An arithmetic valuative criterion for proper maps of tame algebraic stacks

TL;DR

This paper introduces a new valuative criterion for proper maps of tame algebraic stacks, enabling better arithmetic applications by allowing specialization over the same residue field, and proves the Lang-Nishimura theorem in this context.

Contribution

It provides a novel arithmetic-compatible valuative criterion for tame stacks, addressing limitations of previous criteria and extending key theorems like Lang-Nishimura.

Findings

New valuative criterion for tame stacks

Enables specialization over the same residue field

Proves Lang-Nishimura theorem for tame 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.