Class fields generated by coordinates of elliptic curves

Abstract

Let KK 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 KK, let Km{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 xx-coordinate of a certain elliptic curve generates Km{K}_{{\mathfrak{m}}} over KK.

Keywords

• class field theory
• complex multiplication
• modular functions
• 11f03
• 11g15
• 11r37