Generation of class fields by using the Weber function

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 $E$.