Let $X$ be an affine variety and $G$ an affine algebraic group (for example $\operatorname{PGL}\_n$). How do I compute the Selmer set
$$ \operatorname{Sel}\_\zeta(\mathbb{Q},G) = \{\tau \in H^1(\mathbb{Q},G) \ | \ \tau\_\nu \in \zeta(X(\mathbb{Q}\_\nu)) \ \text{for all places} \ \nu\} $$
where $(\zeta \mapsto \zeta(x))$,
$H^1(X,G) \to H^1(\mathbb{Q}\_\nu,G)$ comes from a point $x: \operatorname{Spec}(\mathbb{Q}\_\nu) \to X$ and $(\tau \mapsto \tau\_\nu)$ comes from the map $\mathbb{Q} \to \mathbb{Q}\_\nu$?



How to compute Selmer set?
Assume that $X(\mathbb{Q}\_\nu)$ is nonempty for every place $\nu$. Since $X$ is affine, $H^1(X,G)$ is trivial. This means that $\zeta(x) = e$ for every $x \in X(\mathbb{Q}\_\nu)$, which means that $e \in \operatorname{Sel}\_\zeta(\mathbb{Q},G)$. So the Selmer set is nonempty.


