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Mathematics of Computation

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$p$-adic heights of Heegner points and $\Lambda$-adic regulators

Authors: Jennifer S. Balakrishnan, Mirela Çiperiani and William Stein
Journal: Math. Comp. 84 (2015), 923-954
MSC (2010): Primary 11Y40, 11G50, 11G05
Published electronically: September 11, 2014
MathSciNet review: 3290969
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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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Additional Information

Jennifer S. Balakrishnan
Affiliation: Department of Mathematics, Harvard University, 1 Oxford Street, Cambridge, Massachusetts 02138
MR Author ID: 910890

Mirela Çiperiani
Affiliation: Department of Mathematics, The University of Texas at Austin, 1 University Station, C1200 Austin, Texas 78712
MR Author ID: 838646

William Stein
Affiliation: Department of Mathematics, University of Washington, Box 354350 Seattle, Washington 98195
MR Author ID: 679996

Keywords: Elliptic curve, $p$-adic heights, Heegner points
Received by editor(s): May 21, 2013
Received by editor(s) in revised form: August 2, 2013
Published electronically: September 11, 2014
Additional Notes: The first author was supported by NSF grant DMS-1103831
The second author was supported by NSA grant H98230-12-1-0208
The third author was supported by NSF Grants DMS-1161226 and DMS-1147802.
Article copyright: © Copyright 2014 American Mathematical Society