On the locus of $2$-dimensional crystalline representations with a given reduction modulo $p$

Abstract

We consider the family of irreducible crystalline representations of dimension $2$ of $Gal\left({\bar{\mathbb{Q}}}_p∕{\mathbb{Q}}_p\right)$ given by the $V_{k,a_p}$ for a fixed weight $k\ge 2$. We study the locus of the parameter $a_p$ where these representations have a given reduction modulo $p$. We give qualitative results on this locus and show that for a fixed $p$ and $k$ it can be computed by determining the reduction modulo $p$ of $V_{k,a_p}$ for a finite number of values of the parameter $a_p$. We also generalize these results to other Galois types.