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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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$p$-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


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
  • Email:
  • Mirela Çiperiani
  • Affiliation: Department of Mathematics, The University of Texas at Austin, 1 University Station, C1200 Austin, Texas 78712
  • MR Author ID: 838646
  • Email:
  • William Stein
  • Affiliation: Department of Mathematics, University of Washington, Box 354350 Seattle, Washington 98195
  • MR Author ID: 679996
  • Email:
  • 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.
  • © Copyright 2014 American Mathematical Society
  • Journal: Math. Comp. 84 (2015), 923-954
  • MSC (2010): Primary 11Y40, 11G50, 11G05
  • DOI:
  • MathSciNet review: 3290969