A note on reductions of $2$-dimensional crystalline Galois representations

Abstract:

Let $p$ be an odd prime number, $K_{f}$ the finite unramified extension of $\mathbb {Q} _{p}$ of degree $f$ and $G_{K_{f}}$ its absolute Galois group. We construct analytic families of étale $\left ( \varphi ,\Gamma _{K_{f}}\right )$-modules which give rise to some families of $2$-dimensional crystalline representations of $G_{K_{f}}$ with length of filtration $\geq p.$ As an application we prove that the modulo $p$ reductions of the members of each such family (with respect to appropriately chosen Galois-stable lattices) are constant.