Twists of non-hyperelliptic curves

TL;DR

This paper presents a method to compute twists of non-hyperelliptic curves over perfect fields, utilizing Galois embedding problems, and provides explicit equations for these twists, including an algorithmic approach over finite fields.

Contribution

It introduces a novel method linking twists to Galois embedding problems and details how to compute explicit equations for non-hyperelliptic curve twists, including an algorithm for finite fields.

Findings

Method for computing twists via Galois embedding problems

Explicit equations for twists of non-hyperelliptic curves

Algorithmic solution over finite fields

Abstract

In this paper we show a method for computing the set of twists of a non-singular projective curve defined over an arbitrary (perfect) field $k$. The method is based on a correspondence between twists and solutions to a Galois embedding problem. When in addition, this curve is non-hyperelliptic we show how to compute equations for the twists. If $k = \mathbb{F}_{q}$ the method then becomes an algorithm, since in this case, the Galois embedding problems that appear are known how to be solved. As an example we compute the set of twists of the non-hyperelliptic genus $6$ curve $x^{7}-y^{3}z^{4}-z^{7} = 0$ when we consider it defined over a number field such that $[k(\zeta_{21}):k] = 12$. For each twist equations are exhibited.