On mod local-global compatibility for in the ordinary case

2017/08/25 by Florian Herzig, Daniel Le, Stefano Morra · 1 citation
Mathematics · #Algebraic Geometry and Number Theory #Advanced Algebra and Geometry #Homotopy and Cohomology in Algebraic Topology

paper · doi:10.1112/s0010437x17007357

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 r:Gal(F/F)→ GL₃(\mathbbF_p) is a continuous irreducible Galois representation such that r|_Gal(F_w/F_w) is upper-triangular, maximally non-split, and generic. If r is automorphic, and some suitable technical conditions hold, we show that r|_Gal(F_w/F_w) can be recovered from the GL₃(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 r , show the existence of an ordinary lifting of r , and prove the freeness of certain Taylor–Wiles patched modules in this context. We also show the existence of many Galois representations r to which our main theorem applies.

Cited by

• Serre weights and Breuil's lattice conjecture in dimension three
• Moduli of Fontaine--Laffaille representations and a mod-p local-global compatibility result

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