Torseurs pour les motifs et pour les représentations p-adiques potentiellement de type CM. (Torsors for motives and for p-adic representations potentially of CM type) (Q922605)

The Betti cohomology fibre functor \(H_ B\) and the De Rham cohomology fibre functor \(H_{DR}\) on the category of abelian varieties over \({\mathbb{Q}}\) of CM-type are realizations of the motives over \({\mathbb{Q}}\) with absolute Hodge cycles. Let G be the tensor automorphism group of \(H_ B\) (the Taniyama group) and P the G-torsor of tensor isomorphisms \(H_{DR}\to H_ B\). One determines the cohomology class of P in \(H^ 1({\mathbb{Q}},G)\). A similar cohomology class is determined in a p-adic context, where now G is the group of tensor automorphisms of the fibre functor that associates to a p-adic representation of the Galois group of the algebraic p-adic numbers its representation space and P is the G- torsor of the tensor isomorphisms of P to a fibre functor involving Hodge-Tate modules. The connection of the two situations is indicated.