Towers of function fields and a-numbers

Geometric Iwasawa theory studies the behavior of p-adic towers of curves. Classically, the focus has been on the p-part of class groups, mirroring Iwasawa theory for number fields. However, there are many interesting features of Iwasawa theory for curves that have no number field analogy. The p-part of the class group is only a small part of the p-divisible group, a much more intricate object with no number field analogy. In this talk I will survey various results and conjectures about the behavior of p-divisible groups along towers of curves. I will focus on recent joint work with Jeremy Booher, Bryden Cais, and James Upton where we study the behavior of a-numbers along the tower.