Question

These are parametrized by $H^1(Gal(\mathbb{Q}), Aut X)$, where X is some $\mathbb{Q}$-model of the curve.

It was established in http://mathoverflow.net/questions/22907/confusion-about-how-the-first-cohomology-classifies-torsors that fiber bundles over $B$ with fiber $F$, structure group $G$ and transition maps with property $P$ are classified by $T$-torsors, where $T$ is the sheaf on $B$ of functions to $G$ with property $P$. $T$-torsors, in turn, are classified by $H^1(B, T)$.

Is there a way to interpret the aforementioned classification of $\mathbb{Q}$-models of a curve in these terms?

Answer 1

Yes.

Roughly, the idea is that once your base object $X/B$ is fixed then for any object $Y$ you can consider the sheaf $Iso(X,Y)$ of isomorphisms from $X$ to $Y$. This has an action of the sheaf $Aut(X)$ and on any open cover $U$ where $Y|_U \cong X|_U$, making a choice of such an isomorphism gives an isomorphism of $Aut(X)(U)$-sets $Iso(X,Y)(U) \cong Aut(X)(U)$.

The difference from the standard argument in this case is that you need to use the etale topology instead of the Zariski topology. The etale covers of $\mathbb{Q}$ are finite field extensions $F$ and so the statement that $Y$ is "locally" equivalent to $X$ means that $Y_F \cong X_F$ for a large enough field extension $F$.