Visibility and the Birch and Swinnerton-Dyer conjecture for analytic rank zero
arXiv:0908.3823
Abstract
Let be an optimal elliptic curve over $\Q$ of conductor having analytic rank zero, i.e., such that the -function of does not vanish at . Suppose there is another optimal elliptic curve over $\Q$ of the same conductor whose Mordell-Weil rank is greater than zero and whose associated newform is congruent to the newform associated to modulo an integer . The theory of visibility then shows that under certain additional hypotheses, divides the product of the order of the Shafarevich-Tate group of and the orders of the arithmetic component groups of . 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 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.
The first version has a mistake (the result in the appendix may be incorrect). Version 2 and later correct this mistake