On Serre's conjecture for mod l Galois representations over totally real fields
Abstract
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.
Keywords
Cite
@article{arxiv.0810.2106,
title = {On Serre's conjecture for mod l Galois representations over totally real fields},
author = {Kevin Buzzard and Fred Diamond and Frazer Jarvis},
journal= {arXiv preprint arXiv:0810.2106},
year = {2019}
}
Comments
Version 5: Our Shimura variety conventions differ from Emerton's (see footnote 4) and so (a) some chi's needed to be changed to chi^{-1}'s in section 4 and (b) our Eichler-Shimura relation needed fixing too (also in section 4).