Crystalline liftability of irregular weights

Abstract

Let $p$ be an odd prime. Let $K/{\mathbb{Q}}_p$ be a finite unramified extension. Let $\rho :G_K\to G{L}_2({\overline{\mathbb{F}}}_p)$ be a continuous representation. We prove that $\rho$ has a crystalline lift of small irregular weight if and only if it has multiple crystalline lifts of certain specified regular weights. The inspiration for this result comes from work of Diamond-Sasaki on geometric Serre weight conjectures. Our result provides a way to translate results currently formulated only for regular weights to also include irregular weights. The proof uses results on Kisin and $(\phi ,\hat{G})$-modules obtained from extending recent work of Gee-Liu-Savitt to study crystalline liftability of irregular weights.

arXiv commentComments welcome! arXiv admin note: text overlap with arXiv:1203.2552 by other authors