Vanishing of the Galois-invariant birational cohomology of projective space over finite fields

For a smooth projective variety $X$ over a finite field $\mathbb{F}$, the group $H^{i}(F_{0}\bar{X},\mathbb{Q}_{\ell}(j))^{G_{\mathbb{F}}}$ — the Galois-invariant part of the direct limit of the $\ell$-adic étale cohomology of all dense open subsets of $X_{\bar{\mathbb{F}}}$ — is a birational invariant that depends only on the function field. Balkan and Schreieder proved that the vanishing of $H^{2i}(F_{0}\mathbb{P}^{n}_{\bar{\mathbb{F}}},\mathbb{Q}_{\ell}(i+1))^{G_{\mathbb{F}}}$ for all $i,n\geq 2$ is equivalent to the conjunction of three central open problems of arithmetic geometry: the Tate conjecture, the Beilinson injectivity conjecture, and the Grothendieck–Serre semi-simplicity conjecture for all smooth projective varieties over $\mathbb{F}$. The companion odd-degree vanishing $H^{2i-1}(F_{0}\mathbb{P}^{n}_{\bar{\mathbb{F}}},\mathbb{Q}_{\ell}(i))^{G_{\mathbb{F}}}=0$ is equivalent to the Tate and semi-simplicity conjectures alone. The twist-$i$ analogue $H^{2i}(F_{0}\mathbb{P}^{n}_{\bar{\mathbb{F}}},\mathbb{Q}_{\ell}(i))^{G_{\mathbb{F}}}=0$ is proven unconditionally, but the twist-$(i+1)$ and odd-degree vanishings remain open; the first open case $H^{4}(F_{0}\mathbb{P}^{4}_{\bar{\mathbb{F}}},\mathbb{Q}_{\ell}(3))^{G_{\mathbb{F}}}=0$ is equivalent to the Tate conjecture for divisors on surfaces and the Beilinson conjecture for threefolds over $\mathbb{F}$. The problem is to prove or refute these vanishing statements.

Background

Throughout, $\mathbb{F}$ denotes a finite field of characteristic $p$, $\bar{\mathbb{F}}$ a fixed algebraic closure, and $G_{\mathbb{F}}=\mathrm{Gal}(\bar{\mathbb{F}}/\mathbb{F})$ the absolute Galois group, a procyclic group topologically generated by the Frobenius automorphism. For a smooth projective variety $X$ over $\mathbb{F}$ and a prime $\ell\neq p$, the $\ell$-adic étale cohomology groups $H^{i}(X_{\bar{\mathbb{F}}},\mathbb{Q}_{\ell}(j))$ are finite-dimensional $\mathbb{Q}_{\ell}$-vector spaces, where $\mathbb{Q}_{\ell}(j)$ is the $j$-th Tate twist (the $j$-th tensor power of the one-dimensional representation given by the $\ell$-adic cyclotomic character, the $\ell$-adic analogue of multiplying coefficients by $(2\pi i)^{j}$); they carry a continuous linear action of $G_{\mathbb{F}}$. The Chow group $\mathrm{CH}^{i}(X)$ is the group of algebraic subvarieties of $X$ of codimension $i$ modulo rational equivalence, and there is a cycle class map $\mathrm{cl}\colon\mathrm{CH}^{i}(X)_{\mathbb{Q}_{\ell}}\to H^{2i}(X_{\bar{\mathbb{F}}},\mathbb{Q}_{\ell}(i))^{G_{\mathbb{F}}}$, whose image necessarily lands in the Galois-invariant part.

Three of the central conjectures of arithmetic geometry govern this map. The Tate conjecture (Tate 1966) asserts that $\mathrm{cl}$ is surjective for every smooth projective $X$ over $\mathbb{F}$; it is known essentially only for codimension-one cycles on abelian varieties, K3-type surfaces, and a few related classes (Totaro 2017). The Beilinson conjecture asserts that $\mathrm{cl}$ is injective; it is known for $i\in\{0,1,\dim X\}$, with important special cases due to Soulé (1984). The Grothendieck–Serre semi-simplicity conjecture (Jannsen 1990, Conjecture 12.5) predicts that the $G_{\mathbb{F}}$-action on each $H^{i}(X_{\bar{\mathbb{F}}},\mathbb{Q}_{\ell}(j))$ is semi-simple, i.e. that Frobenius acts diagonalizably over $\bar{\mathbb{Q}}_{\ell}$. All three are open in general; together they underlie the hoped-for Tannakian framework of motives over finite fields (Jannsen 1990).

