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Mathematics of Computation

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Computing rational points
on rank 1 elliptic curves
via $L$-series and canonical heights

Author: Joseph H. Silverman
Journal: Math. Comp. 68 (1999), 835-858
MSC (1991): Primary 11G05, 11Y50
MathSciNet review: 1627825
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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

Keywords: Elliptic curve, canonical height
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.
Article copyright: © Copyright 1999 American Mathematical Society

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