Kahn's question on reducing the Tate conjecture for divisors from characteristic zero to positive characteristic: must an $\ell$-adic class on a surface over $\mathbf{Q}$ whose specializations are algebraic be algebraic?

For a smooth projective surface $S$ over $\mathbf{Q}$ and a prime $\ell$ , consider a class $\alpha$ in the second $\ell$ -adic cohomology group $H^2(S_{\bar{\mathbf{Q}}},\mathbf{Q}_\ell(1))$ . Bruno Kahn (2022, published 2023) asked: if for (almost) every prime $p\neq\ell$ of good reduction the specialization of $\alpha$ in $H^2((S_p)_{\bar{\mathbf{F}}_p},\mathbf{Q}_\ell(1))$ is algebraic (a $\mathbf{Q}_\ell$ -linear combination of divisor classes), must $\alpha$ itself be algebraic over $\bar{\mathbf{Q}}$ ? Kahn asserts — without giving a proof in the source — that this question is equivalent, via proper smooth base change and the Chebotarev density theorem, to whether the Tate conjecture for divisors in characteristic zero can be reduced to its positive-characteristic case. The characteristic-zero Tate conjecture for divisors is a theorem via Hodge theory, so a positive answer is expected, but no specialization-based (Hodge-free) proof is known, and the question remains open as of 2026.

Background

Let $k$ be a field and $X$ a smooth projective variety over $k$ . To $X$ one attaches $\ell$ -adic étale cohomology groups $H^i(X_{\bar{k}},\mathbf{Q}_\ell)$ ( $\ell$ a prime different from the characteristic of $k$ , $\bar{k}$ a separable closure), which carry an action of the absolute Galois group $\mathrm{Gal}(\bar{k}/k)$ . Every algebraic cycle (formal $\mathbf{Q}$ -linear combination of subvarieties) has an $\ell$ -adic cycle class: for divisors this is a map $\mathrm{cl}\colon \mathrm{CH}^1(X)\otimes\mathbf{Q}_\ell\to H^2(X_{\bar{k}},\mathbf{Q}_\ell(1))$ , where $\mathrm{CH}^1$ is the group of divisors modulo rational equivalence, $\mathbf{Q}_\ell(1)$ is the $\ell$ -adic Tate twist that makes degrees integral, and the classes of divisors span the same subspace as the Néron–Severi group $\mathrm{NS}(X_{\bar{k}})$ (divisors modulo algebraic equivalence, a finitely generated group). The Tate conjecture in codimension 1 (Tate, 1965) predicts that $\mathrm{cl}$ maps onto the Galois-invariant subspace $H^2(X_{\bar{k}},\mathbf{Q}_\ell(1))^{\mathrm{Gal}(\bar{k}/k)}$ for every smooth projective $X$ over a finitely generated field $k$ . It is one of the central open problems of arithmetic geometry, open in general even in positive characteristic for surfaces over finite fields, where it is moreover known to be equivalent to the Birch–Swinnerton-Dyer conjecture (the conjecture relating the rank of the group of rational points of an abelian variety to the behaviour of its $L$ -function) for Jacobians of curves over one-variable function fields (Lichtenbaum–Ramachandran–Suzuki). In characteristic zero the conjecture for divisors is a theorem: by the comparison between $\ell$ -adic and singular cohomology and the Lefschetz $(1,1)$ theorem (which identifies the rational $(1,1)$ -classes on the associated complex manifold with divisor classes), Galois-invariant $\ell$ -adic classes are algebraic. Two reduction results shape the positive-characteristic side: Ambrosi (2018), extending an argument of André and using Deligne's global invariant cycle theorem together with the cyclicity of the absolute Galois group of $\mathbf{F}_p$ , reduces the conjecture from finitely generated fields of positive characteristic to finite fields, and Morrow's variational approach (2019) reduces the divisors case to surfaces over the prime field $\mathbf{F}_p$ . Kahn (2022, published 2023) gave a uniform, elementary proof that the Tate conjecture for divisors over any finitely generated field reduces to the case of surfaces over its prime subfield; in characteristic zero the reduction runs through André's specialization theorem for Néron–Severi groups combined with the Lefschetz $(1,1)$ theorem, i.e. through Hodge theory, not through positive characteristic. Kahn closes with the natural question of whether the characteristic-zero case can instead be reduced to the positive-characteristic case (Question 1/A of the paper), and observes that this is equivalent, via proper smooth base change (the theorem that $\ell$ -adic étale cohomology is preserved when a variety with good reduction is reduced to its special fiber) and the Chebotarev density theorem (which realizes Galois group elements as Frobenius elements at primes), to the following concrete local–global question (Question 2/B). For a prime $p$ where $S$ has good reduction (i.e. $S$ extends to a smooth proper scheme over $\mathbf{Z}_{(p)}$ ), proper smooth base change identifies $H^2(S_{\bar{\mathbf{Q}}},\mathbf{Q}_\ell(1))$ with $H^2((S_p)_{\bar{\mathbf{F}}_p},\mathbf{Q}_\ell(1))$ , canonically up to the choice of embedding $\bar{\mathbf{Q}}\hookrightarrow\bar{\mathbf{Q}}_p$ ; call the image of $\alpha$ its specialization at $p$ , and call it algebraic if it lies in the $\mathbf{Q}_\ell$ -span of divisor classes on $(S_p)_{\bar{\mathbf{F}}_p}$ . The subtlety is that $\alpha$ is not assumed Galois-invariant: Frobenius acts on the Néron–Severi group of the reduction only through a finite quotient, so algebraic specializations need not be Frobenius-fixed, and the geometric Picard number can jump under reduction (the algebraic subspace at $p$ may strictly contain the specialization of the algebraic subspace over $\bar{\mathbf{Q}}$ ). Hence the hypothesis gives no direct Galois-invariance constraint on $\alpha$ , blocking a naive appeal to the known characteristic-zero theory; at any single good prime where the Picard number does not jump the answer is elementary and positive, so the open content concerns the possibility of jumps at almost all primes.

