K3 Surfaces of zero entropy admitting an elliptic fibration with only irreducible fibers

TL;DR

This paper classifies certain complex K3 surfaces with zero entropy that admit a specific type of elliptic fibration, providing explicit lattice lists and analyzing their automorphism groups.

Contribution

It offers a complete classification of zero entropy K3 surfaces with irreducible elliptic fibrations and details their automorphism properties and lattice structures.

Findings

Identified 32 Néron-Severi lattices for such surfaces.

Determined which lattices admit a unique genus 1 fibration.

Proved all high Picard rank K3 surfaces with infinite automorphisms have positive entropy.

Abstract

We classify complex K3 surfaces of zero entropy admitting an elliptic fibration with only irreducible fibers. These surfaces are characterized by the fact that they admit a unique elliptic fibration with infinite automorphism group. We furnish an explicit list of 32 N\'eron-Severi lattices corresponding to such surfaces. Incidentally, we are able to decide which of these 32 classes of surfaces admit a unique genus 1 fibration. Finally, we prove that all K3 surfaces with Picard rank >=19 and infinite automorphism group have positive entropy.