p-adic heights of Heegner points and Λ-adic regulators

Let E be an elliptic curve defined over Q. The aim of this paper is to make it possible to compute Heegner L-functions and anticyclotomic Λ-adic regulators of E, which were studied by Mazur-Rubin and Howard. We generalize results of Cohen and Watkins and thereby compute Heegner points of non-fundamental discriminant. We then prove a relationship between the denominator of a point of E defined over a number field and the leading coefficient of the minimal polynomial of its x-coordinate. Using this relationship, we recast earlier work of Mazur, Stein, and Tate to produce effective algorithms to compute p-adic heights of points of E defined over number fields. These methods enable us to give the first explicit examples of Heegner L-functions and anticyclotomic Λ-adic regulators.