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

TL;DR

This paper constructs explicit families of genus 3 curves with Jacobians whose endomorphism algebra contains a specific cubic number field, and computes the zeta function for one such curve, linking it to modular forms.

Contribution

The authors explicitly construct two-dimensional families of genus 3 curves with Jacobians having endomorphisms by a particular cubic field, extending previous work on hyperelliptic cases.

Findings

Constructed explicit nonhyperelliptic genus 3 curves with desired endomorphism properties.

Calculated the zeta function of a specific curve in the family.

Conjecture relating the zeta function to a modular form.

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.