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Uniform bounds for obstructions to the integral Tate conjecture

Anna Cadoret, Alena Pirutka

Abstract

Assuming natural variational realization conjectures, we give uniform bounds for the obstruction to the integral Tate conjecture in 1-dimensional families of algebraic varieties over an infinite finitely generated field.

Uniform bounds for obstructions to the integral Tate conjecture

Abstract

Assuming natural variational realization conjectures, we give uniform bounds for the obstruction to the integral Tate conjecture in 1-dimensional families of algebraic varieties over an infinite finitely generated field.

Paper Structure

This paper contains 31 sections, 10 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Tate conjectures
  3. $\widetilde{C}^{\hbox{\rm \tiny free}}_{\mathbb{Z}_\ell}$ versus $\widetilde{C}_{\mathbb{Z}_\ell}$
  4. $\widetilde{C}_{\mathbb{Q}_\ell}$ versus $\widetilde{C}^{\hbox{\rm \tiny free}}_{\mathbb{Z}_\ell}$
  5. Unramified cohomology
  6. Acknowledgements
  7. Étale cycle class maps in families and global strategy
  8. Étale $\mathbb{Z}_\ell$-local systems
  9. Notational conventions
  10. Specialization and extension of algebraically closed fields
  11. Compatibility with specialization of algebraic cycles
  12. "Invariance" under extension of algebraically closed field
  13. The lattice $\Lambda_{\mathbb{Z}_\ell}$
  14. Strategy for the proof of Theorem \ref{['MainTh1']} and Theorem \ref{['MainTh2']}
  15. $\mathcal{V}_{\mathbb{Q}_\ell}$-generic points
  16. ...and 16 more sections

Key Result

Proposition 2

If $p=0$, one has In general, one always has $\hbox{\rm VEt}^0_{\mathbb{Q}_\ell}(f,i)\Rightarrow \hbox{\rm WVEt}^0_{\mathbb{Q}_\ell}(f,i)$ and $\hbox{\rm Tate}_{\mathbb{Q}_\ell}(X_\eta,i)\Rightarrow \hbox{\rm WVEt}_{\mathbb{Q}_\ell}(f,i)$.

Theorems & Definitions (17)

  • Conjecture 1
  • Proposition 2
  • Theorem A
  • Theorem B
  • Remark 3
  • Corollary 4
  • Lemma 5
  • proof
  • Remark 6
  • Lemma A
  • ...and 7 more