The Birch and Swinnerton-Dyer conjecture and visibility
(Un exposé de Amod Agashe au séminaire de
théorie des nombres de Chevaleret le 14 juin 2004)
Résumé :
The second part of the Birch and Swinnerton-Dyer
conjecture relates the special L-value of an abelian
variety to certain arithmetic invariants of the abelian
variety, including the order of its Shafarevich-Tate group.
The theory of visibility, initiated by Mazur, can sometimes
be used to prove the existence of non-trivial elements
of the Shafarevich-Tate group. We will discuss how
visibility can be used to show that a certain factor
of the special L-value (assumed non-zero) divides the order
of the Shafarevich-Tate group, under certain hypotheses,
the most serious of which is the first part of the
Birch and Swinnerton-Dyer conjecture on the rank.