[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.14403/jcms.2021.34.4.317

GENERATION OF RAY CLASS FIELDS OF IMAGINARY QUADRATIC FIELDS

Jung, Ho Yun (Department of Mathematics Dankook University)

Publication Information

Journal of the Chungcheong Mathematical Society / v.34, no.4, 2021 , pp. 317-326 More about this Journal

Abstract

Let K be an imaginary quadratic field other than ℚ( ${\sqrt{-1}}$ ) and ℚ( ${\sqrt{-3}}$ ), and let 𝒪_K be its ring of integers. Let N be a positive integer such that N = 5 or N ≥ 7. In this paper, we generate the ray class field modulo N𝒪_K over K by using a single x-coordinate of an elliptic curve with complex multiplication by 𝒪_K.

Keywords

Class field theory; complex multiplication; modular functions;

Citations & Related Records

Times Cited By KSCI : 1 (Citation Analysis)

Reference
Cited By KSCI

1	G. J. Janusz, Algebraic Number Fields, 2nd ed., Grad. Studies in Math. 7, Amer. Math. Soc., Providence, RI, 1996.
2	H. Y. Jung, J. K. Koo and D. H. Shin, Ray class invariants over imaginary quadratic fields, Tohoku Math. J. (2) 63 (2011), no. 3, 413-426. DOI
3	H. Y. Jung, J. K. Koo and D. H. Shin, Class fields generated by coordinates of elliptic curves, submitted. https://arxiv.org/abs/2111.01021
4	J. K. Koo and D. H. Shin, Construction of class fields over imaginary quadratic fields using y-coordinates of elliptic curves, J. Korean Math. Soc., 50 (2013), no. 4, 847-864. DOI
5	J. K. Koo, D. H. Shin and D. S. Yoon, On a problem of Hasse and Ramachandra, Open Math., 17 (2019), no. 1, 131-140. DOI
6	S. Lang, Elliptic Functions, With an appendix by J. Tate, 2nd ed., Grad. Texts in Math., 112, Spinger-Verlag, New York, 1987.
7	G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Iwanami Shoten and Princeton University Press, Princeton, NJ, 1971.
8	J. H. Silverman, The Arithmetic of Elliptic Curves, 2nd ed., Grad. Texts in Math., 106, Springer, Dordrecht, 2009.
9	J. Sandor, D. S. Mitrinovic and B. Crstici, Handbook of Number Theory I, Springer, Dordrecht, 1995.
10	G. Frei, F. Lemmermeyer and P. J. Roquette, Emil Artin and Helmut Hassethe Correspondence 1923-1958, Comm App Math Comp Sci., 5 (2014) Springer, Heidelberg.
11	D. A. Cox, Primes of the Form x² + ny²: Fermat, Class Field Theory, and Complex Multiplication, 2nd edition, Pure Appl. Math., (Hoboken), John Wiley & Sons, Inc., Hoboken, NJ, 2013.
12	P. Stevenhagen, Hilbert's 12th problem, complex multiplication and Shimura reciprocity, Adv. Stud. Pure Math., 30, Math. Soc., Japan, Tokyo, 2001.
13	H. Y. Jung, J. K. Koo and D. H. Shin, Generation of ray class fields modulo 2, 3, 4 or 6 by using the Weber function, J. Korean Math. Soc., 55 (2018), no. 2, 343-372. DOI
14	D. Kubert and S. Lang, Modular Units, Grundlehren der mathematischen Wissenschaften 244, Spinger-Verlag, New York-Berlin, 1981.
15	K. Ramachandra, Some applications of Kronecker's limit formula, Ann. of Math., (2) 80 (1964), 104-148. DOI