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 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).

v2: an issue with the definition of the big Hecke algebra has been fixed. Some other minor changes. v3: slightly weakened the local--global compatibility Conjecture (Conjecture 5.1.12). v4: minor changes, mostly to introduction. v5: fixed an issue in Appendix B, some other minor changes

References in corpus (3)

• Derived Galois deformation rings
• The Breuil--Mézard conjecture when $l\ne p$
• Patching and the p-adic Langlands program for GL(2, Q_p)

Cited by in corpus (3)

• Diagrams in the mod $p$ cohomology of Shimura curves
• Conjectures and results on modular representations of ${\mathrm{GL}}_n(K)$ for a $p$-adic field $K$
• The canonical dimension of modules for Iwasawa algebras