Cohomology of Drinfeld Modular Varieties, Part 1, Geometry, Counting of Points and Local Harmonic Analysis

This 1995 book introduces the reader to Drinfeld modular varieties, and is pitched at graduate students.

Gérard Laumon (Author)

9780521172745, Cambridge University Press

Paperback, published 9 December 2010

360 pages
22.9 x 15.2 x 2 cm, 0.53 kg

Review of the hardback: 'This is a very impressive achievement.' H.J. Baues, Mathematika

Originally published in 1995, Cohomology of Drinfeld Modular Varieties aimed 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. 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.

1. Construction of Drinfeld modular varieties
2. Drinfeld A-modules
3. The Lefschetz numbers of Hecke operators
4. The fundamental lemma
5. Very cuspidal Euler–Poincaré functions
6. The Lefschetz numbers as sums of global elliptic orbital integrals
7. Unramified principal series representations
8. Euler-Poincaré functions as pseudocoefficients of the Steinberg relation
Appendices.

Subject Areas: Algebra [PBF]