Explicit coverings of families of elliptic surfaces by squares of curves

We show that, for each $$n>0$$ , there is a family of elliptic surfaces which are covered by the square curve genus $$2n+1$$ and whose Hodge structures have an action $${{\mathbb {Q}}}(\sqrt{-n})$$ . By considering case $$n=3$$ we that one particular K3 squares curves 7. Using this, construct correspondence between 7 general surface in $${\mathbb {P}}^4$$ with 15 ordinary double points up to map finite degree surfaces. This gives explicit proof Kuga–Satake–Deligne these any related them maps degree, further, conjecture conclude motives Kimura-finite. Our analysis birational equivalence moduli space additional data specific fibration.