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From L-functions to "Beyond Endoscopy"

L-functions are a way to encode "local" (prime-by-prime) information about important arithmetic objects, such as Galois representations or modular forms. Their "global" properties (such as analytic continuation and functional equation) are usually studied by analytic techniques that are not yet completely understood.

Starting from a prototype of such techniques, period integrals, I will present a series of reformulations, and results along the way, which suggest a possible way to study constructions of L-functions by generalizing Langlands' functoriality conjecture, and his ambitious "Beyond Endoscopy" program. No prior familiarity with the Langlands program will be assumed.