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

Let $E\mathrm{/}\mathbb{Q}$ 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 $L_{\mathfrak{p}}^{\mathrm{B}\mathrm{D}\mathrm{P}}$ introduced by Bertolini–Darmon–Prasanna. (2) For an algebraic analogue $F_{\mathfrak{p}}^{\mathrm{B}\mathrm{D}\mathrm{P}}$ of $L_{\mathfrak{p}}^{\mathrm{B}\mathrm{D}\mathrm{P}}$, 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 $(F_{\mathfrak{p}}^{\mathrm{B}\mathrm{D}\mathrm{P}})=(L_{\mathfrak{p}}^{\mathrm{B}\mathrm{D}\mathrm{P}})$ is known, our results determine the leading coefficient of $L_{\mathfrak{p}}^{\mathrm{B}\mathrm{D}\mathrm{P}}$ 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.