Patching and the completed homology of locally symmetric spaces

TL;DR

This paper develops a Taylor--Wiles patching method in the derived category for the completed homology of locally symmetric spaces related to GL(n), connecting it to conjectures and the p-adic local Langlands correspondence.

Contribution

It introduces a derived category patching technique for completed homology and links standard conjectures to big R = big T theorems, with special cases relating to p-adic local Langlands.

Findings

Established a derived category patching framework.

Connected conjectures on completed homology to big R = big T theorems.

Related the construction to p-adic local Langlands for GL(2,Q_p).

Abstract

Under an assumption on the existence of p-adic Galois representations, we carry out Taylor--Wiles patching (in the derived category) for the completed homology of the locally symmetric spaces associated to GL(n) over a number field. We use our construction to show that standard conjectures on completed homology imply `big R = big T' theorems. In the case that n=2 and p splits completely in the number field, we relate our construction to the p-adic local Langlands correspondence for GL(2,Q_p).