Class fields generated by coordinates of elliptic curves

Abstract

Let $K$ be an imaginary quadratic field different from $\mathbb{Q}(\sqrt{-1})$ and $\mathbb{Q}(\sqrt{-3})$. For a nontrivial integral ideal $\mathfrak{m}$ of $K$, let $K_{\mathfrak{m}}$ be the ray class field modulo $\mathfrak{m}$. By using some inequalities on special values of modular functions, we show that a single $x$-coordinate of a certain elliptic curve generates $K_{\mathfrak{m}}$ over $K$.