On the Evaluation of Singular Invariants for Canonical Generators of Certain Genus One Arithmetic Groups

Let N be a positive square-free integer such that the discrete group Gamma(0)(N)(+) has genus one. In a previous article, we constructed canonical generators x(N) and y(N) of the holomorphic function field associated with Gamma(0)(N)(+) as well as an algebraic equation P-N(x(N), y(N)) = 0 with integer coefficients satisfied by these generators. In the present paper, we study the singular moduli problem corresponding to x(N) and y(N), by which we mean the arithmetic nature of the numbers x(N)(tau) and y(N)(tau) for any CM point tau in the upper half plane . If tau is any CM point which is not equivalent to an elliptic point of Gamma(0)(N)(+), we prove that the complex numbers x(N)(tau) and y(N)(tau) are algebraic integers. Going further, we characterize the algebraic nature of x(N)(tau) as the generator of a certain ring class field of of prescribed order and discriminant depending on properties of tau and level N. The theoretical considerations are supplemented by computational examples. As a result, several explicit evaluations are given for various N and tau, and further arithmetic consequences of our analysis are presented. In one example, we explicitly construct a set of minimal polynomials for the Hilbert class field of whose coefficients are less than 2.2 x 10(4), whereas the minimal polynomial obtained from the Hauptmodul of have coefficients as large as 6.6 x 10(73).