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June 2019 Genus 3 Curves Whose Jacobians Have Endomorphisms by $\mathbb{Q}(\zeta _7 +\bar{\zeta}_7 )$, II
Jerome William HOFFMAN, Dun LIANG, Zhibin LIANG, Ryotaro OKAZAKI, Yukiko SAKAI, Haohao WANG
Tokyo J. Math. 42(1): 185-218 (June 2019). DOI: 10.3836/tjm/1502179286

Abstract

In this work we consider constructions of genus three curves $Y$ such that $\text{End}(\text{Jac} (Y))\otimes \mathbb{Q}$ contains the totally real cubic number field $\mathbb{Q}(\zeta _7 +\bar{\zeta}_7 )$. We construct explicit three-dimensional families whose general member is a nonhyperelliptic genus 3 curve with this property. The case when $Y$ is hyperelliptic was studied in \textsc{J. W. Hoffman, H. Wang}, $7$-gons and genus $3$ hyperelliptic curves, Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales., Serie A. Matemàticas \textbf{107} (2013), 35--52, and some nonhyperelliptic curves were constructed in \textsc{J. W. Hoffman, Z. Liang, Y. Sakai, H. Wang}, Genus $3$ curves whose Jacobians have endomorphisms by $\mathbb{Q}(\zeta _7 +\bar{\zeta}_7 )$, J. Symb. Comp. \textbf{74} (2016), 561--577.

Citation

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Jerome William HOFFMAN. Dun LIANG. Zhibin LIANG. Ryotaro OKAZAKI. Yukiko SAKAI. Haohao WANG. "Genus 3 Curves Whose Jacobians Have Endomorphisms by $\mathbb{Q}(\zeta _7 +\bar{\zeta}_7 )$, II." Tokyo J. Math. 42 (1) 185 - 218, June 2019. https://doi.org/10.3836/tjm/1502179286

Information

Published: June 2019
First available in Project Euclid: 18 July 2019

zbMATH: 07114905
MathSciNet: MR3982054
Digital Object Identifier: 10.3836/tjm/1502179286

Subjects:
Primary: 14H10
Secondary: 11G15, 14H40, 14H45, 14Q05

Rights: Copyright © 2019 Publication Committee for the Tokyo Journal of Mathematics

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Vol.42 • No. 1 • June 2019
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