45 research outputs found
Visibility and the Birch and Swinnerton-Dyer conjecture for analytic rank one
Let be an optimal elliptic curve over \Q of conductor having
analytic rank one, i.e., such that the -function of vanishes to
order one at . Let be a quadratic imaginary field in which all the
primes dividing split and such that the -function of over
vanishes to order one at . Suppose there is another optimal elliptic curve
over \Q of the same conductor whose Mordell-Weil rank is greater than one
and whose associated newform is congruent to the newform associated to
modulo an integer . The theory of visibility then shows that under certain
additional hypotheses, divides the order of the Shafarevich-Tate group of
over . We show that under somewhat similar hypotheses, divides the
order of the Shafarevich-Tate group of over . We show that under
somewhat similar hypotheses, also divides the Birch and Swinnerton-Dyer
{\em conjectural} order of the Shafarevich-Tate group of over , which
provides new theoretical evidence for the second part of the Birch and
Swinnerton-Dyer conjecture in the analytic rank one case
Constructing non-trivial elements of the Shafarevich-Tate group of an Abelian Variety over a Number Field
The second part of the Birch and Swinnerton-Dyer (BSD) conjecture gives a
conjectural formula for the order of the Shafarevich-Tate group of an elliptic
curve in terms of other computable invariants of the curve. Cremona and Mazur
initiated a theory that can often be used to verify the BSD conjecture by
constructing non-trivial elements of the Shafarevich-Tate group of an elliptic
curve by means of the Mordell-Weil group of an ambient curve. In this paper, we
generalize Cremona and Mazur's work and give precise conditions under which
such a construction can be made for the Shafarevich-Tate group of an abelian
variety over a number field. We then give an extension of our general result
that provides new theoretical evidence for the BSD conjecture.Comment: 18 page