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SERRE WEIGHTS FOR LOCALLY REDUCIBLE TWO-DIMENSIONAL GALOIS REPRESENTATIONS

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  23 May 2014

Fred Diamond  and
David Savitt
Fred Diamond
Affiliation:
Department of Mathematics, King’s College London, UK ([email protected])
David Savitt
Affiliation:
Department of Mathematics, University of Arizona, USA ([email protected])
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Abstract

Let $F$ be a totally real field, and $v$ a place of $F$ dividing an odd prime $p$. We study the weight part of Serre’s conjecture for continuous totally odd representations $\overline{{\it\rho}}:G_{F}\rightarrow \text{GL}_{2}(\overline{\mathbb{F}}_{p})$ that are reducible locally at $v$. Let $W$ be the set of predicted Serre weights for the semisimplification of $\overline{{\it\rho}}|_{G_{F_{v}}}$. We prove that, when $\overline{{\it\rho}}|_{G_{F_{v}}}$ is generic, the Serre weights in $W$ for which $\overline{{\it\rho}}$ is modular are exactly the ones that are predicted (assuming that $\overline{{\it\rho}}$ is modular). We also determine precisely which subsets of $W$ arise as predicted weights when $\overline{{\it\rho}}|_{G_{F_{v}}}$ varies with fixed generic semisimplification.

Keywords

Galois representationsSerre’s conjectureBreuil modules

MSC classification

Primary: 11F80: Galois representations
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 14 , Issue 3 , July 2015 , pp. 639 - 672
Copyright
© Cambridge University Press 2014 

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