Speaker: Samuel Marks

Title: p-adic modular forms à la Serre

Abstract:

The first definition of a p-adic modular form comes from Serre, who identifies a modular form with its q-expansion. Explicitly, Serre defines a p-adic modular form to be a p-adic limit of q-expansions of classical modular forms. This theory provides concrete models for many of the key objects that are defined more abstractly in Katz's geometric approach, the subject to which STAGE turns next. In this talk, we develop Serre's theory of p-adic modular forms and discuss the applications to p-adic L-functions and -- time permitting -- congruences of modular forms.