Construction of Ray class fields by elliptic units

Schertz, 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 -