A note on reductions of 2-dimensional crystalline Galois representations

TL;DR

This paper constructs analytic families of 2-dimensional crystalline Galois representations over unramified extensions of p-adic fields and shows their reductions modulo p are constant within these families.

Contribution

It introduces new analytic families of crystalline Galois representations and proves their mod p reductions are invariant across the family.

Findings

Constructed families of crystalline Galois representations.

Proved the mod p reductions are constant within each family.

Established a link between analytic families and reduction invariance.

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 \'etale $(\varphi,\Gamma)$-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.