## Tohoku Mathematical Journal

### Ray class invariants over imaginary quadratic fields

#### Abstract

Let $K$ be an imaginary quadratic field of discriminant less than or equal to $-7$ and $K_{(N)}$ be its ray class field modulo $N$ for an integer $N$ greater than $1$. We prove that the singular values of certain Siegel functions generate $K_{(N)}$ over $K$ by extending the idea of our previous work. These generators are not only the simplest ones conjectured by Schertz, but also quite useful in the matter of computation of class polynomials. We indeed give an algorithm to find all conjugates of such generators by virtue of the works of Gee and Stevenhagen.

#### Article information

Source
Tohoku Math. J. (2), Volume 63, Number 3 (2011), 413-426.

Dates
First available in Project Euclid: 11 October 2011

https://projecteuclid.org/euclid.tmj/1318338949

Digital Object Identifier
doi:10.2748/tmj/1318338949

Mathematical Reviews number (MathSciNet)
MR2851104

Zentralblatt MATH identifier
1279.11060

#### Citation

Jung, Ho Yun; Koo, Ja Kyung; Shin, Dong Hwa. Ray class invariants over imaginary quadratic fields. Tohoku Math. J. (2) 63 (2011), no. 3, 413--426. doi:10.2748/tmj/1318338949. https://projecteuclid.org/euclid.tmj/1318338949

#### References

• S. Bettner and R. Schertz, Lower powers of elliptic units, J. Théor. Nombres Bordeaux 13 (2001), 339–351.
• B. Cho and J. K. Koo, Constructions of class fields over imaginary quadratic fields and applications, Q. J. Math. 61 (2010), 199–216.
• B. Cho, J. K. Koo and Y. K. Park, On Ramanujan's cubic continued fraction as modular function, Tohoku Math. J. 62 (2010), 579–603.
• D. A. Cox, Primes of the form $x^2+ny^2$, Fermat, class field, and complex multiplication, John Wiley & Sons, Inc., New York, 1989.
• F. Diamond and J. Shurman, A first course in modular forms, Grad. Texts in Math. 228, Springer, New York, 2005.
• A. Gee, Class invariants by Shimura's reciprocity law, J. Théor. Nombres Bordeaux 11 (1999), 45–72.
• B. Gross and D. Zagier, On singular moduli, J. Reine Angew. Math. 355 (1985), 191–220.
• H. Hasse, Neue bergründung der komplexen multiplikation, Teil I, J. für Math. 157 (1927), 115–139, Teil II, ibid. 165 (1931), 64–88.
• H. Y. Jung, J. K. Koo and D. H. Shin, Generation of ray class field by elliptic units, Bull. Lond. Math. Soc. 41 (2009), 935–942.
• J. K. Koo and D. H. Shin, On some arithmetic properties of Siegel functions, Math. Zeit. 264 (2010), 137–177.
• D. Kubert and S. Lang, Modular units, Grundlehren der mathematischen Wissenschaften 244, Spinger-Verlag, New York-Berlin, 1981.
• S. Lang, Elliptic functions, With an appendix by J. Tate, 2nd edition, Grad. Texts in Math. 112, Spinger-Verlag, New York, 1987.
• K. Ramachandra, Some applications of Kronecker's limit formula, Ann. of Math. (2) 80 (1964), 104–148.
• R. Schertz, Construction of ray class fields by elliptic units, J. Théor. Nombres Bordeaux 9 (1997), 383–394.
• G. Shimura, Introduction to the arithmetic theory of automorphic functions, Iwanami Shoten and Princeton University Press, Princeton, N.J., 1971.
• J. H. Silverman, Advanced topics in the arithmetic of elliptic curves, Grad. Texts in Math. 151, Springer-Verlag, New York, 1994.
• P. Stevenhagen, Hilbert's 12th problem, complex multiplication and Shimura reciprocity, Class field theory-its centenary and prospect (Tokyo, 1998), 161–176, Adv. Stud. Pure Math. 30, Math. Soc. Japan, Tokyo, 2001.