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arXiv · 1411.2151

Genus 3 curves whose Jacobians have endomorphisms by $Q(ζ_7 + \overlineζ_7)$

J. William Hoffman·Zhibin Liang·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 $Q(ζ_ 7 + \overlineζ_7)$. 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.

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J. William Hoffman, Zhibin Liang, Yukiko Sakai, Haohao Wang. 2014-11-08. Genus 3 curves whose Jacobians have endomorphisms by $Q(ζ_7 + \overlineζ_7)$. https://arxiv.org/abs/1411.2151

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