On the meromorphic continuation of Eisenstein series

TL;DR

This paper introduces a simplified proof for the meromorphic continuation of Eisenstein series, applicable to a broad class of automorphic forms, using basic Fredholm theory and the principle of meromorphic continuation.

Contribution

It provides a new, more accessible proof of meromorphic continuation that extends to Eisenstein series induced from any automorphic form, unlike traditional methods.

Findings

Proof relies only on rudimentary Fredholm theory in the number field case.

Applicable to Eisenstein series induced from arbitrary automorphic forms.

Close in spirit to Selberg's later proofs.

Abstract

Eisenstein series are ubiquitous in the theory of automorphic forms. The traditional proofs of the meromorphic continuation of Eisenstein series, due to Selberg and Langlands, start with cuspidal Eisenstein series as a special case, and deduce the general case from spectral theory. We present a "soft" proof which relies only on rudimentary Fredholm theory (needed only in the number field case). It is valid for Eisenstein series induced from an arbitrary automorphic form. The proof relies on the principle of meromorphic continuation. It is close in spirit to Selberg's later proofs.