Automorphisms of minimal entropy on supersingular K3 surfaces

TL;DR

This paper develops a method to determine when certain algebraic entropy values, linked to Salem numbers, are realized by automorphisms of supersingular K3 surfaces in positive characteristic, and constructs specific examples.

Contribution

It introduces a strategy to decide realizability of Salem number entropies on supersingular K3 surfaces and constructs explicit automorphisms for specific entropy values.

Findings

Entropy log λ_d realized in characteristic 5 iff d ≤ 22 and d ≠ 18

Constructed automorphism with entropy log λ_{12} in the complex setting

Developed a test for isometries of hyperbolic lattices to be positive

Abstract

In this article we give a strategy to decide whether the logarithm of a given Salem number is realized as entropy of an automorphism of a supersingular K3 surface in positive characteristic. As test case it is proved that $\log \lambda_d$, where $\lambda_d$ is the minimal Salem number of degree $d$, is realized in characteristic $5$ if and only if $d\leq 22$ is even and $d\neq 18$. In the complex projective setting we settle the case of entropy $\log \lambda_{12}$ left open by McMullen, by giving the construction. A necessary and sufficient test is developed to decide whether a given isometry of a hyperbolic lattice, with spectral radius bigger than one, is positive, i.e. preserves a chamber of the positive cone.