The Breuil--M\'ezard conjecture for function fields

TL;DR

This paper establishes a function field analog of the Breuil--Mézard conjecture by constructing a mod p cycle map using the Taylor-Wiles-Kisin method and demonstrates compatibility with the number field case via close fields techniques.

Contribution

It introduces a new approach to the Breuil--Mézard conjecture for function fields using global methods and shows compatibility with existing number field results.

Findings

Constructed a mod p cycle map for function fields.

Proved the function field analog of the Breuil--Mézard conjecture.

Established compatibility with number field cases.

Abstract

Let $K$ be a local function field of characteristic $l$, $\mathbb{F}$ be a finite field over $\mathbb{F}_p$ where $l \ne p$, and $\overline{\rho}: G_K \rightarrow \text{GL}_n (\mathbb{F})$ be a continuous representation. We apply the Taylor-Wiles-Kisin method over certain global function fields to construct a mod $p$ cycle map $\overline{\text{cyc}}$, from mod $p$ representations of $\text{GL}_n (\mathcal{O}_K)$ to the mod $p$ fibers of the framed universal deformation ring $R_{\overline{\rho}}^\square$. This allows us to obtain a function field analog of the Breuil--M\'ezard conjecture. Then we use the technique of close fields to show that our result is compatible with the Breuil-M\'ezard conjecture for local number fields in the case of $l \ne p$, obtained by Shotton.