In 1987 Serre conjectured that any mod -dimensional irreducible odd representation of the absolute Galois group of the rationals came from a modular form in a precise way. We present a generalization of this conjecture to 2-dimensional representations of the absolute Galois group of a totally real field where is unramified. The hard work is in formulating an analogue of the weight part of Serre's conjecture. Serre furthermore asked whether his conjecture could be rephrased in terms of a “mod Langlands philosophy.” Using ideas of Emerton and Vignéras, we formulate a mod local-global principle for the group , where is a quaternion algebra over a totally real field, split above and at or infinite places, and we show how it implies the conjecture.
"On Serre's conjecture for mod Galois representations over totally real fields." Duke Math. J. 155 (1) 105 - 161, 1 October 2010. https://doi.org/10.1215/00127094-2010-052