The filtered Ogus realisation of motives

Abstract

We construct the (filtered) Ogus realisation of Voevodsky motives over a number field K. This realisation extends the functor defined on 1motives by Andreatta, Barbieri-Viale and Bertapelle. As an illustration we note that the analogue of the Tate conjecture holds for K3 surfaces.

The Fontaine–Ogus realization of Laumon 1-motives

“…Remark 3.5. In [5] it is proven that the filtered Ogus realisation T FOg extends to the category of Voevodsky motives. Also the latter functor T MFOg can be extended to Voevodsky motives.…”

Extensions of Filtered Ogus Structures

“…Recently Andreatta, Barbieri-Viale and Bertapelle [1] have defined the filtered Ogus realisation T FOg for 1-motives over a number field K. In fact by [5] there exists a cohomology theory for K-varieties with values in FOg(K) compatible with T FOg . More precisely let DM gm (K) be the Voevodsky's category of geometric motives over K, then there exists a (homological) realisation functor…”