# Number Theory Seminar

Let {X_n} be a Z_p-tower of smooth projective curves over a perfect field k of characteristic p that totally ramifies over a finite, nonempty set of points of X_0 and is unramified elsewhere. In analogy with the case of number fields, Mazur and Wiles studied the growth of the p-parts of the class groups Jac(X_n)[p^infty](kbar) as n-varies, and proved that these naturally fit together to yield a module that is finite and free over the Iwasawa algebra. We introduce a novel perspective by proposing to study growth of the full p-divisible group G_n:=Jac(X_n)[p^infty], which may be thought of as the p-primary part of the *motivic class group* Jac(X_n). One has a canonical decomposition G_n = G_n^et x G_n^m x G_n^ll of G into its etale, multiplicative, and local-local components, as well as an equality G_n(kbar) = G_n^et(kbar). Thus, the work of Mazur and Wiles captures the etale part of G_n, so also (since Jacobians are principally polarized) the multiplicative part: both of these p-divisible subgroups satisfy the expected structural and control theorems in the limit. In contrast, the local-local components G_n^ll are far more mysterious (they can not be captured by kbar points), and indeed the tower they form has no analogue in the number field setting. This talk will survey this circle of ideas, and will present new results and conjectures on the behavior of the local-local part of the tower {G_n}.