Colength one deformation rings

Let K / Q p K/Q_p be a finite unramified extension, ρ ¯ : G a l ( Q ¯ p / K ) → G L n ( F ¯ p ) ρ ̄:Gal(Q̄_p/K)→ GL_n(F̄_p) a continuous representation, and τ τ a tame inertial type of dimension n n . We explicitly determine, under mild regularity conditions on τ τ , the potentially crystalline deformation ring R ρ ¯ η , τ Rη ,τρ ̄ in parallel Hodge–Tate weights η = ( n − 1 , ⋯ , 1 , 0 ) η =(n-1,⋯ ,1,0) and inertial type τ τ when the shape of ρ ¯ ρ ̄ with respect to τ τ 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].