On mod p local–global compatibility for GL 3 ( Q p ) in the non‐ordinary case

Let F / Q be a CM field where p splits completely andr ¯ : Gal ( Q ¯ / F ) → GL 3 ( F ¯ p )a continuous modular Galois representation. Assume that r ¯ is non‐ordinary and non‐split reducible (niveau 2) at a place w above p . We show that the isomorphism class ofr ¯| Gal ( F ¯ w / F w )is determined by theGL 3 ( F w ) ‐action on the space of mod p algebraic automorphic forms using the refined Hecke action of Herzig, Le and Morra [ Compos. Math . 153 (2017) 2215–2286]. We also give a nearly optimal weight elimination result for niveau 2 Galois representations compatible with the explicit conjectures of Gee, Herzig and Savitt [ J. Eur. Math. Soc ., to appear] and Herzig [ Duke Math. J . 149 (2009) 37–116]. Moreover, we prove the modularity of certain Serre weights, in particular, when the Fontaine–Laffaille invariant takes special value ∞ , our methods establish the modularity of a certain shadow weight.