The Poisson summation conjecture of Braverman-Kazhdan, L. Lafforgue, Ngo, and Sakellaridis is an ambitious proposal to prove analytic properties of quite general Langlands L-functions using vast generalizations of the Poisson summation formula. In this talk, we introduce multivariable zeta integrals that unfold to Euler products representing the triple product L-function times a product of L-functions with known analytic properties. Motivated by this integral representation, we formulate a conjectural Poisson summation formula and show it implies the analytic properties of triple product L-functions. Finally, we propose a strategy, the fiber bundle method, to reduce the conjectural formula to two known Poisson summation formulae along with certain local compatibility statements.

This is joint work with Jayce Getz, Chun-Hsien Hsu, and Spencer Leslie.