Braverman and Kazhdan have conjectured the existence of summation formulae that are essentially equivalent to the analytic continuation and functional equation of Langlands L-functions in great generality. Motivated by their conjectures and related conjectures of L. Lafforgue, Ngo, and Sakellaridis, Baiying Liu and I have proven a summation formula analogous to the Poisson summation formula for the subscheme cut out of three quadratic spaces (Vi,Qi) of even dimension by the equation Q1(v1)=Q2(v2)=Q3(v3). I will sketch the proof of this formula in the first portion of the talk. In the second portion, time permitting, I will discuss how these summation formulae lead to functional equations for period integrals for automorphic representations of GLn1 × GLn2 × GLn3 where the ni are arbitrary, and speculate on the relationship between these period integrals and Langlands L-functions.
This video is part of the Institute for Advanced Study‘s Number theory seminar.
