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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Exceptional zeros and $\mathcal {L}$-invariants of Bianchi modular forms
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by Daniel Barrera Salazar and Chris Williams PDF
Trans. Amer. Math. Soc. 372 (2019), 1-34 Request permission


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:
  • Chris Williams
  • Affiliation: Department of Mathematics, South Kensington Campus, Imperial College London, London SW7 2AZ, United Kingdom
  • MR Author ID: 1209792
  • Email:
  • Received by editor(s): July 31, 2017
  • Received by editor(s) in revised form: October 9, 2017
  • Published electronically: April 12, 2019
  • © Copyright 2019 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 372 (2019), 1-34
  • MSC (2010): Primary 11F41, 11F67, 11F85, 11S40; Secondary 11M41
  • DOI:
  • MathSciNet review: 3968760