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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.98.

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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

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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Additional Information
  • Jennifer S. Balakrishnan
  • Affiliation: Department of Mathematics, Harvard University, 1 Oxford Street, Cambridge, Massachusetts 02138
  • MR Author ID: 910890
  • Email: jen@math.harvard.edu
  • Mirela Çiperiani
  • Affiliation: Department of Mathematics, The University of Texas at Austin, 1 University Station, C1200 Austin, Texas 78712
  • MR Author ID: 838646
  • Email: mirela@math.utexas.edu
  • William Stein
  • Affiliation: Department of Mathematics, University of Washington, Box 354350 Seattle, Washington 98195
  • MR Author ID: 679996
  • Email: wstein@uw.edu
  • 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: https://doi.org/10.1090/S0025-5718-2014-02876-7
  • MathSciNet review: 3290969