Does the Tate conjecture imply semisimplicity of Galois representations in positive characteristic?

The strong form of the Tate conjecture over finitely generated fields combines two assertions: (T), that every Tate class in the $\ell$ -adic étale cohomology of a smooth projective variety is a $\mathbb{Q}_\ell$ -linear combination of algebraic cycle classes, and (S), that the Galois representations on such cohomology groups are semisimple. Moonen proved that (T) implies (S) in characteristic $0$ , and that in characteristic $p$ the same implication holds under the additional hypothesis that homological and numerical equivalence of algebraic cycles coincide over finite fields. The present problem asks whether, in characteristic $p$ , the implication (T) $\Rightarrow$ (S) holds without that extra hypothesis. The literature covering Moonen's paper, André's work on the observability of arithmetic monodromy groups, O'Sullivan's super Tannakian hulls, and 2024–2026 work on standard conjecture D shows that the question remains open; the identified obstruction is that no unconditional reductive group of geometric origin is known in positive characteristic.

Background

Let $K$ be a finitely generated field (a field finitely generated over its prime field, such as a number field or the function field of a variety over a finite field), fix a prime $\ell$ and an algebraic closure $\bar{K}$ , and write $\Gamma_K=\mathrm{Gal}(\bar{K}/K)$ . For a smooth projective $K$ -variety $X$ , the $\ell$ -adic étale cohomology $H^i(X_{\bar{K}},\mathbb{Q}_\ell(n))$ carries a continuous action of $\Gamma_K$ . A class in $H^{2n}(X_{\bar{K}},\mathbb{Q}_\ell(n))$ whose stabilizer in $\Gamma_K$ is an open subgroup is called a Tate class (equivalently, a class that becomes Galois-invariant over a finite extension of $K$ ); let $T^n(X)$ denote the space of Tate classes. The strong Tate conjecture over $K$ splits into two parts: (T) the cycle class map $\mathrm{CH}^n(X_{\bar{K}})\otimes\mathbb{Q}_\ell\to T^n(X)$ is surjective for all $n$ , and (S) the $\Gamma_K$ -representations $H^i(X_{\bar{K}},\mathbb{Q}_\ell)$ are semisimple (i.e. every Galois-stable subspace admits a Galois-stable complement); part (S) is also attributed to Grothendieck and Serre. Both are open in general; see Totaro's survey for the state of the art around 2017.

Moonen proved (Theorem 1) that in characteristic $0$ the implication (T) $\Rightarrow$ (S) holds: assuming (T) for all smooth projective varieties over finitely generated fields of characteristic $0$ , all such Galois representations are semisimple. The proof reduces to base field $\mathbb{Q}$ by a specialization argument of Serre, uses Faltings's $p$ -adic Hodge theory (the fact that $\ell$ -adic representations coming from geometry are Hodge–Tate, i.e. that after restriction to a decomposition group at $\ell$ they admit a filtration whose graded pieces are cyclotomic twists) to show that characters of the Zariski closure of the Galois image are, up to finite index, powers of the cyclotomic character (the character describing the Galois action on $\ell$ -power roots of unity), and then concludes via the semisimple Tannakian category of motives over $\mathbb{Q}$ (a Tannakian category is a tensor category equivalent to the category of finite-dimensional representations of an affine group scheme; this one is built from André's motivated cycles, an unconditional construction of motives in characteristic $0$ ), whose motivic Galois group is pro-reductive (an inverse limit of reductive algebraic groups). In positive characteristic neither ingredient is unconditional: the Hodge–Tate input has no direct analogue, and no unconditional semisimple motivic category is available. Moonen states that extending the argument to characteristic $p$ "seems to require further assumptions, e.g., that homological and numerical equivalence agree." Here homological equivalence means vanishing of the $\ell$ -adic cycle class, while numerical equivalence means vanishing of all intersection numbers against complementary-dimensional cycles; their coincidence for smooth projective varieties over finite fields is Grothendieck's standard conjecture D, itself open in general. Under (T) over finite fields of characteristic $p$ plus this coincidence, Moonen's Theorem 2 (based on remarks of D'Addezio) deduces (S) in characteristic $p$ : over a finite field it follows from Milne's analysis of zeta values (semisimplicity of Frobenius, to which (S) is equivalent over finite fields since $\Gamma_K\cong\hat{\mathbb{Z}}$ is topologically generated by Frobenius), and the passage to finitely generated fields uses Fu's theorem on semisimplicity of pure sheaves.

