Reducible Galois representations and arithmetic homology for $\mathrm{GL}\left(4\right)$

Ash, Avner; Doud, Darrin

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 $\mathrm{GL}(4,\mathbb{Z})$, 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 $\mathrm{GL}(n,\mathbb{Q})$ 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 $\mathbb{Z}$ by $\mathrm{GL}(n,\mathbb{Q})$-modules consisting of sums of Steinberg modules for all subspaces of ${\mathbb{Q}}^n$.

Classification : 11F75, 11R80
Keywords: Galois representations, arithmetic homology

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