Spectral Expansions of Overconvergent Modular Functions

The main result of this article is an instance of the conjecture made by Gouvêa and Mazur in [11], 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 [9] and Smithline [16], using the classical LU decomposition and results of Buzzard–Calegari [1]; this implies the conjecture for all $r\hspace{0.4em}\in\hspace{0.4em}(\frac5{12}\hspace{0.4em},\hspace{0.4em}\frac7{12})$. 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 [1].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 [11].