André’s p-curvature/Ogus-type conjecture for flat bundles
Establish that for a variety X over a finitely generated integral \mathbb{Z}-algebra R with fraction field K and a flat vector bundle (\mathscr{E},\nabla) on X/R, the complex bundle (\mathscr{E},\nabla)|_{X_K} is of geometric origin if and only if the p-curvature of (\mathscr{E},\nabla)|_{X_\mathfrak{p}} is nilpotent for almost all closed points \mathfrak{p} of \operatorname{Spec}(R).
References
Conjecture [{\u007f[Appendix to Chapter V]{andre1989g}] Let $X$ be a variety over a finitely generated integral $\mathbb{Z}$-algebra $R$, with fraction field $K$ of characteristic zero. Let $(\mathscr{E},\nabla)$ be a flat vector bundle on $X/R$. Then $(\mathscr{E}, \nabla)|{X_K}$ is of geometric origin if and only if there exists a dense open subset $U\subset \on{Spec}(R)$ such that for all $\mathfrak{p}\in U$, the $p$-curvature of $(\mathscr{E},\nabla)|{X_{\mathfrak{p}}$ is nilpotent.