Unramified principal series representations 8 Euler-Poincare functions as pseudocoefficients of the Steinberg representation Appendices A. Central simple algebras B. Dieudonne's theory : some proofs C. Combinatorial formulas D1 Representations of unimodular, locally compact, totally discontinuous, separated, topological groups References Index Originally published in 1995, Cohomology of Drinfeld Modular Varieties provided an introduction, in two volumes, to both the subject of the title and the Langlands correspondence for function fields. It is based on courses given by the author and will be welcomed by workers in number theory and representation theory. Pt. 1. Geometry, Counting Of Points, And Local Harmonic Analysis -- Pt. 2. Automorphic Forms, Trace Formulas, And Langlands Correspondence. Gérard Laumon. With An Appendix By Jean-loup Waldspurger--pt. 2, T.p. Includes Bibliographical References And Indexes.