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

TL;DR

The paper develops a comprehensive p-adic Birch–Swinnerton-Dyer theory for the Bertolini–Darmon–Prasanna p-adic L-function $L_p^{\rm BDP}$ attached to an elliptic curve $E/\mathbb{Q}$ in the Heegner setting with $p$ splitting in $K$. It formulates a precise p-adic BSD conjecture, introducing a derived p-adic regulator $\mathrm{Reg}_{\mathfrak p,\mathrm{der}}$ built from Howard’s derived p-adic heights, and proves a Leading Coefficient Formula and an Order of Vanishing formula in terms of arithmetic invariants (logarithms of Heegner points, Sha, Tamagawa numbers) up to a $p$-adic unit. The main algebraic input is the anticyclotomic Iwasawa main conjecture, shown in the ordinary case (via existing work) and, under mild hypotheses, in the supersingular case using a signed Euler system approach; this allows determination of the leading term of $L_p^{\rm BDP}$ at $T=0$ and reduction of the BSD conjecture to a conjectural maximal non-degeneracy of the derived height. The work generalizes prior ordinary-case results, removes several technical hypotheses, and provides a new independent framework applicable to all good primes, including supersingular ones. Together, these results connect $p$-adic L-values, Heegner points, Selmer groups, and derived heights to produce precise predictions for the leading coefficient and the order of vanishing of $L_p^{\rm BDP}$, with implications for the $p$-adic BSD conjecture in the anticyclotomic setting.

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.

Theorems & Definitions (82)

• Conjecture 1.1: $p$-adic BSD conjecture for $L_\mathfrak{p}^{\rm BDP}$
• Remark 1.2
• Remark 1.3
• Remark 1.4
• Theorem 1.5
• Theorem 1.6
• Definition 2.1
• Lemma 2.2
• proof
• Remark 2.3