Reducible Galois representations and arithmetic homology for $\mathrm{GL}\left(4\right)$
Annales Mathématiques Blaise Pascal, Volume 25 (2018) no. 2, pp. 207-246.

We prove that a sum of two odd irreducible two-dimensional Galois representations with squarefree relatively prime Serre conductors is attached to a Hecke eigenclass in the homology of a subgroup of $\mathrm{GL}\left(4,ℤ\right)$, with the level, nebentype, and coefficient module of the homology predicted by a generalization of Serre’s conjecture to higher dimensions. To do this we prove along the way that any Hecke eigenclass in the homology of a congruence subgroup of a maximal parabolic subgroup of $\mathrm{GL}\left(n,ℚ\right)$ has a reducible Galois representation attached, where the dimensions of the components correspond to the type of the parabolic subgroup. Our main new tool is a resolution of $ℤ$ by $\mathrm{GL}\left(n,ℚ\right)$-modules consisting of sums of Steinberg modules for all subspaces of ${ℚ}^{n}$.

Published online:
DOI: 10.5802/ambp.375
Classification: 11F75,  11R80
Keywords: Galois representations, arithmetic homology
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publisher = {Universit\'e Clermont Auvergne, Laboratoire de math\'ematiques Blaise Pascal},
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Avner Ash; Darrin Doud. Reducible Galois representations and arithmetic homology for $\protect \mathrm{GL}(4)$. Annales Mathématiques Blaise Pascal, Volume 25 (2018) no. 2, pp. 207-246. doi : 10.5802/ambp.375. https://ambp.centre-mersenne.org/articles/10.5802/ambp.375/

[1] Avner Ash Galois representations attached to mod $p$ cohomology of $\mathrm{GL}\left(n,\mathbf{Z}\right)$, Duke Math. J., Volume 65 (1992) no. 2, pp. 235-255 | Article | MR: 1150586

[2] Avner Ash Unstable cohomology of $\mathrm{SL}\left(n,𝒪\right)$, J. Algebra, Volume 167 (1994) no. 2, pp. 330-342 | Article | MR: 1283290

[3] Avner Ash Direct sums of $\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{4pt}{0ex}}p$ characters of $\mathrm{G}\mathrm{al}\left(\overline{ℚ}/ℚ\right)$ and the homology of $GL\left(n,ℤ\right)$, Commun. Algebra, Volume 41 (2013) no. 5, pp. 1751-1775 | Article | MR: 3062822

[4] Avner Ash Comparison of Steinberg modules for a field and a subfield, J. Algebra, Volume 507 (2018), pp. 200-224 | Article | MR: 3807047

[5] Avner Ash; Darrin Doud Reducible Galois representations and the homology of $\mathrm{GL}\left(3,ℤ\right)$, Int. Math. Res. Not., Volume 5 (2014), pp. 1379-1408 | MR: 3178602

[6] Avner Ash; Darrin Doud Galois representations attached to tensor products of arithmetic cohomology, J. Algebra, Volume 465 (2016), pp. 81-99 | Article | MR: 3537816

[7] Avner Ash; Darrin Doud Relaxation of strict parity for reducible Galois representations attached to the homology of $\mathrm{GL}\left(3,ℤ\right)$, Int. J. Number Theory, Volume 12 (2016) no. 2, pp. 361-381 | Article | MR: 3461437

[8] Avner Ash; Darrin Doud; David Pollack Galois representations with conjectural connections to arithmetic cohomology, Duke Math. J., Volume 112 (2002) no. 3, pp. 521-579 | Article | MR: 1896473

[9] Avner Ash; Paul E. Gunnells; Mark McConnell Torsion in the cohomology of congruence subgroups of $\mathrm{SL}\left(4,ℤ\right)$ and Galois representations, J. Algebra, Volume 325 (2011), pp. 404-415 | Article | MR: 2745546

[10] Avner Ash; Warren Sinnott An analogue of Serre’s conjecture for Galois representations and Hecke eigenclasses in the mod $p$ cohomology of $\mathrm{GL}\left(n,\mathbf{Z}\right)$, Duke Math. J., Volume 105 (2000) no. 1, pp. 1-24 | Article | MR: 1788040

[11] Avner Ash; Glenn Stevens Cohomology of arithmetic groups and congruences between systems of Hecke eigenvalues, J. Reine Angew. Math., Volume 365 (1986), pp. 192-220 | MR: 826158

[12] Avner Ash; Pham Huu Tiep Modular representations of $\mathrm{GL}\left(3,{\mathbf{F}}_{p}\right)$, symmetric squares, and mod-$p$ cohomology of $\mathrm{GL}\left(3,\mathbf{Z}\right)$, J. Algebra, Volume 222 (1999) no. 2, pp. 376-399 | Article | MR: 1727178

[13] Armand Borel; Jean-Pierre Serre Corners and arithmetic groups, Comment. Math. Helv., Volume 48 (1973), pp. 436-491 | Article | MR: 0387495 | Zbl: 0274.22011

[14] Kenneth S. Brown Cohomology of groups, Graduate Texts in Mathematics, Volume 87, Springer, 1994, x+306 pages | MR: 1324339

[15] Stephen R. Doty; Grant Walker The composition factors of ${\mathbf{F}}_{p}\left[{x}_{1},{x}_{2},{x}_{3}\right]$ as a $\mathrm{GL}\left(3,p\right)$-module, J. Algebra, Volume 147 (1992) no. 2, pp. 411-441 | Article | MR: 1161301

[16] Florian Herzig The weight in a Serre-type conjecture for tame $n$-dimensional Galois representations, Duke Math. J., Volume 149 (2009) no. 1, pp. 37-116 | Article | MR: 2541127

[17] James E. Humphreys Modular representations of finite groups of Lie type, London Mathematical Society Lecture Note Series, Volume 326, Cambridge University Press, 2006, xvi+233 pages | MR: 2199819

[18] Chandrashekhar Khare; Jean-Pierre Wintenberger Serre’s modularity conjecture. I, Invent. Math., Volume 178 (2009) no. 3, pp. 485-504 | Article | MR: 2551763

[19] Chandrashekhar Khare; Jean-Pierre Wintenberger Serre’s modularity conjecture. II, Invent. Math., Volume 178 (2009) no. 3, pp. 505-586 | Article | MR: 2551764

[20] Mark Kisin Modularity of 2-adic Barsotti-Tate representations, Invent. Math., Volume 178 (2009) no. 3, pp. 587-634 | Article | MR: 2551765

[21] Derek J. S. Robinson A course in the theory of groups, Graduate Texts in Mathematics, Volume 80, Springer, 1996, xviii+499 pages | Article | MR: 1357169

[22] Peter Scholze On torsion in the cohomology of locally symmetric varieties, Ann. Math., Volume 182 (2015) no. 3, pp. 945-1066 | Article | MR: 3418533

[23] Jean-Pierre Serre Sur les représentations modulaires de degré $2$ de $\mathrm{G}\mathrm{al}\left(\overline{\mathbf{Q}}/\mathbf{Q}\right)$, Duke Math. J., Volume 54 (1987) no. 1, pp. 179-230 | Article | MR: 885783

[24] Goro Shimura Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, Volume 11, Princeton University Press, 1994, xiv+271 pages | MR: 1291394 | Zbl: 0872.11023

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