# Exceptional jumps of Picard ranks of reductions of K3 surfaces over number fields

@article{Shankar2019ExceptionalJO, title={Exceptional jumps of Picard ranks of reductions of K3 surfaces over number fields}, author={Ananth N. Shankar and Arul Shankar and Yunqing Tang and Salim Tayou}, journal={arXiv: Number Theory}, year={2019} }

Given a K3 surface $X$ over a number field $K$, we prove that the set of primes of $K$ where the geometric Picard rank jumps is infinite, assuming that $X$ has everywhere potentially good reduction. The result is a special case of a more general one on exceptional classes for K3 type motives associated to GSpin Shimura varieties and several other applications are given. As a corollary, we give a new proof of the fact that $X_{\overline{K}}$ has infinitely many rational curves.

## 7 Citations

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