Automorphisms of K3 surfaces, signatures, and isometries of lattices

TL;DR

This paper explores the relationship between Salem numbers, automorphisms of non-projective K3 surfaces, and lattice isometries, providing conditions for certain Salem numbers to be realized as dynamical degrees.

Contribution

It introduces a notion of automorphism signature and establishes necessary and sufficient conditions for Salem numbers of degrees 10 and 18 to be dynamical degrees of K3 surface automorphisms.

Findings

Salem numbers of degrees 4, 6, 8, 12, 14, 16 are realizable as automorphism dynamical degrees.

A new signature concept characterizes automorphisms of K3 surfaces.

Conditions for Salem numbers of degrees 10 and 18 are explicitly formulated.

Abstract

Every Salem numbers of degree 4,6,8,12,14 or 16 is the dynamical degree of an automorphism of a non-projective K3 surface. We define a notion of signature of an automorphism, and use it to give a necessary and sufficient condition for Salem numbers of degree 10 and 18 to be realized as the dynamical degree of such an automorphism. The first part of the paper contains results on isometries of lattices.