Highly reducible Galois representations attached to the homology of $\mathrm {GL}(n,\mathbb {Z})$
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- by Avner Ash and Darrin Doud PDF
- Proc. Amer. Math. Soc. 143 (2015), 3801-3813 Request permission
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.References
- Avner Ash, Direct sums of $\bmod p$ characters of $\textrm {GAL}(\overline {\Bbb Q}/{\Bbb Q})$ and the homology of $GL(n,{\Bbb Z})$, Comm. Algebra 41 (2013), no. 5, 1751–1775. MR 3062822, DOI 10.1080/00927872.2011.649508
- Avner Ash and Darrin Doud, Reducible Galois representations and the homology of $\textrm {GL}(3,\Bbb Z)$, Int. Math. Res. Not. IMRN 5 (2014), 1379–1408. MR 3178602, DOI 10.1093/imrn/rns256
- A. Ash and D. Doud, Relaxation of strict parity for Galois representations attached to the cohomology of $\mathrm {GL}(3,\mathbb {Z})$, submitted.
- Avner Ash, Darrin Doud, and David Pollack, Galois representations with conjectural connections to arithmetic cohomology, Duke Math. J. 112 (2002), no. 3, 521–579. MR 1896473, DOI 10.1215/S0012-9074-02-11235-6
- Avner Ash and Lee Rudolph, The modular symbol and continued fractions in higher dimensions, Invent. Math. 55 (1979), no. 3, 241–250. MR 553998, DOI 10.1007/BF01406842
- Avner Ash and Warren Sinnott, An analogue of Serre’s conjecture for Galois representations and Hecke eigenclasses in the mod $p$ cohomology of $\textrm {GL}(n,\textbf {Z})$, Duke Math. J. 105 (2000), no. 1, 1–24. MR 1788040, DOI 10.1215/S0012-7094-00-10511-X
- Florian Herzig, The weight in a Serre-type conjecture for tame $n$-dimensional Galois representations, Duke Math. J. 149 (2009), no. 1, 37–116. MR 2541127, DOI 10.1215/00127094-2009-036
- Chandrashekhar Khare, Serre’s modularity conjecture: the level one case, Duke Math. J. 134 (2006), no. 3, 557–589. MR 2254626, DOI 10.1215/S0012-7094-06-13434-8
- Chandrashekhar Khare and Jean-Pierre Wintenberger, Serre’s modularity conjecture. I, Invent. Math. 178 (2009), no. 3, 485–504. MR 2551763, DOI 10.1007/s00222-009-0205-7
- Chandrashekhar Khare and Jean-Pierre Wintenberger, Serre’s modularity conjecture. II, Invent. Math. 178 (2009), no. 3, 505–586. MR 2551764, DOI 10.1007/s00222-009-0206-6
- Jean-Pierre Serre, Sur les représentations modulaires de degré $2$ de $\textrm {Gal}(\overline \textbf {Q}/\textbf {Q})$, Duke Math. J. 54 (1987), no. 1, 179–230 (French). MR 885783, DOI 10.1215/S0012-7094-87-05413-5
Additional Information
- Avner Ash
- Affiliation: Department of Mathematics, Boston College, Chestnut Hill, Massachusetts 02467
- MR Author ID: 205374
- Email: Avner.Ash@bc.edu
- Darrin Doud
- Affiliation: Department of Mathematics, Brigham Young University, Provo, Utah 84602
- MR Author ID: 634088
- Email: doud@math.byu.edu
- 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
- © Copyright 2015 American Mathematical Society
- 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
- MathSciNet review: 3359572