Elisa Lorenzo García

Twists of non-hyperelliptic curves of genus 3

In this paper we explicitly compute equations for the twists of all the smooth plane quartic curves defined over a number field k. Since the plane quartic curves are non-hyperelliptic curves of genus 3 we can apply the method developed by the author in a previous article. The starting point is a classification due to Henn of the plane quartic curves with non-trivial automorphism group up to C-isomorphism.

On twists of smooth plane curves

Given a smooth curve defined over a field

Bad Reduction of Genus Three Curves with Complex Multiplication

k

Computing twists of hyperelliptic curves

that admits a non-singular plane model over

A bound on the primes of bad reduction for CM curves of genus 3

\overline{k}

Primes dividing invariants of CM Picard curves

, a fixed separable closure of

Modular invariants for genus 3 hyperelliptic curves

k

Reduction type of non-hyperelliptic genus 3 curves

, it does not necessarily have a non-singular plane model defined over the field

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

k

Isogenies for point counting on genus two hyperelliptic curves with maximal real multiplication

. We determine under which conditions this happens and we show an example of such phenomenon. Now, even assuming that such a smooth plane model exists, we wonder about the existence of non-singular plane models over