A note on crystalline liftings in the $\mathbb{Q}_p$ case

TL;DR

This paper investigates conditions under which a reducible mod p crystalline Galois representation of Q_p admits an upper triangular crystalline lift with the same Hodge-Tate weights, extending understanding of crystalline liftings.

Contribution

It proves the existence of upper triangular crystalline lifts with prescribed Hodge-Tate weights for certain reducible mod p representations of G_{Q_p}, building on previous methods and Breuil-Herzig's work.

Findings

Existence of upper triangular crystalline lifts under specific conditions

Extension of previous methods for crystalline liftings

Connection to Breuil-Herzig's work on Galois representations

Abstract

Let $p>2$ be a prime. Let $\rho$ be a crystalline representation of $G_{\mathbb{Q}_p}$ with distinct Hodge-Tate weights in $[0, p]$, such that its reduction $\overline \rho$ is upper triangular. Under certain conditions, we prove that $\overline \rho$ has an upper triangular crystalline lift $\rho'$ such that $\mathrm{HT}(\rho')=\mathrm{HT}(\rho)$. The method is based on the author's previous work, combined with an inspiration from the work of Breuil-Herzig.