Abstract
Let F/Q be a CM field where p splits completely and ¯ r : Gal(Q/F) → GL 3 (Fp) a continuous modular Galois representation. Assume that ¯ r is non-ordinary and nonsplit reducible (niveau 2) at a place w above p. We show that the isomorphism class of ¯ r| Gal(F w /Fw) is determined by the GL 3 (Fw)-action on the space of mod p algebraic automorphic forms by using the refined Hecke action of [HLM]. We also give a nearly optimal weight elimination result for niveau two Galois representations compatible with the explicit conjectures of [Her09] and [GHS]. Moreover, we prove the modularity of certain Serre weights, in particular, when the Fontaine-Laffaille invariant takes special value ∞, our methods provide with the modularity of a certain shadow weight.