Problem Statement

Let $S$ be a smooth projective (geometrically connected) surface over $\mathbf{Q}$ , let $\ell$ be a prime, and let $\alpha\in H^2(S_{\bar{\mathbf{Q}}},\mathbf{Q}_\ell(1)).$ For (almost) every prime $p\neq\ell$ of good reduction, let the specialization of $\alpha$ at $p$ be its image in $H^2((S_p)_{\bar{\mathbf{F}}_p},\mathbf{Q}_\ell(1))$ under the identification given by proper smooth base change (well defined up to the natural action of $\mathrm{Gal}(\bar{\mathbf{F}}_p/\mathbf{F}_p)$ , which does not affect the stated property since the $\mathbf{Q}_\ell$ -span of the divisor classes is stable under this action). Suppose that for (almost) every such $p$ this specialization is algebraic, i.e. lies in the $\mathbf{Q}_\ell$ -span of the cycle classes of divisors on $(S_p)_{\bar{\mathbf{F}}_p}$ . Does it follow that $\alpha$ is algebraic over $\bar{\mathbf{Q}}$ , i.e. that $\alpha$ lies in the image of $\mathrm{cl}\colon\mathrm{NS}(S_{\bar{\mathbf{Q}}})\otimes\mathbf{Q}_\ell\to H^2(S_{\bar{\mathbf{Q}}},\mathbf{Q}_\ell(1))$ ? Equivalently, as asserted by Kahn (via proper smooth base change and the Chebotarev density theorem; the source states this equivalence without proof): can the Tate conjecture for divisors in characteristic zero be reduced to its positive-characteristic case? The parenthetical '(almost)' is in the source and both readings — all primes and all but finitely many primes of good reduction — are admitted; a positive answer is expected because the characteristic-zero statement is known via Hodge theory, but no specialization-based proof is known.

References

Bruno Kahn, Sur la conjecture de Tate pour les diviseurs (English title: On the Tate conjecture for divisors), Essential Number Theory 2 (2023), No. 1, 83–92, DOI 10.2140/ent.2023.2.83, arXiv:2205.05287, https://doi.org/10.2140/ent.2023.2.83
Bruno Kahn, Some remarks on the smash-nilpotence conjecture, Indagationes Mathematicae 37 (2026), No. 1, 137–148, DOI 10.1016/j.indag.2024.05.009, preprint arXiv:2311.14362 (2023), https://doi.org/10.1016/j.indag.2024.05.009
Bruno Kahn, On the injectivity and non-injectivity of the $\ell$ -adic cycle class maps, in: Moduli, Motives and Bundles, Cambridge University Press, 2025, 201–232, DOI 10.1017/9781009497206.008, preprint arXiv:2307.13281 (2023), https://doi.org/10.1017/9781009497206.008
Bruno Kahn, An approach to the Tate conjecture for surfaces over a finite field, preprint (2025), arXiv:2505.07337, https://arxiv.org/abs/2505.07337
Emiliano Ambrosi, A note on the behaviour of the Tate conjecture under finitely generated field extensions, Pure and Applied Mathematics Quarterly 14 (2018), No. 3–4, 515–527, DOI 10.4310/pamq.2018.v14.n3.a4, https://doi.org/10.4310/pamq.2018.v14.n3.a4
Matthew Morrow, A variational Tate conjecture in crystalline cohomology, Journal of the European Mathematical Society 21 (2019), No. 11, 3467–3511, DOI 10.4171/jems/907, https://doi.org/10.4171/jems/907
Yves André, Pour une théorie inconditionnelle des motifs, Publications Mathématiques de l'IHÉS 83 (1996), 5–49, DOI 10.1007/BF02698643, https://doi.org/10.1007/BF02698643
J. Tate, Algebraic cycles and poles of zeta functions, in: Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ. 1963), O. F. G. Schilling (ed.), Harper & Row, New York, 1965, 93–110, Zbl 0213.22804 (zbMATH 3338305), https://zbmath.org/?q=an%3A3338305
S. Lichtenbaum, N. Ramachandran, T. Suzuki, The conjectures of Artin–Tate and Birch–Swinnerton-Dyer, Épijournal de Géométrie Algébrique 6 (2022), article 7, DOI 10.46298/epiga.2022.7482, arXiv:2101.10222, https://doi.org/10.46298/epiga.2022.7482
Yanshuai Qin, On the Brauer groups of fibrations, Mathematische Zeitschrift 307 (2024), No. 1, article 18, DOI 10.1007/s00209-024-03487-8, preprint arXiv:2012.01324, https://doi.org/10.1007/s00209-024-03487-8

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Kahn's question on reducing the Tate conjecture for divisors from characteristic zero to positive characteristic: must an $\ell$ -adic class on a surface over $\mathbf{Q}$ whose specializations are algebraic be algebraic?

Background

Problem Statement

References