Dynamics on supersingular K3 surfaces

TL;DR

This paper constructs a specific automorphism on a supersingular K3 surface in odd characteristic p=2 mod 3, showing it cannot be lifted to characteristic zero, and provides a formula for its characteristic polynomial.

Contribution

It presents an explicit automorphism on supersingular K3 surfaces in certain characteristics and derives its characteristic polynomial as an irreducible Salem polynomial.

Findings

Automorphism does not lift to characteristic zero.

Characteristic polynomial is an irreducible Salem polynomial of degree 22.

Construction uses elliptic fibrations to derive the polynomial.

Abstract

For any odd characteristic p=2 mod 3, we exhibit an explicit automorphism on the supersingular K3 surface of Artin invariant one which does not lift to any characteristic zero model. Our construction builds on elliptic fibrations to produce a closed formula for the automorphism's characteristic polynomial on second cohomology, which turns out to be an irreducible Salem polynomial of degree 22 with coefficients varying with p.