On Serre’s Conjecture for mod ℓ Galois representations over totally real fields

Abstract. In 1987 Serre conjectured that any mod ℓ two-dimensional irreducible odd representation of the absolute Galois group of the rationals came from a modular form in a precise way. We present a generalisation of this conjecture to 2-dimensional representations of the absolute Galois group of a totally real field where ℓ 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 ℓ Langlands philosophy”. Using ideas of Emerton and Vigneras, we formulate a mod ℓ local-global principle for the group D ∗ , where D is a quaternion algebra over a totally real field, split above ℓ and at 0 or 1 infinite places, and show how it implies the conjecture. 1.