# Construction of Ray class fields by elliptic units

Journal de théorie des nombres de Bordeaux (1997)

- Volume: 9, Issue: 2, page 383-394
- ISSN: 1246-7405

## Access Full Article

top## Abstract

top## How to cite

topSchertz, Reinhard. "Construction of Ray class fields by elliptic units." Journal de théorie des nombres de Bordeaux 9.2 (1997): 383-394. <http://eudml.org/doc/248002>.

@article{Schertz1997,

abstract = {From complex multiplication we know that elliptic units are contained in certain ray class fields over a quadratic imaginary number field $K$, and Ramachandra [3] has shown that these ray class fields can even be generated by elliptic units. However the generators constructed by Ramachandra involve very complicated products of high powers of singular values of the Klein form defined below and singular values of the discriminant $\Delta $. It is the aim of this paper to show, that in many cases a generator over $K$ can be constructed as a power of one singular value of the Klein form or as a quotient of two such values. The latter are very suitable for numerical purposes because it turns out that the coefficients of their minimal polynomials are rather small, as it can be seen in the numerical examples at the end of this article.},

author = {Schertz, Reinhard},

journal = {Journal de théorie des nombres de Bordeaux},

keywords = {elliptic unit; imaginary quadratic field; ray class field; normalized Klein form},

language = {eng},

number = {2},

pages = {383-394},

publisher = {Université Bordeaux I},

title = {Construction of Ray class fields by elliptic units},

url = {http://eudml.org/doc/248002},

volume = {9},

year = {1997},

}

TY - JOUR

AU - Schertz, Reinhard

TI - Construction of Ray class fields by elliptic units

JO - Journal de théorie des nombres de Bordeaux

PY - 1997

PB - Université Bordeaux I

VL - 9

IS - 2

SP - 383

EP - 394

AB - From complex multiplication we know that elliptic units are contained in certain ray class fields over a quadratic imaginary number field $K$, and Ramachandra [3] has shown that these ray class fields can even be generated by elliptic units. However the generators constructed by Ramachandra involve very complicated products of high powers of singular values of the Klein form defined below and singular values of the discriminant $\Delta $. It is the aim of this paper to show, that in many cases a generator over $K$ can be constructed as a power of one singular value of the Klein form or as a quotient of two such values. The latter are very suitable for numerical purposes because it turns out that the coefficients of their minimal polynomials are rather small, as it can be seen in the numerical examples at the end of this article.

LA - eng

KW - elliptic unit; imaginary quadratic field; ray class field; normalized Klein form

UR - http://eudml.org/doc/248002

ER -

## References

top- [1] D. Kubert, S. Lang, Modular Units, Grundlehren Math. Wiss., Vol. 244, Springer-Verlag, New-York/ Berlin, (1981). Zbl0492.12002MR648603
- [2] C. Meyer, Die Berechnung der Klassenzahl abelscher Körper über quadratischen Zahlkörpern, Akademie-Verlag, Berlin (1957). Zbl0079.06001MR88510
- [3] K. Ramachandra, Some Applications of Kronecker's limit formula, Ann. Math.80 (1964), 104-148. Zbl0142.29804MR164950
- [4] R. Schertz, Galoismodulstruktur und elliptische Funktionen, Journal of Number Theory, Vol. 39, No. 3, (1991. Zbl0739.11052MR1133558
- [5] R. Schertz, Problèmes de construction en multiplication complexe, Séminaire de Théorie des Nombres de Bordeaux4 (1992), 239-262. Zbl0797.11083MR1208864
- [6] R. Schertz, Zur expliziten Berechnung von Ganzheitsbasen in Strahlklassenkörpern über einem imaginär-quadratischen Zahlkörper, Journal of Number Theory, Vol. 34, No. 1 (1990), 41-53. Zbl0701.11059MR1039766
- [7] H. Stark, L-functions at s = 1, IV, Advances in Math.35 (1980), 197-235. Zbl0475.12018MR563924

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.