This talk is concerned with the radius of convergence of p-adic families of modular forms: q-series over a p-adic disc whose specialization to certain integer points is the q-expansion of a classical Hecke eigenform of level p. Numerical experiments by Gouvêa and Mazur in the 1990s predicted the general existence of such families but also suggested, in spirit, the radius of convergence in terms of an initial member. Buzzard and Calegari showed, ten years later, that the Gouvêa-Mazur prediction was false. It has since remained open question how to salvage it. Here we will present some recent theoretical results towards such a salvage, backed up by numerical data.
This video is part of the Institute for Advanced Study‘s Number theory seminar.
