  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 $(-1)$.
