Singular values of some modular functions and their applications to class fields

Abstract

Since the modular curve \(X(5)=\Gamma(5)\backslash\frak H^{*}\) has genus zero, we have a field isomorphism \(\mathcal{K}(X(5))\approx \mathbb{C}(X_{2}(z))\) where X2(z) is a product of Klein forms. We apply it to construct explicit class fields over an imaginary quadratic field K from the modular function jΔ,25(z):=X2(5z). And, for every integer N≥7 we further generate ray class fields K(N) over K with modulus N just from the two generators X2(z) and X3(z) of the function field \(\mathcal{K}(X_{1}(N))\) , which are also the product of Klein forms without using torsion points of elliptic curves.