Reducible Galois representations and arithmetic homology for $\mathrm{GL}\left(4\right)$
Annales Mathématiques Blaise Pascal, Tome 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 ${ℚ}^{n}$.

Publié le : 2018-11-28
DOI : https://doi.org/10.5802/ambp.375
Classification : 11F75,  11R80
Mots clés: Galois representations, arithmetic homology
@article{AMBP_2018__25_2_207_0,
author = {Avner Ash and Darrin Doud},
title = {Reducible Galois representations and arithmetic homology for $\protect \mathrm{GL}(4)$},
journal = {Annales Math\'ematiques Blaise Pascal},
publisher = {Universit\'e Clermont Auvergne, Laboratoire de math\'ematiques Blaise Pascal},
volume = {25},
number = {2},
year = {2018},
pages = {207-246},
doi = {10.5802/ambp.375},
language = {en},
url = {ambp.centre-mersenne.org/item/AMBP_2018__25_2_207_0/}
}
Ash, Avner; Doud, Darrin. Reducible Galois representations and arithmetic homology for $\protect \mathrm{GL}(4)$. Annales Mathématiques Blaise Pascal, Tome 25 (2018) no. 2, pp. 207-246. doi : 10.5802/ambp.375. https://ambp.centre-mersenne.org/item/AMBP_2018__25_2_207_0/

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