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An approach to the Tate conjecture for surfaces over a finite field

Bruno Kahn

TL;DR

This work reinterprets the Tate conjecture for a smooth projective surface over a finite field in terms of affine opens with Picard triviality, linking the conjecture to the vanishing of $H^3(U,1)$ for these opens. It establishes a bridge through the Brauer group: Tate in codimension $1$ is equivalent to finiteness (or vanishing) of $\operatorname{Br}_l(U_s)_G$ for relevant affine $U$, with $H^3(U,1)$ capturing the obstruction via Hochschild–Serre and purity. The note then outlines three approaches to prove the reformulation, each relying on deep tools like Gabber rigidity, Gersten-type arguments, and a below-ground strategy via ante-Nisnevich neighborhoods; all three face substantial obstacles, including l-dependence, normalization complications, and dimension‑1 counterpoints. The study showcases the intricate interplay between Brauer groups, Galois actions, and cohomological purity in approaching the Tate conjecture, while highlighting the potential and limits of Gersten-style techniques in this arithmetic-geometric context.

Abstract

We give a reformuation of the Tate conjecture for a surface over a finite field in terms of suitable affine open subsets. We then present three attempts to prove this reformulation, each of them falling short. Interestingly, the last two are related to techniques used in proofs of Gersten's conjecture.

An approach to the Tate conjecture for surfaces over a finite field

TL;DR

This work reinterprets the Tate conjecture for a smooth projective surface over a finite field in terms of affine opens with Picard triviality, linking the conjecture to the vanishing of for these opens. It establishes a bridge through the Brauer group: Tate in codimension is equivalent to finiteness (or vanishing) of for relevant affine , with capturing the obstruction via Hochschild–Serre and purity. The note then outlines three approaches to prove the reformulation, each relying on deep tools like Gabber rigidity, Gersten-type arguments, and a below-ground strategy via ante-Nisnevich neighborhoods; all three face substantial obstacles, including l-dependence, normalization complications, and dimension‑1 counterpoints. The study showcases the intricate interplay between Brauer groups, Galois actions, and cohomological purity in approaching the Tate conjecture, while highlighting the potential and limits of Gersten-style techniques in this arithmetic-geometric context.

Abstract

We give a reformuation of the Tate conjecture for a surface over a finite field in terms of suitable affine open subsets. We then present three attempts to prove this reformulation, each of them falling short. Interestingly, the last two are related to techniques used in proofs of Gersten's conjecture.
Paper Structure (18 sections, 16 theorems, 15 equations)

This paper contains 18 sections, 16 theorems, 15 equations.

Table of Contents

  1. The reformulation
  2. The Brauer group of $X$
  3. Cohomology of curves
  4. The Brauer group of $U$
  5. Passing from $\operatorname{Br}_l(U_s)_G$ to $H^3(U,1)$
  6. The condition $\operatorname{Pic}(U)=0$
  7. Proof of $(1)\iff (2)$
  8. From $(2)$ to $(3)$
  9. Going further
  10. Three approaches
  11. The first idea fails
  12. Second idea: from above
  13. First part
  14. Second part
  15. Third part
  16. ...and 3 more sections

Key Result

Theorem 1.1

Suppose $k$ finite, and let $X$ be a smooth projective surface over $k$; assume that $G$ acts trivially on $\operatorname{NS}(X_s)$. Then the following are equivalent:

Theorems & Definitions (28)

  • Theorem 1.1
  • Proposition 1.3
  • proof
  • Proposition 1.4
  • proof
  • Proposition 1.5
  • Proposition 1.6
  • proof
  • Lemma 1.7
  • proof
  • ...and 18 more