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

MathSciNet review:
2904595

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Abstract | References | Similar Articles | Additional Information

Abstract: We study genus one curves that arise as -, - and -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:
https://doi.org/10.1090/S0025-5718-2012-02587-7

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.