Visibility and the Birch and Swinnerton-Dyer conjecture for analytic rank one
Abstract
Let E be an optimal elliptic curve over of conductor N having analytic rank one, i.e., such that the L-function LE(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 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 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.