  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 GL_2 in characteristic p and generalizes a result of
Serre for classical modular forms. Finally we study the fibres 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.
