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A Tate-Type Theorem for Crystalline Classes in the 1-Motivic Category

Mohammadreza Mohajer

Abstract

The Tate conjecture predicts that Galois-invariant classes in $\ell$-adic cohomology, and Frobenius-invariant classes in crystalline cohomology, arise from algebraic cycles. We prove an unconditional p-adic analogue of this principle in the 1-motivic range. Our starting point is a full-faithfulness theorem for Deligne 1-motives: after p-adic scalar extension, the Barsotti-Tate crystal functor identifies Hom-groups of 1-motives with Hom-groups in the category of filtered Dieudonne modules. Using the equivalence between the derived category of 1-motives up to isogeny and the 1-motivic part of Voevodsky's triangulated category of effective motives with rational coefficients, we extend this full-faithfulness result to the entire 1-motivic thick subcategory. More precisely, over a finite field $k = F_q$, we show that every Frobenius-compatible morphism between Barsotti-Tate crystalline realizations is induced by a unique 1-motivic morphism. Consequently, the Frobenius-invariant classes that occur in this range are already motivic and therefore algebraic. This yields an explicit linear-algebraic description of motivic morphisms and extension classes in level at most 1 in terms of Frobenius-equivariant maps preserving the Hodge filtration, giving a crystalline analogue of the principle that Tate classes are algebraic without invoking cycle conjectures in codimension at least 2.

A Tate-Type Theorem for Crystalline Classes in the 1-Motivic Category

Abstract

The Tate conjecture predicts that Galois-invariant classes in -adic cohomology, and Frobenius-invariant classes in crystalline cohomology, arise from algebraic cycles. We prove an unconditional p-adic analogue of this principle in the 1-motivic range. Our starting point is a full-faithfulness theorem for Deligne 1-motives: after p-adic scalar extension, the Barsotti-Tate crystal functor identifies Hom-groups of 1-motives with Hom-groups in the category of filtered Dieudonne modules. Using the equivalence between the derived category of 1-motives up to isogeny and the 1-motivic part of Voevodsky's triangulated category of effective motives with rational coefficients, we extend this full-faithfulness result to the entire 1-motivic thick subcategory. More precisely, over a finite field , we show that every Frobenius-compatible morphism between Barsotti-Tate crystalline realizations is induced by a unique 1-motivic morphism. Consequently, the Frobenius-invariant classes that occur in this range are already motivic and therefore algebraic. This yields an explicit linear-algebraic description of motivic morphisms and extension classes in level at most 1 in terms of Frobenius-equivariant maps preserving the Hodge filtration, giving a crystalline analogue of the principle that Tate classes are algebraic without invoking cycle conjectures in codimension at least 2.
Paper Structure (17 sections, 14 theorems, 147 equations)

This paper contains 17 sections, 14 theorems, 147 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Crystalline Realization for 1-motives
  4. Exploring the Hodge filtration on Tcrysve(M) explicitly
  5. Cartier duality
  6. A semisimplicity input over finite fields
  7. Fully faithfulness of the Barsotti--Tate crystal realization
  8. Semi-abelian dévissage.
  9. Application
  10. Frobenius-invariant crystalline classes and algebraic morphisms
  11. Frobenius on endomorphism ring and its matrix via Barsotti--Tate crystals
  12. Example: the Kummer motive
  13. Example: M=[zto E] and matrix computations for End(M)otimes K0
  14. The 1-motivic thick subcategory of DM(k) and extension of the Barsotti--Tate crystal functor
  15. The thick triangulated subcategory generated by 1-motives
  16. ...and 2 more sections

Key Result

Proposition 2.1

For any $1$-motive $M=[L\xrightarrow{u}G]$, we have Therefore, $\operatorname{T^{\vee}_{crys}}(M)=\operatorname{T_{crys}}(M^{\vee})$.

Theorems & Definitions (38)

  • Definition 2.1
  • Remark 2.1
  • Remark 2.2: The induced morphism $\operatorname{T^{\vee}_{crys}}(f)$ in $\mathcal{FD}_k$
  • Proposition 2.1
  • proof
  • Lemma 2.1: Semisimplicity of $1$-motives up to isogeny over $\mathbb{F}_q$
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 28 more