Orders of automorphisms of K3 surfaces

TL;DR

This paper classifies all possible finite orders of automorphisms of complex and positive characteristic K3 surfaces, establishing bounds and characterizing which integers can occur as automorphism orders.

Contribution

It provides a complete classification of automorphism orders for K3 surfaces in various characteristics, including explicit bounds and conditions.

Findings

A positive integer N is an automorphism order iff φ(N) ≤ 20 for complex K3 surfaces.

66 is the maximum finite order of automorphisms in characteristics p ≠ 2,3.

Bounds are established for finite group actions on K3 surfaces in characteristic p > 7.

Abstract

We determine all posible orders of automorphisms of finite order of complex K3 surfaces or of K3 surfaces in characteristic $p>3$. E.g., a positive integer $N$ is the order of an automorphism of a complex K3 surface if and only if $\phi(N)\le 20$ where $\phi$ is the Euler function. In particular, 66 is the maximum finite order in each characteristic $p\neq 2,3$. As a consequence, we give a bound for the orders of finite groups acting on K3 surfaces in characteristic $p>7$.