p-adic heights of Heegner points on Shimura curves

Disegni, Daniel

doi:10.2140/ant.2015.9.1571

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

Daniel Disegni

Vol. 9 (2015), No. 7, 1571–1646

DOI: 10.2140/ant.2015.9.1571

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 $2 N$ . If $f$ is ordinary at $p$ and $E$ is a CM extension of $F$ of relative discriminant $Δ$ prime to $N p$ , 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 $℘ ∣ p$ are principal ideals of $O_{F}$ that split in $O_{E}$ , 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 $(N, E)$ satisfies the so-called Heegner condition. We deduce applications to both the $p$ -adic and the classical Birch and Swinnerton-Dyer conjectures for $A$ .

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

Milestones

Received: 18 September 2014

Revised: 27 April 2015

Accepted: 11 June 2015

Published: 22 September 2015

Authors

Daniel Disegni
Department of Mathematics and Statistics McGill University 805 Shebrooke Street West Montreal, QC H3A 0B9 Canada

Vol. 9, No. 7, 2015