Journal of the Chungcheong Mathematical Society (충청수학회지)

The Chungcheong Mathematical Society (충청수학회)
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GENERATION OF RAY CLASS FIELDS OF IMAGINARY QUADRATIC FIELDS

  • Jung, Ho Yun (Department of Mathematics Dankook University)
  • Received : 2021.08.12
  • Accepted : 2021.09.23
  • Published : 2021.11.15
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Abstract

Let K be an imaginary quadratic field other than ℚ(${\sqrt{-1}}$) and ℚ(${\sqrt{-3}}$), and let 𝒪K 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 N𝒪K over K by using a single x-coordinate of an elliptic curve with complex multiplication by 𝒪K.

Keywords

Acknowledgement

This research was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT)(No. 2020R1F1A1A01073055).

References

  1. 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.
  2. 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.
  3. G. J. Janusz, Algebraic Number Fields, 2nd ed., Grad. Studies in Math. 7, Amer. Math. Soc., Providence, RI, 1996.
  4. 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. https://doi.org/10.2748/tmj/1318338949
  5. 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. https://doi.org/10.4134/JKMS.j170220
  6. 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
  7. 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. https://doi.org/10.4134/JKMS.2013.50.4.847
  8. D. Kubert and S. Lang, Modular Units, Grundlehren der mathematischen Wissenschaften 244, Spinger-Verlag, New York-Berlin, 1981.
  9. 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. https://doi.org/10.1515/math-2019-0013
  10. S. Lang, Elliptic Functions, With an appendix by J. Tate, 2nd ed., Grad. Texts in Math., 112, Spinger-Verlag, New York, 1987.
  11. K. Ramachandra, Some applications of Kronecker's limit formula, Ann. of Math., (2) 80 (1964), 104-148. https://doi.org/10.2307/1970494
  12. G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Iwanami Shoten and Princeton University Press, Princeton, NJ, 1971.
  13. J. H. Silverman, The Arithmetic of Elliptic Curves, 2nd ed., Grad. Texts in Math., 106, Springer, Dordrecht, 2009.
  14. P. Stevenhagen, Hilbert's 12th problem, complex multiplication and Shimura reciprocity, Adv. Stud. Pure Math., 30, Math. Soc., Japan, Tokyo, 2001.
  15. J. Sandor, D. S. Mitrinovic and B. Crstici, Handbook of Number Theory I, Springer, Dordrecht, 1995.