Ray class fields generated by torsion points of certain elliptic curves

Abstract

We first normalize the derivative Weierstrass ℘′-function appearing in the Weierstrass equations which give rise to analytic parametrizations of elliptic curves, by the Dedekind η-function. And, by making use of this normalization of ℘′, we associate a 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 an elliptic curve. We further construct some ray class invariants of imaginary quadratic fields by utilizing singular values of the normalization of ℘′, as the y-coordinate in the Weierstrass equation of this elliptic curve, which would be a partial result towards the Lang–Schertz conjecture of constructing ray class fields over K by means of the Siegel–Ramachandra invariant.