Cohomology of Drinfeld Modular Varieties, Part 2, Automorphic Forms, Trace Formulas and Langlands Correspondence

by Gérard Laumon

Overview

Cohomology of Drinfeld Modular Varieties aims to provide an introduction 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. This second volume is concerned with the ArthurSHSelberg trace formula, and to the proof in some cases of the Ramanujan-Petersson conjecture and the global Langlands conjecture for function fields. The author uses techniques that 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. This book will be of much interest to all researchers in algebraic number theory and representation theory.

Product Details

ISBN-13:	9780521109901
Publisher:	Cambridge University Press
Publication date:	04/30/2009
Series:	Cambridge Studies in Advanced Mathematics , #56
Pages:	380
Product dimensions:	6.00(w) x 8.90(h) x 1.10(d)

Table of Contents

Preface; 9. Trace of fA on the discrete spectrum; 10. Non-invariant Arthur trace formula: the geometric side; 11. Non-invariant Arthur trace formula: the spectral side; 12. Cohomology with compact supports of Drinfeld modular varieties; 13. Intersection cohomology of Drinfeld modular varieties; Appendix D. Representations of unimodular, locally compact, totally discontinuous, separated topological groups: addendum; Appendix E. Reduction theory and strong approximation; Appendix F. Proof of lemma 10. 6. 4; Appendix G. The decomposition of L2G following the cuspidal data.

Editorial Reviews

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)

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