Heights of rational points on (some) stacks

The study of rational points on algebraic stacks over global fields is in many respects very similar to the familiar world of Diophantine geometry of schemes.  But one key element of that world is missing; a theory of heights.  I will propose such a theory and explain how it recovers many already-in-use notions of complexity for points on stacks, while also generating new ones.  I talked about this project in the Stanford number theory seminar in spring 2018; I will explain the basic ideas again and then talk about some examples we understand better now than we did then, such as points on the moduli stack of abelian varieties, and discuss some questions that remain open for future work.