Integral Tate Conjecture: Advances & Challenges

• Integral Tate Conjecture is an integral-coefficient refinement of the rational Tate conjecture, examining the surjectivity of cycle class maps in étale cohomology over finite and finitely generated fields.
• It employs advanced techniques including Galois cohomology, coniveau filtrations, and cycle map analysis to distinguish between torsion and non-torsion obstructions.
• Its implications span arithmetic geometry and zero-cycle Brauer–Manin obstructions, offering both positive results in codimension one and systematic counterexamples in higher codimensions.

Searching arXiv for recent and foundational papers on the integral Tate conjecture and closely related results. arXiv search query: "all: integral Tate conjecture" The integral Tate conjecture is the integral-coefficient refinement of the rational Tate conjecture. In its most classical form, for a smooth projective variety $X$ over a finite field $k=\mathbb{F}_q$, a prime $\ell\neq \operatorname{char}(k)$, and codimension $i$, it asks whether the $\ell$-adic cycle class map with $\mathbb{Z}_\ell$-coefficients is surjective onto the appropriate Galois-invariant integral cohomology classes. For one-cycles on a smooth projective geometrically irreducible $d$-fold $X/\mathbb{F}_q$, this takes the form

$$\mathrm{CH}_1(X)\otimes \mathbb{Z}_\ell \longrightarrow H^{2d-2}_{\mathrm{\acute{e}t}}(X,\mathbb{Z}_\ell(d-1)),$$

and the conjecture asks for surjectivity of this map; equivalently, one often studies the induced map to

$$H^{2d-2}_{\mathrm{\acute{e}t}}(\overline{X},\mathbb{Z}_\ell(d-1))^{\mathrm{Gal}(\overline{\mathbb{F}}_q/\mathbb{F}_q)}.$$

Over finitely generated fields more generally, the conjecture is formulated as surjectivity onto the union of invariant lattices under open subgroups of the absolute Galois group, or, in the form used for fields inside $k=\mathbb{F}_q$0, as surjectivity onto $k=\mathbb{F}_q$1-invariant integral $k=\mathbb{F}_q$2-adic classes after comparison with Betti cohomology (Tian, 2022, Kok, 2023, Cadoret et al., 2024, Milne, 8 Sep 2025).

1. Formulations and relation to the rational Tate conjecture

The rational Tate conjecture replaces $k=\mathbb{F}_q$3 by $k=\mathbb{F}_q$4. For a smooth projective variety $k=\mathbb{F}_q$5 over a finite field and codimension $k=\mathbb{F}_q$6, it asserts surjectivity of

$k=\mathbb{F}_q$7

The integral Tate conjecture is strictly stronger: integral surjectivity implies rational surjectivity, but the converse can fail. Several of the papers under discussion emphasize that the rational conjecture may remain compatible with the existence of integral obstructions, either torsion or non-torsion (Kameko, 2014, Antieau, 2015).

A standard formulation over an algebraically closed field $k=\mathbb{F}_q$8 equipped with a model $k=\mathbb{F}_q$9 over a finitely generated subfield $\ell\neq \operatorname{char}(k)$0 uses the map

$\ell\neq \operatorname{char}(k)$1

whose image lies in the colimit of Galois invariants

$\ell\neq \operatorname{char}(k)$2

In this language, the integral Tate conjecture asserts equality between the image of the cycle map and this colimit of invariant integral classes (Kok, 2023).

For codimension $\ell\neq \operatorname{char}(k)$3, the integral picture is comparatively rigid. On surfaces over finite fields, the Kummer exact sequence yields

$\ell\neq \operatorname{char}(k)$4

This identifies the obstruction to integral surjectivity for divisors with the $\ell\neq \operatorname{char}(k)$5-adic Tate module of the Brauer group, and it underlies many positive results in codimension $\ell\neq \operatorname{char}(k)$6 (Lodh, 2013).

A further variant, central in recent work on families, separates the rational and integral obstructions by introducing the spaces

$\ell\neq \operatorname{char}(k)$7

their algebraic parts $\ell\neq \operatorname{char}(k)$8, $\ell\neq \operatorname{char}(k)$9, and the “Tate classes” $i$0. The resulting obstruction groups

$i$1

measure precisely the failure of rational and integral Tate, respectively (Cadoret et al., 2024).

2. Positive results: divisors, modular surfaces, and K3 surfaces

For elliptic modular surfaces $i$2 over finite fields, Tate’s conjecture for divisors is established for most primes in a specified congruence class. For $i$3 and primes $i$4 satisfying the partial semisimplicity condition, which holds outside a set of density zero, one has surjectivity

$i$5

together with the zeta-function identity

$i$6

The same framework gives integral consequences for $i$7, including surjectivity of

$i$8

and, under finite height of the formal Brauer group, the stronger conclusion $i$9 (Lodh, 2013).

