Modern Theory of Automorphic Forms ∗

The modern theory of automorphic forms is a response to many 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 complexmultiplication, but so far many of the most powerful techniques are the issue, direct or indirect, of the introduction by Maass and then Selberg of spectral theory into the subject. The spectral theory has two aspects: (i) the spectral decomposition of the spaces L(Γ\G) by means of Eisenstein series; (ii) the trace formula, which can be viewed as a striking extension of the Frobenius reciprocity law to pairs (Γ, G), G a continuous group and Γ a discrete subgroup.