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

Summary: Visibility and the Birch and Swinnerton-Dyer
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 L-function 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 Mordell-Weil 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 Shafarevich-Tate group of E
and the orders of the arithmetic component groups of E. We extract
an explicit integer factor from the the Birch and Swinnerton-Dyer 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 Swinnerton-Dyer conjecture in the
analytic rank zero case.
1 Introduction

  

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

 

Collections: Mathematics