Ray class invariants over imaginary quadratic fields

TL;DR

This paper proves that specific singular values of Siegel functions generate ray class fields over imaginary quadratic fields, extending previous work and providing algorithms for computing conjugates, which aids in class polynomial calculations.

Contribution

It establishes that certain Siegel function singular values generate ray class fields over imaginary quadratic fields, extending prior results and offering computational algorithms.

Findings

Singular values of Siegel functions generate ray class fields.

The generators are the simplest conjectured by Schertz.

An algorithm for conjugate computation is provided.

Abstract

Let $K$ be an imaginary quadratic field of discriminant less than or equal to -7 and $K_{(N)}$ be its ray class field modulo $N$ for an integer $N$ greater than 1. We prove that singular values of certain Siegel functions generate $K_{(N)}$ over $K$ by extending the idea of our previous work. These generators are not only the simplest ones conjectured by Schertz, but also quite useful in the matter of computation of class polynomials. We indeed give an algorithm to find all conjugates of such generators by virtue of Gee and Stevenhagen.