Generation of class fields by using the Weber function

Ja Kyung Koo, Dong Hwa Shin, Dong Sung Yoon

Published 2014-08-25Version 1

Let $K$ be an imaginary quadratic field and $\mathcal{O}_K$ be its ring of integers. Let $h_E$ be the Weber function on a 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 value of $h_E$ at some $N$-torsion point of $E$ generates the ray class field modulo $N\mathcal{O}_K$ over $K$.