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

We consider the family of irreducible crystalline representations ofdimension $2$ of ${\rm Gal}(\overline{\bf Q}_p/{\bf Q}_p)$ given by the $V_{k,a_p}$ for afixed weight integer $k\geq 2$.We study the locus of the parameter $a_p$ where these representationshave a given reduction modulo $p$. We give qualitative results on thislocus and show that for a fixed $p$ and $k$ it can be computed bydetermining the reduction modulo $p$ of $V_{k,a_p}$ for a finite numberof values of the parameter $a_p$