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December 2009 General form of Humbert's modular equation for curves with real multiplication of $\Delta =5$
Kiichiro Hashimoto, Yukiko Sakai
Proc. Japan Acad. Ser. A Math. Sci. 85(10): 171-176 (December 2009). DOI: 10.3792/pjaa.85.171

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.

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Kiichiro Hashimoto. Yukiko Sakai. "General form of Humbert's modular equation for curves with real multiplication of $\Delta =5$." Proc. Japan Acad. Ser. A Math. Sci. 85 (10) 171 - 176, December 2009. https://doi.org/10.3792/pjaa.85.171

Information

Published: December 2009
First available in Project Euclid: 2 December 2009

zbMATH: 1245.11073
MathSciNet: MR2591363
Digital Object Identifier: 10.3792/pjaa.85.171

Subjects:
Primary: 11G10 , 11G15
Secondary: 14H45

Keywords: Curves of genus two , Modular equation , real multiplication

Rights: Copyright © 2009 The Japan Academy

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Vol.85 • No. 10 • December 2009
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