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arXiv · 1007.2307

Ray class fields generated by torsion points of certain elliptic curves

Ja Kyung Koo·Dong Hwa Shin·Dong Sung Yoon
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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.

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Ja Kyung Koo, Dong Hwa Shin, Dong Sung Yoon. 2010-07-14. Ray class fields generated by torsion points of certain elliptic curves. https://arxiv.org/abs/1007.2307

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