Let $ X$ be a finitely presented algebraic stack over a field $ k$ , say of characteristic zero. Let $ k\to \overline{k}$ be an algebraic closure.
Suppose that $ X$ is a gerbe over $ k$ . In particular, there is an fppf cover $ U\to$ Spec $ k$ , a flat finitely presented group scheme $ G$ over $ U$ , and an isomorphism of algebraic stacks $ X_U \cong BG$ over $ U$ . This implies, if I’m not mistaken, that $ X_{\overline{k}}$ is isomorphic to $ BG$ for some finite type group scheme $ G$ over $ \overline{k}$ .
In this case, I’d like to say that $ X$ is a “$ G$ -gerbe” over $ k$ , but this is a bit strange, because $ G$ is not (necessarily) a group scheme over $ k$ .
If $ X$ is proper and etale over $ k$ , then I can choose $ G$ to be an abstract finite group, and one could refer to $ X$ as a $ G$ -gerbe over $ k$ .
My question:
If $ X$ is a gerbe over $ k$ , does there exist an fppf group scheme $ G$ over $ k$ such that $ X$ is isomorphic to $ BG$ over $ \overline{k}$ ?
I realized that I do not know the answer to this question when I bumped into Will Sawin’s answer to this question: Stacks with a small coarse moduli space
The answer is probably positive when the diagonal of $ X$ is affine. Indeed, in this case $ X_{\overline{k}} \cong BG$ for some affine finite type group scheme $ G$ over $ \overline{k}$ . But $ G$ has a model over $ k$ (and even $ \mathbb{Z}$ ).
A possible counterexample could maybe be constructed as follows: Let $ K$ be a number field and let $ E$ be an elliptic curve over $ K$ with $ j(E)\not\in \mathbb{Q}$ . Is it conceivable that the neutral gerbe $ BE$ over $ K$ has a model over $ \mathbb{Q}$ ?
