The Tate conjecture for divisors on surfaces over a finite field in Kahn's affine-open reformulation

The Tate conjecture for divisors predicts that for a smooth projective surface $X$ over a finite field $k$, the rank of the Néron–Severi group $NS(X)$ equals the order of the pole of the zeta function $\zeta(X,s)$ at $s=1$; this is the central open case of the codimension-1 Tate conjecture, to which the general statement over finitely generated fields is known to reduce. Kahn (2025) proved an equivalent formulation: assuming $Gal(k_s/k)$ acts trivially on $NS(X_s)$, the conjecture holds for $X$ if and only if $H^3(U, \mathbb{Q}_l/\mathbb{Z}_l(1)) = 0$ for every affine open $U \subseteq X$ with $Pic(U) = 0$, if and only if for every such $U$ and every smooth irreducible divisor $Z \subset U$ the map $H^3(U,1) \to H^3(U-Z,1)$ is injective - equivalently, in the Gysin exact sequence of the pair, the boundary map $\partial: H^2(V,1) \to H^1(Z,0)$ is surjective. Kahn's three attempted proofs each fail on specific, explicitly isolated obstacles (descent from étale neighbourhoods, non-uniformity of an auxiliary prime bound, presentation constructions obstructed by extra components). The problem is to prove - or refute - the reformulated statement for all smooth projective surfaces over finite fields; as of August 2026 it remains open.

Background

The Tate conjecture, formulated by Tate in the 1960s (see his 1994 survey), is one of the central open problems in the arithmetic theory of algebraic cycles. For a smooth projective variety $X$ over a finite field $k = \mathbb{F}_q$, the zeta function $\zeta(X,s)$ is the generating function encoding the number of $k$-rational points of $X$ over every finite extension of $k$; by the Weil conjectures it is a rational function of $q^{-s}$. The conjecture predicts that the order of the pole of $\zeta(X,s)$ at $s = i$ equals the rank of the group of codimension-$i$ algebraic cycles on $X$ modulo homological equivalence. For divisors ($i = 1$), the case considered here, the cycle group is the Néron–Severi group $NS(X)$ - the group of divisors modulo algebraic equivalence, a finitely generated abelian group - and the conjecture is equivalent, via the Kummer exact sequence (the exact sequence of étale sheaves $1 \to \mu_{l^n} \to \mathbb{G}_m \xrightarrow{\ l^n\ } \mathbb{G}_m \to 1$, where the second map is the $l^n$-th power map, relating Picard groups to cohomology with finite torsion coefficients), to the surjectivity of the $l$-adic cycle class map $NS(X) \otimes \mathbb{Q}_l \to H^2(X_s, \mathbb{Q}_l(1))^G$, where $k_s$ is a separable closure of $k$, $G = Gal(k_s/k)$, $l$ is a prime different from the characteristic of $k$, and $\mathbb{Q}_l(1)$ is the first Tate twist (the one-dimensional $l$-adic representation on which Frobenius acts by $q$, so that classes of divisors defined over $k$ are exactly the Frobenius-invariant classes). The conjecture is known to be independent of $l$.

The codimension-1 Tate conjecture over an arbitrary finitely generated field reduces to the case of surfaces over the prime field $\mathbb{F}_p$ (Ambrosi 2018; Morrow 2019; Kahn 2023, which gives a unified proof), so arbitrary smooth projective surfaces over finite fields are the essential open target. The divisor case is known for abelian varieties and, via their Jacobians, for products of curves (Tate 1994), and for K3 surfaces (Charles 2013; Madapusi Pera 2015); it is not known for a general surface, and no general proof technique is currently available.

