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Cycle conjectures and birational invariants over finite fields

Samet Balkan, Stefan Schreieder

TL;DR

<3-5 sentence high-level summary> The paper investigates a natural birational invariant over finite fields, built from Galois-invariant pieces of etale and Borel–Moore cohomology, and shows that its vanishing on projective space in certain degrees is equivalent to the Tate conjecture, Beilinson conjecture, and the Grothendieck–Serre 1-semi-simplicity conjecture for all smooth projective varieties over the field. It develops a framework based on refined unramified cohomology to relate these conjectures to invariants of the function field and to inductive arguments that reduce global statements to half of the degrees, with effective reductions to specific degrees such as H^4 on P^4. The work also extends to higher Chow groups and motivic cohomology, discusses Tate conjectures for higher Chow groups, and offers a conjectural Milnor K-theory description of the Galois-invariant pieces, tying the birational invariant to fundamental K-theoretic invariants. Together, these results provide a coherent strategy to derive broad arithmetic conjectures from vanishing criteria on projective space and to connect function-field invariants with deep cycle-theoretic questions over finite fields.

Abstract

We study a natural birational invariant for varieties over finite fields and show that its vanishing on projective space is equivalent to the Tate conjecture, the Beilinson conjecture, and the Grothendieck--Serre semi-simplicity conjecture for all smooth projective varieties over finite fields. We further show that the Tate, Beilinson, and 1-semi-simplicity conjecture in half of the degrees implies those conjectures in all degrees.

Cycle conjectures and birational invariants over finite fields

TL;DR

<3-5 sentence high-level summary> The paper investigates a natural birational invariant over finite fields, built from Galois-invariant pieces of etale and Borel–Moore cohomology, and shows that its vanishing on projective space in certain degrees is equivalent to the Tate conjecture, Beilinson conjecture, and the Grothendieck–Serre 1-semi-simplicity conjecture for all smooth projective varieties over the field. It develops a framework based on refined unramified cohomology to relate these conjectures to invariants of the function field and to inductive arguments that reduce global statements to half of the degrees, with effective reductions to specific degrees such as H^4 on P^4. The work also extends to higher Chow groups and motivic cohomology, discusses Tate conjectures for higher Chow groups, and offers a conjectural Milnor K-theory description of the Galois-invariant pieces, tying the birational invariant to fundamental K-theoretic invariants. Together, these results provide a coherent strategy to derive broad arithmetic conjectures from vanishing criteria on projective space and to connect function-field invariants with deep cycle-theoretic questions over finite fields.

Abstract

We study a natural birational invariant for varieties over finite fields and show that its vanishing on projective space is equivalent to the Tate conjecture, the Beilinson conjecture, and the Grothendieck--Serre semi-simplicity conjecture for all smooth projective varieties over finite fields. We further show that the Tate, Beilinson, and 1-semi-simplicity conjecture in half of the degrees implies those conjectures in all degrees.
Paper Structure (33 sections, 45 theorems, 111 equations)

This paper contains 33 sections, 45 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. Cycle conjectures
  3. Birational invariants
  4. Main results
  5. Tate conjecture for higher Chow groups
  6. A conjectural description of $H^i(F_0\bar{X},\mathbb Q_\ell(n))^{G_\mathbb F}$.
  7. Organization of the paper
  8. Preliminaries
  9. Conventions
  10. Étale cohomology
  11. Hochschild-Serre spectral sequence
  12. Borel-Moore cohomology
  13. Refined unramified cohomology
  14. Galois actions
  15. 1-semi-simplicity
  16. ...and 18 more sections

Key Result

Theorem 1.1

The Tate, Beilinson, and semi-simplicity conjecture for all smooth projective varieties over a finite field $\mathbb F$ are equivalent to the vanishing

Theorems & Definitions (105)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Conjecture 1.4: Tate conjecture for function fields
  • Corollary 1.5
  • Conjecture 1.6
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 95 more