6 - Eisenstein Series, the Trace Formula, and the Modern Theory of Automorphic Forms

Abstract

This chapter discusses the general Einstein series, the trace formula, and the modern theory of automorphic forms. The modern theory of automorphic forms is a response to different impulses and influences, above all the work of Hecke, but also class-field theory and the study of quadratic forms, the theory of representations of reductive groups, and of complex multiplication. The chapter further discusses how a larger class of Euler products, the automorphic L-functions, one of the central notions of the modern theory of automorphic forms, arose, a little by accident, from the solution of the problem in analytic continuation. The spectral theory has two aspects: (1) the spectral decomposition of the spaces by means of Eisenstein series; (2) the trace formula, which is an extension of the Frobenius reciprocity law to pairs (Γ, G), G a continuous group and Γ a discrete subgroup. The methods used for the Eisenstein series deal with those that are associated to maximal parabolic subgroups P and involve a single parameter.