Genus 3 curves whose Jacobians have endomorphisms by $Q (ζ_7 +\barζ_7 )$, II

J. W. Hoffman, Dun Liang, Zhibin Liang, Ryotaro Okazaki, Yukiko Sakai, Haohao Wang

Published 2014-11-08Version 1

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.