Automorphisms of supersingular K3 surfaces and Salem polynomials

TL;DR

This paper introduces a method to generate automorphisms of supersingular K3 surfaces in odd characteristic and demonstrates that such automorphisms often have Salem polynomial characteristic polynomials of degree 22, with specific prime bounds.

Contribution

It provides a new technique for constructing automorphisms of supersingular K3 surfaces and establishes the existence of Salem polynomial automorphisms for primes up to certain bounds.

Findings

Automorphisms with Salem polynomial characteristic on Néron-Severi lattice exist for primes ≤ 7919.

For Artin invariant 10, the result extends to primes ≤ 17389.

The method applies broadly to supersingular K3 surfaces in odd characteristic.

Abstract

We present a method to generate many automorphisms of a supersingular K3 surface in odd characteristic. As an application, we show that, if p is an odd prime less than or equal to 7919, then every supersingular K3 surface in characteristic p has an automorphism whose characteristic polynomial on the N\'eron-Severi lattice is a Salem polynomial of degree 22. For a supersingular K3 surface with Artin invariant 10, the same holds for odd primes less than or equal to 17389.