The Local Langlands Conjecture for GL(2) (Grundlehren der mathematischen Wissenschaften)
Edition: 1 edition
Publisher: Springer
Published: September 14, 2006
Language: eng
Format: Hardcover
Number of pages: 354
ISBN-13: 9783540314868
Subjects: Fields & rings; Number Theory; Mathematics; Science/Mathematics; Group Theory; Probability & Statistics - General; Local Langlands correspondence; Mathematics / Number Theory; Weil group; functional equation; smooth representation
First sentence: We work with a non-Archimedean local field F which, we always assume, has finite residue field of characteristic p.
Description: 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.