Primitive and totally primitive Fricke families with applications

TL;DR

This paper introduces the concept of primitivity in Fricke families, constructs generators for modular curve function fields, and uses special values to generate ray class fields of imaginary quadratic fields.

Contribution

It defines primitivity for Fricke families, provides examples, and applies these to generate function fields and ray class fields in number theory.

Findings

Constructed generators of modular curve function fields using Fricke and Siegel functions.

Established the use of totally primitive Fricke families to generate ray class fields.

Provided examples illustrating the primitivity of Fricke families.

Abstract

We introduce the primitivity of Fricke families, and give some examples. As its application, we first construct generators of the function field of the modular curve of level $N$ in terms of Fricke functions and Siegel functions, respectively. Furthermore, we use the special values of a certain function in a totally primitive Fricke family of level $N$ in order to generate ray class fields of imaginary quadratic fields.