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 $\text{PSL}(2,\mathbb{Z})$ has coefficients as large as $6.6\times 10^{73}$.