Reductions of Some Crystalline Representations in the Unramified Setting

Abstract

We determine semi-simple reductions of irreducible, 2-dimensional crystalline representations of absolute Galois groups with unramified base fields. To this end, we provide explicit representatives for the isomorphism classes of the associated weakly admissible filtered phi-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 with labeled Hodge-Tate weights in the range p+1 < k_0 < 2p-3 and 1 < k_i < p-2 for 0 < i < f and certain parameters having p-adic valuation sufficiently large, unless f=2 in which case all reductions in said range are displayed.