On patched completed homology and a conjecture of Venkatesh

Let F be a CM field and Π a regular algebraic cuspidal cohomological representation of G=PGL₂/F. A conjecture of Venkatesh describes the structure of the contribution of Π to the homology of the locally symmetric spaces associated to G. We investigate this conjecture in the setting of p-adic homology with p a totally split prime. Along the way, we elaborate on the relations between Venkatesh's conjecture and completed homology, the Taylor-Wiles method and the p-adic local Langlands correspondence. Our main result is a `big $R=T$' theorem in characteristic 0, from which we deduce a variant of the p-adic realisation of Venkatesh's conjecture, conditional on various natural conjectures and technical assumptions.