Invisibility of Tate-Shafarevich groups in abelian surfaces

We give some examples of elliptic curves over the rationals
whose Tate-Shafarevich groups contain elements of order 6 or 7
that are not visible in an abelian surface. We then show that
some of these elements are visible in an abelian threefold.

Invisibility of Tate-Shafarevich groups in abelian surfaces
(13 pages)