Automorphic Forms and Arithmetic

Euler Systems (ESs) are collections of Galois cohomology classes that verify some co-restriction compatibilities. The key feature of ESs is that they provide a way to bound Selmer groups, thanks to the machinery developed by Rubin, inspired by earlier work of Thaine, Kolyvagin, and Kato. In this talk, I will present joint work with C. Skinner, in which we develop a new method for constructing Euler Systems and apply it to build an ES for the Galois representation attached to the symmetric square of an elliptic modular form. I will stress how this method gives a unifying approach to constructing ESs, in that it can be successfully applied to retrieve most classical ESs (the cyclotomic units ES, the elliptic units ES, Kato's ES, Lei-Loeffler-Zerbes ES for the Rankin-Selberg convolution of two elliptic modular forms...).

Any finite-dimensional p-adic representation of the absolute Galois group of a p-adic local field with imperfect residue field is characterized by its arithmetic and geometric Sen operators defined by Sen and Brinon. We generalize their construction to the fundamental group of a p-adic affine variety with a semi-stable chart, and prove that the module of Sen operators is canonically defined, independently of the choice of the chart. When the representation comes from a Q_p-representation of the fundamental group, we relate the infinitesimal action of inertia subgroups with Sen operators, which is a generalization of a result of Sen and Ohkubo. These Sen operators can be extended continuously to certain infinite-dimensional representations. As an application, we prove that the geometric Sen operators annihilate locally analytic vectors, generalizing a result of Pan.

For a prime number N, Ogg's conjecture states that the torsion in the Jacobian of the modular curve X_0(N) is generated by the cusps. Mazur proved Ogg's conjecture as one of the main theorems in his "Eisenstein ideal" paper. I'll talk about a generalization of Ogg's conjecture for squarefree N and a proof using the Eisenstein ideal. This is joint work with Ken Ribet.

Geometric Iwasawa theory studies the behavior of p-adic towers of curves. Classically, the focus has been on the p-part of class groups, mirroring Iwasawa theory for number fields. However, there are many interesting features of Iwasawa theory for curves that have no number field analogy. The p-part of the class group is only a small part of the p-divisible group, a much more intricate object with no number field analogy. In this talk I will survey various results and conjectures about the behavior of p-divisible groups along towers of curves. I will also discuss what geometric Iwasawa theory for motives should look like and explain new results in this direction.

The $p$-adic Simpson correspondence aims to establish an equivalence between generalized representations and Higgs bundles over a $p$-adic variety. In this talk, we will explain how to upgrade such an equivalence to a twisted isomorphism of moduli stacks in the curve case. This is based on a joint work in progress with Heuer.