Unconditional and integral Brauer–Tate–Shafarevich comparison over global function fields: finiteness of $\operatorname{coker}\rho_d$, $p$-primary control, and Kato homology vanishing beyond degree 3

The Brauer group of a regular proper model $\mathcal{X}$ over a curve over a finite field is finite if and only if the Tate–Shafarevich group of the Albanese variety of the generic fiber $X$ is finite and Tate's conjecture for divisors holds on $X$ (Geisser, 2021, generalizing the Artin–Grothendieck theorem for fibered surfaces to arbitrary relative dimension); under additional hypotheses the orders of these groups are compared by an Artin–Tate-type formula. Three parts of this comparison remain open: the order formula and the exact sequence are proved only up to a power of the characteristic $p$; the formula assumes the finiteness of the kernel of the localization map $\xi^{2d-2}_d$, equivalently of $\operatorname{coker}\rho_d$, which is known (up to $p$-groups) only under Beilinson's conjecture on the Albanese map for zero-cycles over separable closures of global fields; and the key local input, the vanishing of Kato homology $KH_a(\mathcal{Y},\mathbb{Q}_\ell/\mathbb{Z}_\ell)$ for regular proper models, is established only for $a\le 3$. The problem is to remove these conditionalities: prove the finiteness of $\operatorname{coker}\rho_d$ unconditionally, extend the comparison to the full $p$-primary part, and prove or refute the Kato homology vanishing conjecture in all degrees. No resolution of any of these components is known in the literature to date.

Background

Let $\mathbb{F}_q$ be a finite field of characteristic $p$, let $C$ be a smooth proper curve over $\mathbb{F}_q$, and let $K$ be its function field, a global function field. Let $\mathcal{X}$ be a regular scheme, proper and flat over $C$, whose generic fiber $X$ is smooth, projective and geometrically connected of dimension $d$ over $K$; write $X^s$ for the base change to a separable closure $K^s$ and $G_K=\mathrm{Gal}(K^s/K)$, and let $X_v=X\times_K K_v$ for the completions $K_v$. Two global arithmetic invariants are central. The Brauer group $\mathrm{Br}(\mathcal{X})$ is the torsion group classifying Azumaya algebras — sheaves of noncommutative algebras that are étale-locally isomorphic to matrix algebras, the geometric analogue of central simple algebras over a field — and is computable as the direct limit over $n$ (ordered by divisibility) of the étale cohomology groups $H^2_{\mathrm{et}}(\mathcal{X},\mu_n)$, where $\mu_n$ is the group of $n$-th roots of unity and étale cohomology is the cohomology theory for schemes that plays the role of singular cohomology for complex varieties. The Tate–Shafarevich group $\Sha(A)=\ker(H^1(K,A)\to\prod_v H^1(K_v,A))$ of an abelian variety $A$ over $K$ measures the failure of the local-to-global principle for principal homogeneous spaces under $A$. For a fibered surface ($d=1$), a classical theorem of Artin and Grothendieck states that $\mathrm{Br}(\mathcal{X})$ is finite if and only if $\Sha(\mathrm{Jac}\,X)$ is finite, and when both are finite their orders are related by the Artin–Tate formula, involving the index of $X$ (the greatest common divisor of the degrees over $K$ of its closed points) and local correction terms. Geisser's paper, the source of this problem, extends this circle to arbitrary relative dimension. One ingredient is Tate's conjecture for divisors on $X$: the cycle class map $\mathrm{Pic}(X)\otimes\mathbb{Z}_\ell\to H^2_{\mathrm{et}}(X^s,\mathbb{Z}_\ell(1))^{G_K}$ should have torsion (equivalently, finite) cokernel for every prime $\ell\ne p$. Geisser's Theorem 1.1, which is unconditional, states that $\mathrm{Br}(\mathcal{X})$ is finite if and only if $\Sha(\mathrm{Alb}_X)$ is finite and Tate's conjecture for divisors holds on $X$, where $\mathrm{Alb}_X$ is the Albanese variety of $X$, the abelian variety universally receiving morphisms from $X$. The finer comparison of orders uses étale motivic cohomology $H^i_{\mathrm{et}}(-,\mathbb{Z}(n))$: the étale hypercohomology of Bloch's cycle complex, the complex of algebraic cycles on products with affine spaces whose rational cohomology recovers the higher Chow groups (the cycle-theoretic groups connecting algebraic $K$-theory to cohomology), taken with coefficients $\mathbb{Z}'=\mathbb{Z}[1/p]$. Three comparison maps enter: the global-to-geometric map $\rho_d:H^{2d}_{\mathrm{et}}(X,\mathbb{Z}'(d))\to H^{2d}_{\mathrm{et}}(X^s,\mathbb{Z}'(d))^{G_K}$; the localization map $\xi^{2d-2}_d:H^2(K,H^{2d-2}_{\mathrm{et}}(X^s,\mathbb{Q}/\mathbb{Z}'(d)))\to\bigoplus_v H^2(K_v,H^{2d-2}_{\mathrm{et}}(X^s,\mathbb{Q}/\mathbb{Z}'(d)))$ for the étale cohomology of $X^s$ viewed as a $G_K$-module; and $l^{2d+1}_d:H^{2d+1}_{\mathrm{et}}(X,\mathbb{Z}'(d))\to\prod_v H^{2d+1}_{\mathrm{et}}(X_v,\mathbb{Z}'(d))$. Geisser proves an exact sequence relating $\operatorname{coker}\rho_d$, $\ker\xi^{2d-2}_d$, $\ker l^{2d+1}_d$, $\Sha(\mathrm{Alb}_X)$, the cokernel of the Tate cycle map, and the Pontryagin duals of the Tate modules (inverse limits of the $\ell^n$-torsion) of $\mathrm{Br}(\mathcal{X})$ and of the Tate–Shafarevich group of the Picard variety (Theorem 1.3), and, under the additional hypothesis that $\ker\xi^{2d-2}_d$ (equivalently $\operatorname{coker}\rho_d$) is finite, an Artin–Tate-type order formula (Theorem 1.2) — but both only modulo the Serre subcategory (the full subcategory closed under subobjects, quotients and extensions) generated by finite groups and $p$-power torsion groups, that is, up to a power of $p$. The proofs rest on Kato homology $KH_a(\mathcal{Y},\mathbb{Q}_\ell/\mathbb{Z}_\ell)$ for $\mathcal{Y}$ regular, flat and proper over a henselian discrete valuation ring (a discrete valuation ring satisfying Hensel's lemma, so that its finite étale extensions are controlled by the residue field): the homology of the Gersten-type complex built from the Galois cohomology of the residue fields at the points of $\mathcal{Y}$, which measures the failure of the cohomological Hasse principle (the local-to-global principle for étale cohomology). Saito and Sato conjectured that for residue field $k$ separably closed, $KH_a(\mathcal{Y},\mathbb{Q}_\ell/\mathbb{Z}_\ell)$ vanishes for all $a$, while for $k$ finite, $KH_a=0$ for $a\ne 1$ and $KH_1\cong(\mathbb{Q}_\ell/\mathbb{Z}_\ell)^I$ with $I$ the number of irreducible components of the special fiber; they proved the vanishing for $a\le 3$ when the reduced special fiber is a strict normal crossing scheme (its irreducible components are regular and meet like coordinate hyperplanes), and Geisser's Theorem 3.2 removes this restriction, extending the $a\le 3$ case to arbitrary regular proper $\mathcal{Y}$ using Gabber's alteration theorem (every integral quasi-excellent scheme admits a proper, surjective, generically finite map of degree prime to $\ell$ from a regular scheme) and a Kato-homology pullback construction of Kerz–Saito. Finally, the finiteness of $\operatorname{coker}\rho_d$ is reduced by Geisser to a conjecture of Beilinson (Conjecture 5.2 of the source, stated in 1987 for number fields): for $X^s$ smooth and projective over the separable closure of a global field, the Albanese map $CH_0(X^s)^0\to\mathrm{Alb}_{X^s}(K^s)$, from the degree-zero part of the Chow group of zero-cycles (finite integral combinations of closed points modulo rational equivalence) to the $K^s$-rational points of the Albanese, is an isomorphism.

