Finiteness theorems for K3 surfaces over arbitrary fields

TL;DR

This paper extends finiteness theorems for automorphism groups of K3 surfaces from algebraically closed fields to arbitrary fields, highlighting differences and providing new examples.

Contribution

It generalizes known finiteness results for K3 surfaces to arbitrary fields and illustrates how automorphism behavior can differ from the algebraically closed case.

Findings

Finiteness results hold over arbitrary fields with modifications.

Examples show automorphism groups can behave differently over non-closed fields.

Theoretical framework for automorphisms over general fields.

Abstract

Over an algebraically closed field, various finiteness results are known regarding the automorphism group of a K3 surface and the action of the automorphisms on the Picard lattice. We formulate and prove versions of these results over arbitrary base fields, and give examples illustrating how behaviour can differ from the algebraically closed case.