K3 surfaces with an order 50 automorphism

TL;DR

This paper proves that in characteristics other than 2 and 5, any K3 surface with a cyclic automorphism of order 50 is isomorphic to Kondō's explicit example, confirming its uniqueness.

Contribution

It establishes the uniqueness of Kondō's K3 surface with order 50 automorphism across all characteristics not equal to 2 or 5.

Findings

Any such K3 surface is isomorphic to Kondō's example.

The automorphism of order 50 is purely non-symplectic.

The result holds in all characteristics p ≠ 2, 5.

Abstract

In any characteristic different from 2 and 5, Kond\=o gave an example of a K3 surface with a purely non-symplectic automorphism of order 50. The surface was explicitly given as a double plane branched along a smooth sextic curve. In this note we show that, in any characteristic $p\neq 2, 5$, a K3 surface with a cyclic action of order 50 is isomorphic to the example of Kond\=o.