This section is devoted to a rapid review of some of the basic analysis that is necessary in representation theory and the basic theory of automorphic forms. Even though the material below …

Here is an outline for the sections one might cover for a semester long introduction to automorphic representation theory: It is useful to say in words what these chapters cover. Given a global field F …

It relates the automorphic representations at-tached to GL(2) to those attached to its twists. There is a local version of the correspondence. Note that 1 is the dis-crete series representation of GL2(R) of …

The C-vector space of automorphic functions of weightkforΓis denotedΩk(Γ). We note the following (easily checked) facts: •Ifkis odd and −1 ∈ΓthenΩk(Γ) = {0}, sincefk|−1 = −fwhenkis odd.

Introduction Historically, the main motivation for studying modular forms came from their connnection to representation numbers of qua. ratic forms. Let us make this a little . ore precise. Given a positive de …

3 Automorphic forms and L-functions for SL(2;Z) Key idea: Automorphicity is equivalent to existence of functional equation for certain L-functions - this is the idea of converse theorems. Hecke operators: …

Automorphic forms for G with coefficients in R ; this is a generalization of the notion of modular form. Generalized Galois representations with targets intoG_(R), whereG_is the dual group to G.