p-adic Modular Forms: An Introduction

Serre in the 1970s was the first to formalize such a question on the way to constructing p-adic L-functions, by way of developing the notion of a p-adic modular form to be the p-adic limit of some compatible family of q-expansions of classical modular forms. Katz came along fairly soon afterwards and generalized the theory to a much more geometric context, and showed that Serre’s p-adic forms exist as a special case of a much wider family of p-adic objects. The theory has been further refined since then by Dwork, Hida, and most recently, Coleman, leading to the modern theory of overconvergent modular forms.