On Characteristic Polynomials of Automorphisms of Enriques Surfaces

Abstract

Let $f$ be an automorphism of a complex Enriques surface $Y$ and let $p_f$ denote the characteristic polynomial of the isometry $f^{\ast }$ of the numerical Néron–Severi lattice of $Y$ induced by $f$. We combine a modification of McMullen’s method with Borcherds’ method to prove that the modulo-$2$ reduction $(p_f(x)\mathrm{mod}2)$ is a product of modulo-$2$ reductions of (some of) the five cyclotomic polynomials ${\Phi }_m$, where $m\le 9$ and $m$ is odd. We study Enriques surfaces that realizevmodulo-$2$ reductions of ${\Phi }_7$, ${\Phi }_9$ and show that each of the five polynomials $({\Phi }_m(x)\mathrm{mod}2)$ is a factor of the modulo-$2$ reduction $(p_f(x)\mathrm{mod}2)$ for a complex Enriques surface.