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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.

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