The Hauptmodul at elliptic points of certain arithmetic groups

Jay Jorgenson, Lejla Smajlović, H. Then

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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 languageEnglish
Number of pages22
JournalInternational Journal of Number Theory
Early online date17 Apr 2019
DOIs
Publication statusE-pub ahead of print - 17 Apr 2019

Keywords

  • Hauptmoduli
  • Class fields
  • Singular moduli

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