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$p$ -adic $L$ -functions for $text{GL}_{2}$

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Title: $p$ -adic $L$ -functions for $text{GL}_{2}$
Authors: Barrera Salazar, D
Williams, CD
Item Type: Journal Article
Abstract: Since Rob Pollack and Glenn Stevens used overconvergent modular symbols to construct p-adic L-functions for non-critical slope rational modular forms, the theory has been extended to construct p-adic L-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 there is a canonical way of constructing a ray class distribution from the corresponding overconvergent symbol, that moreover interpolates critical values of the L-function of the eigenform. We thus define its p-adic L-function to be this distribution.
Issue Date: Oct-2019
Date of Acceptance: 20-Dec-2017
URI: http://hdl.handle.net/10044/1/55611
DOI: https://doi.org/10.4153/CJM-2017-062-0
ISSN: 0008-414X
Publisher: Canadian Mathematical Society
Start Page: 1019
End Page: 1059
Journal / Book Title: Canadian Journal of Mathematics
Volume: 71
Issue: 5
Copyright Statement: © Canadian Mathematical Society 2018. This is the post-print (ie final draft post-refereeing) version of the following published article: Barrera Salazar, D., & Williams, C. (2019). $p$ -adic $L$ -functions for $text{GL}_{2}$. Canadian Journal of Mathematics, 71(5), 1019-1059. doi:10.4153/CJM-2017-062-0
Keywords: math.NT
11F41, 11F67, 11F85, 11S40, 11M41
0101 Pure Mathematics
General Mathematics
Notes: In this paper, we blended the methods and results of our respective PhD theses to construct p-adic L-functions for small slope cohomological automorphic forms for GL(2) over general number fields. The main novelties over our previous papers were a more general definition of 'automorphic cycles', adapted to the number field case, and a canonicity result for the resulting distribution. In particular, in the completely general setting p-adic L-functions are not uniquely determined by an interpolation property, unlike in our previous papers; we proved that our construction is canonical. The results were a significant generalisation of previously known constructions of p-adic L-functions in this setting, which largely restricted to the ordinary setting.
Publication Status: Published
Online Publication Date: 2019-01-07
Appears in Collections:Faculty of Natural Sciences