#### Vol. 9, No. 7, 2015

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

### Daniel Disegni

Vol. 9 (2015), No. 7, 1571–1646
##### 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 ${L}_{p}\left({f}_{E},\cdot \phantom{\rule{0.3em}{0ex}}\right)$. When the sign of its functional equation is $-1$, we show, under the assumption that all primes $\wp \mid p$ are principal ideals of ${\mathsc{O}}_{F}$ that split in ${\mathsc{O}}_{E}$, 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 $F=ℚ$ 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 $A$.

##### Keywords
Gross–Zagier, Heegner points, $p$-adic $L$-functions, Hilbert modular forms, $p$-adic heights, Birch and Swinnerton-Dyer conjecture
##### Mathematical Subject Classification 2010
Primary: 11G40
Secondary: 11F41, 11G18, 11F33, 11G50