School of Mathematical & Statistical Sciences Faculty Publications and Presentations
Document Type
Article
Publication Date
5-9-2025
Abstract
Let E/Q be an elliptic curve and let p be an odd prime of good reduction for E. Let K be an imaginary quadratic field satisfying the classical Heegner hypothesis and in which p splits. The goal of this paper is two-fold: (1) we formulate a p-adic BSD conjecture for the p-adic L-function LBDP p introduced by Bertolini–Darmon–Prasanna [Duke Math. J. 162 (2013), pp. 1033–1148]; and (2) for an algebraic analogue F BDP p of LBDP p , we show that the “leading coefficient” part of our conjecture holds, and that the “order of vanishing” part follows from the expected “maximal non-degeneracy” of an anticyclotomic p-adic height. In particular, when the Iwasawa–Greenberg Main Conjecture (F BDP p )=(LBDP p ) is known, our results determine the leading coefficient of LBDP p at T = 0 up to a p-adic unit. Moreover, by adapting the approach of Burungale–Castella–Kim [Algebra Number Theory 15 (2021), pp. 1627–1653], we prove the main conjecture for supersingular primes p under mild hypotheses.
In the p-ordinary case, and under some additional hypotheses, similar results were obtained by Agboola–Castella [J. Th´eor. Nombres Bordeaux 33 (2021), pp 629–658], but our method is new and completely independent from theirs, and apply to all good primes.
Recommended Citation
Castella, Francesc, Chi-Yun Hsu, Debanjana Kundu, Yu-Sheng Lee, and Zheng Liu. "Derived 𝑝-adic heights and the leading coefficient of the Bertolini–Darmon–Prasanna 𝑝-adic 𝐿-function." Transactions of the American Mathematical Society, Series B 12, no. 20 (2025): 748-788. https://doi.org/10.1090/btran/227
Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.
Publication Title
Transactions of the AMS, Series B
DOI
10.1090/btran/227
Comments
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