P-adic L-functions of Bianchi modular forms
File(s)
Author(s)
Williams, CD
Type
Journal Article
Abstract
The theory of overconvergent modular symbols, developed by Rob Pollack and Glenn Stevens, gives a beautiful and effective construction of the p-adic L-function of a modular form. In this paper, we give an analogue of their results for Bianchi modular forms, that is, modular forms over
imaginary quadratic fields. In particular, we prove control theorems that say that the canonical specialisation map from overconvergent to classical Bianchi modular symbols is an isomorphism on small slope eigenspaces of suitable Hecke operators. We also give an explicit link between the
classical modular symbol attached to a Bianchi modular form and critical values of its L-function, which then allows us to construct p-adic L-functions of Bianchi modular forms.
imaginary quadratic fields. In particular, we prove control theorems that say that the canonical specialisation map from overconvergent to classical Bianchi modular symbols is an isomorphism on small slope eigenspaces of suitable Hecke operators. We also give an explicit link between the
classical modular symbol attached to a Bianchi modular form and critical values of its L-function, which then allows us to construct p-adic L-functions of Bianchi modular forms.
Date Issued
2017-04
Date Acceptance
2016-11-22
Citation
Proceedings of the London Mathematical Society, 2017, 114 (4), pp.614-656
ISSN
1460-244X
Publisher
London Mathematical Society
Start Page
614
End Page
656
Journal / Book Title
Proceedings of the London Mathematical Society
Volume
114
Issue
4
Copyright Statement
© 2017 The Authors. The Proceedings of the London Mathematical Society is copyright © London Mathematical Society
This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.
This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.
Identifier
https://londmathsoc.onlinelibrary.wiley.com/doi/full/10.1112/plms.12020
Subjects
Science & Technology
Physical Sciences
Mathematics
IMAGINARY QUADRATIC FIELDS
ELLIPTIC-CURVES
COHOMOLOGY
SYMBOLS
VALUES
SLOPE
math.NT
math.NT
11F41, 11F67
0101 Pure Mathematics
0104 Statistics
Notes
In this paper, I give an explicit (and constructive) construction of the p-adic L-function of a Bianchi modular form (that is, an automorphic form for GL(2) over an imaginary quadratic field) under a 'small slope' condition. The construction was inspired by the methods of Pollack and Stevens, who carried out such a construction for classical modular forms. In particular, I developed the theory of overconvergent modular symbols in explicit detail. As part of this, I gave very hands-on proofs (and more concrete statements) of classical integral formulae for the L-function of a Bianchi modular form. In the Bianchi case, the small slope condition is stricter than in the classical case, and one of the main novelties of this paper are refined results in the case where p splits in the imaginary quadratic field, allowing a much more general construction. Previous constructions of such objects applied only in special cases, such as trivial weight under an ordinarity assumption; this paper provided a significant generalisation.
Publication Status
Published
Date Publish Online
2017-04-04