$y$-coordinates of elliptic curves

TL;DR

This paper introduces new $y$-coordinates for elliptic curves, uses them as modular functions, and constructs ray class fields over imaginary quadratic fields, advancing the understanding of elliptic and modular functions.

Contribution

It develops new $y$-coordinates for elliptic curves and demonstrates their use in generating modular function fields and constructing ray class fields over imaginary quadratic fields.

Findings

New $y$-coordinates of elliptic curves are obtained.

These coordinates generate modular function fields of level N.

Ray class fields are constructed using singular values of the $y$-coordinates.

Abstract

By a change of variables we obtain new $y$-coordinates of elliptic curves. Utilizing these $y$-coordinates as modular functions, together with the elliptic modular function, we generate the modular function fields of level $N\geq3$. Furthermore, by means of the singular values of the $y$-coordinates we construct the ray class fields modulo $N$ over imaginary quadratic fields as well as normal bases of these ray class fields.