Capitulation and transfer kernels

If K/k is a finite Galois extension of number fields with Galois group G, then the kernel of the capitulation map Clk ~ ClK 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. Let K/k be a finite unramified Galois extension of number fields with Galois group G. The capitulation kernel for is the kernel of the natural homomorphism of ideal class groups Clk e CIK. Suzuki [S] proved that when G is abelian, its order G~ divides the order of the capitulation kernel. This remarkable result encapsulates much of the information previously available about capitulation. We refer to the surveys [J] and [M] Manuscrit requ le 23 septembre 1999. The authors gratefully acknowledge financial help from EPSRC and NSERC.