Capitulation and transfer kernels

K. W. Gruenberg; A. Weiss

Journal de théorie des nombres de Bordeaux (2000)

  • Volume: 12, Issue: 1, page 219-226
  • ISSN: 1246-7405

Abstract

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If K / k is a finite Galois extension of number fields with Galois group G , then the kernel of the capitulation map C l k C l K of ideal class groups is isomorphic to the kernel X ( H ) of the transfer map H / H ' A , where H = Gal ( K ˜ / k ) , A = Gal ( K ˜ / K ) and 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 X ( H ) for some group extension A H 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.

How to cite

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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 -

References

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  1. [G] K.W. Gruenberg, Relation Modules of Finite Groups. CBMS Monograph 25, Amer. Math. Soc., Providence, R.I., 1976. Zbl0327.20019MR457538
  2. [GW] K.W. Gruenberg, A. Weiss, Galois invariants for units. Proc. London Math. Soc.70 (1995), 264-284. Zbl0828.11062MR1309230
  3. [J] J.-F. Jaulent, L'état actuel du problème de la capitulation. Sem. de Théorie des Nombres de Bordeaux, 1987-1988, exposé no. 17. Zbl0704.11046
  4. [L] S. Lang, Algebraic Number Theory. Springer, New York, 1994. Zbl0811.11001MR1282723
  5. [M] K. Miyake, Algebraic investigations of Hilbert's Theorem 94, the principal ideal theorem and the capitulation problem. Expo. Math.7 (1989), 289-346. Zbl0704.11048MR1018712
  6. [S] H. Suzuki, A generalization of Hilbert's Theorem 94. Nagoya Math. J.121 (1991), 161-169. Zbl0728.11061MR1096472

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