Twists of non-hyperelliptic curves

Abstract

In this paper we present 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, it is known how to solve the Galois embedding problems that appear. As an example we compute the set of twists of the non-hyperelliptic genus 6 curve $x^7-y^3-1=0$ when we consider it defined over a number field such that $[k({\zeta }_{21}):k]=12$. For each twist equations are exhibited.