Tate conjecture for abelian varieties over finite fields: algebraicity of Tate classes in codimension at least 2, and the rationality of the field of motivated cycles

The Tate conjecture predicts that on a smooth projective variety over a finite field, every $\ell$-adic cohomology class fixed by the Galois group (a Tate class) is a $\mathbb{Q}_\ell$-linear combination of classes of algebraic cycles. For abelian varieties the conjecture is proved only for divisors (codimension 1, Tate 1966); in codimension at least 2 it remains one of the central open problems of arithmetic geometry. In 2006 Yves André proved unconditionally that every Tate class on an abelian variety in characteristic $p$ is motivated, i.e. generated from algebraic classes by the standard Lefschetz and Künneth operations, thereby establishing a weaker version of the conjecture that describes all Tate classes in terms of algebraic ones. This record targets the two linked questions left open by that work: (a) the Tate conjecture proper, namely whether every Tate class on an abelian variety over a finite field (equivalently over $\overline{\mathbb{F}}_p$) is a $\mathbb{Q}_\ell$-combination of algebraic cycle classes in codimension $r\ge 2$; and (b) André's rationality question, namely whether the endomorphism field $Q_{(\ell)}$ of the unit object in his category of motivated motives over $\overline{\mathbb{F}}_p$ equals $\mathbb{Q}$ (and in particular is formally real), equivalently Milne's rationality conjecture on the existence of a good theory of rational Tate classes. Both parts remain open; unconditional results are confined to divisors, ordinary abelian varieties, and special families.

Background

An abelian variety over a field $k$ is a projective algebraic group variety, a higher-dimensional analogue of an elliptic curve. Let $A$ be defined over a finite field $\mathbb{F}_q$ ($q$ a power of the prime $p$), let $\bar{k}$ be an algebraic closure, and let $\ell\ne p$ be a prime. The $\ell$-adic étale cohomology group $H^{2r}(A_{\bar{k}},\mathbb{Q}_\ell(r))$ (the Tate twist $(r)$ normalizes the weight of Frobenius eigenvalues) carries a continuous action of $\mathrm{Gal}(\bar{k}/\mathbb{F}_q)$, topologically generated by the Frobenius automorphism. A Tate class is a class fixed by an open subgroup of the Galois group, equivalently by the Galois group of a model of $A$ over some finite extension of $\mathbb{F}_q$; the $\mathbb{Q}_\ell$-space of Tate classes is denoted $T^r_\ell(A)$. Algebraic cycles of codimension $r$ (finite $\mathbb{Q}$-linear combinations of subvarieties of codimension $r$ on $A_{\bar{k}}$) have cycle classes in this cohomology group, and the Tate conjecture asserts the converse: the cycle class map $\mathrm{cl}:CH^r(A)\otimes\mathbb{Q}_\ell\to T^r_\ell(A)$ from cycles modulo rational equivalence to Tate classes is surjective. For divisors ($r=1$) this is a theorem of Tate (1966), whose method identifies Galois-invariant homomorphisms between Tate modules with homomorphisms of abelian varieties; Zarhin and Faltings extended the framework to finitely generated fields. For $r\ge 2$ the conjecture is open for abelian varieties, and a theorem of Milne reduces it to the single case $r=2$.