The paper of Balkan and Schreieder studies a birational invariant through which these conjectures can be expressed. For an irreducible variety $X$ over $\mathbb{F}$, let

$$H^{i}(F_{0}\bar{X},\mathbb{Q}_{\ell}(j)):=\varinjlim_{\emptyset\neq\bar{U}\subseteq\bar{X}}H^{i}(\bar{U},\mathbb{Q}_{\ell}(j)),$$

the direct limit of the étale cohomology of all nonempty (equivalently, dense) open subsets of $\bar{X}=X_{\bar{\mathbb{F}}}$ — a form of classical unramified cohomology. This group depends only on the function field $\bar{\mathbb{F}}(X)$, so it is a birational invariant of $X$, and it carries an induced $G_{\mathbb{F}}$-action.

For $X=\mathbb{P}^{n}$ the authors prove unconditionally that $H^{2i}(F_{0}\mathbb{P}^{n}_{\bar{\mathbb{F}}},\mathbb{Q}_{\ell}(i))^{G_{\mathbb{F}}}=0$ for all $i,n\geq 2$ (Theorem 6.1). Their main theorem (Theorem 1.1) shows that the neighboring twist-$(i+1)$ statement is exactly as hard as the three classical conjectures: $H^{2i}(F_{0}\mathbb{P}^{n}_{\bar{\mathbb{F}}},\mathbb{Q}_{\ell}(i+1))^{G_{\mathbb{F}}}=0$ for all $i,n\geq 2$ holds if and only if the Tate, Beilinson, and semi-simplicity conjectures hold for all smooth projective varieties over $\mathbb{F}$. A companion result (Theorem 1.2) shows that the odd-degree vanishing $H^{2i-1}(F_{0}\mathbb{P}^{n}_{\bar{\mathbb{F}}},\mathbb{Q}_{\ell}(i))^{G_{\mathbb{F}}}=0$ for all $i,n\geq 2$ is equivalent to the Tate and semi-simplicity conjectures alone. The equivalences are effective (Theorem 7.1, Remark 7.2): for instance, $H^{4}(F_{0}\mathbb{P}^{4}_{\bar{\mathbb{F}}},\mathbb{Q}_{\ell}(3))^{G_{\mathbb{F}}}=0$ is equivalent to the Tate conjecture for divisors on surfaces together with the Beilinson conjecture for threefolds over $\mathbb{F}$, and $H^{3}(F_{0}\mathbb{P}^{3}_{\bar{\mathbb{F}}},\mathbb{Q}_{\ell}(2))^{G_{\mathbb{F}}}=0$ is equivalent to the Tate conjecture for surfaces over $\mathbb{F}$.

The authors also conjecture a complete description (Conjecture 1.6): $H^{i}(F_{0}\bar{X},\mathbb{Q}_{\ell}(j))^{G_{\mathbb{F}}}$ should be isomorphic to $K^{M}_{i}(\mathbb{F}(X))\otimes_{\mathbb{Z}}\mathbb{Q}_{\ell}$ when $i=j$ and should vanish otherwise, where $K^{M}_{i}$ denotes Milnor K-theory, the abelian group generated by symbols $\{a_{1},\dots,a_{i}\}$ with $a_{k}\in\mathbb{F}(X)^{\times}$ modulo multilinearity and the Steinberg relation $\{a,1-a\}=0$. This is related to the Beilinson–Parshin conjecture that the motivic cohomology $H^{p}_{M}(\mathrm{Spec}\,\bar{\mathbb{F}}(X),\mathbb{Q}(q))$ of a function field vanishes for $p\neq q$, and to the Tate conjecture for higher Chow groups (higher Chow groups $\mathrm{CH}^{q}(X,2q-p)$ compute motivic cohomology and generalize classical Chow groups). The authors isolate a further missing ingredient, a hard Lefschetz isomorphism for Chow groups — the statement that intersection with powers of a hyperplane class induces $\mathrm{CH}^{i}(X)_{\mathbb{Q}_{\ell}}\cong\mathrm{CH}^{\dim X-i}(X)_{\mathbb{Q}_{\ell}}$ — which is known only for abelian varieties (Soulé 1984) and wide open in general. Establishing the twist-$(i+1)$ vanishing for the function field of projective space would resolve the three classical conjectures, and the authors state this explicitly as an open research direction.

Problem Statement

The problem, as isolated by Balkan–Schreieder's Theorems 1.1 and 1.2, is to prove or refute the vanishing of the Galois-invariant birational cohomology of projective space over finite fields.

