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 $\operatorname{GL}_{2}(\mathbb{Q}_{p})$ .

Ihara’s Lemma for Shimura curves over totally real fields via patching

“…The following is a mild generalisation of the "miracle flatness criterion", for which see [28] Theorem 23.1 or [37, Lemma 00R4]. A similar generalisation, in the setting of noncommutative completed group rings, also appears in [20].…”

Gelfand-Kirillov dimension and the $p$-adic Jacquet-Langlands correspondence

“…The second issue is that Scholze in [55, Section 9] patches cohomology instead of homology and to get the objects analogous to the patched module in [17] we would like to patch homology, so we spend some time in Section 7.1 discussing how Pontryagin duality interacts with localization with respect to an ideal defined by an ultrafilter. We patch following Dotto-Le [29], who in turn follow Gee-Newton [34]; both of these papers use a variation of Scholze's idea to patch using ultrafilters. The third issue is that in Kisin's paper on the Fontaine-Mazur conjecture [43] there is a problem, fixed in [33,Appendix B], when p = 3 and the image of Galois representation is not 'big' enough.…”

Shimura varieties at level and Galois representations

“…[Hil10]. This conjecture is motivated by the Langlands reciprocity conjecture and is expected to play an important role in the development of the classical and -adic Langlands programs; see, for example, [Eme14] and [GN16]. When the locally symmetric spaces do not admit an algebraic structure, the Calegari–Emerton conjecture seems out of reach at the moment, outside the case of low-dimensional examples such as arithmetic hyperbolic -manifolds.…”

“…[CGH20], or local–global compatibility for the -adic local Langlands correspondence, cf. [GN16]. These potential applications would also need local–global compatibility at for the Galois representations , which is still open.…”