N\'eron blowups and low-degree cohomological applications

We define dilatations of general schemes and study their basic properties. Dilatations of group schemes are – in favorable cases – again group schemes, called N'eron blowups. We give two applications to their cohomology in degree zero (integral points) and degree one (torsors): we prove a canonical Moy-Prasad isomorphism that identifies the graded pieces in the congruent filtration of $G$ with the graded pieces in its Lie algebra $\mathfrak{g}$, and we show that many level structures on moduli stacks of $G$-bundles are encoded in torsors under N'eron blowups of $G$.