  The second part of the Birch and Swinnerton-Dyer (BSD) conjecture gives a
conjectural formula for the order of the Shafarevich-Tate group of an elliptic
curve in terms of other computable invariants of the curve. Cremona and Mazur
initiated a theory that can often be used to verify the BSD conjecture by
constructing non-trivial elements of the Shafarevich-Tate group of an elliptic
curve by means of the Mordell-Weil group of an ambient curve. In this paper, we
generalize Cremona and Mazur's work and give precise conditions under which
such a construction can be made for the Shafarevich-Tate group of an abelian
variety over a number field. We then give an extension of our general result
that provides new theoretical evidence for the BSD conjecture.
