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

TL;DR

This paper constructs explicit families of genus 3 curves with Jacobians whose endomorphism algebra contains a specific totally real cubic field, expanding understanding of such curves beyond hyperelliptic cases.

Contribution

It provides explicit three-dimensional families of nonhyperelliptic genus 3 curves with specified endomorphism properties, extending previous work on hyperelliptic and some nonhyperelliptic cases.

Findings

Constructed explicit families of nonhyperelliptic genus 3 curves with desired endomorphism properties.

Demonstrated the existence of three-dimensional families with generic members having the specified endomorphism algebra.

Extended the classification of genus 3 curves with special Jacobian endomorphisms.

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.