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Highly reducible Galois representations attached to the homology of $ \mathrm{GL}(n,\mathbb{Z})$


Authors: Avner Ash and Darrin Doud
Journal: Proc. Amer. Math. Soc. 143 (2015), 3801-3813
MSC (2010): Primary 11F75; Secondary 11F67, 11F80
DOI: https://doi.org/10.1090/S0002-9939-2015-12559-2
Published electronically: March 18, 2015
MathSciNet review: 3359572
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Abstract: Let $ n\ge 1$ and $ \mathbb{F}$ be an algebraic closure of a finite field of characteristic $ p>n+1$. Let $ \rho :G_{\mathbb{Q}}\to \mathrm {GL}(n,\mathbb{F})$ be a Galois representation that is isomorphic to a direct sum of a collection of characters and an odd $ m$-dimensional representation $ \tau $. We assume that $ m=2$ or $ m$ is odd, and that $ \tau $ is attached to a homology class in degree $ m(m-1)/2$ of a congruence subgroup of $ \mathrm {GL}(m,\mathbb{Z})$ in accordance with the main conjecture of an earlier work of the authors and Pollack. We also assume a certain compatibility of $ \tau $ with the parity of the characters and that the Serre conductor of $ \rho $ is square-free. We prove that $ \rho $ is attached to a Hecke eigenclass in $ H_t(\Gamma ,M)$, where $ \Gamma $ is a subgroup of finite index in $ \rm {SL}$ $ (n,\mathbb{Z})$, $ t=n(n-1)/2$ and $ M$ is an $ \mathbb{F}\Gamma $-module. The particular $ \Gamma $ and $ M$ are as predicted by the main conjecture of an earlier work. The method uses modular cosymbols, as in a recent work of the first author.


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Additional Information

Avner Ash
Affiliation: Department of Mathematics, Boston College, Chestnut Hill, Massachusetts 02467
Email: Avner.Ash@bc.edu

Darrin Doud
Affiliation: Department of Mathematics, Brigham Young University, Provo, Utah 84602
Email: doud@math.byu.edu

DOI: https://doi.org/10.1090/S0002-9939-2015-12559-2
Keywords: Serre's conjecture, Hecke operator, Galois representation, character
Received by editor(s): March 4, 2014
Received by editor(s) in revised form: June 4, 2014
Published electronically: March 18, 2015
Additional Notes: The first author thanks the NSA for support of this research through NSA grant H98230-13-1-0261. This manuscript is submitted for publication with the understanding that the United States government is authorized to reproduce and distribute reprints.
Communicated by: Romyar T. Sharifi
Article copyright: © Copyright 2015 American Mathematical Society

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