On $\psi$-lattices in modular $(\varphi,\Gamma)$-modules

TL;DR

This paper generalizes the concept of $ ext{psi}$-lattices in étale $( ext{phi}, ext{Gamma})$-modules from one variable to multiple variables over certain power series rings, expanding their theoretical framework.

Contribution

It introduces and studies $ ext{psi}$-lattices in higher-dimensional étale $( ext{phi}, ext{Gamma})$-modules, extending Colmez's original constructions to multiple variables.

Findings

Defined $ ext{psi}$-lattices in multivariable settings.

Provided examples illustrating the new $ ext{psi}$-lattices.

Extended the theoretical framework of $( ext{phi}, ext{Gamma})$-modules.

Abstract

Let $F/{\mathbb Q}_p$ be a finite field extension, let $k$ be a finite field extension of the residue field of $F$. Generalizing the $\psi$-lattices which Colmez constructed in \'{e}tale $(\varphi,\Gamma)$-modules over $k[[t]][t^{-1}]$, we define, study and exemplify $\psi$-lattices in \'{e}tale $(\varphi,\Gamma)$-modules over $k[[t_1,\ldots,t_d]][\prod_it_i^{-1}]$ for arbitrary $d\in{\mathbb N}$.