An affine model of X-0(mn)

Abstract

We show that the modular equation Phi(Tn)(m)(X, Y) for the Thompson series T-n corresponding to Gamma(0)(n) gives an affine model of the modular curve X-0(mn) which has better properties than the one derived from the modular j invariant. Here, m and n are relative prime. As an application, we construct a ring class field over an imaginary quadratic field K by singular values of T-n(z) and T-n (mz).