p-adic heights of Heegner points on Shimura curves.

摘要

Let f be a primitive Hilbert modular form of weight 2 and level N for the totally real field F, and let p be an odd rational prime such that f is ordinary at all primes dividing p. When 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,·) and prove that when the sign of its functional equation is –1, its central derivative is given by the p-adic height of a Heegner point on the abelian variety A associated to f. This p-adic Gross-Zagier formula generalises the result obtained by Perrin-Riou when F = Q and I satisfies the so-called Heegner condition. We deduce applications to both the p-adic and the classical Birch and Swinnerton-Dyer conjectures for A.