Bryden Cais


University of Arizona


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



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.