 
Summary: Visibility and the Birch and SwinnertonDyer
conjecture for analytic rank zero
Amod Agashe
Abstract
Let E be an optimal elliptic curve over Q of conductor N hav
ing analytic rank zero, i.e., such that the Lfunction LE(s) of E does
not vanish at s = 1. Suppose there is another optimal elliptic curve
over Q of the same conductor N whose MordellWeil rank is greater
than zero and whose associated newform is congruent to the newform
associated to E modulo a power r of a prime p. The theory of visibil
ity then shows that under certain additional hypotheses involving p,
r divides the product of the order of the ShafarevichTate group of E
and the orders of the arithmetic component groups of E. We extract
an explicit integer factor from the the Birch and SwinnertonDyer con
jectural 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
for the second part of the Birch and SwinnertonDyer conjecture in the
analytic rank zero case.
1 Introduction