Kahn (2025) gives a new equivalent formulation for surfaces. Write $H^i(S, j)$ for the étale cohomology $H^i_{\text{ét}}(S, \mathbb{Q}_l/\mathbb{Z}_l(j))$ with torsion coefficients, and $Br_l(S)$ for the $l$-primary part of the cohomological Brauer group $H^2(S, \mathbb{G}_m)$. Two facts drive the reformulation: first, by the Kummer sequence and Galois descent, the Tate conjecture for a smooth projective variety over a finitely generated field is equivalent to the finiteness of the group of invariants $Br_l(X_s)^G$; second, by a theorem of M. Artin, an affine scheme of dimension 2 over a separably closed field has $l$-primary étale cohomological dimension at most 2. For an affine open $U \subseteq X$ with $Pic(U) = 0$ (such $U$ exist because $Pic(X)$ is finitely generated), Artin's theorem and the Hochschild–Serre spectral sequence (relating the cohomology of $U$ to that of its geometric fibre $U_s$ through the Galois cohomology of $G$) identify $H^3(U,1)$ with the group of coinvariants $Br_l(U_s)_G$; moreover (Proposition 1.6 of the note, using divisibility and the procyclicity of $G$) the invariants $Br_l(U_s)^G$ are finite if and only if the coinvariants $Br_l(U_s)_G$ are finite. Kahn's Theorem 1.1 then states: assuming $G$ acts trivially on $NS(X_s)$ - harmless after replacing $k$ by a finite extension, since $NS(X_s)$ is finitely generated and this reduction suffices for the conjecture - the following are equivalent: (1) the Tate conjecture holds for $X$; (2) for every affine open $U \subseteq X$ with $Pic(U) = 0$, one has $H^3(U,1) = 0$; (3) for every such $U$ and every smooth irreducible divisor $Z \subset U$, the restriction map $H^3(U,1) \to H^3(U-Z,1)$ is injective.

Condition (3) is attacked through the Gysin exact sequence (the étale localization sequence) of the pair $U \supset V = U - Z$,

$H^2(V,1) \xrightarrow{\ \partial\ } H^1(Z,0) \xrightarrow{\ i_*\ } H^3(U,1) \to H^3(V,1),$

in which the condition is equivalent to the vanishing of the Gysin pushforward $i_*$ and to the surjectivity of the boundary (residue) map $\partial$. Proposition 2.1 of the note shows that the image of $i_*$ is contained in the finite group $H^1(Z_s,0)^G$ and vanishes for all primes $l \ge l_0$ where $l_0$ depends on $Z$. Three strategies are then attempted, each imitating techniques from proofs of Gersten's conjecture (the degeneracy statement for the coniveau spectral sequence in K-theory), and each fails on a specific obstacle. (i) The $l$-independence route fails because $l_0$ is not bounded independently of the divisor $Z$. (ii) The "from above" strategy (Gillet's method for discrete valuation rings): Gabber's rigidity theorem - torsion étale cohomology is unchanged when passing from a henselian pair (a pair $(A,I)$ whose étale covers split over $A/I$; here obtained as the filtering colimit of Nisnevich neighbourhoods, i.e. étale maps $q: U_1 \to U$ inducing an isomorphism above $Z$) to its closed subscheme - yields an étale neighbourhood $U_1$ on which the analogue $\partial_1$ of $\partial$ is surjective; passing to the normalisation $\overline{U}_1$ of $U$ in $U_1$ makes the structural map finite and flat, so that trace maps in étale cohomology are available, and one obtains $\mathrm{Im}\, \partial \supseteq \mathrm{Im}(\overline{q}_* \circ \overline{\partial}_1)$; but completing the argument would additionally require either the surjectivity of the composition $H^1(\overline{U}_1, 0) \to H^1(Z, 0)$ (Question 3.5 asks this already over the normalisation $\overline{U}_h$ of $U$ in its henselisation) or a function $f_1$ regular on $\overline{U}_1$ with $Z$ principal of equation $f_1$ and $f_1 \equiv 1 \pmod T$, where $T$ is the extra part of the boundary - neither is established. (iii) The "from below" strategy (Gabber's geometric presentation lemma, in the finite-field version of Hogadi–Kulkarni): one would want a morphism $v: U \to U_1$, with $U_1$ an affine open of a surface for which the Tate conjecture is known, such that $Z = v^{-1}(v(Z))$ scheme-theoretically; functoriality of Gysin maps would then force $i_* = 0$. But finite covers of generic degree $> 1$ necessarily acquire extra codimension-1 components above $v(Z)$, and Proposition 3.7 gives a definite dimension-1 counterexample: for a smooth affine curve $U$ with $Pic(U) = 0$ whose completion has genus $> 0$ and a closed point of degree exceeding the number of geometric points at infinity, no such diagram with $\deg(v) > 1$ exists. Whether these obstructions can be overcome in dimension 2 is precisely what the note leaves open.

