Workshop

It is often useful to regard modular forms as in the larger space of p-adic overconvergent modular forms, so that p-adic analytic techniques can be used to study them. The geometric interpretation of this is an eigenvariety, which is a rigid analytic space parametrizing finite-slope overconvergent Hecke eigenforms. For Hilbert modular forms, Andreatta-Iovita-Pilloni constructed p-adic families of modular sheaves as well as the eigenvariety. Moreover, for Hilbert modular forms, it makes sense to consider an intermediate notion - the partially classical overconvergent forms. I will talk about the construction of the eigenvariety in this scenario following the approach of AIP. As an application, it can be proved that the Galois representation associated to a partially classical Hilbert Hecke eigenform is partially de Rham.

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'll explore generalizations of Mazur's result to squarefree level, focusing on recent work, joint with P. Wake and C. Wang-Erickson, about a non-optimal level N that is the product of two distinct primes and where the Galois deformation ring is not expected to be Gorenstein. First, we will outline a Galois-theoretic criterion for the deformation ring to be as small as possible, and when this criterion is satisfied, deduce an R=T theorem. Then we'll discuss some of the techniques required to computationally verify the criterion.

The supersingular locus of a unitary Shimura variety parametrizes supersingular abelian varieties with an action of a quadratic imaginary field, meeting a certain signature condition. In low-dimensional cases these supersingular loci have a particular kind of structure: for example, every irreducible component is isomorphic to a Deligne-Lusztig variety. We’ll consider these low-dimensional cases for motivation, and then see how the structure of the supersingular locus can be very different in general.

I will discuss duality theorems for p-adic pro-étale cohomology of analytic curves. This is a joint work with Pierre Colmez and Sally Giles.