Twists of genus three curves over finite fields

In this article we recall how to describe the twists of a curve over a finite field and we show how to compute the number of rational points on such a twist by methods of linear algebra. We illustrate this in the case of plane quartic curves with at least 16 automorphisms. In particular we treat the...

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Bibliographic Details
Published in:Finite fields and their applications 2010-09, Vol.16 (5), p.347-368
Main Authors: Meagher, Stephen, Top, Jaap
Format: Article
Language:English
Subjects:
Curve
Dyck–Fermat quartic
Jacobian
Klein quartic
Non-Abelian Galois cohomology
Twist
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Summary:In this article we recall how to describe the twists of a curve over a finite field and we show how to compute the number of rational points on such a twist by methods of linear algebra. We illustrate this in the case of plane quartic curves with at least 16 automorphisms. In particular we treat the twists of the Dyck–Fermat and Klein quartics. Our methods show how in special cases non-Abelian cohomology can be explicitly computed. They also show how questions which appear difficult from a function field perspective can be resolved by using the theory of the Jacobian variety.
ISSN:1071-5797
1090-2465
DOI:10.1016/j.ffa.2010.06.001