The Tate conjecture for divisors over finite fields, reduced to the case of surfaces with $H^1(X,\mathcal{O}_X)=0$

The Tate conjecture for divisors asserts that on a smooth projective variety over a finite field, every Galois-invariant $\ell$-adic cohomology class in degree two is a $\mathbb{Q}_\ell$-linear combination of classes of algebraic divisors. Work of de Jong, Morrow, and Yuan has reduced this conjecture, for all smooth projective varieties over all finite fields, to the case of smooth projective surfaces with vanishing first coherent cohomology $H^1(X,\mathcal{O}_X)=0$. The problem, isolated in Ertl–Keller–Qin's study of the relations among Tate's conjectures, is to prove the conjecture in this reduced surface case; by the reduction it is equivalent to the full codimension-one Tate conjecture in positive characteristic, and would imply the Birch–Swinnerton-Dyer (BSD) rank conjecture equivalences and Tate–Shafarevich and Brauer finiteness statements that Ertl–Keller–Qin establish conditionally. The problem remains open as of August 2026: Kahn (2025) and Schreieder (2024) reformulate it without proving it, and known cases (abelian varieties, K3 surfaces in all characteristics, surfaces of geometric genus one, generic-in-moduli $h^{2,0}=1$ results) cover only special geometries.

Background

For a smooth projective variety $X$ over a finite field $k$ and a prime $\ell\neq\operatorname{char}k$, étale cohomology (the $\ell$-adic analogue of singular cohomology for varieties in positive characteristic) provides the group $\operatorname{H}^2(X_{\bar k},\mathbb{Q}_\ell(1))$, where $X_{\bar k}$ denotes base change to a separable closure $\bar k$ and $(1)$ is the Tate twist that normalizes the weights of the Frobenius action. Every algebraic divisor on $X$ carries a cycle class in this group, and the classes arising from divisors are stable under the absolute Galois group $G_k=\operatorname{Gal}(\bar k/k)$; the classes fixed by $G_k$ are called Tate classes. The Tate conjecture for divisors, denoted $T^1(X,\ell)$ and posed by Tate in the 1960s, asserts that every Tate class in $\operatorname{H}^2(X_{\bar k},\mathbb{Q}_\ell(1))$ is a $\mathbb{Q}_\ell$-linear combination of divisor classes; equivalently, the cycle class map $A^1(X)\otimes_{\mathbb{Q}}\mathbb{Q}_\ell\longrightarrow\operatorname{H}^2(X_{\bar k},\mathbb{Q}_\ell(1))^{G_k}$ is surjective, where $A^1(X)$ is the group of codimension-one cycle classes with rational coefficients modulo homological equivalence. Through the Kummer exact sequence (the short exact sequence of étale sheaves $0\to\mu_{\ell^n}\to\mathbb{G}_m\xrightarrow{t\mapsto t^{\ell^n}}\mathbb{G}_m\to 0$ relating units to roots of unity, which links cohomology in degree three to Brauer groups) the conjecture is equivalent to finiteness of the $\ell$-primary part of the cohomological Brauer group invariants $\operatorname{Br}(X_{\bar k})^{G_k}(\ell)$, where $\operatorname{Br}(X)=\operatorname{H}^2_{\mathrm{\acute{e}t}}(X,\mathbb{G}_m)$ is the Brauer group of $X$ (the torsion group classifying twisted forms of $\mathbb{P}^n$-bundles); it is also equivalent, via the Artin–Tate formula, to the statement that the order of the pole of the zeta function of $X$ at the critical point equals the rank of the Néron–Severi group $\operatorname{NS}(X)$, the finitely generated group of divisors modulo algebraic equivalence. The conjecture is classical for abelian varieties and varieties dominated by them, by Tate's theorem relating divisors on abelian varieties to Galois-invariant endomorphisms. For K3 surfaces (simply connected surfaces with trivial canonical bundle and $H^1(\mathcal{O}_X)=0$) it was proved in all characteristics through the Kuga–Satake construction, which attaches an abelian variety to a K3 surface so that divisors on the surface correspond to endomorphisms of the abelian variety: Charles settled characteristic $p>3$, Madapusi Pera odd characteristic, and Kim–Madapusi Pera characteristic $2$. Surfaces of geometric genus one (the geometric genus $p_g=\dim H^2(X,\mathcal{O}_X)$ being the number of independent holomorphic 2-forms) have been largely completed by Guo–Yang, and Hamacher–Yang–Zhao proved the conjecture generically in moduli for varieties with $h^{2,0}=1$. No method is known for general surfaces of general type (surfaces of maximal Kodaira dimension, the generic case in the classification of algebraic surfaces) with zero irregularity. Three reductions make the surface case with $H^1(X,\mathcal{O}_X)=0$ the decisive one. First, an unpublished note of de Jong (as credited by Ertl–Keller–Qin and independently by Kahn) and Theorem 4.3 of Morrow's work on the variational Tate conjecture in crystalline cohomology — a $p$-adic cohomology theory for varieties in positive characteristic, playing the role that de Rham cohomology plays over the complex numbers — which characterizes, for a smooth proper family in characteristic $p$, whether a divisor extends cohomologically across the family in terms of its crystalline cycle class, reduce $T^1$ over a finite field from arbitrary dimension to the case of surfaces. Second, Yuan's theorem on the positivity of Hodge bundles of abelian varieties over global function fields (Theorem 1.6 of his Compositio paper) — the Hodge bundles being the vector bundles on the base of a family of abelian varieties whose fibers are the spaces of holomorphic differential forms — reduces $T^1$ for all surfaces over a fixed finite field to those with $H^1(X,\mathcal{O}_X)=0$, using the Albanese fibration (the canonical map from a variety to its Albanese variety, the abelian variety that receives all morphisms from the variety to abelian varieties) and the Artin–Tate theorem. Combining them: $T^1$ holds for all smooth projective varieties over finite fields if and only if it holds for smooth projective surfaces with $H^1(X,\mathcal{O}_X)=0$ over finite fields. Third, Ambrosi and Kahn extended the reduction to finitely generated ground fields of positive characteristic, reducing them to surfaces over the prime field. The problem originates in the paper of Ertl–Keller–Qin comparing the interrelated conjectures over finitely generated fields that the paper labels $T^i$, $E^i$, B+S-D, Conjecture 2, and B+S-D+2 (not all of these were originally formulated by Tate; B+S-D+2, the analytic Tate conjecture, predicts the order of the pole of the L-function at its central point). They prove that for an abelian variety $A$ over a finitely generated field $K$ of characteristic $p>0$ the Birch–Swinnerton-Dyer rank conjecture — equality of the Mordell–Weil rank, i.e. the rank of the group $A(K)$ of $K$-rational points (finitely generated by the Mordell–Weil theorem), with the order of vanishing of the Hasse–Weil $L$-function of $A$ (the Euler product over the finite places of $K$ encoding the point counts of the reductions of $A$) — is equivalent to finiteness of the $\ell$-primary part of the Tate–Shafarevich group $\Sha(A/K)$ (the group measuring the failure of the local-to-global principle for torsors under $A$), and their Corollary 4.12 states that $T^1$ for all smooth projective varieties over finite fields of characteristic $p$ implies the whole suite of Tate's conjectures for divisors, L-function pole, BSD rank, and zeta-function order statements over every finitely generated field of characteristic $p$. Their Remark 4.13 records that the conjecture has been reduced to surfaces with $H^1(X,\mathcal{O}_X)=0$ by de Jong, Morrow, and Yuan, and explicitly leaves the verification of that reduced statement open.

