On Néron-Raynaud class groups of tori and the capitulation problem

Abstract

We discuss the Capitulation Problem for Néron-Raynaud class groups of tori over global fields F and obtain generalizations of the main results of [González-Avilés, J. reine angew. Math. 613: 75–97, 2007]. We also show that short exact sequences of F-tori induce long exact sequences involving the corresponding Néron-Raynaud class groups. For example, we show that the Néron-Raynaud class group of any F-torus which is split by a metacyclic extension of F can be “resolved” in terms of classical ideal class groups of global fields.