Order 3 symplectic automorphisms on K3 surfaces

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