Ray class fields generated by torsion points of certain elliptic curves

TL;DR

This paper constructs infinite families of ray class fields over imaginary quadratic fields by associating elliptic curves with normalized Weierstrass functions and torsion points, advancing the explicit class field theory.

Contribution

It introduces a normalization of the Weierstrass $ ext{wp}'$-function and uses it to generate ray class fields via torsion points of associated elliptic curves, providing partial progress on the Lang-Schertz conjecture.

Findings

Generated infinite ray class fields over imaginary quadratic fields.

Constructed ray class invariants using singular values of normalized $ ext{wp}'$.

Connected elliptic curve parametrizations with class field theory.

Abstract

We first normalize the derivative Weierstrass $\wp'$-function appearing in Weierstrass equations which give rise to analytic parametrizations of elliptic curves by the Dedekind $\eta$-function. And, by making use of this normalization of $\wp'$ we associate certain elliptic curve to a given imaginary quadratic field $K$ and then generate an infinite family of ray class fields over $K$ by adjoining to $K$ torsion points of such elliptic curve. We further construct some ray class invariants of imaginary quadratic fields by utilizing singular values of the normalization of $\wp'$, as the $y$-coordinate in the Weierstrass equation of this elliptic curve, which would be a partial result for the Lang-Schertz conjecture of constructing ray class fields over $K$ by means of the Siegel-Ramachandra invariant.