The Hauptmodul at elliptic points of certain arithmetic groups

TL;DR

This paper proves that Hauptmodul values at elliptic points for certain genus zero arithmetic groups are algebraic integers, and provides explicit evaluations and generating polynomials for related class fields.

Contribution

It establishes the algebraic integrality of Hauptmodul values at elliptic points and explicitly computes these values along with their minimal polynomials.

Findings

Hauptmodul at elliptic points are algebraic integers.

Explicit formulas for Hauptmodul values at elliptic points.

Generating polynomials for class fields related to elliptic points.

Abstract

Let $N$ be a square-free integer such that the arithmetic group $\Gamma_0(N)^+$ has genus zero; there are $44$ such groups. Let $j_N$ denote the associated Hauptmodul normalized to have residue equal to one and constant term equal to zero in its $q$-expansion. In this article we prove that the Hauptmodul at any elliptic point of the surface associated to $\Gamma_0(N)^+$ is an algebraic integer. Moreover, for each such $N$ and elliptic point $e$, we show how to explicitly evaluate $j_{N}(e)$ and provide the list of generating polynomials (with small coefficients) of the class fields or their subfields corresponding to the orders over the imaginary quadratic extension of rationals stemming from the elliptic points under consideration.