Spectral expansions of overconvergent modular functions

Details Spectral expansions of overconvergent modular functions Spectral expansions of overconvergent modular functions

2007-01-05

arxiv.org/abs/math/0701168v3

International Mathematics Research Notices Vol. 2007 no. 16 (rnm050)

Mathematics

Number Theory

Updated version fixing some minor typos

Scientific paper

10.1093/imrn/rnm050

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.

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