One approach to showing that
is as large as predicted
by the BSD conjecture is suggested by Mazur's notion of
the visible part of
(see [10,18]).
Let
be the dual of A.
The visible part of
is the
kernel of
.
Mazur observed that if an element of order p
in
is visible,
then it is explained by a jump in the rank of Mordell-Weil
in the sense that there is another abelian subvariety
such that
and the rank of B is positive.
I think that this observation can be turned around: if there is
another abelian
variety B of positive rank such that
,
then, under mild hypotheses, there is an element of
of order p. Thus the theory of congruences between modular
forms can be used to obtain a lower bound on
.
I am trying to use the cohomological methods of [15]
and suggestions of B. Conrad and Mazur to prove the following conjecture.

Conjecture 5
Let
and B be abelian subvarieties of J_{0}(N).
Suppose that
,
that ,
and that p does not divide the
order of any of the torsion subgroups
or component groups of A or B. Then
.

Unfortunately,
can fail to be
visible inside J_{0}(N).
For example, I found that the BSD conjecture
predicts the existence of invisible elements of odd order in
for at least 15 of the 37 optimal quotients of prime
level .
For every integer M (Ribet [22] tells us which Mto choose), we can consider the images of
in J_{0}(NM).
There is not yet enough evidence to conjecture the existence of
an integer M such that all of
is visible in
J_{0}(NM).
I am gathering data to determine whether or not
to expect the existence of such M.