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

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...

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Bibliographic Details
Published in:Mathematics of computation 1999-04, Vol.68 (226), p.835-858
Main Author: Silverman, Joseph H.
Format: Article
Language:English
Subjects:
Algebra
Algorithms
Coefficients
Curves
Discriminants
Exact sciences and technology
Integers
Mathematics
Number theory
Numbers
Rationality
Sciences and techniques of general use
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Summary: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.
ISSN:0025-5718
1088-6842
DOI:10.1090/S0025-5718-99-01068-6