Exceptional jumps of Picard ranks of reductions of K3 surfaces over number fields. (26th September 2022)

• Record Type: Journal Article
• Authors: Shankar, Ananth N.; Shankar, Arul; Tang, Yunqing; Tayou, Salim
• Is Part Of: Forum of mathematics. Volume 10(2022)
• Publication Date: 2022-09-26
• Subjects: 11G18 -- 14J28 -- 11G10; Mathematics -- Periodicals; 510
• DOI: 10.1017/fmp.2022.14
• Languages: English
• ISSNs: 2050-5086

Abstract

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 specialisations. 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 .