We establish a version of "level-lowering" for mod p Galois representations arising from the reductions of representations associated to Hilbert modular forms. In particular, we show that level-lowering can be easily achieved if one replaces the base field with a suitable solvable extension. This is often enough for applications to proving the modularity of p-adic representations.
C. M. Skinner. A. J. Wiles. "Base change and a problem of Serre." Duke Math. J. 107 (1) 15 - 25, 1 March 2001. https://doi.org/10.1215/S0012-7094-01-10712-6