Torsion representations arising from (φ,Gˆ)-modules

The notion of a (φ,Gˆ)-module is defined by Tong Liu in 2010 to classify lattices in semi-stable representations. In this paper, we study torsion (φ,Gˆ)-modules, and torsion p-adic representations associated with them, including the case where p=2. First we prove that the category of torsion p-adic representations arising from torsion (φ,Gˆ)-modules is an abelian category. Secondly, we construct a maximal (minimal) theory for (φ,Gˆ)-modules by using the theory of étale (φ,Gˆ)-modules, essentially proved by Xavier Caruso, which is an analogue of Fontaineʼs theory of étale (φ,Γ)-modules. Non-isomorphic two maximal (minimal) objects give non-isomorphic two torsion p-adic representations.