### Philadelphia Area Number Theory Seminar

Wednesday, February 13, 2019 - 2:15pm

#### John Bergdall

Bryn Mawr College

The purpose of this talk is to discuss two related problems about modular forms. The ﬁrst is a conjecture stated by Gouvˆea and Mazur in the early 1990’s. Their conjecture aims to predict a speciﬁc local constancy result for the multiplicity of (the p-adic norm of) a certain Hecke eigenvalue appearing in spaces of modular forms, as the weight varies. (Caveat: their conjecture was disproven!) Their conjecture was an attempt to nail down the as-of-then undiscovered general theory of p-adic modular forms. Later, Coleman proved families of p-adic modular forms exist as q-expansions converging on p-adic discs. The second problem, a variation of the Gouvˆea-Mazur conjecture, is to ask for the radius of convergence of a given family. Our discussion will highlight new results on this second problem, but we will start by making precise both problems.