The dihedral group $\Dh_5$ as group of symplectic automorphisms on K3 surfaces

Abstract

We prove that if a K3 surface $X$ admits $\mathbb{Z}/5\mathbb{Z}$ as group of symplectic automorphisms, then it actually admits $\Dh_5$ as group of symplectic automorphisms. The orthogonal complement to the $\Dh_5$-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 $L$.

arXiv comment11 pages. Arguments revised, results unchanged. Final version, to appear in Proc. Amer. Math. Soc

References in corpus (2)

• $E{E}_8$-lattices and dihedral groups
• Symplectic automorphisms and the Picard group of a K3 surface

Cited by in corpus (1)

• Symplectic automorphisms and the Picard group of a K3 surface