On mod p local-global compatibility for GL_3 in the ordinary case

Abstract

Suppose that F/F+ is a CM extension of number fields in which the prime p splits completely and every other prime is unramified. Fix a place w|p of F. Suppose that rbar : Gal(F-bar/F) -> GL_3(Fp-bar) is a continuous irreducible Galois representation such that rbar|_{Gal(F_w-bar/F_w)} is upper-triangular, maximally non-split, and generic. If rbar is automorphic, and some suitable technical conditions hold, we show that rbar|_{\Gal(F_w-bar/F_w)} can be recovered from the GL_3(F_w)-action on a space of mod p automorphic forms on a compact unitary group. On the way we prove results about weights in Serre's conjecture for rbar, show the existence of an ordinary lifting of rbar, and prove the freeness of certain Taylor-Wiles patched modules in this context. We also show the existence of many Galois representations rbar to which our main theorem applies.

revised, 72 pages

References in corpus (3)

• On the automorphy of $l$-adic Galois representations with small residual image
• Potentially crystalline deformation rings and Serre weight conjectures
• Lattices in the cohomology of $U(3)$ arithmetic manifolds

Cited by in corpus (6)

• Diagrams in the mod $p$ cohomology of Shimura curves
• Weight elimination in Serre-type conjectures
• Serre weight conjectures for $p$-adic unitary groups of rank 2
• Serre weights and Breuil's lattice conjecture in dimension three
• Moduli of Fontaine--Laffaille representations and a mod-$p$ local-global compatibility result
• Serre weights for three-dimensional wildly ramified Galois representations