The Poisson summation formula for the space of \(n \times n\) matrices was used by Godement and Jacquet to prove the analytic continuation and functional equation of the standard \(L\)-function of a cuspidal automorphic representation of \(\mathrm{GL}_n(\mathbb{A}_F)\). Braverman and Kazhdan conjectured that this was the first case of a general phenomenon. Ngo, L. Lafforgue, and Sakellaridis later refined and extended their conjecture to general spherical varieties. We refer to this collection of conjectures as the Poisson summation conjecture. The import of the conjecture is that it implies the meromorphic continuation and functional equation of general Langlands \(L\)-functions. By the converse theorem of Cogdell and Piatetski-Shapiro, this implies a broad swath of Langlands functoriality.
This TRT has obtained the first new global cases of Poisson summation conjecture since it was first introduced 2000. The ultimate goal is to prove it in special cases crafted to imply key cases of Langlands functoriality.
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