On the meromorphic continuation of Eisenstein series

Bernstein, Joseph; Lapid, Erez

doi:10.1090/jams/1020

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On the meromorphic continuation of Eisenstein series

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Bernstein, Joseph
Max Planck Institute for Mathematics, Max Planck Society;

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https://doi.org/10.48550/arXiv.1911.02342
(Preprint)

https://doi.org/10.1090/jams/1020
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1911.02342.pdf
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Citation

Bernstein, J., & Lapid, E. (2024). On the meromorphic continuation of Eisenstein series. Journal of the American Mathematical Society, 37(1), 187-234. doi:10.1090/jams/1020.

Cite as: https://hdl.handle.net/21.11116/0000-000F-2372-9

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.