Local solubility and height bounds for coverings of elliptic curves

2011/03/25 by Tom Fisher, Fisher, Tom, Graham Sills +1
Mathematics · #11G05 #11G07 #11G50 #11Y50 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11G05 #msc:11G07 #msc:11G50 #msc:11Y50

paper · pdf · doi:10.48550/arxiv.1103.4944

40 pages

arxiv created 2011/03/25 · arxiv updated 2011/03/28

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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