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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Local solubility and height bounds for coverings of elliptic curves


Authors: T. A. Fisher and G. F. Sills
Journal: Math. Comp. 81 (2012), 1635-1662
MSC (2010): Primary 11G05; Secondary 11G07, 11G50, 11Y50
Published electronically: February 21, 2012
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Abstract: We study genus one curves that arise as $ 2$-, $ 3$- and $ 4$-coverings of elliptic curves. We describe efficient algorithms for testing local solubility and modify the classical formulae for the covering maps so that they work in all characteristics. These ingredients are then combined to give explicit bounds relating the height of a rational point on one of the covering curves to the height of its image on the elliptic curve. We use our results to improve the existing methods for searching for rational points on elliptic curves.


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Additional Information

T. A. Fisher
Affiliation: University of Cambridge, DPMMS, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, United Kingdom
Email: T.A.Fisher@dpmms.cam.ac.uk

G. F. Sills
Affiliation: University of Cambridge, DPMMS, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, United Kingdom
Email: gs300@cantab.net

DOI: http://dx.doi.org/10.1090/S0025-5718-2012-02587-7
PII: S 0025-5718(2012)02587-7
Received by editor(s): October 21, 2010
Received by editor(s) in revised form: March 28, 2011
Published electronically: February 21, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.