On a cohomological generalization of the Shafarevich conjecture for K3 surfaces

Takamatsu, Teppei

doi:10.2140/ant.2020.14.2505

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On a cohomological generalization of the Shafarevich conjecture for K3 surfaces

Teppei Takamatsu

Vol. 14 (2020), No. 9, 2505–2531

DOI: 10.2140/ant.2020.14.2505

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. 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.

Keywords

K3 surfaces, Shafarevich conjecture, good reduction

Mathematical Subject Classification 2010

Primary: 14J28

Secondary: 11F80, 11G18, 11G25, 11G35

Milestones

Received: 15 October 2019

Revised: 11 March 2020

Accepted: 11 May 2020

Published: 13 October 2020

Authors

Teppei Takamatsu
Graduate School of Mathematical Sciences The University of Tokyo Tokyo Japan

Vol. 14, No. 9, 2020