The dihedral group $\mathcal D_{5}$ as a group of symplectic automorphisms on K3 surfaces

Abstract

We prove that if a K3 surface X admits Z/5Z as a group of symplectic automorphisms, then it actually admits D 5 as a group of symplectic automorphisms. The orthogonal complement to the D 5 -invariants in the second cohomology group of X is a rank 16 lattice, L. It is known that L does not depend on X: we prove that it is isometric to a lattice recently described by R. L. Griess Jr. and C. H. Lam. We also give an elementary construction of L.