K3 surfaces with an automorphism of order 11

TL;DR

This paper investigates K3 surfaces with automorphisms of order 11 across different characteristics, establishing their Picard numbers, constructing examples, and exploring supersingular cases.

Contribution

It provides the first comprehensive study of order 11 automorphisms on K3 surfaces in arbitrary characteristic, including existence and Picard number results.

Findings

General such K3 surfaces in characteristic 11 have Picard number 2

Constructs K3 surfaces with automorphisms of order 11 in all characteristics

Identifies supersingular K3 surfaces with these automorphisms where possible

Abstract

This paper concerns K3 surfaces with automorphisms of order 11 in arbitrary characteristic. Specifically we study the wild case and prove that a general such surface in characteristic 11 has Picard number 2. We also construct K3 surfaces with an automorphism of order 11 in every characteristic, and supersingular K3 surfaces whenever possible.