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

Abstract

Abstract. Let N be a positive square-free integer such that the discrete group Γ 0 (N ) + has genus one. In a previous article, we constructed canonical generators x N and y N of the holomorphic function field associated to Γ 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 (τ ) and y N (τ ) for any CM point τ in the upper half plane H. If τ is any CM point which is not equivalent to an elliptic point of Γ 0 (N ) + , we prove that the complex numbers x N (τ ) and y N (τ ) are algebraic integers. Going further, we characterize the algebraic nature of x N (τ ) as the generator of a certain ring class field of Q(τ ) of prescribed order and discriminant depending on properties of τ and level N . The theoretical considerations are supplemented by computational examples. As a result, several explicit evaluations are given for various N and τ , 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 Q( √ −74) whose coefficients are less than 2.2 × 10 4 , whereas the minimal polynomials obtained from the Hauptmodul of PSL(2, Z) has coefficients as large as 6.6 × 10 73 .