On the generalized principal ideal theorem of complex multiplication

Schertz, Reinhard. "On the generalized principal ideal theorem of complex multiplication." Journal de Théorie des Nombres de Bordeaux 18.3 (2006): 683-691. <http://eudml.org/doc/249659>.

@article{Schertz2006,
abstract = {In the $p^n$-th cyclotomic field $\mathbb\{Q\}_\{p^n\},~p$ a prime number, $n\in \mathbb\{N\}$, the prime $p$ is totally ramified and the only ideal above $p$ is generated by $\omega _n=\zeta _\{p^n\}-1$, with the primitive $p^n$-th root of unity $\zeta _\{p^n\}=e^\{\frac\{2\pi i\}\{p^n\}\}$. Moreover these numbers represent a norm coherent set, i.e. $\mbox \{\text\{\large \textbf\{N\}\}\}_\{\mathbb\{Q\}_\{p^\{n+1\}\}\}/\mathbb\{Q\}_\{p^n\}(\omega _\{n+1\})=\omega _n$. It is the aim of this article to establish a similar result for the ray class field $K_\{\{\mathfrak\{p\}\}^n\} $ of conductor $\{\mathfrak\{p\}\}^n$ over an imaginary quadratic number field $K$ where $\{\mathfrak\{p\}\}^n$ is the power of a prime ideal in $K$. Therefore the exponential function has to be replaced by a suitable elliptic function.},
affiliation = {Institut für Mathematik der Universität Augsburg Universitätsstraße 8 86159 Augsburg, Germany},
author = {Schertz, Reinhard},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {class field theory; cyclotomic extensions; quadratic extensions; complex multiplication},
language = {eng},
number = {3},
pages = {683-691},
publisher = {Université Bordeaux 1},
title = {On the generalized principal ideal theorem of complex multiplication},
url = {http://eudml.org/doc/249659},
volume = {18},
year = {2006},
}

TY - JOUR
AU - Schertz, Reinhard
TI - On the generalized principal ideal theorem of complex multiplication
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2006
PB - Université Bordeaux 1
VL - 18
IS - 3
SP - 683
EP - 691
AB - In the $p^n$-th cyclotomic field $\mathbb{Q}_{p^n},~p$ a prime number, $n\in \mathbb{N}$, the prime $p$ is totally ramified and the only ideal above $p$ is generated by $\omega _n=\zeta _{p^n}-1$, with the primitive $p^n$-th root of unity $\zeta _{p^n}=e^{\frac{2\pi i}{p^n}}$. Moreover these numbers represent a norm coherent set, i.e. $\mbox {\text{\large \textbf{N}}}_{\mathbb{Q}_{p^{n+1}}}/\mathbb{Q}_{p^n}(\omega _{n+1})=\omega _n$. It is the aim of this article to establish a similar result for the ray class field $K_{{\mathfrak{p}}^n} $ of conductor ${\mathfrak{p}}^n$ over an imaginary quadratic number field $K$ where ${\mathfrak{p}}^n$ is the power of a prime ideal in $K$. Therefore the exponential function has to be replaced by a suitable elliptic function.
LA - eng
KW - class field theory; cyclotomic extensions; quadratic extensions; complex multiplication
UR - http://eudml.org/doc/249659
ER -