On the connected components of moduli spaces of finite flat models

TL;DR

This paper proves the connectedness of the non-ordinary component in moduli spaces of finite flat models for 2D local Galois representations, confirming a conjecture by Kisin and applying it to modularity theorems.

Contribution

It establishes the connectedness of the non-ordinary component in these moduli spaces, confirming Kisin's conjecture and linking local properties to global modularity results.

Findings

Non-ordinary component is connected in the moduli space

Confirmed Kisin's conjecture on connectedness

Applied to compare deformation and Hecke rings in global Galois representations

Abstract

We prove that the non-ordinary component is connected in the moduli spaces of finite flat models of two-dimensional local Galois representations over finite fields. This was conjectured by Kisin. As an application to global Galois representations, we prove a theorem on the modularity comparing a deformation ring and a Hecke ring.