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

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 $Q({\zeta }_7+{\overline{\zeta }}_7)$. We construct explicit two-dimensional families defined over $Q(s;t)$ whose generic member is a nonhyperelliptic genus 3 curve with this property. The case when X is hyperelliptic was studied by the authors Hoffman and Wang in a previous work. We calculate the zeta function of one of these curves. Conjecturally this zeta function is described by a modular form.