General form of Humbert's modular equation for curves with real multiplication of $\Delta =5$
Hashimoto, Kiichiro ; Sakai, Yukiko
Proc. Japan Acad. Ser. A Math. Sci., Tome 85 (2009) no. 2, p. 171-176 / Harvested from Project Euclid
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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.
Publié le : 2009-12-15
Classification:  Curves of genus two,  modular equation,  real multiplication,  11G10,  11G15,  14H45 
@article{1259763079,
     author = {Hashimoto, Kiichiro and Sakai, Yukiko},
     title = {General form of Humbert's modular equation for curves with real multiplication of $\Delta =5$},
     journal = {Proc. Japan Acad. Ser. A Math. Sci.},
     volume = {85},
     number = {2},
     year = {2009},
     pages = { 171-176},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1259763079}
}

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., Tome 85 (2009) no. 2, pp.  171-176. http://gdmltest.u-ga.fr/item/1259763079/