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

Abstract

Since the modular curve X(5) = Gamma(5)\h* has genus zero, we have a field isomorphism K(X(5)) approximate to 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(Delta,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 K(X(1)(N)), which are also the product of Klein forms without using torsion points of elliptic curves.