On Characteristic Polynomials of Automorphisms of Enriques Surfaces

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 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 $(p\_f(x) \bmod 2)$ 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 realizevmodulo-$2$ 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.