Étale Brauer–Manin Obstruction

• Étale Brauer–Manin obstruction is a refined cohomological tool that incorporates all finite étale covers to explain why the Hasse principle may fail for algebraic varieties.
• Key constructions by Poonen and Smeets illustrate that even with refined étale information, local adelic points can exist while global rational points remain absent.
• Applications extend to varieties with trivial Albanese and simply connected cases, highlighting limitations and motivating the search for even finer nonabelian descent obstructions.

The étale Brauer–Manin obstruction is a refinement of the classical Brauer–Manin obstruction for explaining failures of the Hasse principle and weak approximation on algebraic varieties and more general arithmetic spaces over global fields. While the classical Brauer–Manin set considers only the cohomological Brauer group and its evaluation on adelic points, the étale refinement accounts for all finite étale covers, capturing obstructions not visible to the algebraic or even transcendental Brauer group alone. Crucially, recent constructions have demonstrated that even the étale Brauer–Manin obstruction is not always sufficient to account for all instances of Hasse principle failure, including in simply connected contexts (Smeets, 2014, Balestrieri et al., 2015, Colliot-Thélène et al., 2013).

1. Definitions and Theoretical Framework

Given a number field $k$ with set of places $\Omega_k$ and a smooth, projective, geometrically integral $k$-variety $X$, the cohomological Brauer group is

$$\operatorname{Br} X = H_{\text{ét}}^2(X, \mathbb{G}_m).$$

For each place $v\in\Omega_k$, the local invariant map $$\operatorname{inv}_v: \operatorname{Br} k_v \to \mathbb{Q}/\mathbb{Z}$$ induces the Brauer–Manin pairing

$$\langle (P_v), \alpha \rangle = \sum_{v\in\Omega_k} \operatorname{inv}_v(\alpha(P_v)), \quad (P_v)\in X(\mathbb{A}_k),\;\; \alpha\in \operatorname{Br} X.$$

The classical Brauer–Manin set is

$$X(\mathbb{A}_k)^{\operatorname{Br}} = \{ (P_v) \mid \langle (P_v), \alpha \rangle = 0,\;\; \forall \alpha \in \operatorname{Br} X \}.$$

The étale Brauer–Manin set is defined as

$$X(\mathbb{A}_k)^{\operatorname{et}, \operatorname{Br}} = \bigcap_{\pi: Y \to X \text{ finite étale}} \pi(Y(\mathbb{A}_k)^{\operatorname{Br}}).$$

Whenever $$X(\mathbb{A}_k)^{\operatorname{et}, \operatorname{Br}} = \emptyset$$ but $X(\mathbb{A}_k) \neq \emptyset$, the étale Brauer–Manin obstruction explains the failure of the Hasse principle (Smeets, 2014).

2. Classical Examples and Counterexamples

Poonen’s construction provided the first explicit example of a smooth, projective threefold $X$ over a number field $k$ with $X(k)=\emptyset$ but $$X(\mathbb{A}_k)^{\operatorname{et}, \operatorname{Br}} \neq \emptyset$$. This method uses a fibration $X\to C$, where $C/k$ is a curve of genus $g \ge 1$ with $C(k)=\emptyset$ but $C(\mathbb{A}_k)\neq\emptyset$. Over $C$ one builds a pencil of Châtelet surfaces such that all local fibers have points, yet the global rational locus is empty; all rational points on étale covers are forced to lie above rational points of $C$, which do not exist (Smeets, 2014).

Subsequent examples employed similar "Poonen's trick", constructing varieties as fibrations over higher genus curves with nontrivial Albanese variety. All known examples prior to (Smeets, 2014) and (Balestrieri et al., 2015) had this fibration structure and nontrivial Albanese.

3. New Constructions: Trivial Albanese and Simply Connected Varieties

Smeets (Smeets, 2014) provided the first examples of smooth, projective, geometrically integral varieties $X$ with trivial Albanese variety and

$$X(k)=\emptyset,\qquad X(\mathbb{A}_k)^{\operatorname{et},\operatorname{Br}}\neq\emptyset.$$

This construction uses Beauville surfaces $S=(C\times D)/G$, with $C$ and $D$ of genus 5 and 3, and $G\simeq (\mathbb{Z}/2)^3$ acting freely, so that $\operatorname{Alb}(S)=0$. Then, building a suitable family of Châtelet surfaces $X_S$ as a fiber product over $S$, one obtains a fourfold $X_S$ with the desired properties. Importantly, the restriction map $\operatorname{Br} S\to\operatorname{Br} X_S$ is an isomorphism [(Smeets, 2014), Proposition 3.2].

Assuming the $abc$ conjecture and using Campana's orbifold theory, Smeets also produced a simply connected example via a fibered construction where all étale covers become isomorphic, so $$X(\mathbb{A}_k)^{\operatorname{et},\operatorname{Br}}=X(\mathbb{A}_k)^{\operatorname{Br}}$$ (Smeets, 2014).

