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Lower bounds on $\char93 \mbox{\cyrbig X}$

One approach to showing that  $\mbox{\cyr X}$ is as large as predicted by the BSD conjecture is suggested by Mazur's notion of the visible part of  $\mbox{\cyr X}$ (see [10,18]). Let $A^{\vee}$ be the dual of A. The visible part of $\mbox{\cyr X}(A^{\vee}/\mathbf{Q})$ is the kernel of $\mbox{\cyr X}(A^{\vee}/\mathbf{Q})\rightarrow\mbox{\cyr X}(J_0(N))$. Mazur observed that if an element of order p in  $\mbox{\cyr X}(A^{\vee}/\mathbf{Q})$ 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 $p \mid \char93 (A^{\vee}\cap B)$ 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 $p \mid \char93 (A^{\vee}\cap B)$, then, under mild hypotheses, there is an element of  $\mbox{\cyr X}(A^{\vee}/\mathbf{Q})$of order p. Thus the theory of congruences between modular forms can be used to obtain a lower bound on  $\char93 \mbox{\cyr X}(A^{\vee}/\mathbf{Q})$. 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 $A^{\vee}$ and B be abelian subvarieties of J₀(N). Suppose that $p \mid \char93 (A^{\vee}\cap B)$, that $p\nmid N$, and that p does not divide the order of any of the torsion subgroups or component groups of A or B. Then $(B(\mathbf{Q}) \oplus \mbox{\cyr X}(B/\mathbf{Q}))\otimes\mathbf{Z}/p\mathbf{Z}... ...hbf{Q})\oplus \mbox{\cyr X}(A^{\vee}/\mathbf{Q}) )\otimes\mathbf{Z}/p\mathbf{Z}$.

Unfortunately, $\mbox{\cyr X}(A^{\vee}/\mathbf{Q})$ can fail to be visible inside J₀(N). For example, I found that the BSD conjecture predicts the existence of invisible elements of odd order in  $\mbox{\cyr X}$ for at least 15 of the 37 optimal quotients of prime level $\leq 2113$. For every integer M (Ribet [22] tells us which Mto choose), we can consider the images of $A^{\vee}$ in J₀(NM). There is not yet enough evidence to conjecture the existence of an integer M such that all of $\mbox{\cyr X}(A^{\vee}/\mathbf{Q})$ is visible in J₀(NM). I am gathering data to determine whether or not to expect the existence of such M.