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Twists of genus three curves over finite fields. (English) Zbl 1254.11064
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. It is also shown how questions which appear difficult from a function field perspective can be resolved by using the theory of the Jacobian variety.

MSC:
11G20 Curves over finite and local fields
14G05 Rational points
14G15 Finite ground fields in algebraic geometry
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