2016/02/24 by Jorgenson, Jay, Smajlović, Lejla, Then, Holger · 1 citation
#FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1602.07426
Let N be a square-free integer such that the arithmetic group Γ0(N)+ has genus zero; there are 44 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)+ is an algebraic integer. Moreover, for each such N and elliptic point e, we show how to explicitly evaluate jN(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.