Vol. 9, No. 7, 2015

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
$p$-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 Δ prime to Np, we give an explicit construction of the p-adic Rankin–Selberg L-function Lp(fE,). When the sign of its functional equation is 1, we show, under the assumption that all primes p are principal ideals of OF that split in OE, 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 = 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