A mod p Jacquet-Langlands relation and Serre filtration via the geometry of Hilbert modular varieties

We consider Hilbert modular varieties in characteristic p with Iwahori level at p and construct a geometric Jacquet- Langlands relation showing that the irreducible components are isomorphic to products of projective bundles over quaternionic Shimura varieties of level prime to p. We use this to establish a relation between mod p Hilbert and quaternionic modular forms that reflects the representation theory of GL² in characteristic p and generalizes a result of Serre for classical modular forms. Finally we study the fibers of the degeneracy map to level prime to p and prove a cohomological vanishing result that is used to associate Galois representations to mod p Hilbert modular forms.