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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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Computing rational points on rank 1 elliptic curves via $L$-series and canonical heights
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by Joseph H. Silverman PDF
Math. Comp. 68 (1999), 835-858 Request permission

Abstract:

Let $E/\mathbb {Q}$ be an elliptic curve of rank 1. We describe an algorithm which uses the value of $L’(E,1)$ and the theory of canonical heghts to efficiently search for points in $E(\mathbb {Q})$ and $E(\mathbb {Z}_{S})$. For rank 1 elliptic curves $E/\mathbb {Q}$ of moderately large conductor (say on the order of $10^{7}$ to $10^{10}$) and with a generator having moderately large canonical height (say between 13 and 50), our algorithm is the first practical general purpose method for determining if the set $E(\mathbb {Z}_{S})$ contains non-torsion points.
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Additional Information
  • Joseph H. Silverman
  • Affiliation: Mathematics Department, Box 1917, Brown University, Providence, RI 02912 USA
  • MR Author ID: 162205
  • ORCID: 0000-0003-3887-3248
  • Email: jhs@gauss.math.brown.edu
  • Received by editor(s): May 8, 1996
  • Received by editor(s) in revised form: March 3, 1997
  • Additional Notes: Research partially supported by NSF DMS-9424642.
  • © Copyright 1999 American Mathematical Society
  • Journal: Math. Comp. 68 (1999), 835-858
  • MSC (1991): Primary 11G05, 11Y50
  • DOI: https://doi.org/10.1090/S0025-5718-99-01068-6
  • MathSciNet review: 1627825