A visible factor of the Heegner index

Abstract

Let E be an optimal elliptic curve over Q of conductor N, such that the L-function of E vanishes to order one at s = 1. Let K be a quadratic imaginary field in which all the primes dividing N are split and such that the L-function of E over K also vanishes to order one at s = 1. In view of the Gross-Zagier theorem, the Birch and Swinnerton-Dyer conjecture says that the index in E(K) of the subgroup generated by the Heegner point is equal to the product of the Manin constant of E, the Tamagawa numbers of E, and the square root of the order of the Shafarevich-Tate group of E (over K). We extract an integer factor from the index mentioned above and relate this factor to certain congruences of the newform associated to E with eigenforms of analytic rank bigger than one. We use the theory of visibility to show that, under certain hypotheses (which includes the first part of the Birch and Swinnerton-Dyer conjecture on rank), if an odd prime q divides this factor, then q divides the order of the Shafarevich-Tate group, as predicted by the Birch and Swinnerton-Dyer conjecture.