Explicit n-descent on elliptic curves. III. Algorithms

TL;DR

This paper develops practical algorithms for computing the n-Selmer group of elliptic curves, focusing on n=3 over rationals, with implementations in Magma and applications to trivialising central simple algebras.

Contribution

It introduces new algorithms for representing elements of the n-Selmer group as degree n curves and for trivialising central simple algebras, with practical implementation.

Findings

Algorithms are practical for n=3 over rationals.

Implemented in Magma for real-world use.

Potential applications in parametrising Brauer-Severi surfaces.

Abstract

This is the third in a series of papers in which we study the n-Selmer group of an elliptic curve, with the aim of representing its elements as curves of degree n in P^{n-1}. The methods we describe are practical in the case n=3 for elliptic curves over the rationals, and have been implemented in Magma. One important ingredient of our work is an algorithm for trivialising central simple algebras. This is of independent interest: for example, it could be used for parametrising Brauer-Severi surfaces.