Bryden Cais

Institution:

University of Arizona

Time:

Thursday, June 6, 2019 - 3:00pm to 3:50pm

Location:

RH 306

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.