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 $\mathrm{Cl}_k$ of $k$: the capitulation map $\mathrm{Cl}_k \to \mathrm{Cl}_K$ and the Artin map $\mathrm{Cl}_k \twoheadrightarrow G^{\mathrm{ab}}$ onto the abelianization $G^{\mathrm{ab}}$ of $G$. We call (ker (capitulation), $\mathrm{Cl}_k$, Artin) the capitulation triple of $K / k$.

Artin's transition to group theory shows that any triple $(X, Y, \zeta)$ 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 $A \rightarrowtail H \twoheadrightarrow G$ with $A$ finite abelian and an isomorphism $\eta : Y \stackrel{\sim}{\to} H^{\mathrm{ab}}$ such that $\eta (X)$ is the kernel of the transfer homomorphism $H^{\mathrm{ab}}\to A$, and $\zeta$ is the composite of $\eta$ with $H^{\mathrm{ab}}\to G^{\mathrm{ab}}$.

When $G$ is abelian, we show that a triple $(X, Y, \zeta)$ 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.