Local Galois Deformation Rings and A Function Field Breuil--Mezard Conjecture

Abstract

In this dissertation we study the geometry of Galois deformation rings for local function fields in view of the inertial local Langlands correspondence. More precisely, let $K$ be a local function field of characteristic $p$, $\mathbb F$ be a finite field over $\mathbb{F}_l$ with $l \ne p$. Let $\bar \rho: \textup{Gal}_K \rightarrow \textup{GL}_n (\mathbb F)$ be a continuous residual Galois representation of the absolute Galois group of $K$. We apply the Taylor--Wiles--Kisin patching method over certain global function fields to construct a cycle map, from mod $l$ representations of $\textup{GL}n (O_K)$ to the mod $l$ fibers of the framed deformation ring $R{\bar \rho}^{\square, \chi}$. This allows us to obtain a function field analog of the Breuil--M'ezard conjecture, which essentially predicts a certain form of ``mod $l$ inertial local Langlands correspondence'' that is compatible with the inertial local Langlands correspondence. We then use the technique of close fields (due to Deligne--Kazhdan) to compare our result with the Breuil--M'ezard conjecture for local number fields obtained by Shotton.