Capitulation and transfer kernels

Gruenberg, K. W., and Weiss, A.. "Capitulation and transfer kernels." Journal de théorie des nombres de Bordeaux 12.1 (2000): 219-226. <http://eudml.org/doc/248487>.

@article{Gruenberg2000,
abstract = {If $K/k$ is a finite Galois extension of number fields with Galois group $G$, then the kernel of the capitulation map $Cl_k \rightarrow Cl_K$ of ideal class groups is isomorphic to the kernel $X (H)$ of the transfer map $H/H^\{\prime \} \rightarrow A,$ where $H = \text\{ Gal\}(\tilde\{K\}/k), A = \text\{ Gal\}(\tilde\{K\}/K)$ and $\tilde\{K\}$ is the Hilbert class field of $K$. H. Suzuki proved that when $G$ is abelian, $|G|$ divides $|X(H)|$. We call a finite abelian group $X$ a transfer kernel for $G$ if $X \cong X(H)$ for some group extension $A \hookrightarrow H \twoheadrightarrow G$. After characterizing transfer kernels in terms of integral representations of $G$, we show that $X$ is a transfer kernel for the abelian group $G$ if and only if $|G|X = 0$ and $|G|$ divides $|X|$. Our arguments give a new proof of Suzuki’s result.},
author = {Gruenberg, K. W., Weiss, A.},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {capitulation; transfer kernel; Hilbert class field; Suzuki's theorem},
language = {eng},
number = {1},
pages = {219-226},
publisher = {Université Bordeaux I},
title = {Capitulation and transfer kernels},
url = {http://eudml.org/doc/248487},
volume = {12},
year = {2000},
}

TY - JOUR
AU - Gruenberg, K. W.
AU - Weiss, A.
TI - Capitulation and transfer kernels
JO - Journal de théorie des nombres de Bordeaux
PY - 2000
PB - Université Bordeaux I
VL - 12
IS - 1
SP - 219
EP - 226
AB - If $K/k$ is a finite Galois extension of number fields with Galois group $G$, then the kernel of the capitulation map $Cl_k \rightarrow Cl_K$ of ideal class groups is isomorphic to the kernel $X (H)$ of the transfer map $H/H^{\prime } \rightarrow A,$ where $H = \text{ Gal}(\tilde{K}/k), A = \text{ Gal}(\tilde{K}/K)$ and $\tilde{K}$ is the Hilbert class field of $K$. H. Suzuki proved that when $G$ is abelian, $|G|$ divides $|X(H)|$. We call a finite abelian group $X$ a transfer kernel for $G$ if $X \cong X(H)$ for some group extension $A \hookrightarrow H \twoheadrightarrow G$. After characterizing transfer kernels in terms of integral representations of $G$, we show that $X$ is a transfer kernel for the abelian group $G$ if and only if $|G|X = 0$ and $|G|$ divides $|X|$. Our arguments give a new proof of Suzuki’s result.
LA - eng
KW - capitulation; transfer kernel; Hilbert class field; Suzuki's theorem
UR - http://eudml.org/doc/248487
ER -