Problem Statement

Let $k$ be a finite field and let $X$ be a smooth projective surface over $k$ with $H^1(X,\mathcal{O}_X)=0$ (zero irregularity, i.e. $q=\dim_k H^1(X,\mathcal{O}_X)=0$: the surface admits no nonzero holomorphic 1-forms). For a prime $\ell\neq\operatorname{char}k$, let $G_k=\operatorname{Gal}(\bar k/k)$ act on $\operatorname{H}^2(X_{\bar k},\mathbb{Q}_\ell(1))$, where $X_{\bar k}$ is the base change to a separable closure.

Problem: prove that every Tate divisor class on $X$ is algebraic, that is, that the $\ell$-adic cycle class map $A^1(X)\otimes_{\mathbb{Q}}\mathbb{Q}_\ell\longrightarrow\operatorname{H}^2(X_{\bar k},\mathbb{Q}_\ell(1))^{G_k}$ is surjective — where $A^1(X)$ is the group of codimension-one algebraic cycles (divisors) on $X$ with rational coefficients, modulo homological equivalence — equivalently, that every $G_k$-invariant class in $\operatorname{H}^2(X_{\bar k},\mathbb{Q}_\ell(1))$ is a $\mathbb{Q}_\ell$-linear combination of classes of divisors on $X$ — for every smooth projective surface $X$ with $H^1(X,\mathcal{O}_X)=0$ over every finite field $k$, for some (equivalently, by the known independence results for divisors, all) primes $\ell\neq\operatorname{char}k$.

This is the reduced form, recorded as Remark 4.13 of Ertl–Keller–Qin, of the Tate conjecture for divisors $T^1$ for all smooth projective varieties over finite fields: by Theorem 4.3 of Morrow and Theorem 1.6 of Yuan (together with an unpublished note of de Jong), the statement above for surfaces with $H^1(X,\mathcal{O}_X)=0$ is equivalent to $T^1(Y,\ell)$ for every smooth projective variety $Y$ over every finite field. Equivalently, it asserts finiteness of the $\ell$-primary part of $\operatorname{Br}(Y_{\bar k})^{G_k}$ for all such $Y$, and the corresponding Artin–Tate and L-function pole statements. A solution must be uniform in the surface, the finite field, and the characteristic (including $p=2$), and must cover surfaces of general type with $q=0$ and arbitrary geometric genus, which no existing case-by-case method reaches. A disproof would exhibit a specific surface $X$ with $H^1(X,\mathcal{O}_X)=0$, a finite field, and a prime $\ell$, together with a Tate class on $X$ no $\mathbb{Q}_\ell$-combination of divisor classes represents.

References

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