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 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 6F that split in 6E, 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.