Isometries of lattices and automorphisms of K3 surfaces

TL;DR

This paper characterizes when an integral polynomial can be the characteristic polynomial of a semi-simple isometry of an even unimodular lattice and applies this to show that certain Salem numbers are dynamical degrees of automorphisms of non-projective K3 surfaces.

Contribution

It provides necessary and sufficient conditions for such polynomials and demonstrates that specific Salem numbers occur as dynamical degrees of K3 surface automorphisms.

Findings

Characterization of characteristic polynomials for semi-simple isometries.

Existence of automorphisms of K3 surfaces with given Salem numbers.

Every Salem number of degree 4, 6, 8, 12, 14, or 16 is realized as a dynamical degree.

Abstract

The aim of this paper is to give necessary and sufficient conditions for an integral polynomial to be the characteristic polynomial of a semi-simple isometry of some even unimodular lattice of given signature. This result has applications applications to automorphisms of K3 surfaces; in particular, we show that every Salem number of degree 4, 6, 8,1 2, 14 or 16 is the dynamical degree of an automorphism of a non-projective K3 surface.