Heights of Rational Points on Stacks

I'll report on a project with Matt Satriano and David Zureick-Brown (who will be a Research Member later this semester) which in some sense started out in the MSRI "Rational and Integral Points on Higher-Dimensional Varieties" semester in 2006. Some natural arithmetic objects (like numbers) are parametrized by an algebraic variety, but many others (like elliptic curves, or number fields) are parametrized by more general algebraic objects called "stacks." Rational points on stacks are just as interesting as rational points on varieties, but a very useful tool for the study of rational points, a theory of *heights*, was missing in this case. I'll talk about the definition of heights we developed for this situation, make the case that it's a good definition, and discuss some of the refinements that have come in the time since we proposed it.

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