Twists of non-hyperelliptic curves

2015/03/11 by Elisa Lorenzo Garcia, Garcia, Elisa Lorenzo
Mathematics · #FOS: Mathematics #Number Theory (math.NT) #math.NT

paper · pdf · doi:10.48550/arxiv.1503.03281

arxiv created 2015/03/11 · arxiv updated 2015/03/12

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 = \mathbbF_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 x⁷-y³z⁴-z⁷ = 0 when we consider it defined over a number field such that [k(ζ₂₁):k] = 12. For each twist equations are exhibited.

Related

• Twists of non-hyperelliptic curves of genus 3
• Twists of superelliptic curves without rational points
• On twists of smooth plane curves
• Families of twists of tuples of hyperelliptic curves
• Counting hyperelliptic curves