Derived pp-adic heights and the leading coefficient of the Bertolini--Darmon--Prasanna pp-adic LL-function

Let $E/\mathbf{Q}$ 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. In a previous work, Agboola--Castella formulated an analogue of the Birch--Swinnerton-Dyer conjecture for the $p$-adic $L$-function $L_{\mathfrak{p}}^{\rm BDP}$ of Bertolini--Darmon--Prasanna attached to $E/K$, assuming the prime $p$ to be ordinary for $E$. The goal of this paper is two-fold: (1) We formulate a $p$-adic BSD conjecture for $L_{\mathfrak{p}}^{\rm BDP}$ for all odd primes $p$ of good reduction. (2) For an algebraic analogue $F_{\overline{\mathfrak{p}}}^{\rm BDP}$ 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 $(F_{\overline{\mathfrak{p}}}^{\rm BDP})=(L_{\mathfrak{p}}^{\rm BDP})$ 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 in the $p$-ordinary case, we prove the main conjecture for supersingular primes $p$ under mild hypotheses.Comment: 34 page

Exceptional zero formulae and a conjecture of Perrin-Riou

Let $A/\mathbb{Q}$ be an elliptic curve with split multiplicative reduction at a prime $p$. We prove (an analogue of) a conjecture of Perrin-Riou, relating $p$-adic Beilinson$-$Kato elements to Heegner points in $A(\mathbb{Q})$, and a large part of the rank-one case of the Mazur$-$Tate$-$Teitelbaum exceptional zero conjecture for the cyclotomic $p$-adic $L$-function of $A$. More generally, let $f$ be the weight-two newform associated with $A$, let $f_{\infty}$ be the Hida family of $f$, and let $L_{p}(f_{\infty},k,s)$ be the Mazur$-$Kitagawa two-variable $p$-adic $L$-function attached to $f_{\infty}$. We prove a $p$-adic Gross$-$Zagier formula, expressing the quadratic term of the Taylor expansion of $L_{p}(f_{\infty},k,s)$ at $(k,s)=(2,1)$ as a non-zero rational multiple of the extended height-weight of a Heegner point in $A(\mathbb{Q})$

On the elliptic stark conjecture in higherweight

We study the special values of the triple product p-adic L-function constructedby Darmon and Rotger at all classical points outside the region of interpolation.We propose conjectural formulas for these values that can be seen as extendingthe Elliptic Stark Conjecture, and we provide theoretical evidence for them by provingsome particular cases