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 ${\mathrm{\Lambda }}_{K3}$, isometric to the second cohomology group of a K3 surface, by a symplectic automorphism of order 3; we exhibit the maps ${\pi }_{\ast }$ and ${\pi }^{\ast }$ induced in cohomology by the rational quotient map $\pi :X⇢Y$, where $X$ is a K3 surface admitting an order 3 symplectic automorphism $\sigma$ and $Y$ is the minimal resolution of the quotient $X\mathrm{/}\sigma$; we deduce the relation between the N'eron–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.