An arithmetic valuative criterion for proper maps of tame algebraic stacks

Abstract

The valuative criterion for proper maps of schemes has many applications in arithmetic, e.g. specializing ${\mathbb{Q}}_p$-points to ${\mathbb{F}}_p$-points. For algebraic stacks, the usual valuative criterion for proper maps is ill-suited for these kind of arguments, since it only gives a specialization point defined over an extension of the residue field, e.g. a ${\mathbb{Q}}_p$-point will specialize to an ${\mathbb{F}}_{p^n}$-point for some $n$. We give a new valuative criterion for proper maps of tame stacks which solves this problem and is well-suited for arithmetic applications. As a consequence, we prove that the Lang-Nishimura theorem holds for tame stacks.