Galois representation of partially classical Hilbert modular forms

Abstract: Let F be a totally real field. A Hilbert modular form is a section of a modular sheaf, defined over the whole Hilbert modular variety associated to F, while a p-adic overconvergent form is defined only over a strict neighborhood of the ordinary locus. For each subset I of the primes of F above p, one has the intermediate notion of I-classical Hilbert modular forms by replacing ordinary by I-ordinary. Given an overconvergent Hecke eigenform f, we have the associated Galois representation $\rho$, which is well-known to be de Rham at p when f is classical. We prove that $\rho$ is I-de Rham when f is I-classical. The idea is to p-adically deform f in the weight direction of the complement of I, and knowing that classical points are dense and I-de Rham points are closed when the I-Hodge Tate weights are fixed.

algebraic geometrynumber theory

Audience: researchers in the topic

Séminaire de géométrie arithmétique et motivique (Paris Nord)