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

Kiichiro Hashimoto, Yukiko Sakai

Research output: Contribution to journalArticle

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 languageEnglish
Pages (from-to)171-176
Number of pages6
JournalProceedings of the Japan Academy Series A: Mathematical Sciences
Volume85
Issue number10
DOIs
Publication statusPublished - 2009 Dec 1

Keywords

  • Curves of genus two
  • Modular equation
  • Real multiplication

ASJC Scopus subject areas

  • Mathematics(all)

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