# Lower powers of elliptic units

Stefan Bettner; Reinhard Schertz

Journal de théorie des nombres de Bordeaux (2001)

- Volume: 13, Issue: 2, page 339-351
- ISSN: 1246-7405

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topBettner, Stefan, and Schertz, Reinhard. "Lower powers of elliptic units." Journal de théorie des nombres de Bordeaux 13.2 (2001): 339-351. <http://eudml.org/doc/248707>.

@article{Bettner2001,

abstract = {In the previous paper [Sch2] it has been shown that ray class fields over quadratic imaginary number fields can be generated by simple products of singular values of the Klein form defined below. In the present article the second named author has constructed more general products that are contained in ray class fields thereby correcting Theorem 2 of [Sch2]. An algorithm for the computation of the algebraic equations of the numbers in Theorem 1 of this paper has been implemented in a KASH program by the first named author, who also calculated the list of examples at the end of this article. As in the special cases treated in [Sch2] these examples exhibit again that the coefficients of the algebraic equations are rather small. Moreover, apart from trivial exceptions, all numbers computed so far turn out to be generators of the corresponding ray class field, thereby suggesting the conjecture formulated more precisely after Theorem 1.},

author = {Bettner, Stefan, Schertz, Reinhard},

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

keywords = {products of singular values; Klein form; ray class fields},

language = {eng},

number = {2},

pages = {339-351},

publisher = {Université Bordeaux I},

title = {Lower powers of elliptic units},

url = {http://eudml.org/doc/248707},

volume = {13},

year = {2001},

}

TY - JOUR

AU - Bettner, Stefan

AU - Schertz, Reinhard

TI - Lower powers of elliptic units

JO - Journal de théorie des nombres de Bordeaux

PY - 2001

PB - Université Bordeaux I

VL - 13

IS - 2

SP - 339

EP - 351

AB - In the previous paper [Sch2] it has been shown that ray class fields over quadratic imaginary number fields can be generated by simple products of singular values of the Klein form defined below. In the present article the second named author has constructed more general products that are contained in ray class fields thereby correcting Theorem 2 of [Sch2]. An algorithm for the computation of the algebraic equations of the numbers in Theorem 1 of this paper has been implemented in a KASH program by the first named author, who also calculated the list of examples at the end of this article. As in the special cases treated in [Sch2] these examples exhibit again that the coefficients of the algebraic equations are rather small. Moreover, apart from trivial exceptions, all numbers computed so far turn out to be generators of the corresponding ray class field, thereby suggesting the conjecture formulated more precisely after Theorem 1.

LA - eng

KW - products of singular values; Klein form; ray class fields

UR - http://eudml.org/doc/248707

ER -

## References

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- [La] S. Lang, Elliptic functions. Addison Wesley, 1973. Zbl0316.14001MR409362
- [Me] C. Meyer, Über einige Anwendungen Dedekindscher Summen. Journal Reine Angew. Math.198 (1957), 143-203. Zbl0079.10303MR104643
- [Ro] G. Robert, La racine 12-ième canonique de Δ(L)[L:L]/Δ(L)Sém. de th. des nombresParis, 1989-90.
- [Sch1] R. Schertz, Niedere Potenzen elliptischer Einheiten. Proc. of the International Conference on Class Numbers and Fundamental Units of Algebraic Number Fields, Katata, Japan (1986), 67-87. Zbl0615.12013MR891888
- [Sch2] R. Schertz, Construction of Ray Class Fields by Elliptic Units. J. Théor. Nombres Bordeaux9 (1997), 383-394. Zbl0902.11047MR1617405
- [St] H. Stark, L-functions at s = 1, IV. Adv. Math.35 (1980), 197-235. Zbl0475.12018MR563924

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