Mod $p$ local-global compatibility for $\mathrm{GSp}_4(\mathbb{Q}_p)$ in the ordinary case

TL;DR

This paper proves a local-global compatibility result for mod p Galois representations associated with automorphic forms on GSp_4 over a totally real field, under certain genericity and Taylor-Wiles assumptions.

Contribution

It establishes the mod p local-global compatibility for GSp_4 in the ordinary case, linking the Galois representation to the Hecke action on automorphic forms.

Findings

GSp_4 mod p automorphic forms determine local Galois representations.

Under genericity conditions, the local Galois representation is recovered from automorphic data.

The result applies to totally real fields where p splits completely.

Abstract

Let $F$ be a totally real field of even degree in which $p$ splits completely. Let $\overline{r}:G_F \rightarrow \mathrm{GSp}_4(\overline{\mathbb{F}}_p)$ be a modular Galois representation unramified at all finite places away from $p$ and upper-triangular, maximally nonsplit, and of parallel weight at places dividing $p$. Fix a place $w$ dividing $p$. Assuming certain genericity conditions and Taylor--Wiles assumptions, we prove that the $\mathrm{GSp}_4(F_w)$-action on the corresponding Hecke-isotypic part of the space of mod $p$ automorphic forms on a compact mod center form of $\mathrm{GSp}_4$ with infinite level at $w$ determines $\overline{r}|_{G_{F_w}}$.