For K3 surfaces, the theorem proved in characteristic $\ell$0 concerns the rational Tate conjecture rather than the full integral form. If $\ell$1 is a K3 surface over a finitely generated field and $\ell$2, the cycle class map on divisors induces

$\ell$3

The characteristic-$\ell$4 case is obtained by combining 2-adic integral canonical models for Shimura varieties of abelian type, E. Lau’s classification of 2-divisible groups via displays, and the Kuga–Satake transfer of special endomorphisms to divisor classes on K3 surfaces. The result establishes the remaining characteristic-$\ell$5 case for the rational Tate conjecture on divisors, but it does not claim the $\ell$6 integral statement (Kim et al., 2015).

These codimension-$\ell$7 results illustrate a persistent asymmetry. Integral Tate for divisors is closely tied to Néron–Severi theory and the Brauer group, whereas in higher codimension the obstruction can survive even when rational Tate is expected or known. A recurring misconception is therefore that codimension-$\ell$8 behavior extrapolates to all codimensions; the counterexamples discussed below show that this extrapolation fails.

3. One-cycles, separable rational connectedness in codimension one, and geometric criteria

A major positive development for higher codimension is a geometric criterion for the integral Tate conjecture for one-cycles on smooth projective varieties that are separably rationally connected in codimension one. For a morphism $\ell$9 with image in the smooth locus and splitting

$\mathbb{Z}_\ell$0

one says that $\mathbb{Z}_\ell$1 is separably rationally connected if it contains a very free curve, i.e. all $\mathbb{Z}_\ell$2, and separably rationally connected in codimension one if it contains a free curve with exactly one vanishing $\mathbb{Z}_\ell$3 and all others positive (Tian, 2022).

This condition is tailored to the theory of one-cycles for two reasons. Structurally, it implies that the rational Chow group of $\mathbb{Z}_\ell$4-cycles is universally supported on a curve and yields a decomposition of the diagonal

$\mathbb{Z}_\ell$5

with $\mathbb{Z}_\ell$6 supported on $\mathbb{Z}_\ell$7, where $\mathbb{Z}_\ell$8 is $\mathbb{Z}_\ell$9-dimensional, and $d$0 supported on $d$1 for a divisor $d$2. Deformation-theoretically, it allows algebraic equivalence of one-cycles to be realized by families of nodal curves using moving, bending, and comb constructions of Kollár–Tian type (Tian, 2022).

For a smooth projective geometrically integral variety $d$3 of dimension $d$4, separably rationally connected in codimension one, the criterion assumes the following hypotheses on $d$5:

$d$6

$d$7

and either the surjectivity of the higher cycle class map

$d$8

or the coniveau condition

$d$9

Under (A) and (B), the geometric part of the integral Tate conjecture for one-cycles holds; under (C) or (D), the arithmetic part is surjective: $X/\mathbb{F}_q$0 The proof proceeds through the Hochschild–Serre exact sequence

$X/\mathbb{F}_q$1

and the identification $X/\mathbb{F}_q$2 via Kollár–Tian (Tian, 2022).

A central technical consequence is the equality of the coniveau and strong coniveau filtrations on degree-$X/\mathbb{F}_q$3 homology. If $X/\mathbb{F}_q$4 is smooth projective of dimension $X/\mathbb{F}_q$5 over an algebraically closed field and separably rationally connected in codimension one, then

$X/\mathbb{F}_q$6

By Poincaré duality this is an equality on $X/\mathbb{F}_q$7, and for threefolds it yields

$X/\mathbb{F}_q$8

This filtration control is the key input in the arithmetic part of the integral Tate criterion (Tian, 2022).

4. Arithmetic consequences: local–global principles and zero-cycles

The integral Tate conjecture for one-cycles enters arithmetic through Brauer–Manin theory for zero-cycles. For a smooth projective variety $X/\mathbb{F}_q$9 over a global field and a prime $$\mathrm{CH}_1(X)\otimes \mathbb{Z}_\ell \longrightarrow H^{2d-2}_{\mathrm{\acute{e}t}}(X,\mathbb{Z}_\ell(d-1)),$$0, Colliot-Thélène’s Conjecture E predicts exactness of

$$\mathrm{CH}_1(X)\otimes \mathbb{Z}_\ell \longrightarrow H^{2d-2}_{\mathrm{\acute{e}t}}(X,\mathbb{Z}_\ell(d-1)),$$1

so that the Brauer–Manin obstruction is the only obstruction to the local–global principle for zero-cycles. The local pairing is

$$\mathrm{CH}_1(X)\otimes \mathbb{Z}_\ell \longrightarrow H^{2d-2}_{\mathrm{\acute{e}t}}(X,\mathbb{Z}_\ell(d-1)),$$2

When the sum of local invariants vanishes for a family of local zero-cycles, Conjecture E predicts the existence of a global cycle with prescribed degree data (Tian, 2022).

