Statement copy in the gpt 6 astra pro collection (Math-165)

Brauer–Manin Obstruction for Rationally Connected Varieties

Definitions and mathematical statement

For a number field $k$, write ${\mathbb{A}}_k$ for its ring of adeles and $\operatorname{Br}(X)=H^2_{\mathrm{et}}(X,\mathbf G_m)_{\mathrm{tors}}$. Evaluation of a Brauer class at local points, followed by the local invariant maps, defines \[ X({\mathbb{A}}_k)^{\operatorname{Br}}=\{(x_v):\ \sum_v\operatorname{inv}_v(b(x_v))=0\text{ for every }b\in\operatorname{Br}(X)\}. \] A geometrically rationally connected variety has rational curves joining two general geometric points. For every smooth proper geometrically irreducible such $X$, the assertion is $\overline{X(k)}=X({\mathbb{A}}_k)^{\operatorname{Br}}$ in the adelic topology. Thus a compatible adelic point should be approximable by rational points at any prescribed finite collection of places. The precise adelic convention, including archimedean factors, is that of the cited formulation.

Formulation and concept references: [S1].

Short English statement

For a rationally connected variety over a number field, do the Brauer compatibility conditions account for every failure of rational points to exist or approximate local solutions?

Sources

• S1 Olivier Wittenberg, "Rational points and zero-cycles on rationally connected varieties over number fields," arXiv:1604.08543; Proc. Sympos. Pure Math. 97.2 $2018$, 597-635. https://arxiv.org/pdf/1604.08543. Catalogue formulation source.
• E1 J.-L. Colliot-Thélène and A. N. Skorobogatov, Good reduction of the Brauer--Manin obstruction, Transactions of the American Mathematical Society 365 $2013$, 579--590; Lemmas 1.1 and 1.3, §1 and Theorem 3.1. https://arxiv.org/abs/1009.0247. Added research source; primary manuscript consulted 24 September 2026.
• E2 Yonatan Harpaz and Olivier Wittenberg, On the fibration method for zero-cycles and rational points, Annals of Mathematics 183 $2016$, 229--295; Theorems 1.3, 1.5--1.6, Conjecture 9.1 and Corollary 9.25. https://arxiv.org/abs/1409.0993. Added research source; the rational-point theorem is used with its explicit conditional hypotheses. Consulted 24 September 2026.
• E3 Nguyen Manh Linh, On the descent conjecture for rational points and zero-cycles, Journal of the Institute of Mathematics of Jussieu 25$1$ $2026$, 515--563, published online 19 November 2025; Theorem A and §3.4. https://doi.org/10.1017/S1474748025101400. Author manuscript: https://arxiv.org/html/2305.13228v3. Added research source; publication metadata and author manuscript consulted 24 September 2026.
• E4 Yisheng Tian, A Simpler Approach to a Descent Conjecture of Wittenberg, arXiv:2604.08146 $2026$, consulted HTML version, abstract and Theorem 1.4. https://arxiv.org/html/2604.08146. Added frontier reference; preprint, not an independent proof endorsement. The consulted abstract restricts the result to certain connected-group torsors. Consulted 24 September 2026.