Twists of genus three curves over finite fields

Abstract

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 DyckFermat 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.

Maximal hyperelliptic curves of genus three

“…The beautiful survey by Elkies [3] does not mention this result which is probably well known; a recent reference is the PhD thesis [10]. A second well-known example is given by the Fermat curve with equation…”

Sato–Tate distributions of twists of the Fermat and the Klein quartics

“…In this section, for any twist C of C 0 , we explicitly construct ST(Jac(C)), which to simplify notation we denote by ST(C). The first step is to compute a (non-canonical) embedding ι : Aut(C 0 M ) → USp(6) (see [MT10] for a very similar approach). Let…”

Fully maximal and fully minimal abelian varieties

“…Similarly, if A ′ = A and τ ∈ Aut k (A), let F r K τ denote the (twisted) automorphism, which acts on x ∈Ā(k) by , (see also [26,Propositions 5,9] for curves) Given φ :Ā ≃ →Ā ′ as in (3), consider the cocycle ξ φ :…”