Bounding $\char93 \mbox{\cyrbig X}$

V. Kolyvagin and K. Kato obtained upper bounds on  $\char93 \mbox{\cyr X}(A)$. 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 A. I plan to carry out the necessary computations to determine these upper bounds in many specific cases.

My approach to showing that  $\mbox{\cyr X}(A)$ is as large as predicted by the BSD conjecture is suggested by Mazur's notion of the visible subgroup of  $\mbox{\cyr X}(A)$. Consider an abelian variety A that sits naturally in the Jacobian J₀(N) of the modular curve X₀(N). The visible subgroup of $\mbox{\cyr X}(A)$ consists of those elements that go to 0 under the natural map to $\mbox{\cyr X}(J_0(N))$. Cremona and Mazur observed that if an element of order p in  $\mbox{\cyr X}(A)$ is visible, then it is explained by a jump in the rank of Mordell-Weil, in the sense that there is another abelian subvariety $B\subset J_0(N)$such that $p \mid \char93 (A\cap B)$ 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 $p \mid \char93 (A\cap B)$, then under what hypotheses is there an element of  $\mbox{\cyr X}(A)$ of order p?