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

Abstract

Let F/Q be a CM field where p splits completely and (r) over bar : Gal((Q) over bar /F) -> GL(3)((F) over bar (p)) a continuous modular Galois representation. Assume that (r) over bar is non-ordinary and non-split reducible (niveau 2) at a place w above p. We show that the isomorphism class of (r) over bar vertical bar(Gal((F) over barw/Fw)) is determined by the GL(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 infinity, our methods establish the modularity of a certain shadow weight.