On the meromorphic continuation of Eisenstein series

Bernstein, Joseph; Lapid, Erez

doi:10.1090/jams/1020

On the meromorphic continuation of Eisenstein series
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by Joseph Bernstein and Erez Lapid;

J. Amer. Math. Soc. 37 (2024), 187-234

DOI: https://doi.org/10.1090/jams/1020

Published electronically: April 27, 2023

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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.

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