On special values of generators of modular function fields

Abstract

Let N be a positive integer and ζN be a primitive Nth root of unity. Let X(N) be the modular curve of level N and FX(N),Q(ζN) be its field of meromorphic functions with Fourier coefficients in Q(ζN). Let u and v be nonconstant functions in FX(N),Q(ζN) which generate a sufficiently large subfield of FX(N),Q(ζN) over Q(ζN). In this paper, we give sufficient conditions for generating the ray class field of an imaginary quadratic field K modulo N over K(ζN) by the special values of u and v.