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arXiv · 1411.2152

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

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

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J. W. Hoffman, Dun Liang, Zhibin Liang, Ryotaro Okazaki, Yukiko Sakai, Haohao Wang. 2014-11-08. Genus 3 curves whose Jacobians have endomorphisms by $Q (\zeta _7 +\bar{\zeta}_7 )$, II. https://arxiv.org/abs/1411.2152

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