Higher congruence companion forms

For a rational prime p≥3 we consider p-ordinary, Hilbert modular newforms f of weight k≥2 with associated p-adic Galois representations \rho_f and mod p^n reductions \rho_{f,n}. Under suitable hypotheses on the size of the image, we use deformation theory and modularity lifting to show that if the restrictions of \rho_{f,n} to decomposition groups above p split then f has a companion form g modulo pn (in the sense that \rho_{f,n} \sim \rho_{g,n}\otimes \chi^{k−1}).