Derived $p$-adic heights and the leading coefficient of the Bertolini--Darmon--Prasanna $p$-adic $L$-function

TL;DR

This paper formulates a $p$-adic BSD conjecture for a special $p$-adic $L$-function associated with elliptic curves over imaginary quadratic fields, and proves key parts of it, including the leading coefficient, under certain conditions.

Contribution

It introduces a new formulation of the $p$-adic BSD conjecture for Bertolini--Darmon--Prasanna's $p$-adic $L$-function and proves the leading coefficient and vanishing order results, including for supersingular primes.

Findings

Proved the leading coefficient of the $p$-adic $L$-function up to a $p$-adic unit.

Established the order of vanishing based on non-degeneracy of $p$-adic heights.

Extended results to supersingular primes under mild hypotheses.

Abstract

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.