Existence of invariant norms in $p$-adic representations of $GL_2(F)$ of large weights

Abstract

In [BS07] Breuil and Schneider formulated a conjecture on the equivalence of the existence of invariant norms on certain $p$-adically locally algebraic representations of $GL_n(F)$ and the existence of certain de-Rham representations of $Gal(\bar{F}/F)$, where $F$ is a finite extension of $\mathbb{Q}_p$. In [Bre03b, DI13] Breuil and de Ieso proved that in the case $n = 2$ and under some restrictions, the existence of certain admissible filtrations on the $\phi$-module associated to the two-dimensional de-Rham representation of $Gal(\bar{F}/F)$ implies the existence of invariant norms on the corresponding locally algebraic representation of $GL_2(F)$. In [Bre03b, DI13], there is a significant restriction on the weight - it must be small enough. In [CEG+13] the conjecture is proved in greater generality, but the weights are still restricted to the extended Fontaine-Laffaille range. In this paper we prove that in the case $n = 2$, even with larger weights, under some restrictions, the existence of certain admissible filtrations implies the existence of invariant norms.