Cohomology of the Weil group of a p -adic field

Abstract

We establish duality and vanishing results for the cohomology of the Weil group of a p -adic field. Among them is a duality theorem for finitely generated modules, which implies Tate–Nakayama Duality. We prove comparison results with Galois cohomology, which imply that the cohomology of the Weil group determines that of the Galois group. When the module is defined by an abelian variety, we use these comparison results to establish a duality theorem analogous to Tateʼs duality theorem for abelian varieties over p -adic fields. ► We prove a duality theorem for the cohomology of the Weil group of a p -adic field. ► We prove results comparing Weil and Galois cohomology of tori and abelian varieties. ► We deduce Tate–Nakayama Duality for p -adic fields from our duality theorem. ► We establish a Weil analogue of Tateʼs duality theorem for abelian varieties.