Singular moduli of rth Roots of modular functions

Abstract

When singular moduli of Hauptmodules generate ring class fields (resp. ray class fields) of imaginary quadratic fields, using the theory of Shimura reciprocity law, we determine a necessary and sufficient condition for singular moduli of r th roots of the Hauptmodules to generate the same ring class fields (resp. ray class fields) as do the singular moduli of the Hauptmodules.