PATCHING AND THE COMPLETED HOMOLOGY OF LOCALLY SYMMETRIC SPACES

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 with $\operatorname{GL}_{n}$ over a number field. We use our construction, and some new results in non-commutative algebra, to show that standard conjectures on completed homology imply ‘big $R=\text{big}~\mathbb{T}$ ’ theorems in situations where one cannot hope to appeal to the Zariski density of classical points (in contrast to all previous results of this kind). In the case where $n=2$ and  $p$ splits completely in the number field, we relate our construction to the $p$ -adic local Langlands correspondence for  $\operatorname{GL}_{2}(\mathbb{Q}_{p})$ .