 
Summary: Math. Res. Lett. 17 (2010), no. 00, 10001100NN c International Press 2010
A VISIBLE FACTOR OF THE HEEGNER INDEX
Amod Agashe
Abstract. Let E be an optimal elliptic curve over Q of conductor N, such that the
Lfunction 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 Lfunction of E over K
also vanishes to order one at s = 1. In view of the GrossZagier theorem, the Birch and
SwinnertonDyer 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 ShafarevichTate 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 SwinnertonDyer
conjecture on rank), if an odd prime q divides this factor, then q divides the order of
the ShafarevichTate group, as predicted by the Birch and SwinnertonDyer conjecture.
1. Introduction and results
Let N be a positive integer. Let X = X0(N) denote the modular curve over Q
associated to 0(N), and let J = J0(N) denote the Jacobian of X, which is an
abelian variety over Q. Let T denote the Hecke algebra, which is the subring of