André subsequently found the unconditional fact behind Moonen's argument: in any characteristic, the Zariski closure of the Galois image in $\mathrm{GL}(H^*(X_{\bar{K}},\mathbb{Q}_\ell))$ is an observable subgroup (in the sense of Bialynicki-Birula, Hochschild and Mostow: a closed subgroup $H$ of an algebraic group $G$ such that every finite-dimensional representation of $H$ occurs inside a representation of $G$ ), equivalently it is the largest closed subgroup fixing the Tate classes on all powers $X^n$ . In characteristic $0$ this sharpens Moonen's theorem: (T) for all powers of $X$ alone implies (S) for $X$ . In characteristic $p$ , however, André notes that the natural candidate group, constructed from O'Sullivan's theory of super Tannakian hulls (a universal procedure embedding a tensor category of geometric origin into the representation category of an affine group scheme, analogous to the passage from an integral domain to its field of fractions) and whose invariants are the "fractionally algebraic" classes (Tate classes that become algebraic after tensoring with a suitable algebraic Tate class on an auxiliary smooth projective variety), is not known to be reductive, so the semisimplicity conclusion cannot be derived. This is precisely the gap the present problem targets.

Problem Statement

Let $p$ and $\ell$ be prime numbers with $\ell\neq p$ . For a smooth projective variety $X$ over a finitely generated field $K$ of characteristic $p$ , let $T^n(X)\subset H^{2n}(X_{\bar{K}},\mathbb{Q}_\ell(n))$ be the space of Tate classes (classes whose stabilizer in $\Gamma_K=\mathrm{Gal}(\bar{K}/K)$ is open), and consider the two conjectures: (T) for every such $X$ and every $n\geq 0$ , the cycle class map $\mathrm{CH}^n(X_{\bar{K}})\otimes\mathbb{Q}_\ell\to T^n(X)$ is surjective; (S) for every such $X$ , the $\Gamma_K$ -representation $H^i(X_{\bar{K}},\mathbb{Q}_\ell)$ is semisimple for all $i$ . Question: does (T) imply (S) in positive characteristic, without any additional hypothesis on cycle equivalence relations — in particular without assuming that homological and numerical equivalence coincide over finite fields? A known reduction (Moonen's Theorem 2, via Milne and Fu) shows that a positive answer would follow from the finite-field case: does (T) over finite fields of characteristic $p$ imply that Frobenius acts semisimply on $H^i(X_{\bar{\mathbb{F}}_q},\mathbb{Q}_\ell)$ for every smooth projective variety over every finite field of characteristic $p$ , and hence (S) for all finitely generated fields of characteristic $p$ (via Fu's semisimplicity theorem)? The sharpened variant, open in the same way, asks whether (T) for all powers $X^m$ of a single variety $X$ implies (S) for $X$ , as André established in characteristic $0$ . Known partial results that do not answer the question: (T) $\Rightarrow$ (S) holds in characteristic $0$ (Moonen, sharpened by André), and in characteristic $p$ under the additional assumption of hom=num over finite fields (Moonen, Theorem 2, via Milne and Fu).

References

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Yves André, "On the observability of Galois representations and the Tate conjecture", Journal of Number Theory 270 (2025), 6–16. DOI: 10.1016/j.jnt.2023.07.010, https://doi.org/10.1016/j.jnt.2023.07.010 (preprint arXiv:2303.06549, https://arxiv.org/abs/2303.06549)
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