On the distribution of the Picard ranks of the reductions of a $K3$   surface
  We report on our results concerning the distribution of the geometric Picard ranks of $K3$ surfaces under reduction modulo various primes. In the situation that $\rk \Pic S_{\overline{K}}$ is even, we introduce a quadratic character, called the jump character, such that $\rk \Pic S_{\overline\bbF_{\!\frakp}} > \rk \Pic S_{\overline{K}}$ for all good primes, at which the character evaluates to $(-1)$. 
Edgar Costa, Andreas-Stephan Elsenhans, J\"org Jahnel
math.AG