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 $j_{N}(e)$. Furthermore, we provide a list of generating polynomials (with small cefficients) of the class fields of the orders over the imaginary quadratic extension of rationals corresponding to the elliptic points under consideration.
Original language | English |
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Number of pages | 22 |
Journal | International Journal of Number Theory |
Early online date | 17 Apr 2019 |
DOIs | |
Publication status | E-pub ahead of print - 17 Apr 2019 |
Keywords
- Hauptmoduli
- Class fields
- Singular moduli