The Hauptmodul at elliptic points of certain arithmetic groups

Abstract

Let N be a square-free integer such that the arithmetic group Γ0(N)+ or Γ0(N)⁎ has genus zero; there are 52 such groups. Let jN+,⁎ 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 Γ0(N)+ or Γ0(N)⁎ is an algebraic integer. Moreover, for each such N and elliptic point e, we show how to explicitly evaluate jN+,⁎(e), and we provide the list of generating polynomials (with small coefficients) of the class fields or their subfields corresponding to the associated orders over the imaginary quadratic extension of rationals.