Motivated cycles (André 1996) provide an unconditional substitute for the still-conjectural category of pure motives: starting from algebraic classes on powers of $A$ and formally adjoining the Lefschetz operators (cup product with the cohomology class of an ample line bundle), the Künneth projectors (the graded components of the diagonal acting on the cohomology of each factor), and the inverses of the Lefschetz involutions, one obtains a Tannakian $\mathbb{Q}$-linear category (a rigid abelian tensor category equivalent to the category of representations of an affine group scheme) of motivated classes containing all algebraic classes. André (2006) proved that over $\overline{\mathbb{F}}_p$ every Tate class on an abelian variety is motivated, a weakened form of the Tate conjecture describing Tate classes in terms of algebraic cycles. However, in characteristic $p$ the endomorphism ring of the unit object (the identity object for the tensor product) of this category is only known to be a countable subfield $Q_{(\ell)}$ of $\mathbb{Q}_\ell$, possibly strictly larger than $\mathbb{Q}$; consequently the motivated description neither produces a $\mathbb{Q}$-structure on the space of Tate classes nor produces algebraic cycles. Milne recast the question $Q_{(\ell)}=\mathbb{Q}$ as the rationality conjecture, the existence of a good theory of rational Tate classes (a system of $\mathbb{Q}$-structures on the $T^r_\ell$ compatible with the four standard operations), proved that such a theory is unique if it exists and exists unconditionally for ordinary abelian varieties, and showed that the Hodge conjecture for CM abelian varieties (abelian varieties with commutative endomorphism ring of maximal rank over $\mathbb{C}$) would imply the Tate conjecture for abelian varieties over finite fields. Related frameworks include Grothendieck's standard conjectures (Milne proved that the Lefschetz standard conjecture over finite fields implies the full Tate conjecture for abelian varieties over $\overline{\mathbb{F}}_p$), the theorem of Clozel and of Clozel–Deligne that numerical and homological equivalence coincide for abelian varieties over finite fields, and Milne's program on abelian motives and Shimura varieties in nonzero characteristic, whose November 2025 incarnation states the corresponding rationality conjectures as open. The rational statement studied here must not be confused with the integral Tate conjecture, which fails for some abelian varieties.

Problem Statement

Let $p$ be a prime, $k=\overline{\mathbb{F}}_p$, and $\ell\ne p$ a prime.

(a) Tate conjecture for abelian varieties. For every abelian variety $A$ over $k$ (equivalently, for every abelian variety over a finite field, passing to the Tate classes of a model over a finite subfield), every codimension $r\ge 1$, and every $\ell\ne p$, is the cycle class map $\mathrm{cl}:CH^r(A)\otimes_{\mathbb{Q}}\mathbb{Q}_\ell\to T^r_\ell(A)$ surjective, where $T^r_\ell(A)$ is the $\mathbb{Q}_\ell$-space of Tate classes in $H^{2r}(A,\mathbb{Q}_\ell(r))$? The case $r=1$ is Tate's theorem; the open content is $r\ge 2$, where by Milne's reduction it suffices to decide $r=2$. Equivalently: is every Tate class on $A$ a $\mathbb{Q}_\ell$-linear combination of classes of algebraic cycles, with no exceptional (non-algebraic) Tate classes in any codimension?

(b) Rationality of the motivated description. In André's Tannakian category of motivated motives over $k$ with $\ell$-adic realization, let $Q_{(\ell)}=\mathrm{End}(\mathbf{1})$, the endomorphism field of the unit object (the tensor identity of the category), a countable subfield of $\mathbb{Q}_\ell$. Is $Q_{(\ell)}=\mathbb{Q}$? Is $Q_{(\ell)}$ formally real, i.e. is $-1$ not a sum of squares in $Q_{(\ell)}$? Equivalently (Milne): does there exist a good theory of rational Tate classes for abelian varieties over $k$, that is, $\mathbb{Q}$-structures on the $T^r_\ell$ compatible with cup product, pullback, pushforward, and the Künneth components of the diagonal (axioms R1–R4)?

The two parts are coupled: a positive answer to (b) supplies the missing $\mathbb{Q}$-structure on Tate classes, and its formal-reality strengthening would imply the Hodge standard conjecture for abelian varieties over finite fields; together with the Hodge conjecture for CM abelian varieties it would imply (a). A resolution is a proof or disproof of (a) or of (b). A disproof of (a) consists of a single abelian variety $A$ over a finite field, one codimension $r\ge 2$, and one prime $\ell\ne p$ for which the cycle class map $\mathrm{cl}:CH^r(A)\otimes_{\mathbb{Q}}\mathbb{Q}_\ell\to T^r_\ell(A)$ is not surjective; a proof of (a) must establish surjectivity for every abelian variety over a finite field, every codimension $r\ge 2$, and every $\ell\ne p$ (by Milne's reduction it suffices to decide the case $r=2$, but $\ell$-independence of the Tate conjecture is not known, so all $\ell\ne p$ must still be covered). In (b) the mandatory claim is $Q_{(\ell)}=\mathbb{Q}$ (equivalently, the existence of a good theory of rational Tate classes); a proof that $Q_{(\ell)}=\mathbb{Q}$ automatically yields formal reality, whereas a proof of formal reality alone does not resolve (b).

References

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