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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 a fixed weight integer $k\geq 2$. We study the locus of theparameter $a_p$ 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 afinite number of values of the parameter $a_p$. We also generalize theseresults to other Galois types.

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