K3 surfaces with a symplectic automorphism of order 4

Given X$X$, a K3 surface admitting a symplectic automorphism tau$\tau$ of order 4, we describe the isometry tau*$\tau <^>*$ on H2(X,Z)$H<^>2(X,\mathbb {Z})$. Having called Z similar to$\tilde{Z}$ and Y similar to$\tilde{Y}$, respectively, the minimal resolutions of the quotient surfaces Z=X/tau 2$Z=X/\tau <^>2$ and Y=X/tau$Y=X/\tau$, we also describe the maps induced in cohomology by the rational quotient maps X -> Z similar to,X -> Y similar to$X\rightarrow \tilde{Z},\ X\rightarrow \tilde{Y}$ and Y similar to -> Z similar to$\tilde{Y}\rightarrow \tilde{Z}$: With this knowledge, we are able to give a lattice-theoretic characterization of Z similar to$\tilde{Z}$, and find the relation between the Neron-Severi lattices of X,Z similar to$X,\tilde{Z}$ and Y similar to$\tilde{Y}$ in the projective case. We also produce three different projective models for X,Z similar to$X,\tilde{Z}$ and Y similar to$\tilde{Y}$, each associated to a different polarization of degree 4 on X$X$.