On a cohomological generalization of the Shafarevich conjecture for K3 surfaces

Abstract

The Shafarevich conjecture for K3 surfaces asserts the finiteness of isomorphism classes of K3 surfaces over a fixed number field admitting good reduction away from a fixed finite set of finite places. André proved this conjecture for polarized K3 surfaces of fixed degree, and recently She proved it for polarized K3 surfaces of unspecified degree. In this paper, we prove a certain generalization of their results, which is stated by the unramifiedness of l-adic étale cohomology groups for K3 surfaces over finitely generated fields of characteristic 0. As a corollary, we get the original Shafarevich conjecture for K3 surfaces without assuming the extendability of polarization, which is stronger than the results of André and She. Moreover, as an application, we get the finiteness of twists of K3 surfaces via a finite extension of characteristic 0 fields.