The Breuil--Mézard conjecture for function fields

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 $\overline{\rho }:G_K\to {\text{GL}}_n(\mathbb{F})$ 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--Mézard conjecture. Then we use the technique of close fields to show that our result is compatible with the Breuil-Mézard conjecture for local number fields in the case of $l\ne p$, obtained by Shotton.