Tokyo Journal of Mathematics

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, and Haohao WANG

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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.

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Tokyo J. Math., Volume 42, Number 1 (2019), 185-218.

First available in Project Euclid: 18 July 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14H10: Families, moduli (algebraic)
Secondary: 14H40: Jacobians, Prym varieties [See also 32G20] 11G15: Complex multiplication and moduli of abelian varieties [See also 14K22] 14H45: Special curves and curves of low genus 14Q05: Curves


HOFFMAN, Jerome William; LIANG, Dun; LIANG, Zhibin; OKAZAKI, Ryotaro; SAKAI, Yukiko; WANG, Haohao. Genus 3 Curves Whose Jacobians Have Endomorphisms by $\mathbb{Q}(\zeta _7 +\bar{\zeta}_7 )$, II. Tokyo J. Math. 42 (2019), no. 1, 185--218.

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