## Duke Mathematical Journal

### On Serre's conjecture for mod $\ell$ Galois representations over totally real fields

#### Abstract

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.

#### Article information

Source
Duke Math. J. Volume 155, Number 1 (2010), 105-161.

Dates
First available: 23 September 2010

http://projecteuclid.org/euclid.dmj/1285247220

Digital Object Identifier
doi:10.1215/00127094-2010-052

Zentralblatt MATH identifier
05816156

Mathematical Reviews number (MathSciNet)
MR2730374

#### Citation

Buzzard, Kevin; Diamond, Fred; Jarvis, Frazer. On Serre's conjecture for mod ℓ Galois representations over totally real fields. Duke Mathematical Journal 155 (2010), no. 1, 105--161. doi:10.1215/00127094-2010-052. http://projecteuclid.org/euclid.dmj/1285247220.

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