On the exceptional zeros of 𝑝-non-ordinary 𝑝-adic 𝐿-functions and a conjecture of Perrin-Riou

Benois, Denis; Büyükboduk, Kâzım

doi:10.1090/tran/8704

On the exceptional zeros of $p$-non-ordinary $p$-adic $L$-functions and a conjecture of Perrin-Riou
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Trans. Amer. Math. Soc. 376 (2023), 231-284

Abstract:

Our goal in this article is to prove a form of $p$-adic Birch and Swinnerton-Dyer formula for the second derivative of the $p$-adic $L$-function associated to a newform $f$ which is non-crystalline semistable at $p$ at its central critical point, by expressing this quantity in terms of a $p$-adic (cyclotomic) regulator defined on an extended trianguline Selmer group. We also prove a two-variable version of this result for height pairings we construct by considering infinitesimal deformations afforded by a Coleman family passing through $f$. This, among other things, leads us to a proof of an appropriate version of Perrin-Riou’s conjecture in this set up.

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Additional Information

Denis Benois
Affiliation: Institut de Mathématiques, Université de Bordeaux 351, Cours de la Libération, 33405 Talence, France
MR Author ID: 306510
Email: denis.benois@math.u-bordeaux.fr
Kâzım Büyükboduk
Affiliation: UCD School of Mathematics and Statistics, University College Dublin, Ireland
ORCID: 0000-0001-5010-4949
Email: kazim.buyukboduk@ucd.ie
Received by editor(s): December 20, 2016
Received by editor(s) in revised form: July 7, 2017, September 1, 2020, December 24, 2020, April 6, 2021, February 21, 2022, February 27, 2022, and March 15, 2022
Published electronically: October 24, 2022
Additional Notes: The second author was partially supported by the TÜBİTAK Grant 113F059, Science Academy Young Investigator Award (BAGEP) and EU Horizon 2020 MC-GF Grant CriticalGZ
Journal: Trans. Amer. Math. Soc. 376 (2023), 231-284
DOI: https://doi.org/10.1090/tran/8704
MathSciNet review: 4510110