Globally realizable components of local deformation rings

TL;DR

This paper demonstrates that certain local deformation ring components are globally realizable independently of the global context, and improves potential automorphy results by replacing a key condition.

Contribution

It shows the independence of local deformation ring components from the global setting and enhances automorphy theorems by weakening the diagonalizability assumption.

Findings

Components are independent of global situation under certain conditions.

Replaces 'potentially diagonalizable' with 'potentially globally realizable' in automorphy results.

Applicable to n-dimensional p-adic potentially semistable local Galois deformation rings.

Abstract

Let n be either 2, or an odd integer greater than 1, and fix a prime p > 2(n + 1). Under standard "adequate image" assumptions, we show that the set of components of n-dimensional p-adic potentially semistable local Galois deformation rings that are seen by potentially automorphic compatible systems of polarizable Galois representations over some CM field is independent of the particular global situation. We also (under the same assumption on n) improve on the main potential automorphy result of [BLGGT14b], replacing "potentially diagonalizable" by "potentially globally realizable".