Minimisation and reduction of 2-, 3- and 4-coverings of elliptic curves

TL;DR

This paper develops algorithms for minimizing and reducing models of genus one curves of degrees 2, 3, and 4, arising in explicit n-descent on elliptic curves, over general number fields and specifically over Q.

Contribution

It proves the existence of minimal models with invariants matching the Jacobian and provides explicit minimization and reduction algorithms for these models.

Findings

Algorithms for minimising models over number fields

Explicit reduction algorithms over Q for n=2,3,4

Theoretical proof of minimal model existence with matching invariants

Abstract

In this paper we consider models for genus one curves of degree n for n = 2, 3 and 4, which arise in explicit n-descent on elliptic curves. We prove theorems on the existence of minimal models with the same invariants as the minimal model of the Jacobian elliptic curve and provide simple algorithms for minimising a given model, valid over general number fields. Finally, for genus one models defined over Q, we develop a theory of reduction and again give explicit algorithms for n = 2, 3 and 4.