Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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

 
 

 

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
Full-text PDF Free Access

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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Mathematics of Computation with MSC (1991): 11G05, 11Y50

Retrieve articles in all journals with MSC (1991): 11G05, 11Y50


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