Let $\mathbb{F}$ be a finite field of characteristic $p$, $\ell\neq p$ a prime, $G_{\mathbb{F}}=\mathrm{Gal}(\bar{\mathbb{F}}/\mathbb{F})$ the absolute Galois group, and

$$H^{i}(F_{0}\mathbb{P}^{n}_{\bar{\mathbb{F}}},\mathbb{Q}_{\ell}(j)):=\varinjlim_{\emptyset\neq\bar{U}\subseteq\mathbb{P}^{n}_{\bar{\mathbb{F}}}}H^{i}(\bar{U},\mathbb{Q}_{\ell}(j))$$

the direct limit over the dense open subsets of $\mathbb{P}^{n}_{\bar{\mathbb{F}}}$, with $(-)^{G_{\mathbb{F}}}$ denoting Galois invariants. Determine whether

$$H^{2i}\bigl(F_{0}\mathbb{P}^{n}_{\bar{\mathbb{F}}},\mathbb{Q}_{\ell}(i+1)\bigr)^{G_{\mathbb{F}}}=0\quad\text{for all integers }i,n\geq 2,$$

and whether

$$H^{2i-1}\bigl(F_{0}\mathbb{P}^{n}_{\bar{\mathbb{F}}},\mathbb{Q}_{\ell}(i)\bigr)^{G_{\mathbb{F}}}=0\quad\text{for all integers }i,n\geq 2.$$

The question is asked for every finite field $\mathbb{F}$ and every prime $\ell\neq\mathrm{char}\,\mathbb{F}$. By the published equivalences, the first statement is equivalent to the conjunction of the Tate, Beilinson, and Grothendieck–Serre semi-simplicity conjectures for all smooth projective varieties over $\mathbb{F}$ (Balkan–Schreieder, Theorems 1.1 and 1.2), and it implies the second statement, which is itself equivalent to the Tate and semi-simplicity conjectures alone. The first open instances are $H^{4}(F_{0}\mathbb{P}^{4}_{\bar{\mathbb{F}}},\mathbb{Q}_{\ell}(3))^{G_{\mathbb{F}}}=0$, equivalent to the Tate conjecture for divisors on surfaces together with the Beilinson conjecture for threefolds over $\mathbb{F}$, and $H^{3}(F_{0}\mathbb{P}^{3}_{\bar{\mathbb{F}}},\mathbb{Q}_{\ell}(2))^{G_{\mathbb{F}}}=0$, equivalent to the Tate conjecture for surfaces over $\mathbb{F}$.

References

1. Samet Balkan and Stefan Schreieder, 'Cycle conjectures and birational invariants over finite fields', Selecta Mathematica (New Series) 32 (2026), DOI 10.1007/s00029-026-01142-0; arXiv:2406.14438, https://arxiv.org/abs/2406.14438
2. Stefan Schreieder, 'Remarks on Milnor K-theory and Tate's conjecture for divisors', arXiv:2406.18706 (2024), https://arxiv.org/abs/2406.18706
3. Jean-Louis Colliot-Thélène and Stefan Schreieder, 'Divisibility phenomena in motivic Bloch–Ogus theory', arXiv:2605.22494 (2026), https://arxiv.org/abs/2605.22494
4. John Tate, 'Endomorphisms of abelian varieties over finite fields', Inventiones Mathematicae 2 (1966), 134–144, DOI 10.1007/BF01404549, https://doi.org/10.1007/BF01404549
5. Uwe Jannsen, 'Mixed Motives and Algebraic K-Theory', Lecture Notes in Mathematics 1400, Springer (1990), DOI 10.1007/BFb0085080, https://doi.org/10.1007/BFb0085080
6. Christophe Soulé, 'Groupes de Chow et K-théorie de variétés sur un corps fini', Mathematische Annalen 268 (1984), 317–345, DOI 10.1007/BF01457062, https://doi.org/10.1007/BF01457062
7. Burt Totaro, 'Recent progress on the Tate conjecture', Bulletin of the American Mathematical Society 54 (2017), 575–590, DOI 10.1090/bull/1588, https://doi.org/10.1090/bull/1588
8. A. A. Beĭlinson, 'Notes on absolute Hodge cohomology', in Applications of Algebraic K-Theory to Algebraic Geometry and Number Theory, Contemporary Mathematics 55, Part 1, American Mathematical Society (1986), pp. 35–68, DOI 10.1090/conm/055.1/862628, https://doi.org/10.1090/conm/055.1/862628