Derived $p$-adic heights and the leading coefficient of the Bertolini--Darmon--Prasanna $p$-adic $L$-function
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866913741295058944 |
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| author | Castella, Francesc Hsu, Chi-Yun Kundu, Debanjana Lee, Yu-Shen Liu, Zheng |
| author_facet | Castella, Francesc Hsu, Chi-Yun Kundu, Debanjana Lee, Yu-Shen Liu, Zheng |
| contents | Let $E/\mathbb{Q}$ 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 $L_{\mathfrak{p}}^{\rm BDP}$ introduced by Bertolini--Darmon--Prasanna.
(2) For an algebraic analogue $F_{\mathfrak{p}}^{\rm BDP}$ of $L_{\mathfrak{p}}^{\rm BDP}$, 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 $(F_{\mathfrak{p}}^{\rm BDP})=(L_{\mathfrak{p}}^{\rm BDP})$ is known, our results determine the leading coefficient of $L_{\mathfrak{p}}^{\rm BDP}$ at $T=0$ up to a $p$-adic unit. Moreover, by adapting the approach of Burungale--Castella--Kim, 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, but our method is new and completely independent from theirs, and apply to all good primes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_10474 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Derived $p$-adic heights and the leading coefficient of the Bertolini--Darmon--Prasanna $p$-adic $L$-function Castella, Francesc Hsu, Chi-Yun Kundu, Debanjana Lee, Yu-Shen Liu, Zheng Number Theory Let $E/\mathbb{Q}$ 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 $L_{\mathfrak{p}}^{\rm BDP}$ introduced by Bertolini--Darmon--Prasanna. (2) For an algebraic analogue $F_{\mathfrak{p}}^{\rm BDP}$ of $L_{\mathfrak{p}}^{\rm BDP}$, 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 $(F_{\mathfrak{p}}^{\rm BDP})=(L_{\mathfrak{p}}^{\rm BDP})$ is known, our results determine the leading coefficient of $L_{\mathfrak{p}}^{\rm BDP}$ at $T=0$ up to a $p$-adic unit. Moreover, by adapting the approach of Burungale--Castella--Kim, 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, but our method is new and completely independent from theirs, and apply to all good primes. |
| title | Derived $p$-adic heights and the leading coefficient of the Bertolini--Darmon--Prasanna $p$-adic $L$-function |
| topic | Number Theory |
| url | https://arxiv.org/abs/2308.10474 |