K3 surfaces with a symplectic automorphism of order 4

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