On characteristic polynomials of automorphisms of Enriques surfaces

TL;DR

This paper investigates the characteristic polynomials of automorphisms of complex Enriques surfaces, showing their modulo-2 reductions factor into products of reductions of certain cyclotomic polynomials, and constructs examples for specific cases.

Contribution

It introduces a modified method to analyze the modulo-2 reductions of characteristic polynomials and demonstrates the realization of these reductions for specific cyclotomic polynomials on Enriques surfaces.

Findings

Modulo-2 reduction of characteristic polynomials factors into cyclotomic polynomial reductions.

Each of the five relevant cyclotomic polynomials' reductions appears as a factor.

Constructed examples of Enriques surfaces realizing these polynomial factors.

Abstract

Let $f$ be an automorphism of a complex Enriques surface $Y$ and let $p_f$ denote the characteristic polynomial of the isometry $f^*$ of the numerical N\'eron-Severi lattice of $Y$ induced by $f$. We apply a modification of McMullen's method to prove that the modulo-$2$ reduction $(p_f(x) \bmod 2)$ is a product of modulo-$2$ reductions of (some of) the five cyclotomic polynomials $\Phi_m$, where $m \leq 9$ and $m$ is odd. We study Enriques surfaces that realize modulo-$2$ reductions of $\Phi_7$, $\Phi_9$ and show that each of the five polynomials $(\Phi_m(x) \bmod 2)$ is a factor of the modulo-$2$ reduction $(p_f(x) \bmod 2)$ for a complex Enriques surface.