Lattices in potentially semi-stable representations and weak $(\varphi,\hat{G})$-modules

TL;DR

This paper establishes an anti-equivalence between weak $(,g)$-modules of a certain height and Galois stable lattices in potentially semi-stable $p$-adic representations, advancing understanding of their categorical relationships.

Contribution

It proves an anti-equivalence between weak $(,g)$-modules and Galois stable lattices, answering a key question about the functor's essential image in this context.

Findings

Established anti-equivalence between categories

Classified lattices in potentially semi-stable representations

Extended understanding of weak $(,g)$-modules in $p$-adic Hodge theory

Abstract

Let $p$ be a prime number and $r$ a non-negative integer. In this paper, we prove that there exists an anti-equivalence between the category of weak $(\varphi,\hat{G})$-modules of height $r$ and a certain subcategory of the category of Galois stable lattices in potentially semi-stable $p$-adic representations with Hodge-Tate weights in $[0,r]$. This gives an answer to a Tong Liu's question about the essential image of a functor on weak $(\varphi,\hat{G})$-modules. For a proof, following Liu's methods, we construct linear algebraic data which classify lattices in potentially semi-stable representations.