Spectral expansions of overconvergent modular functions

Abstract

The main result of this paper is an instance of the conjecture made by Gouvea and Mazur (Math. Res. Lett., 1995) 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. Using methods adapted from the work of Buzzard and Calegari I show that for p=2, k = 0, N = 1, this holds for all r in (5/12, 7/12). The proof relies on a certain factorisation of the U_p operator which is known in this case but I conjecture also holds for p = 3 and 5; this conjecture also implies exact formulae for the set of slopes similar to those proved for p=2 by Buzzard and Calegari. The same methods also provide an efficient approach to explicit computations of q-expansions of small slope overconvergent eigenforms, extending the computations of Gouvea and Mazur.

Updated version fixing some minor typos

References in corpus (2)

• The 2-adic Eigencurve is Proper
• Bounding slopes of $p$-adic modular forms

Cited by in corpus (6)

• Syntomic regulators of Asai--Flach classes
• Slopes of eigencurves over boundary disks
• Slopes of modular forms and the ghost conjecture (unabridged version)
• On the $U_p$ operator acting on $p$-adic overconvergent modular forms when $X_0(p)$ has genus 1
• On a $p$-adic extension of the Jacquet-Langlands correspondence to weight 1
• A modular proof of the properness of the Coleman-Mazur eigencurve