For smooth projective geometrically rational surfaces over a global function field $$\mathrm{CH}_1(X)\otimes \mathbb{Z}_\ell \longrightarrow H^{2d-2}_{\mathrm{\acute{e}t}}(X,\mathbb{Z}_\ell(d-1)),$$3, the paper proves that Conjecture E holds. Consequently, Brauer–Manin is the only obstruction to the local–global principle for zero-cycles on all geometrically rational surfaces over such fields. A second theorem establishes that for del Pezzo surfaces of degree $$\mathrm{CH}_1(X)\otimes \mathbb{Z}_\ell \longrightarrow H^{2d-2}_{\mathrm{\acute{e}t}}(X,\mathbb{Z}_\ell(d-1)),$$4 over a global function field of odd characteristic, the Brauer–Manin obstruction is the only obstruction to the Hasse principle for rational points (Tian, 2022).

The mechanism is indirect but precise. If $$\mathrm{CH}_1(X)\otimes \mathbb{Z}_\ell \longrightarrow H^{2d-2}_{\mathrm{\acute{e}t}}(X,\mathbb{Z}_\ell(d-1)),$$5 is a projective flat family of surfaces over a smooth projective curve $$\mathrm{CH}_1(X)\otimes \mathbb{Z}_\ell \longrightarrow H^{2d-2}_{\mathrm{\acute{e}t}}(X,\mathbb{Z}_\ell(d-1)),$$6, with $$\mathrm{CH}_1(X)\otimes \mathbb{Z}_\ell \longrightarrow H^{2d-2}_{\mathrm{\acute{e}t}}(X,\mathbb{Z}_\ell(d-1)),$$7 smooth and geometric generic fiber a smooth rational surface, then for every $$\mathrm{CH}_1(X)\otimes \mathbb{Z}_\ell \longrightarrow H^{2d-2}_{\mathrm{\acute{e}t}}(X,\mathbb{Z}_\ell(d-1)),$$8 one has

$$\mathrm{CH}_1(X)\otimes \mathbb{Z}_\ell \longrightarrow H^{2d-2}_{\mathrm{\acute{e}t}}(X,\mathbb{Z}_\ell(d-1)),$$9

Thus the integral Tate conjecture for one-cycles holds for the total space $$H^{2d-2}_{\mathrm{\acute{e}t}}(\overline{X},\mathbb{Z}_\ell(d-1))^{\mathrm{Gal}(\overline{\mathbb{F}}_q/\mathbb{F}_q)}.$$0. This surjectivity, combined with the general implication from integral Tate to the Colliot-Thélène local–global sequence after Saito and Colliot-Thélène–Kahn, yields Conjecture E for the generic fiber (Tian, 2022).

This arithmetic application clarifies a point that can otherwise be obscured by the terminology: the integral Tate conjecture is not only a statement about cycle classes on varieties over finite fields. In the one-cycle setting, it becomes a structural input for zero-cycles over global function fields, via Hochschild–Serre, Leray, unramified cohomology, and Brauer–Manin duality.

5. Counterexamples and failure mechanisms

The failure of the integral Tate conjecture is by now systematic. One major source of counterexamples arises from approximations to classifying spaces. For every prime $$H^{2d-2}_{\mathrm{\acute{e}t}}(\overline{X},\mathbb{Z}_\ell(d-1))^{\mathrm{Gal}(\overline{\mathbb{F}}_q/\mathbb{F}_q)}.$$1, there exist a finite field $$H^{2d-2}_{\mathrm{\acute{e}t}}(\overline{X},\mathbb{Z}_\ell(d-1))^{\mathrm{Gal}(\overline{\mathbb{F}}_q/\mathbb{F}_q)}.$$2 with $$H^{2d-2}_{\mathrm{\acute{e}t}}(\overline{X},\mathbb{Z}_\ell(d-1))^{\mathrm{Gal}(\overline{\mathbb{F}}_q/\mathbb{F}_q)}.$$3, a smooth projective variety $$H^{2d-2}_{\mathrm{\acute{e}t}}(\overline{X},\mathbb{Z}_\ell(d-1))^{\mathrm{Gal}(\overline{\mathbb{F}}_q/\mathbb{F}_q)}.$$4, and an open subgroup $$H^{2d-2}_{\mathrm{\acute{e}t}}(\overline{X},\mathbb{Z}_\ell(d-1))^{\mathrm{Gal}(\overline{\mathbb{F}}_q/\mathbb{F}_q)}.$$5 such that

