General form of Humbert’s modular equation for curves with real multiplication of \(\Delta =5\). (English) Zbl 1245.11073

Summary: 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 \(x_1,\dots ,x_6\) the coordinate of the Weierstrass points, and show that it is invariant under a transitive permutation group of degree 6 isomorphic to \(\mathfrak S_5\). We also prove the rationality of the hypersurface in \(\mathbb P^5\) defined by the generalized modular equation.


11G10 Abelian varieties of dimension \(> 1\)
11G15 Complex multiplication and moduli of abelian varieties
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