I will introduce higher Hida theory for Hilbert modular forms. This is a theory of higher coherent cohomological ordinary p-adic modular forms, which can be thought of as p-adically interpolating non holomorphic Hilbert modular forms. Then I will explain a Jacquet-Langlands type correspondence with certain quaternionic modular forms, which is explained by the existence of "exotic Hecke correspondences" which exist modulo powers of p. This is joint work with Vincent Pilloni.
