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

Let $K$ be a local function field of characteristic $l$, $\mathbb{F}$ be a finite field over ${\mathbb{F}}_p$ where $l\mathrm{\ne }p$, 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 $R_{\bar{\rho }}^{\mathrm{\square }}$. 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 $l\mathrm{\ne }p$, obtained by Shotton.