Patching and the p-adic Langlands program for GL(2, Q_p)

TL;DR

This paper introduces a novel construction of the p-adic local Langlands correspondence for GL(2, Q_p) using patching methods, providing new insights and proofs related to local-global compatibility in the p-adic Langlands program.

Contribution

It offers a new patching-based construction of the p-adic Langlands correspondence for GL(2, Q_p), improving understanding and simplifying proofs of local-global compatibility.

Findings

New construction of p-adic Langlands correspondence for GL(2, Q_p)

Simplified proof of local-global compatibility results

Relaxed hypotheses on local mod p representations

Abstract

We present a new construction of the p-adic local Langlands correspondence for GL(2, Q_p) via the patching method of Taylor--Wiles and Kisin. This construction sheds light on the relationship between the various other approaches to both the local and global aspects of the p-adic Langlands program; in particular, it gives a new proof of many cases of the second author's local-global compatibility theorem, and relaxes a hypothesis on the local mod p representation in that theorem.