Problem Statement

Let $k$ be a finite field and let $X$ be a smooth projective surface over $k$. The problem is to prove the Tate conjecture for divisors on $X$ - that the order of the pole of the zeta function $\zeta(X,s)$ (the generating function of the numbers of rational points of $X$ over the finite extensions of $k$) at $s = 1$ equals the rank of the Néron–Severi group $NS(X)$ (the group of divisors on $X$ modulo algebraic equivalence, a finitely generated abelian group) - for every smooth projective surface over every finite field.

In the equivalent reformulation proven in Kahn (2025, Theorem 1.1), which is the source of this problem and which a solution may exploit: after replacing $k$ by a finite extension so that $G = Gal(k_s/k)$ acts trivially on $NS(X_s)$, where $k_s$ is a separable closure of $k$, $X_s = X \times_k k_s$, and $NS(X_s)$ is the Néron–Severi group of $X_s$, prove that for every prime $l \ne \mathrm{char}(k)$ and every affine open subset $U \subseteq X$ with trivial Picard group ($Pic(U) = 0$, where $Pic(U)$ is the group of divisors on $U$ modulo linear equivalence),

$H^3_{\text{ét}}(U, \mathbb{Q}_l/\mathbb{Z}_l(1)) = 0,$

where $H^3_{\text{ét}}$ denotes étale cohomology and $\mathbb{Q}_l/\mathbb{Z}_l(1)$ is the group of $l$-power roots of unity, viewed as an étale sheaf (the first Tate twist of $\mathbb{Q}_l/\mathbb{Z}_l$); the notation $H^i(S, j)$ below abbreviates $H^i_{\text{ét}}(S, \mathbb{Q}_l/\mathbb{Z}_l(j))$.

Equivalently, for every such $U$ and every smooth irreducible divisor $Z \subset U$ with $V = U - Z$, the restriction map $H^3(U,1) \to H^3(V,1)$ is injective; equivalently, in the Gysin exact sequence

$H^2(V,1) \xrightarrow{\ \partial\ } H^1(Z,0) \xrightarrow{\ i_*\ } H^3(U,1) \to H^3(V,1)$

the boundary map $\partial$ is surjective (equivalently, $i_* = 0$).

The statement under consideration ranges over every smooth projective surface over every finite field, with no restriction to classes where the conjecture is already known (abelian surfaces, K3 surfaces, products of curves, elliptic surfaces, surfaces dominated by products of curves, etc.), no restriction on the prime $l$ beyond $l \ne \mathrm{char}(k)$, and no restriction on the affine open $U$ beyond $Pic(U) = 0$. The source identifies intermediate milestones - a bound on the threshold $l_0$ in Proposition 2.1(c) uniform in the divisor $Z$; a positive answer to Question 3.5 (surjectivity of $H^1(\overline{U}_h, 0) \to H^1(Z, 0)$ over the normalisation $\overline{U}_h$ of $U$ in its henselisation); the construction of "from below" presentations $v: U \to U_1$ with $Z = v^{-1}(v(Z))$ scheme-theoretically and $U_1$ of a type for which the conjecture is known - but these milestones alone do not constitute a solution. A disproof would be an explicit $X$ with $G$ acting trivially on $NS(X_s)$ for which the Tate conjecture fails (equivalently, by Theorem 1.1, an affine open $U$ with $Pic(U) = 0$ and $H^3(U, \mathbb{Q}_l/\mathbb{Z}_l(1)) \ne 0$).

References

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