Reducible Galois representations and the homology of \(\mathrm{GL}(3, \mathbb{Z})\) (Q2878711)

Let \(\rho: \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\rightarrow \mathrm{GL}(3,\overline{\mathbb{F}}_p)\), \(p>3 \) prime, be a continuous Galois representation which is isomorphic to a direct sum of a character and a two-dimensional, odd, irreducible representation. The authors prove that under the condition that the Serre conductor \(N\) of \(\rho\) is squarefree, there is a Hecke eigenclass in the homology \(H_3(\Gamma_0(3, N), V\otimes \varepsilon)\) that is attached to \(\rho\), where \(\varepsilon\) is the nebentype of \(\rho\) and \(V\) is an irreducible admissible \(\overline{\mathbb{F}}_p[\mathrm{GL}(3,\mathbb{F}_p]\)-module. In addition, for any such \(\rho\), there may be several predicted values for \(V\), the authors show that all of them yield eigenclasses with \(\rho\) attached.