Certain CM class fields with smaller generators

TL;DR

This paper presents an efficient algorithm for computing explicit class fields of imaginary quadratic fields using special values of Siegel functions, advancing computational number theory methods.

Contribution

It introduces a novel algorithm that improves the efficiency of computing class fields and proves that specific Siegel function quotients generate these fields.

Findings

The algorithm outperforms classical methods in efficiency.

Certain Siegel function quotients are proven to generate ray class fields.

The approach enhances explicit class field construction techniques.

Abstract

We introduce an algorithm that computes explicit class fields of an imaginary quadratic field $K$ for a given modulus $\mathfrak{f}\subset\mathcal{O}_K$ more efficiently than the use of their classical counterparts. Therein, we prove the fact that certain values of a simple quotient of Siegel $\phi$-function are elements in the ray class field $K_{\mathfrak{f}}$ of $K$.