Limit of torsion semi-stable Galois 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 $n\ge 1$, 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.

Final version. arXiv admin note: text overlap with arXiv:1606.07216