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L-functions, which are generalizations of the Riemann zeta function, play a role in some of the deepest conjectures in mathematics. A huge step in their understanding was undertaken in the late ’60s and ’70s by Robert Langlands, who gave a general definition of automorphic L-functions, and conjectured that they contain all the L-functions arising from Galois theory and arithmetic geometry. Still, the true nature of these complex-analytic functions which contain crucial arithmetic information remained a mystery.

In this talk, I will review various incarnations of L-functions associated to number fields or function fields, premised on 2 paradigms: On one hand, the cohomological interpretation of the zeta functions of varieties over finite fields (“the Galois side”), and, on the other, Riemann’s interpretation of the zeta function as a Mellin transform of a theta series (“the automorphic side”). I will then sketch a correspondence between the two paradigms, based on joint work with David Ben-Zvi and Akshay Venkatesh, based on a conjectural duality between Hamiltonian spaces, with ramifications that extend into mathematical physics.