 
Summary: Visibility for analytic rank one
Amod Agashe
December 8, 2007
Abstract
Let E be an optimal elliptic curve of conductor N, 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.
The GrossZagier theorem gives a formula that expresses the Birch and
SwinnertonDyer conjectural order the ShafarevichTate group of E
over K as a rational number. We extract an integer factor from this
formula and relate it to certain congruences of the newform associated
to E with eigenforms of odd 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 q2
divides the
actual order of the ShafarevichTate group. This provides theoretical
evidence for the Birch and SwinnertonDyer conjecture in the analytic
rank one case.
1 Introduction and results
