Colength one deformation rings (Q6571616)

To questions of local-global compatibility in the \(\bmod p\) and \(p\)-adic Langlands program, including the weight part of Serre's conjecture several of authors have noticed that calculations of various potentially crystalline deformation spaces can be applied [\textit{M. Emerton} et al., Invent. Math. 200, No. 1, 1--96 (2015; Zbl 1396.11089); \textit{D. Le} et al., ``Extremal weights and a tameness criterion for mod $p$ Galois representations'', Preprint, \url{arXiv:2206.06442}; \textit{D. Le} et al., Invent. Math. 231, No. 3, 1277--1488 (2023; Zbl 1522.11047); \textit{A. Dotto} and \textit{D. Le}, Compos. Math. 157, No. 8, 1653--1723 (2021; Zbl 1480.11081)]. Following the work of [\textit{D. Le} et al., ``Extremal weights and a tameness criterion for mod $p$ Galois representations'', Preprint, \url{arXiv:2206.06442}], the authors of the paper under review present the following main result for representations of \(\mathrm{Gal}((\overline{\mathbb Q}_{p})/(\mathbb Q_{p}))\). Theorem. Let \(E\) be a finite extension of \(\mathbb Q_{p}\) with ring of integers \(O\) and residue field \(F\). Let \(\rho:\mathrm{Gal}((\overline{\mathbb Q}_{p})/(\mathbb Q_{p}))\to\mathrm{GL}_{n}(F)\) a continuous Galois representation, \(\tau\) a \(3n\)-7-generic tame inertial type, \(\eta=(n-1,\cdots,1,0)\in Z^n\), and \(R_\rho^{\eta,\tau}\) the lifting ring for potentially crystalline representations of type \((\eta,\tau)\). If \(R_\rho^{\eta,\tau}\) is nonzero and the colength of the shape \(w(\rho,\tau)\) is at most one, then \(R_\rho^{\eta,\tau}\) is formally smooth over \(O\). As they say in their paper under review ``Our principal motivation in writing this paper was to apply the above Theorem to prove the weight elimination and \(\bmod p\) multiplicity one results necessary to make unconditional the local-global compatibility result of [\textit{C. Park}, ``On \(\operatorname{mod}\, p\) local-global compatibility for \(\operatorname{GL}_n(\mathbb{Q}_p)\) in the ordinary case'', Mém. Soc. Math. Fr. (N.S.), 150 p. (2022)] which states roughly that the local mod p Galois representation at \(p\) can be recovered from the \(\mathrm{GL}_{n}(\mathbb Q_{p})\)-action on the Hecke isotypic part of the mod p completed cohomology of a definite unitary group.'' The paper under review is well written excellent research and an interesting source for interested researchers in the field.