Spectral Expansions of Overconvergent Modular Functions

The main result of this paper is an instance of the conjecture made by Gouvêa and Mazur in [GM95], which asserts that for certain values of r the space of r-overconvergent p-adic modular forms of tame level N and weight k should be spanned by the finite slope Hecke eigenforms. For N = 1, p = 2 and k = 0 we show that this follows from the combinatorial approach initiated by Emerton [Eme98] and Smithline [Smi00], using the classical LU decomposition and results of Buzzard–Calegari [BC05]; this implies the conjecture for all r ∈ ( 5 12 , 7 12 ). Similar results follow for p = 3 and p = 5 with the assumption of a plausible conjecture, which would also imply formulae for the slopes analogous to those of [BC05]. We also show that (for general p and N) the space of weight 0 overconvergent forms carries a natural inner product with respect to which the Hecke action is self-adjoint. When N = 1 and p ∈ {2, 3, 5, 7, 13}, combining this with the combinatorial methods allows easy computations of the q-expansions of small slope overconvergent eigenfunctions; as an application we calculate the q-expansions of the first 20 eigenfunctions for p = 5, extending the data given in [GM95]. 1 Background Let Sk(Γ1(N)) denote the space of classical modular cusp forms of weight k and level N . It has long been known that these objects satisfy many interesting congruence relations. One very powerful method for studying the congruences obeyed by modular forms modulo powers of a fixed prime p is to embed this space into the p-adic Banach space Sk(Γ1(N), r) of r-overconvergent p-adic cusp forms, defined as in [Kat73] using sections of ω on certain affinoid subdomains of X1(N) obtained by removing discs of radius p −r around the supersingular points; this space has been used to great effect by Coleman and others ([Col96, Col97]). It is known that there is a Hecke action on Sk(Γ1(N), r), as with the classical spaces, and these operators are continuous; and moreover, at least for 0 < r < p p+1 , the Atkin-Lehner operator U is compact. There is a rich spectral theory for