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The Quarterly Journal of Mathematics Advance Access published online on January 7, 2009
The Quarterly Journal of Mathematics, doi:10.1093/qmath/han035
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© The author 2009. Published by Oxford University Press. All rights reserved. For permissions, please email: journals.permissions@oxfordjournals.org

CONSTRUCTION OF CLASS FIELDS OVER IMAGINARY QUADRATIC FIELDS AND APPLICATIONS

Bumkyu Cho {dagger} and Ja Kyung Koo{ddagger}

Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, 373-1 Guseong-dong, Yuseong-gu, Daejeon 305-701, Korea

{ddagger} Corresponding author. E-mail: jkkoo{at}math.kaist.ac.kr

Received 18 July 2008; revised 13 November 2008
  Abstract

Let K be an imaginary quadratic field, HO the ring class field of an order O in K and K(N) be the ray class field modulo N over K for a positive integer N. In this paper we provide certain general techniques of finding HO and K(N) by using the theory of Shimura's canonical models via his reciprocity law, from which we partially extend some results of Schertz (Remark 4.2), Chen-Yui (Remark 4.2, Corollary 4.4), Cox–McKay–Stevenhagen (Corollary 4.5) and Cais–Conrad (Remark 5.3). And, we further reilluminate the classical result of Hasse by means of such a method (Corollary 5.4), and discover how to get one ray class invariant over K from Hasse's two generators (Corollary 5.5) which is different from Ramachandra's invariant [K. Ramachandra, Some applications of Kronecker's limit formulas, Ann. Math. 80 (1964), 104–148].

{dagger} E-mail: bam{at}math.kaist.ac.kr


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