Joint IAS/Princeton University Number Theory Seminar

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 $(V_i,Q_i)$ of even dimension by the equation

$Q_1(v_1)=Q_2(v_2)=Q_3(v_3)$.

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

$GL_{n_1} \times GL_{n_2} \times \GL_{n_3}$

where the $n_i$ are arbitrary, and speculate on the relationship between these period integrals and Langlands L functions.