Problem Statement

Let $X$ be smooth and projective of dimension $d$ over a global function field $K$ of characteristic $p$, admitting a regular proper flat model $\mathcal{X}$ over the smooth proper curve $C$ with function field $K$, and keep the notation of the background. The source paper explicitly leaves the following issues unresolved; their common objective is to make the comparison between $\mathrm{Br}(\mathcal{X})$, $\Sha(\mathrm{Alb}_X)$ and Tate's conjecture for divisors unconditional and integral in arbitrary relative dimension over global function fields.

(a) Unconditional finiteness of the comparison kernel: prove, without assuming Beilinson's conjecture on the Albanese map for zero-cycles or any other unproved hypothesis, that $\operatorname{coker}\rho_d$ — equivalently $\ker\xi^{2d-2}_d$ — is finite (at least up to $p$-groups; ideally integrally) for every such $X$; or exhibit a specific $X$ for which this group is infinite.

(b) $p$-primary control: extend the exact sequence of Theorem 1.3 and the order formula of Theorem 1.2 of the source beyond the Serre subcategory of finite groups and $p$-power torsion groups, i.e. remove the 'up to a power of $p$' qualification and control the $p$-primary parts of the groups and of the formula.

(c) Kato homology vanishing in all degrees: prove, or refute, that for $\mathcal{Y}$ regular, flat and proper over a henselian excellent discrete valuation ring with residue field $k$ and $\ell\ne\mathrm{char}\,k$: if $k$ is separably closed, $KH_a(\mathcal{Y},\mathbb{Q}_\ell/\mathbb{Z}_\ell)=0$ for all $a$ (in particular for all $a\ge 4$); if $k$ is finite, $KH_a(\mathcal{Y},\mathbb{Q}_\ell/\mathbb{Z}_\ell)=0$ for $a\ne 1$ and $KH_1\cong(\mathbb{Q}_\ell/\mathbb{Z}_\ell)^I$ with $I$ the number of irreducible components of the special fiber (Conjecture 3.1 of the source, strengthening Conjectures 2.11–2.12 of Saito–Sato).

The three questions are scientifically linked: (c) is the local input through which the methods of the source would extend past the current $a\le 3$ restriction; (a) is the remaining conditional hypothesis of the order formula Theorem 1.2, currently discharged only via Beilinson's conjecture; and (b) is the precision upgrade of the same theorems.

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