Higher congruence companion forms

Abstract

For a rational prime p � 3 we consider p-ordinary, Hilbert modular newforms f of
weight k � 2 with associated p-adic Galois representations �f and mod pn reductions �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 �f;n to decomposition groups above p split then f has a
companion form g modulo pn (in the sense that �f;n � �g;n
�k