Generation of class fields by using the Weber function

TL;DR

This paper demonstrates that the Weber function evaluated at specific torsion points of an elliptic curve with complex multiplication can generate the corresponding ray class field over an imaginary quadratic field, extending class field theory methods.

Contribution

It proves that the Weber function alone suffices to generate ray class fields over imaginary quadratic fields for certain moduli, providing a new explicit class field generation method.

Findings

Weber function evaluated at N-torsion points generates the ray class field.

The result applies for integers N > 1 coprime to 6.

Provides explicit class field generation technique.

Abstract

Let $K$ be an imaginary quadratic field and $\mathcal{O}_K$ be its ring of integers. Let $h_E$ be the Weber function on certain elliptic curve $E$ with complex multiplication by $\mathcal{O}_K$. We show that if $N$ ($>1$) is an integer prime to $6$, then the function $h_E$ alone generates the ray class field modulo $N\mathcal{O}_K$ over $K$ when evaluated at some $N$-torsion point of $E$.