A topology on points on stacks

Abstract

For a variety over certain topological rings R, like Z[subscript p] or 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 X over many topological rings R, we define a topology on the isomorphism classes of R-points of 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.