Reductions of Some Crystalline Representations in the Unramified Setting

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.