On K3 surfaces with hyperbolic automorphism groups

TL;DR

This paper proves the finiteness of Néron-Severi lattices for certain K3 surfaces with hyperbolic automorphism groups, using genus one fibrations and recent mathematical results.

Contribution

It establishes the optimal Picard number threshold (≥6) for finiteness and provides explicit descriptions of automorphism groups.

Findings

Finiteness of Néron-Severi lattices for specified K3 surfaces.

Optimal Picard number threshold identified as 6.

Utilizes genus one fibrations and recent research for proof.

Abstract

We show the finiteness of the N\'eron-Severi lattices of complex projective K3 surfaces whose automorphism groups are non-elementary hyperbolic with explicit descriptions, under the assumption that the Picard number $\ge 6$ which is optimal to ensure the finiteness. Our proof of finiteness is based on the study of genus one fibrations on K3 surfaces and recent work of Kikuta and Takatsu.