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The search for p-adic automorphic forms

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Modular forms have been a central object of study in number theory for many years. In the 1970s, Serre introduced p-adic modular forms. These are more general objects which, unlike classical modular forms, live in p-adic families. Since then, the theory of p-adic modular forms has seen many applications to classical problems in number theory, especially in the Langlands program. Modular forms are special cases of more general objects: automorphic forms and automorphic representations. However, the concept of a “p-adic automorphic form” is more elusive. In this talk, I will explain the basics of the theory of (p-adic) modular forms, what properties we look for in the more general p-adic automorphic forms, and some of their proposed constructions.

This talk is part of the Junior Algebra and Number Theory seminar series.

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