Liu: Filtration associated to torsion semi-stable representations

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Tong Liu
Filtration associated to torsion semi-stable representations
(Filtration associée aux représentations semi-stables de torsion)
Annales de l'institut Fourier, 65 no. 5 (2015), p. 1999-2035, doi: 10.5802/aif.2980
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Class. Math.: 14F30, 14L05
Keywords: semi-stable representations, filtration

Résumé - Abstract

Let $p$ be an odd prime, $K$ a finite extension of $\mathbb{Q}_p$ and $G:= \operatorname{Gal} (\overline{\mathbb{Q}}_p/K)$ the Galois group. We construct and study filtration structures associated torsion semi-stable representations of $G$. In particular, we prove that two semi-stable representations share the same $p$-adic Hodge-Tate type if they are congruent modulo $p^n$ with $n \ge c^{\prime }$, where $c^{\prime }$ is a constant only depending on $K$ and the differences between the maximal and minimal Hodge-Tate weights of two representations. As an application, we reprove a part of Kisin’s result: the existence of a quotient of the universal Galois deformation ring which parameterizes semi-stable representations with a fixed $p$-adic Hodge-Tate type.

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