Constructing non-trivial elements of the Shafarevich-Tate group of an Abelian variety

Let E/Q be an elliptic curve over the rationals Q. The Shafarevich-Tate group of E/Q, denoted by III(E/Q), is an important group invariant of E, which plays a role in finding the generators of E(Q). The second part of the Birch and Swinnerton-Dyer conjecture gives a formula for the order of III(E/ Q) in terms of other invariants of E that are often computable. Cremona and Mazur initiated a theory that can be sometimes be used to verify the Birch and Swinnerton-Dyer conjectural order of III( E/Q). In particular, given any elliptic curve E/Q of rank zero for which the conjectural order of III(E/Q) is non-trivial, Cremona and Mazur observed that one often finds another elliptic curve F/Q of rank greater than zero such that the elements of the Mordell-Weil group F(Q) can be used to construct non-trivial elements of III(E/Q).;In this thesis, we first extract a general theorem out of the work of Cremona and Mazur. We show that if E/Q and F/Q are elliptic curves such that E/ Q has rank 0 and E[n] ≅ F[n] over Q for some integer n, then under certain additional hypotheses on n, we have an injection F(Q)/nF( Q) ↪ III(E/Q). In particular, if F/Q has rank bigger than zero, then this gives us non-trivial elements of III(E/Q). We also give an extension of our general theorem and show that if rF (the rank of F/Q) is greater than r E (the rank of E/Q), then in the situation mentioned above, nrF--r E divides the product of the order of III( E/Q) and the Tamagawa numbers of E/ Q. Under our hypotheses, this divisibility is predicted by the second part of the Birch and Swinnerton-Dyer (BSD) conjecture, in view of work of Agashe. Thus our result gives new theoretical evidence for the BSD conjecture. We also prove a theorem that gives an alternative method to potentially construct non-trivial elements of III(E/Q) by using the component groups of F/Q.;Finally, we mention that the main results in this thesis have been proved in the more general setting of abelian varieties over a number field, of which elliptic curves over the rationals are a particular case.