Borcherds lattices and K3 surfaces of zero entropy

TL;DR

This paper classifies Borcherds lattices and applies the results to classify K3 surfaces with zero entropy and infinite automorphism groups, revealing that most special K3 surfaces admit positive entropy automorphisms.

Contribution

It provides a complete classification of Borcherds lattices and applies this to classify certain K3 surfaces with zero entropy and infinite automorphism groups.

Findings

194 Borcherds lattices identified

193 K3 lattices of zero entropy with infinite automorphism group classified

Most special K3 surfaces admit automorphisms of positive entropy

Abstract

Let $L$ be an even, hyperbolic lattice with infinitely many simple $(-2)$-roots. We call $L$ a Borcherds lattice if it admits an isotropic vector with bounded inner product with all the simple $(-2)$-roots. We show that this is the case if and only if $L$ has zero entropy, or equivalently if and only if all symmetries of $L$ preserve some isotropic vector. We obtain a complete classification of Borcherds lattices, consisting of $194$ lattices. In turn this provides a classification of hyperbolic lattices of rank $\ge 5$ with virtually solvable symmetry group. Finally, we apply these general results to the case of K3 surfaces. We obtain a classification of Picard lattices of K3 surfaces of zero entropy and infinite automorphism group, consisting of $193$ lattices. In particular we show that all Kummer surfaces, all supersingular K3 surfaces and all K3 surfaces covering an Enriques surface (with one exception) admit an automorphism of positive entropy.