Higher congruence companion forms

Abstract

For a rational prime $p \geq 3$ we consider $p$-ordinary, Hilbert modular newforms $f$ of weight $k\geq 2$ 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 $p^n$ (in the sense that $\rho_{f,n}\sim \rho_{g,n}\otimes\chi^{k-1}$).