# Limit of torsion semi-stable Galois representations with unbounded weights

## TL;DR

This paper extends the understanding of torsion semi-stable Galois representations by relaxing the boundedness condition on Hodge-Tate weights, allowing for unbounded growth while preserving semi-stability.

## Contribution

It generalizes Liu's theorem by replacing the uniform boundedness condition with a more flexible unbounded growth condition on weights.

## Key findings

- Semi-stability is preserved under unbounded weight growth.
- The relaxation broadens the class of Galois representations that can be analyzed.
- The results apply to torsion semi-stable representations with unbounded weights.

## Abstract

Let $K$ be a complete discrete valuation field of characteristic $(0, p)$ with perfect residue field, and let $T$ be an integral $\mathbb{Z}_p$-representation of $\mathrm{Gal}(\overline{K}/K)$. A theorem of T. Liu says that if $T/p^n T$ is torsion semi-stable (resp. crystalline) of uniformly bounded Hodge-Tate weights for all $n \geq 1$, then $T$ is also semi-stable (resp. crystalline). In this note, we show that we can relax the condition of "uniformly bounded Hodge-Tate weights" to an unbounded (log-)growth condition.