Local solubility and height bounds for coverings of elliptic curves

TL;DR

This paper develops algorithms for testing local solubility of genus one curves covering elliptic curves and derives explicit height bounds to enhance methods for finding rational points.

Contribution

It introduces efficient algorithms for local solubility testing and modifies classical covering map formulas to work in all characteristics, improving rational point searches.

Findings

Algorithms for local solubility testing of coverings

Modified covering map formulas for all characteristics

Explicit height bounds relating covering curves and elliptic curves

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.