General form of Humbert's modular equation for curves with real multiplication of Δ = 5

Kiichiro Hashimoto; Yukiko Sakai

doi:10.3792/pjaa.85.171

General form of Humbert's modular equation for curves with real multiplication of Δ = 5

Kiichiro Hashimoto^*, Yukiko Sakai

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

3 Citations (Scopus)

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 x1;.. .; x6 the coordinate of the Weierstrass points, and show that it is invariant under a transitive permutation group of degree 6 isomorphic to S{fraktur}5. We also prove the rationality of the hypersurface in P⁵ defined by the generalized modular equation.

Original language	English
Pages (from-to)	171-176
Number of pages	6
Journal	Proceedings of the Japan Academy Series A: Mathematical Sciences
Volume	85
Issue number	10
DOIs	https://doi.org/10.3792/pjaa.85.171
Publication status	Published - 2009 Dec
Externally published	Yes

Keywords

Curves of genus two
Modular equation
Real multiplication

ASJC Scopus subject areas

Mathematics(all)

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10.3792/pjaa.85.171

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AB - 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 x1;.. .; x6 the coordinate of the Weierstrass points, and show that it is invariant under a transitive permutation group of degree 6 isomorphic to S{fraktur}5. We also prove the rationality of the hypersurface in P5 defined by the generalized modular equation.

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KW - Real multiplication

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General form of Humbert's modular equation for curves with real multiplication of Δ = 5

Abstract

Keywords

ASJC Scopus subject areas

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