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 $L_p(f_E,\cdot )$. When the sign of its functional equation is $-1$, we show, under the assumption that all primes $\wp |p$ are principal ideals of ${\mathcal{O}}_F$ which 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 $F=\mathbf{Q}$ and $(N,E)$ satisfies the so-called Heegner condition. We deduce applications to both the $p$-adic and the classical Birch and Swinnerton-Dyer conjectures for~$A$.

69 pages, final version (minor corrections)

Cited by in corpus (4)

• The p-adic Gross-Zagier formula on Shimura curves
• The universal $p$-adic Gross-Zagier formula
• The $p$-adic Gross-Zagier formula on Shimura curves, II: nonsplit primes
• On the p-adic Birch and Swinnerton-Dyer conjecture for elliptic curves over number fields