$$H^{2d-2}_{\mathrm{\acute{e}t}}(\overline{X},\mathbb{Z}_\ell(d-1))^{\mathrm{Gal}(\overline{\mathbb{F}}_q/\mathbb{F}_q)}.$$6

is not surjective. Equivalently, there exists a non-torsion $$H^{2d-2}_{\mathrm{\acute{e}t}}(\overline{X},\mathbb{Z}_\ell(d-1))^{\mathrm{Gal}(\overline{\mathbb{F}}_q/\mathbb{F}_q)}.$$7-invariant class in degree $$H^{2d-2}_{\mathrm{\acute{e}t}}(\overline{X},\mathbb{Z}_\ell(d-1))^{\mathrm{Gal}(\overline{\mathbb{F}}_q/\mathbb{F}_q)}.$$8 that is not algebraic in codimension $$H^{2d-2}_{\mathrm{\acute{e}t}}(\overline{X},\mathbb{Z}_\ell(d-1))^{\mathrm{Gal}(\overline{\mathbb{F}}_q/\mathbb{F}_q)}.$$9. This extends earlier work for $k=\mathbb{F}_q$00 to all primes by using

$k=\mathbb{F}_q$01

and the existence of a class $k=\mathbb{F}_q$02 such that

$k=\mathbb{F}_q$03

after restriction to a suitable elementary abelian $k=\mathbb{F}_q$04-subgroup (Kameko, 2014).

A distinct representation-theoretic mechanism produces counterexamples for groups of type $k=\mathbb{F}_q$05. For

$k=\mathbb{F}_q$06

one has

$k=\mathbb{F}_q$07

but the image of $k=\mathbb{F}_q$08 is $k=\mathbb{F}_q$09 in the first case and $k=\mathbb{F}_q$10 in the second. After Totaro-style approximation by smooth projective varieties over finite fields, the pulled-back class $k=\mathbb{F}_q$11 is Frobenius-invariant and non-torsion, $k=\mathbb{F}_q$12 is algebraic, but $k=\mathbb{F}_q$13 is not. Here the mod-$k=\mathbb{F}_q$14 reduction is invisible to the standard Milnor operations $k=\mathbb{F}_q$15: all $k=\mathbb{F}_q$16 vanish, and non-algebraicity is instead detected by a higher differential in a $k=\mathbb{F}_q$17-theoretic spectral sequence (Antieau, 2015).

Another failure mechanism is geometric rather than representation-theoretic. For an Enriques surface $k=\mathbb{F}_q$18 over an algebraically closed field of characteristic $k=\mathbb{F}_q$19, and a Lefschetz pencil of odd-dimensional hypersurfaces of degree $k=\mathbb{F}_q$20 with geometric generic fiber $k=\mathbb{F}_q$21 of dimension $k=\mathbb{F}_q$22, the integral Tate conjecture fails for $k=\mathbb{F}_q$23 in codimension $k=\mathbb{F}_q$24 at $k=\mathbb{F}_q$25. Concretely, there exists a 2-torsion class

$k=\mathbb{F}_q$26

which is Galois-invariant after choosing a suitable model over a finitely generated subfield, but which is not in the image of the integral cycle class map. Equivalently,

$k=\mathbb{F}_q$27

The proof combines Colliot-Thélène’s specialization method for Lefschetz pencils, a class $k=\mathbb{F}_q$28 with nonzero residue at a nodal degeneration, and the special class $k=\mathbb{F}_q$29 on the Enriques surface that does not lift to $k=\mathbb{F}_q$30. Schreieder’s refined unramified cohomology then identifies the resulting non-liftable class with a nontrivial element in the cokernel of the integral cycle map (Kok, 2023).

The Enriques construction generalizes to products with multiple Enriques surfaces and to Fano varieties of lines on odd-dimensional smooth cubic hypersurfaces. In the latter case, the cylinder isomorphism

$k=\mathbb{F}_q$31

transfers the non-algebraic torsion from $k=\mathbb{F}_q$32 to $k=\mathbb{F}_q$33 (Kok, 2023).

