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I will have to wait for a more reasonable hour to give a complete answer, but I believe this paper 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.