Crystalline liftability of irregular weights

TL;DR

This paper characterizes when certain Galois representations over local fields admit crystalline lifts of irregular weights, extending the understanding of Serre weight conjectures to include irregular weights.

Contribution

It establishes a criterion linking irregular crystalline lifts to multiple regular lifts, expanding the scope of known results from regular to irregular weights.

Findings

Provides a necessary and sufficient condition for irregular crystalline liftability.

Extends methods from regular to irregular weights using advanced module theory.

Connects geometric Serre weight conjectures with new liftability criteria.

Abstract

Let $p$ be an odd prime. Let $K/\mathbb{Q}_p$ be a finite unramified extension. Let $\rho: G_K \to GL_2(\overline{\mathbb{F}}_p)$ 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 $(\varphi,\hat{G})$-modules obtained from extending recent work of Gee-Liu-Savitt to study crystalline liftability of irregular weights.