Potentially crystalline deformation rings for certain shapes of colength one

Abstract

For a given mod-p representation R of the absolute Galois group of a p-adic eld K, one can dene a mod-p automoprhic representation P of GLn(K) by a certain space of mod-p algebraic automoprhic forms on a unitary group. We wish that P corresponds to R for a mod-p Langlands correspondence, but the structure of P is quite mysterious as a representation. It is natural to ask if P determines R, and our approach towards this question, in the case that R is Fontaine--Laffaille, requires constructing certain potentially crystalline deformation rings. In this talk, we will discuss how we construct these deformation rings. This is a joint work with Daniel Le, Bao Le Hung, Stefano Morra, and Zicheng Qian.