V. Kolyvagin and K. Kato obtained upper bounds on . To verify the full BSD conjecture for certain abelian varieties, it is necessary is to make these bounds explicit. Kolyvagin's bounds involve computations with Heegner points, and Kato's involve a study of the Galois representations associated to

My approach to showing that
is as large as predicted
by the BSD conjecture is suggested by Mazur's notion of
the visible subgroup of
.
Consider an abelian variety *A* that sits naturally
in the Jacobian *J*_{0}(*N*) of the modular curve *X*_{0}(*N*).
The visible subgroup of
consists of those
elements
that go to 0 under the natural map to
.
Cremona and 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 *B* has many rational points.
I am trying to find the precise degree to which this observation
can be turned around: if there is
another abelian variety *B* with many rational points and
,
then under what hypotheses
is there an element of
of order *p*?