The Library
Spectral expansions of overconvergent modular functions
Tools
Tools
Tools
Loeffler, David (2007) Spectral expansions of overconvergent modular functions. International Mathematics Research Notices, 2007 . doi:10.1093/imrn/rnm050 ISSN 1073-7928.
|
PDF
WRAP_Loeffler_eigenfunctions.pdf - Accepted Version - Requires a PDF viewer. Download (523Kb) | Preview |
Official URL: http://dx.doi.org/10.1093/imrn/rnm050
Abstract
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 [Formula]. 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].
| Item Type: | Journal Article | ||||||
|---|---|---|---|---|---|---|---|
| Subjects: | Q Science > QA Mathematics | ||||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||
| Library of Congress Subject Headings (LCSH): | Modular functions, Number theory, Eigenfunctions | ||||||
| Journal or Publication Title: | International Mathematics Research Notices | ||||||
| Publisher: | Oxford University Press | ||||||
| ISSN: | 1073-7928 | ||||||
| Official Date: | 7 September 2007 | ||||||
| Dates: |
|
||||||
| Volume: | 2007 | ||||||
| Number of Pages: | 17 | ||||||
| DOI: | 10.1093/imrn/rnm050 | ||||||
| Status: | Peer Reviewed | ||||||
| Publication Status: | Published | ||||||
| Access rights to Published version: | Restricted or Subscription Access | ||||||
| Date of first compliant deposit: | 21 January 2016 | ||||||
| Date of first compliant Open Access: | 21 January 2016 | ||||||
| Funder: | Engineering and Physical Sciences Research Council (EPSRC) |
Request changes or add full text files to a record
Repository staff actions (login required)
 |
View Item |
Downloads
Downloads per month over past year
