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 $\Gamma_{0}(N)^{+}$ has genus one. In a previous article, we constructed canonical generators $x_{N}$ and $y_{N}$ of the holomorphic function field associated to $\Gamma_{0}(N)^{+}$ 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 $\mathbb{H}$. 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 $\mathbb{Q}(\tau)$ 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 $\mathbb{Q}(\sqrt{-74})$ whose coefficients are less than $2.2\times 10^{4}$, whereas the minimal polynomials obtained from the Hauptmodul of $\textrm{PSL}(2,\mathbb{Z})$ has coefficients as large as $6.6\times 10^{73}$.