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P-adic L-functions for GL(2)

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Barrera Salazar, Daniel and Williams, Christopher David (2019) P-adic L-functions for GL(2). Canadian Journal of Mathematics, 71 (5). pp. 1019-1059. doi:10.4153/CJM-2017-062-0

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Official URL: https://doi.org/10.4153/CJM-2017-062-0

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Abstract

Since Rob Pollack and Glenn Stevens used overconvergent modular symbols to construct -adic -functions for non-critical slope rational modular forms, the theory has been extended to construct -adic -functions for non-critical slope automorphic forms over totally real and imaginary quadratic fields by the first and second authors, respectively. In this paper, we give an analogous construction over a general number field. In particular, we start by proving a control theorem stating that the specialisation map from overconvergent to classical modular symbols is an isomorphism on the small slope subspace. We then show that if one takes the modular symbol attached to a small slope cuspidal eigenform, then one can construct a ray class distribution from the corresponding overconvergent symbol, which moreover interpolates critical values of the -function of the eigenform. We prove that this distribution is independent of the choices made in its construction. We define the -adic -function of the eigenform to be this distribution.

Item Type: Journal Article
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH): Algebraic number theory, L-functions
Journal or Publication Title: Canadian Journal of Mathematics
Publisher: Cambridge University Press
ISSN: 0008-414X
Official Date: October 2019
Dates:
DateEvent
October 2019Published
7 January 2019Available
29 December 2017Accepted
Volume: 71
Number: 5
Page Range: pp. 1019-1059
DOI: 10.4153/CJM-2017-062-0
Status: Peer Reviewed
Publication Status: Published
Publisher Statement: This article has been published in a revised form in Canadian Journal of Mathematics. https://doi.org/10.4153/CJM-2017-062-0 This version is free to view and download for private research and study only. Not for re-distribution, re-sale or use in derivative works. © Canadian Mathematical Society 2018
Access rights to Published version: Restricted or Subscription Access
Copyright Holders: © Canadian Mathematical Society 2018
RIOXX Funder/Project Grant:
Project/Grant IDRIOXX Funder NameFunder ID
UNSPECIFIED[EPSRC] Engineering and Physical Sciences Research Councilhttp://dx.doi.org/10.13039/501100000266
682152[ERC] Horizon 2020 Framework Programmehttp://dx.doi.org/10.13039/100010661
UNSPECIFIEDUniversité de Montréal. ‎Centre de recherches mathématiqueshttp://viaf.org/viaf/142520504
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