Kyoto Journal of Mathematics

Construction of class fields over cyclotomic fields

Ja Kyung Koo and Dong Sung Yoon

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Let and p be odd primes. For a positive integer μ, let kμ be the ray class field of k=Q(e2πi/) modulo 2pμ. We present certain class fields Kμ of k such that kμKμkμ+1, and we provide a necessary and sufficient condition for Kμ=kμ+1. We also construct, in the sense of Hilbert, primitive generators of the field Kμ over kμ by using Shimura’s reciprocity law and special values of theta constants.

Article information

Kyoto J. Math., Volume 56, Number 4 (2016), 803-829.

Received: 31 August 2015
Revised: 26 October 2015
Accepted: 4 November 2015
First available in Project Euclid: 7 November 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11R37: Class field theory 11F46: Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
Secondary: 11G15: Complex multiplication and moduli of abelian varieties [See also 14K22] 14H42: Theta functions; Schottky problem [See also 14K25, 32G20]

class field theory Siegel modular forms complex multiplication theta functions


Koo, Ja Kyung; Yoon, Dong Sung. Construction of class fields over cyclotomic fields. Kyoto J. Math. 56 (2016), no. 4, 803--829. doi:10.1215/21562261-3664923.

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