COLENGTH ONE DEFORMATION RINGS

Let K/Qp Q p be a finite unramified extension, rho : Gal(Qp/K) Q p /K ) -> GLn(Fp) n ( F p ) a continuous representation, and tau a tame inertial type of dimension n . We explicitly determine, under mild regularity conditions on tau , the potentially crystalline deformation ring R eta,tau rho in parallel Hodge-Tate weights eta = (n n - 1, , <middle dot> <middle dot> <middle dot>,1, 1 , 0) and inertial type tau when the shape of rho with respect to 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 [Me<acute accent>m. Soc. Math. Fr. (N.S.) 173 (2022), pp. vi+150].