Duality of automorphic periods

Groups Arithmetic and Algebraic Geometry

The study of automorphic integrals has a long history which began with Hecke's Mellin transform of a modular form, and more recently it has provided an indispensable language with which to describe functoriality phenomena in the Langlands program. In the first half of this presentation, I will explain these developments via emblematic examples while rephrasing them in terms of the recent framework of relative Langlands duality proposed by Ben-Zvi--Sakellaridis--Venkatesh. In the second half, I will present joint work with Venkatesh in which we establish relative duality in certain "singular" examples with the aid of new numerical invariants of Galois representations that we call "nonabelian L-functions".