Explicit coverings of families of elliptic surfaces by squares of elliptic curves

We show that, for each n>0, there is a family of elliptic surfaces which are covered by the square of a curve of genus 2n+1, and whose Hodge structures have an action by Q(−n−−−√). By considering the case n=3, we show that one particular family of K3 surfaces are covered by the squares of curves of genus 7. Using this, we construct a correspondence between the square of a curve of genus 7 and a general K3 surface in P4 with 15 ordinary double points up to a map of finite degree of K3 surfaces. This gives an explicit proof of the Kuga–Satake–Deligne correspondence for these K3 surfaces and any K3 surfaces related to them by maps of finite degree, and further, a proof of the Hodge conjecture for the squares of these surfaces. We conclude that the motives of these surfaces are Kimura-finite. Our analysis gives a birational equivalence between a moduli space of curves with additional data and the moduli space of these K3 surfaces with a specific elliptic fibration.