Capitulation and Transfer Triples

Abstract

Let K/k be a finite unramified Galois extension of number fields with Galois group G. This determines two homomorphisms from the ideal class group Clk of k: the capitulation map Clk → CK and the Artin map Ck↠Gab onto the abelianization Gab of G. We call (ker (capitulation), Clk, Artin) the capitulation triple of K/k. Artin's transition to group theory shows that any triple (X, Y, ζ) which arises in this way satisfies the group-theoretic property of being a transfer triple for G, defined as follows: there exist a group extension A ↣ H ↠ G with A finite abelian and an isomorphism η : Y → ∼ H a b such that η(X) is the kernel of the transfer homomorphism Hab → A, and ζ is the composite of η with Hab → Gab. When G is abelian, we show that a triple (X, Y, ζ) is a transfer triple for G if and only if |G| X = 0 and |G| divides |X|. Whether all transfer triples for G can be realized arithmetically remains an unsolved problem. 2000 Mathematics Subject Classification 11R33, 20J99.