Recent work extends the failure phenomenon to abelian varieties over finitely generated subfields of $k=\mathbb{F}_q$34. If $k=\mathbb{F}_q$35 is an abelian variety over such a field and the integral Hodge conjecture fails on $k=\mathbb{F}_q$36, then after a finite extension $k=\mathbb{F}_q$37 there exists a prime $k=\mathbb{F}_q$38 such that the induced class in

$k=\mathbb{F}_q$39

does not lie in the image of the integral cycle map. The argument uses Deligne’s theorem that Hodge classes on abelian varieties are absolute Hodge classes, together with Betti–étale comparison and the torsion-freeness of cohomology for abelian varieties (Milne, 8 Sep 2025).

These counterexamples show that the integral Tate conjecture can fail in two essentially different ways: through torsion classes not generated by cycles, and through non-torsion integral Tate classes whose rational multiples are algebraic. The integral obstruction therefore cannot be reduced to torsion bookkeeping alone.

6. Families, obstruction groups, and current directions

Recent work studies not only whether integral Tate fails, but how the failure varies in families. Let $k=\mathbb{F}_q$40 be a smooth projective morphism over a smooth geometrically connected curve $k=\mathbb{F}_q$41, with $k=\mathbb{F}_q$42 infinite and finitely generated over its prime subfield. For each fiber $k=\mathbb{F}_q$43, the obstruction to integral Tate is measured by the torsion subgroup of

$k=\mathbb{F}_q$44

where $k=\mathbb{F}_q$45 denotes the integral Tate classes and $k=\mathbb{F}_q$46 the image of the cycle class map. The size

$k=\mathbb{F}_q$47

quantifies the integral obstruction on the fiber (Cadoret et al., 2024).

Under variational realization hypotheses—formulated as $k=\mathbb{F}_q$48 in characteristic $k=\mathbb{F}_q$49, and $k=\mathbb{F}_q$50, $k=\mathbb{F}_q$51, and $k=\mathbb{F}_q$52 in mixed or positive characteristic—the obstruction admits uniform bounds in one-dimensional families. In characteristic $k=\mathbb{F}_q$53, for every $k=\mathbb{F}_q$54,

$k=\mathbb{F}_q$55

and in fact $k=\mathbb{F}_q$56 for $k=\mathbb{F}_q$57. In positive characteristic, assuming the local system $k=\mathbb{F}_q$58 is GLU and one of the stated realization hypotheses holds, one obtains a uniform bound over $k=\mathbb{F}_q$59-rational points: $k=\mathbb{F}_q$60 The proof isolates a fiber-independent lattice $k=\mathbb{F}_q$61, generated by global algebraic cycles via the Leray edge map, and bounds the obstruction by the finite torsion group

$k=\mathbb{F}_q$62

This makes the failure of integral Tate uniform on the monodromy-generic locus (Cadoret et al., 2024).

In codimension $k=\mathbb{F}_q$63, the obstruction is identified with unramified cohomology. Specifically,

$k=\mathbb{F}_q$64

and by the theorem of Colliot-Thélène–Kahn,

$k=\mathbb{F}_q$65

Hence uniform bounds for the integral Tate obstruction yield uniform bounds for the non-divisible part of degree-$k=\mathbb{F}_q$66 unramified cohomology in families (Cadoret et al., 2024).

Several open directions are explicit in the current literature. One concerns the verification of the hypotheses (A)–(D) in the separably rationally connected in codimension one setting, where Lawson homology formulations over $k=\mathbb{F}_q$67 and expected positive-characteristic analogues would imply the required filtration and higher-cycle surjectivity statements (Tian, 2022). Another is the search for explicit arithmetic examples of abelian varieties over number fields that fail the integral Tate conjecture together with an explicit non-algebraic integral Tate class (Milne, 8 Sep 2025). A third concerns the extent to which integral Tate modulo torsion, full integral Tate, and the Brauer–Manin local–global principles can be unified through a common theory of variation in families (Tian, 2022, Cadoret et al., 2024).

The resulting picture is sharply bifurcated. On one side stand substantial positive results for divisors, modular surfaces, K3 surfaces in the rational form, one-cycles on separably rationally connected varieties in codimension one, and rational surface fibrations. On the other stand broad counterexample mechanisms: classifying-space approximations, representation-theoretic failures for type $k=\mathbb{F}_q$68, refined unramified obstructions on products with Enriques surfaces, and absolute-Hodge-theoretic failures for abelian varieties. The integral Tate conjecture is therefore not a straightforward strengthening of rational Tate, but a substantially finer and more delicate statement about the integral lattice of Tate classes.