Reductions of Some Crystalline Representations in the Unramified Setting

TL;DR

This paper explicitly determines the semisimple reductions of certain 2-dimensional crystalline Galois representations over unramified extensions, providing concrete classifications and bounds for reductions based on lattice structures.

Contribution

It introduces explicit representatives for associated filtered $ ext{φ}$-modules and constructs Kisin modules to compute reductions in the unramified setting.

Findings

Explicit classification of semisimple reductions.

Construction of Kisin modules for Galois stable lattices.

Establishment of bounds for reductions based on valuations.

Abstract

We determine semisimple reductions of irreducible, 2-dimensional crystalline representations of the absolute Galois group $\text{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_{p^f})$. To this end, we provide explicit representatives for the isomorphism classes of the associated weakly admissible filtered $\varphi$-modules by concretely describing the strongly divisible lattices which characterize the structure of the aforementioned modules. Using these representatives, we construct Kisin modules canonically associated to Galois stable lattice representations inside our crystalline representations. This allows us to compute the reduction of such crystalline representations for arbitrary labeled Hodge-Tate weights so long as the $p$-adic valuations of certain parameters are sufficiently large. Hence, we provide a Berger-Li-Zhu type bound in the unramified setting.