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

Abstract

Given a K3 surface $X$ over a number field $K$ with potentially good reduction everywhere, we prove that the set of primes of $K$ where the geometric Picard rank jumps is infinite. As a corollary, we prove that either $X_{\overline{K}}$ has infinitely many rational curves or $X$ has infinitely many unirational specializations. Our result on Picard ranks is a special case of more general results on exceptional classes for K3 type motives associated to GSpin Shimura varieties. These general results have several other applications. For instance, we prove that an abelian surface over a number field $K$ with potentially good reduction everywhere is isogenous to a product of elliptic curves modulo infinitely many primes of $K$.

arXiv commentFinal version. To appear in Forum of Mathematics, Pi

References in corpus (9)

• Automorphic forms with singularities on Grassmannians
• Integral canonical models for Spin Shimura varieties
• Some cases of the Mumford--Tate conjecture and Shimura varieties
• Constructing rational curves on K3 surfaces
• Frobenius distribution for pairs of elliptic curves and exceptional isogenies
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Cited by in corpus (4)

• Picard ranks of K3 surfaces over function fields and the Hecke orbit conjecture
• Picard rank jumps for K3 surfaces with bad reduction
• Just-likely intersections on Hilbert modular surfaces
• Arithmetic Deformation of Line Bundles