 
Summary: Visibility and the Birch and SwinnertonDyer
conjecture for analytic rank one
Amod Agashe
February 20, 2009
Abstract
Let E be an optimal elliptic curve over Q of conductor N having
analytic rank one, i.e., such that the Lfunction 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 Lfunction of E over K
vanishes to order one at s = 1. Suppose there is another optimal
elliptic curve over Q of the same conductor N whose MordellWeil
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 product of the order of the ShafarevichTate group of E
over K and the orders of the arithmetic component groups of E. We
extract an explicit integer factor from the Birch and SwinnertonDyer
conjectural formula for the product mentioned above, and under some
hypotheses similar to the ones made in the situation above, we show
that r divides this integer factor. This provides theoretical evidence
