Class fields generated by coordinates of elliptic curves

TL;DR

This paper demonstrates that the coordinates of specific elliptic curves can generate ray class fields over imaginary quadratic fields, expanding understanding of explicit class field theory.

Contribution

It shows that a single coordinate of a particular elliptic curve can generate the entire ray class field over certain imaginary quadratic fields, using inequalities on modular functions.

Findings

Single elliptic curve coordinate generates the ray class field

Uses inequalities on special values of modular functions

Applicable to imaginary quadratic fields excluding $\

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 $K$.