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On the distribution of the Picard ranks of the reductions of a $K3$ surface
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abstract
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 $(-1)$.
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