Brauer–Manin Density in Rationally Connected Fibrations

Definitions and mathematical statement

Let $k$ be a number field, let $\Omega_k$ be its set of places, let $k_v$ denote its completion at $v$, and let $\mathbb A_k$ be its ring of adeles. For a proper $k$-variety $X$, equip $X(\mathbb A_k)=\prod_{v\in\Omega_k}X(k_v)$ with the usual adelic topology. Write \[ \operatorname{Br}(X)=H^2_{\mathrm{\acute et}}(X,\mathbb G_m)_{\mathrm{tors}}. \] For $\alpha\in\operatorname{Br}(X)$ and $x_v\in X(k_v)$, let $\alpha(x_v)\in\operatorname{Br}(k_v)$ be its evaluation, and let $\operatorname{inv}_v:\operatorname{Br}(k_v)\to{\mathbb{Q}}/{\mathbb{Z}}$ be the local invariant map. Define the Brauer--Manin set by \[ X(\mathbb A_k)^{\operatorname{Br}(X)} = \left\{(x_v)_v\in X(\mathbb A_k): \sum_{v\in\Omega_k}\operatorname{inv}_v\left(\alpha(x_v)\right)=0 \text{ for every }\alpha\in\operatorname{Br}(X)\right\}. \] The sum has only finitely many nonzero terms. Rational points are embedded diagonally, and density below means density in the subspace topology on this set.

Use the geometric convention for rational connectedness: a variety over a field $K$ is rationally connected if, after extension to an algebraic closure of $K$, it is integral and two general points lie on the image of a morphism from ${\mathbb{P}}^1$.

Question. Let $p:Z\to B$ be a dominant morphism of smooth proper varieties over $k$. Let $\eta=\operatorname{Spec}k(B)$ and $Z_\eta=Z\times_B\eta$. Assume that $B$ and $Z_\eta$ are rationally connected, the latter over $k(B)$. Suppose that \[ \overline{B(k)}^{\,B(\mathbb A_k)^{\operatorname{Br}(B)}} =B(\mathbb A_k)^{\operatorname{Br}(B)}, \] and that there exists a Zariski dense open subset $U\subseteq B$ such that, for every $b\in U(k)$, the fiber $Z_b=Z\times_B\operatorname{Spec}k$ over $b$ satisfies \[ \overline{Z_b(k)}^{\,Z_b(\mathbb A_k)^{\operatorname{Br}(Z_b)}} =Z_b(\mathbb A_k)^{\operatorname{Br}(Z_b)}. \] Must it follow that \[ \overline{Z(k)}^{\,Z(\mathbb A_k)^{\operatorname{Br}(Z)}} =Z(\mathbb A_k)^{\operatorname{Br}(Z)}? \]

The phrase concerning rational fibers is interpreted as the existence of one dense open $U$ on which the assertion holds for every rational point. The source does not explicitly specify the adelic topology or the geometric convention for rational connectedness; the conventions used above require confirmation against the original problem. The supplied registry labels the problem open and its statement corrected_verified; these are supplied curation labels, not independent verification.

Short English statement

For a dominant morphism of smooth proper varieties over a number field, assume that the base and generic fiber are rationally connected. If rational points are dense in the Brauer--Manin set of the base and of every rational fiber over some dense open subset, must the same density hold for the total space? This is a statement-only draft requiring mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003770$, title: "Brauer--Manin Density in Rationally Connected Fibrations." Collection: Oberwolfach Reports Open Problems. Exact source problem code: catalogue entry. No individual proposer is supplied.
• Supplied report citation: Mini-Workshop: Arithmetic Geometry and Symmetries around Galois and Fundamental Groups (2019), Oberwolfach Reports, pp. 1021--1080, DOI 10.4171/owr/2018/17. https://doi.org/10.4171/owr/2018/17. The registry gives proposed year 2018 and also labels this reference "OWR 2018 report"; the year discrepancy is unresolved.
• Supplied literature source: Wittenberg, "zero cycles and fibrations." https://arxiv.org/abs/1010.1883.
• Supplied literature source: Harpaz--Wittenberg, "homogeneous spaces." https://arxiv.org/abs/1802.09605.
• Supplied literature source: "OWR 2018 report." https://doi.org/10.4171/owr/2018/17.
• Literature assessment checked at 2026-08-21: the supplied curation labels the problem open and states that fibration results settle rational-base and several special zero-cycle cases, while the stated rationally connected-base implication for rational points remains open. This is supplied dated literature triage derived from the OWR curation, not an independently verified assessment; its status and cited claims require independent checking.