A visible factor of the Heegner index

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.

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Published 1 January 2010