A Topology on Points on Stacks

Abstract

For a variety over certain topological rings $R$, like $\mathbb{Z}_p$ or $\mathbb{C}$, there is a well-studied way to topologize the $R$-points on the variety. In this paper, we generalize this definition to algebraic stacks. For an algebraic stack $\mathfrak{X}$ over many topological rings $R$, we define a topology on the isomorphism classes of $R$-points of $\mathfrak{X}$. We prove expected properties of the resulting topological spaces including functoriality. Then, we extend the definition to the case when $R$ is the ring of adeles of some global field. Finally, we use this last definition to strengthen the local-global compatibility for stacky curves of Bhargava--Poonen to a strong approximation result.