Colength one deformation rings

Abstract

Let $K/{\mathbf{Q}}_p$ be a finite unramified extension, $\overline{\rho }:\mathrm{Gal}({\overline{\mathbf{Q}}}_p/K)\to {\mathrm{GL}}_n({\overline{\mathbf{F}}}_p)$ 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 $R_{\overline{\rho }}^{\eta ,\tau }$ in parallel Hodge--Tate weights $\eta =(n-1,\cdots ,1,0)$ and inertial type $\tau$ when the \emph{shape} of $\overline{\rho }$ 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 \cite{PQ} and further study the geometry of moduli spaces of Fontaine--Laffaille representations in terms of colength one weights.

Major revision: shortened some of the arguments, removed the appendix and fixed a number of typos. Comments welcome!