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Philadelphia Area Number Theory Seminar

Wednesday, February 13, 2019 - 2:15pm

John Bergdall

Bryn Mawr College


Bryn Mawr College

Park Science Building, Room 328

Talk is scheduled to begin at 2:40 PM. Tea and refreshments will be served at 2:20 PM in the Math Lounge, Park Science Building, Room 361.

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.