Visibility of ideal classes

Cremona, Mazur, and others have studied what they call visibility of elements of Shafarevich–Tate groups of elliptic curves. The analogue for an abelian number field K is capitulation of ideal classes of K in the minimal cyclotomic field containing K. We develop a new method to study capitulation and use it and classical methods to compute data with the hope of gaining insight into the elliptic curve case. For example, the numerical data for number fields suggests that visibility of non-trivial Shafarevich–Tate elements might be much more common for elliptic curves of positive rank than for curves of rank 0.