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 X ₂(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 j _Δ,25(z):=X ₂(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 X ₂(z) and X ₃(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.