Congruences between Siegel modular forms on the level of group cohomology

Vector-valued Siegel modular forms may be found in certain cohomology groups with coefficients lying in an irreducible representation of the symplectic group. Using functoriality in the coefficients, we show that the ordinary components of the cohomology are independent of the weight parameter. The meaning of ordinary depends on a choice of parabolic subgroup of GSp(4), giving a particular direction in the change of weight. Our results complement those of Taylor and Tilouine-Urban for the two other possible classes of parabolic subgoups.