Variation of Néron–Severi Ranks of Reductions of K3 Surfaces

Abstract

We study the behavior of geometric Picard ranks of K3 surfaces over under reduction modulo primes. We compute these ranks for reductions of smooth quartic surfaces modulo all primes p < 2¹⁶ in several representative examples and investigate the resulting statistics.

2000 AMS Subject Classification::

• Primary 14J28
• Secondary 14C22

Keywords:

• K3 surfaces
• variation of Picard ranks
• variation of geometric Picard number
• Kedlaya’s algorithm

Notes

¹FLINT, written by William Hart, Fredrik Johansson, and Sebastian Pancratz, is available at http://flintlib.org. NTL, by Victor Shoup, is available at http://www.shoup.net/ntl/. The raw data of all experiments are available at www.cims.nyu.edu/~costa.