On mod $p$ local-global compatibility for$\text{GL}_{3}$ in the ordinary case. (25th August 2017)

• Record Type: Journal Article
• Authors: Herzig, Florian; Le, Daniel; Morra, Stefano
• Is Part Of: Compositio mathematica. Volume 153:Number 11(2017)
• Page Start: 2215
• Page End: 2286
• Publication Date: 2017-08-25
• Subjects: 11F80 (primary), -- 11S37, -- 22E50 (secondary); Galois representations, -- mod p Langlands program, -- Serre-type conjectures; Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X17007357
• Languages: English
• ISSNs: 0010-437X

Abstract

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 $\overline{r}:\operatorname{Gal}(\overline{F}/F)\rightarrow \text{GL}_{3}(\overline{\mathbb{F}}_{p})$ is a continuous irreducible Galois representation such that $\overline{r}|_{\operatorname{Gal}(\overline{F}_{w}/F_{w})}$ is upper-triangular, maximally non-split, and generic. If $\overline{r}$ is automorphic, and some suitable technical conditions hold, we show that $\overline{r}|_{\operatorname{Gal}(\overline{F}_{w}/F_{w})}$ can be recovered from the $\text{GL}_{3}(F_{w})$ -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 $\overline{r}$, show the existence of an ordinary lifting of $\overline{r}$, and prove the freeness of certain Taylor–Wiles patched modules in this context. We also show the existence of many Galois representations $\overline{r}$ to which our main theorem applies.