Can every connected reductive group over a char $p$ field be defined over $\mathbb F_p$?

Question

If I have a connected reductive group $G$ over a field with characteristic $p>0$, can it always be defined over $\mathbb F_p$? For split groups like $GL_n, GSp_{2n}$ it's trivial, how about general case?

Answer 1

No. Here's a simple example. Let $T:=\mathsf{Res}^1_{\mathbb{F}_{p^4}/\mathbb{F}_{p^2}}\mathbb{G}_{m,\mathbb{F}_{p^4}}$. Then, $T$ is a non-split one-dimensional torus over $\mathbb{F}_{p^2}$ which does not have a model over $\mathbb{F}_p$. Indeed, to say that $T$ has a model over $\mathbb{F}_p$ would mean that there was some torus $T'$ over $\mathbb{F}_p$ such that $T'_{\mathbb{F}_{p^2}}\cong T$. But evidently $\dim T'=1$ and, up to isomorphism, the only one-dimensional tori over $\mathbb{F}_p$ are $\mathsf{Res}^1_{\mathbb{F}_{p^2}/\mathbb{F}_p}\mathbb{G}_{m,\mathbb{F}_{p^2}}$ and $\mathbb{G}_{m,\mathbb{F}_p}$. Both of these split over $\mathbb{F}_{p^2}$ so can't be models of $T$.

EDIT: To be clear, I was answering the question in the first sentence of the body of the post. The answer to the question in the title is yes, as Tobias Kildetoft pointed out in the above comments. Every group over $\overline{\mathbb{F}_p}$ is split, and every split group has a model over $\mathrm{Spec}(\mathbb{Z})$.