$p$-adic heights of Heegner points on Shimura curves

Abstract

Let $f$ be a primitive Hilbert modular form of parallel weight $2$ and level $N$ for the totally real field $F$, and let $p$ be a rational prime coprime to $2N$. If $f$ is ordinary at $p$ and $E$ is a CM extension of $F$ of relative discriminant $\Delta$ prime to $Np$, we give an explicit construction of the $p$-adic Rankin–Selberg $L$-function $L_p\left(f_E,\cdot \right)$. When the sign of its functional equation is $-1$, we show, under the assumption that all primes $\wp \mid p$ are principal ideals of ${\mathcal{O}}_F$ that split in ${\mathcal{O}}_E$, that its central derivative is given by the $p$-adic height of a Heegner point on the abelian variety $A$ associated with $f$.

This $p$-adic Gross–Zagier formula generalises the result obtained by Perrin-Riou when $F=\mathbb{Q}$ and $\left(N,E\right)$ satisfies the so-called Heegner condition. We deduce applications to both the $p$-adic and the classical Birch and Swinnerton-Dyer conjectures for $A$.