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

DOI:
https://doi.org/10.1090/S0025-5718-99-01068-6

MathSciNet review:
1627825

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Abstract | References | Similar Articles | Additional Information

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

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