Class fields generated by coordinates of elliptic curves

Abstract

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