Approaches to computing overconvergent p-adic modular forms

Let p be a prime number. It’s well known that classical modular Hecke eigenforms can satisfy nontrivial congruence relations modulo powers of p; for example the standard Eisenstein seriesEk satisfiesEp−1 = 1 (mod p), and more generally E(p−1)pr = 1 (mod p) for all integers r ≥ 0. We’d like to construct some kind of p-adic space of modular forms in which this really represents a limiting process, with E(p−1)pr being a sequence of modular forms tending towards the constant form 1 in weight 0. An early attempt to construct such a space dates back to work of Serre in the early 1970’s [Ser73]. Serre considered the space of formal q-expansions that are uniform p-adic limits of q-expansions of modular forms of fixed level N and arbitrary weight. This space, the space of convergent p-adic modular forms, is infinite-dimensional but nonetheless has many good properties; for example, such forms have a well-defined weight, and one can define an action of Hecke operators by the usual formulae on qexpansions and these turn out to be continuous. However, this space has too many eigenforms: a construction due to Gouvea shows that one may construct eigenfunctions for the Up operator with arbitrary eigenvalue, so in particular these p-adic eigenfunctions contain no interesting arithmetical information. We shall see in the remainder of this section how to construct a space which remedies this, and it is the elements of this space which we will attempt to compute in the following sections.