Order eight non-symplectic automorphisms on elliptic K3 surfaces

TL;DR

This paper classifies elliptic K3 surfaces with order eight non-symplectic automorphisms, detailing their Picard ranks, fixed loci, and providing explicit examples for various configurations.

Contribution

It provides a complete classification of such automorphisms on elliptic K3 surfaces, including fixed locus structures and Picard group ranks.

Findings

Picard rank is 10, 14, or 18

Fixed locus consists of elliptic, rational curves, and points

Examples are given for different fixed locus types

Abstract

In this paper we classify complex K3 surfaces with non-symplectic automorphism of order 8 that leaves invariant a smooth elliptic curve. We show that the rank of the Picard group is either 10, 14 or 18 and the fixed locus is the disjoint union of elliptic curves, rational curves and points, whose number does not exceed 1, 2, respectively 14. We give examples corresponding to several types of fixed locus in the classification.