Abstract
Let C be a 4-cover of an elliptic curve E, written as a quadric intersection in \({\mathbb P}^3\). 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 \({\mathbb P}^5\). 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 \({\mathbb P}^5\) with exactly 16 singular points. This allows us to give examples of elliptic curves over \({\mathbb Q}\) that have elements of order 4 in their Tate–Shafarevich group that are not visible in a principally polarized abelian surface.
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Bruin, N., Fisher, T. Visibility of 4-covers of elliptic curves. Res. number theory 4, 11 (2018). https://doi.org/10.1007/s40993-018-0106-1
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DOI: https://doi.org/10.1007/s40993-018-0106-1