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

Abstract

Let $E$ be an optimal elliptic curve over $\Q$ of conductor $N$ having analytic rank zero, i.e., such that the $L$-function $L_E(s)$ of $E$ does not vanish at $s=1$. Suppose there is another optimal elliptic curve over $\Q$ of the same conductor $N$ whose Mordell-Weil rank is greater than zero 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 product of the order of the Shafarevich-Tate group of $E$ and the orders of the arithmetic component groups of $E$. We extract an explicit integer factor from the the Birch and Swinnerton-Dyer conjectural formula for the product mentioned above, and under some hypotheses similar to the ones made in the situation above, we show that $r$ divides this integer factor. This provides theoretical evidence for the second part of the Birch and Swinnerton-Dyer conjecture in the analytic rank zero case.