A basic but difficult question in the analytic theory of automorphic forms is: given a reductive group G and a representation r of its L-group, how many automorphic representations of bounded analytic conductor are there? In this talk I will present an answer to this question in the case that G is a torus over a number field.

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.

Abstract: Motivated by questions of Zagier, Lorscheid develops in his Ph. D thesis the theory of graphs of Hecke operators. These graphs encode the action of Hecke operators on automorphic forms over a global function field. In this talk, I intend to present this theory and connect it with the Hall algebras theory. This connection allow us to describe these graphs for an elliptic curve.  

In his seminal work on modular curves and the Eisenstein ideal, Mazur studied the existence of congruences between certain Eisenstein series and newforms, proving that Eisenstein ideals associated to weight 2 cusp forms of prime level are locally principal. In this talk, we re-examine Eisenstein congruences, incorporating a notion of "depth of congruence," in order to understand the local structure of Eisenstein ideals associated to weight 2 cusp forms of squarefree level.

In this talk we will discuss an arithmetic analogue of the gonality of a curve over a number field: the smallest positive integer e such that the points of residue degree bounded by e are infinite.  By work of Faltings, Harris--Silverman and Abramovich--Harris, it is well-understood when this invariant is 1, 2, or 3; by work of Debarre--Fahlaoui these criteria do not generalize to e at least 4.  We will study this invariant using the auxiliary geometry of a surface containing the curve and devote particular attention to scenarios under which we can guarantee that this invariant is actually equal to the gonality . This is joint work with Geoffrey Smith.