The Local Langlands Conjecture for GL(2)

Colin J. Bushnell, Guy Henniart (auth.)9783540314868, 3-540-31486-5

If F is a non-Archimedean local field, local class field theory can be viewed as giving a canonical bijection between the characters of the multiplicative group GL(1,F) of F and the characters of the Weil group of F. If n is a positive integer, the n-dimensional analogue of a character of the multiplicative group of F is an irreducible smooth representation of the general linear group GL(n,F). The local Langlands Conjecture for GL(n) postulates the existence of a canonical bijection between such objects and n-dimensional representations of the Weil group, generalizing class field theory.

This conjecture has now been proved for all F and n, but the arguments are long and rely on many deep ideas and techniques. This book gives a complete and self-contained proof of the Langlands conjecture in the case n=2. It is aimed at graduate students and at researchers in related fields. It presupposes no special knowledge beyond the beginnings of the representation theory of finite groups and the structure theory of local fields. It uses only local methods, with no appeal to harmonic analysis on adele groups.

Table of contents :
Smooth Representations….Pages 7-41
Finite Fields….Pages 43-48
Induced Representations of Linear Groups….Pages 49-83
Cuspidal Representations….Pages 85-122
Parametrization of Tame Cuspidals….Pages 123-136
Functional Equation….Pages 137-177
Representations of Weil Groups….Pages 179-209
The Langlands Correspondence….Pages 211-224
The Weil Representation….Pages 225-250
Arithmetic of Dyadic Fields….Pages 251-266
Ordinary Representations….Pages 267-283
The Dyadic Langlands Correspondence….Pages 285-324
The Jacquet-Langlands Correspondence….Pages 325-337