On the evaluation of singular invariants for canonical generators of certain genus one arithmetic groups

TL;DR

This paper investigates the algebraic properties of canonical generators for certain genus one arithmetic groups at CM points, establishing their algebraic integrality and connection to class fields, with explicit computational examples.

Contribution

It introduces a detailed study of singular moduli for canonical generators of genus one groups, linking their values at CM points to ring class fields and providing explicit polynomial examples.

Findings

Proves algebraic integrality of generator values at non-elliptic CM points.

Characterizes these values as generators of specific class fields.

Provides explicit minimal polynomials for class field generators.

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 $6.6\times 10^{73}$.