On Serre's conjecture for mod $\ell$ Galois representations over totally real fields
Buzzard, Kevin ; Diamond, Fred ; Jarvis, Frazer
Duke Math. J., Tome 151 (2010) no. 1, p. 105-161 / Harvested from Project Euclid
In 1987 Serre conjectured that any mod $\ell$ $2$ -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 $\ell$ 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 $\ell$ Langlands philosophy.” Using ideas of Emerton and Vignéras, we formulate a mod $\ell$ local-global principle for the group $D^*$ , where $D$ is a quaternion algebra over a totally real field, split above $\ell$ and at $0$ or $1$ infinite places, and we show how it implies the conjecture.
Publié le : 2010-10-01
Classification:  11F41,  11F33
@article{1285247220,
     author = {Buzzard, Kevin and Diamond, Fred and Jarvis, Frazer},
     title = {On Serre's conjecture for mod $\ell$ Galois representations over totally real fields},
     journal = {Duke Math. J.},
     volume = {151},
     number = {1},
     year = {2010},
     pages = { 105-161},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1285247220}
}
Buzzard, Kevin; Diamond, Fred; Jarvis, Frazer. On Serre's conjecture for mod $\ell$ Galois representations over totally real fields. Duke Math. J., Tome 151 (2010) no. 1, pp.  105-161. http://gdmltest.u-ga.fr/item/1285247220/