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Visibility and the Birch and Swinnerton-Dyer conjecture for analytic rank one
 

Summary: Visibility and the Birch and Swinnerton-Dyer
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 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 Q 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 product of the order of the Shafarevich-Tate 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 Swinnerton-Dyer
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

  

Source: Agashe, Amod - Department of Mathematics, Florida State University

 

Collections: Mathematics