Let $X$ be a smooth projective geometrically integral variety over a number field $k$. We begin with some established notions in the theory of descent obstruction to the local-global principle, otherwise known as the Hasse principle.
Let $f:Y \rightarrow X$ be a torsor under a smooth affine algebraic group $G$, and thus the class $[f]$ belongs to the cohomology group $H^1(X,G)$. We denote by $X(\mathbb{A}_k)^f$ the set of adelic points whose image in $\prod_vH^1(k_v,G)$ comes from $H^1(k,G)$. Alternatively, we have the convenient result $$X(\mathbb{A}_k)^f = \bigcup _{\tau \in H^1(k,G)}f_\tau(Y_\tau(\mathbb{A}_k)).$$
Remark 1. Since $X$ is proper, the number of class of torsors $\tau$ in $H^1(k,G)$ such that the twist $f_\tau$ has points everywhere locally is finite. Later on, we will drop the properness condition and the (lack of) finiteness would arise as an issue.
We define the set $$X(\mathbb{A}_k)^\mathrm{descent} := \bigcap _{\mathrm{all \, smooth \, affine}\,G}\, \bigcap_{\mathrm{all}\,G-\mathrm{torsors}\,f:Y \rightarrow X} \bigcup_{\tau \in H^1(k,G)}f_\tau(Y_\tau(\mathbb{A}_k)).$$
This further introduces constraints on the possible locations of rational points, i.e., points in $X(k)$. One says that there is descent obstruction to the local-global principle if $$X(k) = \emptyset \implies X(\mathbb{A}_k)^\mathrm{descent} = \emptyset.$$
In the definition of $X(\mathbb{A}_k)^\mathrm{descent}$, we can instead consider the second intersection to be taken over all finite étale groups $G$ to obtain the set $X(\mathbb{A}_k)^\mathrm{et}$.
Remark 2. This set was introduced in the paper Finite Descent and Rational Points on Curves by M. Stoll as $X(\mathbb{A}_k)^{\mathrm{f}-\mathrm{cov}}$ to stand for finite (étale) covers.
Similarly, one says that there is finite étale descent obstruction to the local-global principle if $$ X(k) = \emptyset \implies X(\mathbb{A}_k)^\mathrm{et} = \emptyset.$$
With these two descent obstruction sets, we can "combine" them by defining the set $$X(\mathbb{A}_k)^\mathrm{et,descent}:=\bigcap _{\mathrm{all \, finite \, étale}\,G}\, \bigcap_{\mathrm{all}\,G-\mathrm{torsors}\,f:Y \rightarrow X} \bigcup_{\tau \in H^1(k,G)}f_\tau(Y_\tau(\mathbb{A}_k)^\mathrm{descent}).$$
Of course, when talking about obstructions, we always relate to the classical Brauer-Manin obstruction, and thus define the étale-Brauer set
$$X(\mathbb{A}_k)^\mathrm{et,Br}:=\bigcap _{\mathrm{all \, finite \, étale}\,G}\, \bigcap_{\mathrm{all}\,G-\mathrm{torsors}\,f:Y \rightarrow X} \bigcup_{\tau \in H^1(k,G)}f_\tau(Y_\tau(\mathbb{A}_k)^\mathrm{Br}),$$
where $Y_\tau(\mathbb{A}_k)^\mathrm{Br}$ is the Brauer set of $Y_\tau$, i.e., the set of points of $Y_\tau(\mathbb{A}_k)$ that vanish under the Brauer pairing $$Y_\tau(\mathbb{A}_k) \times \mathrm{Br}(Y_\tau) \rightarrow \mathbb{Q}/\mathbb{Z}.$$
We define $X(\mathbb{A}_k)^\mathrm{et,et}$ similarly, this set is an example of an iterated descent obstruction.
The first result of note is
Theorem 1. Let $X$ be a smooth projective geometrically integral variety over a number field $k$, then $$X(\mathbb{A}_k)^\mathrm{et,Br} = X(\mathbb{A}_k)^\mathrm{et,descent} = X(\mathbb{A}_k)^\mathrm{descent}.$$
Part of the proof of Theorem 1 comes from Theorem 1.1 of Skorobogatov's paper Descent Obstruction is Equivalent to Étale Brauer-Manin Obstruction, which generalizes Proposition 5.17 of the abovementioned paper by Stoll, stating that we have the equality $$X(\mathbb{A}_k)^\mathrm{et,et} = X(\mathbb{A}_k)^\mathrm{et}.$$
In Proposition 5.18 that follows, it says that Skorobogatov's paper Beyond the Manin Obstruction constructed a surface $Z$ such that the analogue for finite abelian étale group schemes does not hold, i.e., if $$Z(\mathbb{A}_k)^\mathrm{desc-ab} := \bigcap _{\mathrm{all \, finite \, abelian \,étale}\,G}\, \bigcap_{\mathrm{all}\,G-\mathrm{torsors}\,f:Y \rightarrow Z} \bigcup_{\tau \in H^1(k,G)}f_\tau(Y_\tau(\mathbb{A}_k)),$$ then one DOES NOT have $$Z(\mathbb{A}_k)^\mathrm{desc-ab,desc-ab} = Z(\mathbb{A}_k)^\mathrm{desc-ab}.$$
However, in general for any smooth projective geometrically integral $k$-variety $X$, we can (easily) say that we have the inclusion
$$X(\mathbb{A}_k)^\mathrm{desc-ab,desc-ab} \subset X(\mathbb{A}_k)^\mathrm{desc-ab}.$$
Question 1. Would the equality be true for the case of curves? What about for affine curves? (I believe the affine case is hard because the class of torsors $\tau$ such that the twist of the $X$-torsor $Y$ by $\tau$ having points everywhere locally is not finite, so we cannot follow the proof in Stoll's paper.)
The motivation for the latter question is due to the fact that by the joint work of Cao, Demarche and Xu as well as a separate work by the former, we have the results
Theorem 2. If $X$ is a smooth quasi-projective variety over a number field $k$, then $$X(\mathbb{A}_k)^\mathrm{descent} = X(\mathbb{A}_k)^\mathrm{et,Br}.$$
and
Theorem 3. If $X$ is a smooth quasi-projective variety over a number field $k$, then $$X(\mathbb{A}_k)^\mathrm{descent,descent} = X(\mathbb{A}_k)^\mathrm{descent}.$$
Therefore, they are true, in particular, when $X$ is a smooth affine curve. Furthermore, Theorem 7.1 of the abovementioned paper by Stoll and Proposition 3.1 of the paper Descent Theory for Open Varieties by Harari and Skorobogatov gives us
Theorem 4. For a smooth geometrically integral variety $X$ over a number field $k$, one has $$X(\mathbb{A}_k)^{\mathrm{Br}_1} \subset X(\mathbb{A}_k)^\mathrm{desc-ab}.$$
When $X$ is a curve, Tsen's theorem tells us that the former set is simply the Brauer set of $X$.
Question 2. Again, it seems like the information provided by studying torsors under finite abelian étale group schemes is very little, in particular when the variety $X$ is open. Does anyone know how to better understand this set? It would help if there are more references about it!
