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Visible 2-torsion in the Tate-Shafarevich group of an elliptic curve

We show that every pair of 2-torsion elements in the Tate-Shafarevich group of an elliptic curve are visible in the same abelian surface. This was previously only known for a single 2-torsion element. Our result explains some of the original observations on visibility in the paper of Cremona and Mazur.

Visible 2-torsion in the Tate-Shafarevich group of an elliptic curve   (18 pages)     dvi   ps   ps.gz   pdf

A Magma file checking some of the calculations and examples in this paper is available here.