On Serre's conjecture for mod l Galois representations over totally real fields

Details On Serre's conjecture for mod l Galois representations over totally real fields On Serre's conjecture for mod l Galois representations over totally real fields

2008-10-13

arxiv.org/abs/0810.2106v5

Mathematics

Number Theory

Version 5: Our Shimura variety conventions differ from Emerton's (see footnote 4) and so (a) some chi's needed to be changed t

Scientific paper

In 1987 Serre conjectured that any mod l ("ell", not "1") 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.

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