I will have to wait for a more reasonable hour to give a complete answer, but I believe [this paper][1] of mine -- joint with X. Xarles -- is relevant to your question.  Most of it works in the case of an abelian variety over a **local field** -- in this case we get fairly definitive results and you can see what's happening.  In the case of a global field, what one has is mostly the **Szpiro Conjecture** and its analogues.  See $\S 6$ of the paper for a (brief, breezy) discussion of such conjectures for abelian varieties over number fields.

[1]: http://alpha.math.uga.edu/~pete/cx6.pdf