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

Balakrishnan, Jennifer; Çiperiani, Mirela; Stein, William

doi:10.1090/S0025-5718-2014-02876-7

$p$-adic heights of Heegner points and $\Lambda$-adic regulators
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by Jennifer S. Balakrishnan, Mirela Çiperiani and William Stein PDF

Math. Comp. 84 (2015), 923-954 Request permission

Abstract:

Let $E$ be an elliptic curve defined over $\mathbb {Q}$. The aim of this paper is to make it possible to compute Heegner $L$-functions and anticyclotomic $\Lambda$-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 $\Lambda$-adic regulators.

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