Exceptional zeros and L-invariants of Bianchi modular forms
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Accepted version
Author(s)
Barrera Salazar, D
Williams, CD
Type
Journal Article
Abstract
Let f be a Bianchi modular form, that is, an automorphic form for GL(2) over an imaginary quadratic field F. In this paper, we prove an exceptional zero conjecture in the case where f is new at a prime above p. More precisely, for each prime p of F above p we prove the existence of an L-invariant Lp, depending only on p and f, such that when the p-adic L-function of f has an exceptional zero at p, its derivative can be related to the classical L-value multiplied by Lp. The proof uses cohomological methods of Darmon and Orton, who proved similar results for GL(2) over the rationals. When p is not split and f is the base-change of a classical modular form F, we relate Lp to the L-invariant of F, resolving a conjecture of Trifkovi\'{c} in this case.
Editor(s)
Darmon, H
Date Issued
2019-07-01
Date Acceptance
2017-10-11
ISSN
0002-9947
Publisher
American Mathematical Society
Start Page
1
End Page
34
Journal / Book Title
Transactions of the American Mathematical Society
Volume
372
Issue
1
Copyright Statement
First published in Transactions of the American Mathematical Society, published by the American Mathematical Society. © 2017 American Mathematical Society.
Source Database
manual-entry
Subjects
General Mathematics
0101 Pure Mathematics
0102 Applied Mathematics
Notes
In this paper, we prove the existence of L-invariants attached to Bianchi modular forms, and use them to prove the 'weak exceptional zero conjecture' in this setting. In the Bianchi case, the study of L-invariants is hindered greatly by an absence of underlying algebraic geometry, but we bypass this by using 'modular symbols', elements in the cohomology of arithmetic manifolds. We combined several older techniques -- such as Darmon and Orton's work on the Bruhat-Tits tree for Q_p -- with more modern methods (including the overconvergent modular symbols from my thesis). This allowed us to go beyond previous research in the area, treating forms of non-trivial weight. In the base-change case, we were able to prove stronger results, giving arithmetic descriptions of the L-invariants we obtained.
Publication Status
Published online
Date Publish Online
2019-04-12