Shifting Hecke Eigensystems in Positive Characteristic

We study congruences modulo p between modular forms arising from different contexts. In the first part of the dissertation we use geometric methods to show that the (mod p) PEL Hecke eigensystems associated to a reductive group G coincide with the (mod p) algebraic Hecke eigensystems associated to an inner form of G. In the second part of the dissertation we use cohomological methods to construct weight shiftings for (mod p) automorphic forms associated to a definite quaternion algebra over a totally real field in which p is unramified. In particular, we construct cohomological avatars of the partial Hasse invariants.