Visibility of 4-covers of elliptic curves

05 Apr 2018

Let C be a 4-cover of an elliptic curve E, written as a quadric intersection in P^3. Let E' be another elliptic curve with 4-torsion isomorphic to that of E. We show how to write down the 4-cover C' of E' with the property that C and C' are represented by the same cohomology class on the 4-torsion. In fact we give equations for C' as a curve of degree 8 in P^5. We also study the K3-surfaces fibred by the curves C' as we vary E'. In particular we show how to write down models for these surfaces as complete intersections of quadrics in P^5 with exactly 16 singular points. This allows us to give examples of elliptic curves over Q that have elements of order 4 in their Tate-Shafarevich group that are not visible in a principally polarized abelian surface.