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Capitulation and transfer kernels

K. W. Gruenberg; A. Weiss — 2000

Journal de théorie des nombres de Bordeaux

If $K / k$ is a finite Galois extension of number fields with Galois group $G$ , then the kernel of the capitulation map $C l_{k} \to C l_{K}$ of ideal class groups is isomorphic to the kernel $X (H)$ of the transfer map $H / H^{'} \to A,$ where $H = Gal (\tilde{K} / k), A = Gal (\tilde{K} / K)$ 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 $X ≅ X (H)$ 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...

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