### 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 P^{5} defined by the generalized modular equation.

Original language | English |
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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 | |

Publication status | Published - 2009 Dec |

### Fingerprint

### Keywords

- Curves of genus two
- Modular equation
- Real multiplication

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**General form of Humbert's modular equation for curves with real multiplication of Δ = 5.** / Hashimoto, Kiichiro; Sakai, Yukiko.

Research output: Contribution to journal › Article

*Proceedings of the Japan Academy Series A: Mathematical Sciences*, vol. 85, no. 10, pp. 171-176. https://doi.org/10.3792/pjaa.85.171

}

TY - JOUR

T1 - General form of Humbert's modular equation for curves with real multiplication of Δ = 5

AU - Hashimoto, Kiichiro

AU - Sakai, Yukiko

PY - 2009/12

Y1 - 2009/12

N2 - 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.

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.

KW - Curves of genus two

KW - Modular equation

KW - Real multiplication

UR - http://www.scopus.com/inward/record.url?scp=77949336899&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77949336899&partnerID=8YFLogxK

U2 - 10.3792/pjaa.85.171

DO - 10.3792/pjaa.85.171

M3 - Article

AN - SCOPUS:77949336899

VL - 85

SP - 171

EP - 176

JO - Proceedings of the Japan Academy Series A: Mathematical Sciences

JF - Proceedings of the Japan Academy Series A: Mathematical Sciences

SN - 0386-2194

IS - 10

ER -