Proceedings of the Japan Academy, Series A, Mathematical Sciences

General form of Humbert's modular equation for curves with real multiplication of $\Delta =5$

Kiichiro Hashimoto and Yukiko Sakai

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Abstract

We study Humbert's modular equation which characterizes curves of genus two having real multiplication by the quadratic order of discriminant 5. We give it a simple, but general expression as a polynomial in $x_1,\ldots ,x_6$ the coordinate of the Weierstrass points, and show that it is invariant under a transitive permutation group of degree 6 isomorphic to $\frak S_5$. We also prove the rationality of the hypersurface in P5 defined by the generalized modular equation.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 85, Number 10 (2009), 171-176.

Dates
First available in Project Euclid: 2 December 2009

Permanent link to this document
https://projecteuclid.org/euclid.pja/1259763079

Digital Object Identifier
doi:10.3792/pjaa.85.171

Mathematical Reviews number (MathSciNet)
MR2591363

Zentralblatt MATH identifier
1245.11073

Subjects
Primary: 11G10: Abelian varieties of dimension > 1 [See also 14Kxx] 11G15: Complex multiplication and moduli of abelian varieties [See also 14K22]
Secondary: 14H45: Special curves and curves of low genus

Keywords
Curves of genus two modular equation real multiplication

Citation

Hashimoto, Kiichiro; Sakai, Yukiko. General form of Humbert's modular equation for curves with real multiplication of $\Delta =5$. Proc. Japan Acad. Ser. A Math. Sci. 85 (2009), no. 10, 171--176. doi:10.3792/pjaa.85.171. https://projecteuclid.org/euclid.pja/1259763079


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