Weak abelian direct summands and irreducibility of Galois representations - Böckle, Gebhard; Hui, Chun Yin (Mathematische Annalen, 2025)
Let $$ρ _\ell $$be a semisimple $$\ell $$-adic representation of a number field K that is unramified almost everywhere. We introduce a new notion called weak abelian direct summands of $$ρ _\ell $$and completely characterize them, for example, if the algebraic monodromy of $$ρ _\ell $$is connected. If $$ρ _\ell $$is in addition E-rational for some number field E, we prove that the weak abelian direct summands are locally algebraic (and thus de Rham). We also show that the weak abelian parts of a connected semisimple Serre compatible system form again such a system. Using our results on weak abelian direct summands, when K is totally real and $$ρ _\ell $$is the three-dimensional $$\ell $$-adic representation attached to a regular algebraic cuspidal automorphic, not necessarily polarizable representation $$π $$of $$GL_3(\mathbb {A}_K)$$together with an isomorphism $$\mathbb {C}\simeq {\overline{\mathbb {Q}}}_\ell $$, we prove that $$ρ _\ell $$is irreducible. We deduce in this case also some $$\ell $$-adic Hodge theoretic properties of $$ρ _\ell $$if $$\ell $$belongs to a Dirichlet density one set of primes.