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1 October 2010 On Serre's conjecture for mod Galois representations over totally real fields
Kevin Buzzard, Fred Diamond, Frazer Jarvis
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Duke Math. J. 155(1): 105-161 (1 October 2010). DOI: 10.1215/00127094-2010-052

Abstract

In 1987 Serre conjectured that any mod 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 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 Langlands philosophy.” Using ideas of Emerton and Vignéras, we formulate a mod local-global principle for the group D*, where D is a quaternion algebra over a totally real field, split above and at 0 or 1 infinite places, and we show how it implies the conjecture.

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Kevin Buzzard. Fred Diamond. Frazer Jarvis. "On Serre's conjecture for mod Galois representations over totally real fields." Duke Math. J. 155 (1) 105 - 161, 1 October 2010. https://doi.org/10.1215/00127094-2010-052

Information

Published: 1 October 2010
First available in Project Euclid: 23 September 2010

zbMATH: 1227.11070
MathSciNet: MR2730374
Digital Object Identifier: 10.1215/00127094-2010-052

Subjects:
Primary: 11F41
Secondary: 11F33

Rights: Copyright © 2010 Duke University Press

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Vol.155 • No. 1 • 1 October 2010
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