ON SERRE'S CONJECTURE FOR MOD GALOIS REPRESENTATIONS OVER TOTALLY REAL FIELDS

摘要

In 1987 Serre conjectured that any mod l 2-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 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 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 we show how it implies the conjecture.