On a cohomological generalization of the Shafarevich conjecture for K3 surfaces

TL;DR

This paper generalizes the Shafarevich conjecture for K3 surfaces by proving unramifiedness of l-adic étale cohomology groups over finitely generated fields, leading to broader finiteness results without polarization assumptions.

Contribution

It extends previous results by proving a cohomological unramifiedness property, enabling the original conjecture's proof without polarization extendability assumptions.

Findings

Proves unramifiedness of l-adic étale cohomology for K3 surfaces over finitely generated fields.

Establishes the original Shafarevich conjecture for K3 surfaces without polarization extendability.

Demonstrates finiteness of twists of K3 surfaces via finite field extensions.

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. Andr\'{e} 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 \'{e}tale 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 Andr\'{e} and She. Moreover, as an application, we get the finiteness of twists of K3 surfaces via a finite extension of characteristic 0 fields.