In positive characteristic, Kebekus–Pereira–Smeets constructed an unconditional, simply connected fourfold $Z$ over a global function field $K$ with $\pi_1^{\operatorname{et}}(Z)=1$, $\operatorname{Br}(Z)=0$, $$Z(\mathbb{A}_K)^{\operatorname{Br}}=Z(\mathbb{A}_K)^{\operatorname{et}, \operatorname{Br}}\neq \emptyset$$, yet $Z(K)=\emptyset$ (Kebekus et al., 2019).

4. The Case of Enriques and K3 Surfaces

The phenomenon of insufficiency also occurs for Enriques surfaces. There exists an Enriques surface $X/\mathbb{Q}$ and a transcendental Brauer class $$\alpha\in \operatorname{Br}(X_{\overline{\mathbb{Q}}})\setminus \operatorname{Br}_1(X)$$ with

$$X(\mathbb{A}_\mathbb{Q})^{\operatorname{Br}_1}=X(\mathbb{A}_\mathbb{Q})\neq\emptyset,\qquad X(\mathbb{A}_\mathbb{Q})^\alpha=\emptyset,$$

confirming that the étale Brauer–Manin obstruction is strictly stronger than the algebraic Brauer–Manin obstruction in this context (Balestrieri et al., 2015). The construction proceeds through careful construction of a K3 double cover and explicit descent theory for the relevant Brauer classes.

This shows that purely transcendental elements in $\operatorname{Br}(X_{\bar{k}})$ can obstruct the Hasse principle even when the classical and algebraic Brauer–Manin set does not, necessitating a full analysis in the étale setting.

5. Failures and Explanatory Mechanisms

Geometric and cohomological analysis of these counterexamples reveals that the failure of the étale Brauer–Manin obstruction is fundamentally linked to their structure as fibrations over a base with $B(k)$ finite and $B(\mathbb{A}_k)$ nonempty. Any finite étale cover $Y\to X$ factors through pullback from the base, so no global points exist on $Y$ either and no étale torsor can detect the missing $k$-rational points. The surjectivity $\operatorname{Br}(S)\to\operatorname{Br}(X_S)$ and rigidity of the Albanese variety further ensure these phenomena [(Smeets, 2014), Proposition 3.2 and Lemma 3.4].

Other constructions, such as conic or quadric bundles over curves of positive genus, confirm that no further finiteness assumption on $\pi_1(\bar X)$ nor restriction to the algebraic part of the Brauer group suffices to guarantee that the étale Brauer–Manin obstruction can account for all failures of the Hasse principle (Colliot-Thélène et al., 2013).

6. Broader Implications and Open Questions

These findings underline that even the refined étale Brauer–Manin obstruction does not resolve the Hasse principle for all smooth projective varieties:

• There exist surfaces and higher-dimensional varieties $X$ with $$X(\mathbb{A}_k)^{\operatorname{et}, \operatorname{Br}}\neq\emptyset$$ but $X(k)=\emptyset$ (Smeets, 2014, Colliot-Thélène et al., 2013, Harpaz et al., 2012).
• For simply connected varieties in positive characteristic, similar failures can be constructed unconditionally (Kebekus et al., 2019).
• On curves, no counterexample to sufficiency is known; the conjecture that the (étale) Brauer–Manin obstruction always suffices for curves remains open (Harpaz et al., 2012).

Significant open directions include an unconditional proof of Campana’s orbifold Mordell conjecture (to avoid the $abc$ hypothesis), systematic paper of non-abelian descent obstructions, and explicit construction of low-dimensional varieties with trivial Albanese where all known obstructions fail (Smeets, 2014). The search for even finer obstructions—incorporating nonabelian cohomology or étale homotopy—remains active.

7. Summary Table: Key Examples of Inadequacy

Example	Type	Invariant properties	Main obstruction failure
Poonen (2010), (Smeets, 2014)	3-fold fibration over $g\ge1$	$\operatorname{Alb}(X)\neq0$	Étale Brauer–Manin not detecting failure
Smeets (2014), (Smeets, 2014)	4-fold over Beauville surface	$\operatorname{Alb}(X)=0$, $X$ not simply connected	Same, but with trivial Albanese
Smeets (2014) under $abc$, (Smeets, 2014)	4-fold, simply connected	$\pi_1^{\operatorname{et}}(X)=1$	Same; conditional on $abc$
Enriques surface, (Balestrieri et al., 2015)	Surface, $K3$ double cover	$\operatorname{Br}(X)/\operatorname{Br}_1(X)\neq0$	Purely transcendental class cuts out rational points
CPTS (2013), (Colliot-Thélène et al., 2013)	Conic/quadric bundle over $g\ge 1$ curve	varies	Étale BM does not suffice
Kebekus–Pereira–Smeets (2019), (Kebekus et al., 2019)	Simply connected $4$-fold, pos. char	$\pi_1^{\operatorname{et}}=1, \operatorname{Br}=0$	Same, unconditional in char $p$

The continued discovery of such examples demonstrates both the power and the limits of the étale Brauer–Manin obstruction, motivating further work into more sophisticated arithmetic and cohomological obstructions (Smeets, 2014, Balestrieri et al., 2015, Kebekus et al., 2019, Colliot-Thélène et al., 2013).