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.
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J.K. Koo and D.S. Yoon were partially supported by the NRF of Korea grant funded by MEST (2012-0000798). The second named author was supported by Hankuk University of Foreign Studies Research Fund of 2012.
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Koo, J.K., Shin, D.H. & Yoon, D.S. Ray class fields generated by torsion points of certain elliptic curves. Ramanujan J 28, 341–360 (2012). https://doi.org/10.1007/s11139-012-9396-4
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DOI: https://doi.org/10.1007/s11139-012-9396-4