Primitive and totally primitive Fricke families with applications (II)

Abstract

We find necessary and sufficient conditions for a Fricke family of level N (≥2) to be primitive or totally primitive. Let K be an imaginary quadratic field of discriminant d K other than Q ( − 1 ) and Q ( − 3 ) . As applications of Fricke families, we show that if | d K | is sufficiently large, then the special values of a primitive Fricke family generate the ray class field K ( N ) modulo N over K . Moreover, we construct a primitive generator of K ( N ) over K in terms of the special values of classical Fricke functions for every K which would be a partial answer to a question of Hasse and Ramachandra.