Torsion representations arising from $(\varphi,\hat{G})$-modules

TL;DR

This paper explores torsion $(,hat)$-modules and their associated p-adic representations, establishing their categorical properties and developing maximal and minimal theories inspired by Fontaine's framework.

Contribution

It introduces a categorical framework for torsion $(,hat)$-modules and constructs maximal and minimal theories using étale modules, extending Fontaine's classical theory.

Findings

The category of torsion p-adic representations from torsion $(,hat)$-modules is abelian.

Constructed maximal and minimal theories for $(,hat)$-modules.

Non-isomorphic maximal/minimal objects correspond to non-isomorphic torsion p-adic representations.

Abstract

The notion of a $(\varphi,\hat{G})$-module is defined by Tong Liu in 2010 to classify lattices in semi-stable representations. In this paper, we study torsion $(\varphi,\hat{G})$-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 $(\varphi,\hat{G})$-modules is an abelian category. Secondly, we construct a maximal (minimal) theory for $(\varphi,\hat{G})$-modules by using the theory of \'etale $(\varphi, \hat{G})$-modules, essentially proved by Xavier Caruso, which is an analogue of Fontaine's theory of \'etale $(\varphi,\Gamma)$-modules. Non-isomorphic two maximal (minimal) objects give non-isomorphic two torsion p-adic representations.