Counting rational points of bounded height on certain stacks
Soumya Sankar (MSRI - Mathematical Sciences Research Institute)
Given a variety over a global field K, one can ask how many K-rational points of bounded height it has. However, a lot of interesting spaces are not varieties. For instance, one might want to formulate the same question for stacks, but there are roadblocks to even defining heights on them. This talk explores heights on stacks in the context of counting rational points on moduli spaces of elliptic curves, which can be described classically as quotients of the upper half plane by congruence subgroups. I will talk about height functions on varieties and how they generalize (or don't generalize) to stacks. I will also explain how one can use height machinery developed recently by Ellenberg, Satriano and Zureick-Brown to answer, for certain integers N, the classical question: how many elliptic curves over Q have a rational N isogeny? This is based on joint work with Brandon Boggess. The talk assumes no prior knowledge of stacks or of the arithmetic of elliptic curves.