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Korean Mathematical Society JKMS

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J. Korean Math. Soc. 2013; 50(4): 847-864

Printed July 1, 2013

https://doi.org/10.4134/JKMS.2013.50.4.847

Copyright © The Korean Mathematical Society.

Construction of class fields over imaginary quadratic fields using $y$-coordinates of elliptic curves

Ja Kyung Koo and Dong Hwa Shin

KAIST, Hankuk University of Foreign Studies

Abstract

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

Keywords: class field theory, complex multiplication, elliptic and modular units, modular functions

MSC numbers: Primary 11G16; Secondary 11F03, 11G15, 11R37

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