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arXiv · 1610.07823

On the distribution of the Picard ranks of the reductions of a $K3$ surface

Edgar Costa·Andreas-Stephan Elsenhans·Jörg Jahnel
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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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Edgar Costa, Andreas-Stephan Elsenhans, Jörg Jahnel. 2016-10-25. On the distribution of the Picard ranks of the reductions of a $K3$ surface. https://arxiv.org/abs/1610.07823

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