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http://dx.doi.org/10.4134/JKMS.2014.51.6.1269

ON SOME THETA CONSTANTS AND CLASS FIELDS

Shin, Dong Hwa (Department of Mathematics Hankuk University of Foreign Studies)

Publication Information

Journal of the Korean Mathematical Society / v.51, no.6, 2014 , pp. 1269-1289 More about this Journal

Abstract

We first find a sufficient condition for a product of theta constants to be a Siegel modular function of a given even level. And, when $K_{(2p)}$ denotes the ray class field of $K=\mathbb{Q}(e^{2{\pi}i/5})$ modulo 2p for an odd prime p, we describe a subfield of $K_{(2p)}$ generated by the special value of a certain theta constant by using Shimura's reciprocity law.

Keywords

CM-fields; Shimura's reciprocity law; theta functions;

Citations & Related Records

Reference

1	E. de Shalit and E. Z. Goren, On special values of theta functions of genus two, Ann. Inst. Fourier (Grenoble) 47 (1997), no. 3, 775-799. DOI ScienceOn
2	J.-I. Igusa, On the graded ring of theta-constants. II, Amer. J. Math. 88 (1966), no. 1, 221-236. DOI ScienceOn
3	H. Y. Jung, J. K. Koo, and D. H. Shin, Ray class invariants over imaginary quadratic fields, Tohoku Math. J. 63 (2011), no. 3, 413-426. DOI ScienceOn
4	H. Klingen, Introductory Lectures on Siegel Modular Forms, Cambridge Studies in Advanced Mathematics, 20, Cambridge University Press, Cambridge, 1990.
5	K. Komatsu, Construction of a normal basis by special values of Siegel modular functions, Proc. Amer. Math. Soc. 128 (2000), no. 2, 315-323. DOI ScienceOn
6	D. Kubert and S. Lang, Modular Units, Grundlehren der mathematischen Wissenschaften 244, Spinger-Verlag, 1981.
7	S. Lang, Elliptic Functions, 2nd edition, Grad. Texts in Math. 112, Spinger-Verlag, New York, 1987.
8	G. Shimura, On canonical models of arithmetic quotients of bounded symmetric domains, Ann. of Math. 91 (1970), 144-222. DOI
9	G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Iwanami Shoten and Princeton University Press, Princeton, N. J., 1971.
10	G. Shimura, Theta functions with complex multiplication, Duke Math. J. 43 (1976), no. 4, 673-696. DOI
11	G. Shimura, On certain reciprocity-laws for theta functions and modular forms, Acta Math. 141 (1978), no. 1-2, 35-71. DOI
12	G. Shimura, Abelian Varieties with Complex Multiplication and Modular Functions, Princeton Mathematical Series, 46., Princeton University Press, Princeton, N. J., 1998.