Colength one deformation rings

Let K / Q p K/\mathbb {Q}_p be a finite unramified extension, ρ ¯ : G a l ( Q ¯ p / K ) → G L n ( F ¯ p ) \overline {\rho }:\mathrm {Gal}(\overline {\mathbb {Q}}_p/K)\rightarrow \mathrm {GL}_n(\overline {\mathbb {F}}_p) a continuous representation, and τ \tau a tame inertial type of dimension n n . We explicitly determine, under mild regularity conditions on τ \tau , the potentially crystalline deformation ring R ρ ¯ η , τ R^{\eta ,\tau }_{\overline {\rho }} in parallel Hodge–Tate weights η = ( n − 1 , ⋯ , 1 , 0 ) \eta =(n-1,\cdots ,1,0) and inertial type τ \tau when the shape of ρ ¯ \overline {\rho } with respect to τ \tau has colength at most one. This has application to the modularity of a class of shadow weights in the weight part of Serre’s conjecture. Along the way we make unconditional the local-global compatibility results of Park and Qian [Mém. Soc. Math. Fr. (N.S.) 173 (2022), pp. vi+150].