King's College London

On the reductions of some crystalline representations

Student thesis: Doctoral ThesisDoctor of Philosophy

In this work we study a semilinear category $\operatorname{Mod}^{\operatorname{SD}}_k$ which appears as a full subcategory of the category of $p$-torsion Breuil--Kisin modules. We view $\operatorname{Mod}^{\operatorname{SD}}_k$ as extending Fontaine--Laffaille theory (for $p$-torsion coefficients) to weights contained in the range $[0,p]$. As an application we relate the restriction to inertia of a residual representation $\overline{\rho}$, with the weights $\subset [0,p]$ for which $\overline{\rho}$ has a crystalline lift. This allows us to deduce some new cases of the weight part of Serre's conjecture for unitary groups of rank $n$.
Original language English King's College London Fred Irvin Diamond (Supervisor) 2018