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

ABSTRACT

Let N be a positive square-free integer such that the discrete group Γ₀(N)⁺ has genus one. In a previous article, we constructed canonical generators x_N and y_N of the holomorphic function field associated with Γ₀(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 $\mathbb{H}$. If τ is any CM point which is not equivalent to an elliptic point of Γ₀(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 $\mathbb{Q}\left(\tau \right)$ 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 $\mathbb{Q}\left(\sqrt{-74}\right)$ whose coefficients are less than 2.2 × 10⁴, whereas the minimal polynomial obtained from the Hauptmodul of $PSL(2,\mathbb{Z})$ have coefficients as large as 6.6 × 10⁷³.

Keywords:

• singular moduli
• holomorphic functions
• Hilbert class fields

2010 AMS Subject Classification:

• 11R37
• 11R58
• 11G15
• 14K22
• 14H45

Additional information

Funding

PSC-CUNY [60657-00 48].