Visibility and the Birch and Swinnerton-Dyer conjecture for analytic rank one

Let $E$ be an optimal elliptic curve over \Q of conductor $N$ having analytic rank one, i.e., such that the $L$-function $L_E(s)$ of $E$ vanishes to order one at $s=1$. Let $K$ be a quadratic imaginary field in which all the primes dividing $N$ split and such that the $L$-function of $E$ over $K$ vanishes to order one at $s=1$. Suppose there is another optimal elliptic curve over \Q of the same conductor $N$ whose Mordell-Weil rank is greater than one and whose associated newform is congruent to the newform associated to $E$ modulo an integer $r$. The theory of visibility then shows that under certain additional hypotheses, $r$ divides the order of the Shafarevich-Tate group of $E$ over $K$. We show that under somewhat similar hypotheses, $r$ divides the order of the Shafarevich-Tate group of $E$ over $K$. We show that under somewhat similar hypotheses, $r$ also divides the Birch and Swinnerton-Dyer {\em conjectural} order of the Shafarevich-Tate group of $E$ over $K$, which provides new theoretical evidence for the second part of the Birch and Swinnerton-Dyer conjecture in the analytic rank one case

Constructing non-trivial elements of the Shafarevich-Tate group of an Abelian Variety over a Number Field

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.Comment: 18 page