Order 3 symplectic automorphisms on K3 surfaces
The aim of this paper is to generalize results known for the symplectic involutions on K3 surfaces to the order 3 symplectic automorphisms on K3 surfaces. In particular, we will explicitly describe the action induced on the lattice $Λ_{K3}$, isometric to the second cohomology group of a K3 surface, by a symplectic automorphism of order 3; we exhibit the maps $π_*$ and $π^*$ induced in cohomology by the rational quotient map $π:X\dashrightarrow Y$, where $X$ is a K3 surface admitting an order 3 symplectic automorphism $σ$ and $Y$ is the minimal resolution of the quotient $X/σ$; we deduce the relation between the Néron--Severi group of $X$ and the one of $Y$. Applying these results we describe explicit geometric examples and generalize the Shioda--Inose structures, relating Abelian surfaces admitting order 3 endomorphisms with certain specific K3 surfaces admitting particular order 3 symplectic automorphisms.