Genus 3 curves whose Jacobians have endomorphisms by $Q({\zeta }_7+{\bar{\zeta }}_7)$, II

Abstract

In this work we consider constructions of genus three curves $X$ such that $\mathrm{End}(\mathrm{Jac}(X))\otimes Q$ contains the totally real cubic number field $Q({\zeta }_7+{\bar{\zeta }}_7)$. We construct explicit three-dimensional families whose generic member is a nonhyperelliptic genus 3 curve with this property. The case when $X$ is hyperelliptic was studied in a previous work by Hoffman and Wang and some nonhyperelliptic curves were constructed in a previous paper by Hoffman, Z. Liang. Sakai and Wang.