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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 $Jac(X_n)[p^infty](\overline{k})$ 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

$G_n = G_n^{et} \times G_n^{m} \times G_n^{ll}$ of $G$ into its etale, multiplicative, and

local-local components, as well as an equality $G_n(\overline{k}) = G_n^{et}(\overline{k})$.

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 $\overline{k}$-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.