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The filtered Ogus realisation of motives

Bruno Chiarellotto, Christopher Lazda, Nicola Mazzari

Abstract

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

The filtered Ogus realisation of motives

Abstract

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

Paper Structure

This paper contains 12 sections, 8 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Notations and conventions
  3. The (filtered) Ogus category
  4. The basic construction
  5. Realisation à la Nori
  6. Nori category
  7. The representability theorem of Déglise--Nizioł
  8. Construction of FOg-valued cohomology
  9. Cohomology of varieties with values in FOg
  10. Cohomology of pairs with values in FOg
  11. Compatibility with the realisation for 1-motives
  12. The FOg avatar of the Tate conjecture

Key Result

Theorem 1.1

There exists a (homological) realisation functor compatible with $T_{\mathbf{FOg}}$.

Theorems & Definitions (15)

  • Theorem 1.1
  • Definition 2.1
  • Lemma 2.2: ABVB16, Lemma 1.3.2
  • Remark 3.1
  • Definition 4.1
  • Proposition 4.2
  • proof
  • Corollary 4.3
  • Proposition 4.4
  • proof
  • ...and 5 more