On characteristic polynomials of automorphisms of Enriques surfaces

Let f be an automorphism of a complex Enriques surface Y and let pf​ 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 (pf​(x)mod2) is a product of modulo-2 reductions of (some of) the five cyclotomic polynomials Φm​, where m≤9 and m is odd. We study Enriques surfaces that realizevmodulo-2 reductions of Φ7​, Φ9​ and show that each of the five polynomials (Φm​(x)mod2) is a factor of the modulo-2 reduction (pf​(x)mod2) for a complex Enriques surface