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On mod $p$ local-global compatibility for $\text{GL}_{3}$ in the ordinary case

Published online by Cambridge University Press:  25 August 2017

Florian Herzig
Affiliation:
Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, ON M5S 2E4, Canada email herzig@math.toronto.edu
Daniel Le
Affiliation:
Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, ON M5S 2E4, Canada email le@math.toronto.edu
Stefano Morra
Affiliation:
Institut Montpelliérain Alexander Grothendieck, Université de Montpellier, Case courrier 051, Place Eugène Bataillon, 34095 Montpellier Cedex, France email stefano.morra@umontpellier.fr

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.

Type
Research Article
Copyright
© The Authors 2017 

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On mod $p$ local-global compatibility for $\text{GL}_{3}$ in the ordinary case
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