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Transactions of the American Mathematical Society

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Exceptional zeros and $\mathcal {L}$-invariants of Bianchi modular forms


Authors: Daniel Barrera Salazar and Chris Williams
Journal: Trans. Amer. Math. Soc. 372 (2019), 1-34
MSC (2010): Primary 11F41, 11F67, 11F85, 11S40; Secondary 11M41
DOI: https://doi.org/10.1090/tran/7436
Published electronically: April 12, 2019
MathSciNet review: 3968760
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Abstract: Let $f$ be a Bianchi modular form, that is, an automorphic form for $\mathrm {GL}_2$ 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 $\mathfrak {p}$ of $F$ above $p$ we prove the existence of an $\mathcal {L}$-invariant $\mathcal {L}_{\mathfrak {p}}$, depending only on $\mathfrak {p}$ and $f$, such that when the $p$-adic $L$-function of $f$ has an exceptional zero at $\mathfrak {p}$, its derivative can be related to the classical $L$-value multiplied by $\mathcal {L}_{\mathfrak {p}}$. The proof uses cohomological methods of Darmon and Orton, who proved similar results for $\mathrm {GL}_2/\mathbb {Q}$. When $p$ is not split and $f$ is the base-change of a classical modular form $\tilde {f}$, we relate $\mathcal {L}_{\mathfrak {p}}$ to the $\mathcal {L}$-invariant of $\tilde {f}$, resolving a conjecture of Trifković in this case.


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Additional Information

Daniel Barrera Salazar
Affiliation: Department of Mathematics, Campus Nord, Universitat Politècnica de Catalunya, Calle Jordi Girona, 1-3, 08034 Barcelona, Spain
Address at time of publication: Departmento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile, Alameda 3363, Estación Central, Santiago, Chile
MR Author ID: 1217503
Email: danielbarreras@hotmail.com

Chris Williams
Affiliation: Department of Mathematics, South Kensington Campus, Imperial College London, London SW7 2AZ, United Kingdom
MR Author ID: 1209792
Email: christopher.williams@imperial.ac.uk

Received by editor(s): July 31, 2017
Received by editor(s) in revised form: October 9, 2017
Published electronically: April 12, 2019
Article copyright: © Copyright 2019 American Mathematical Society