Following the papers [DZ] and [DZ2], we continue the investigation of a local-global principle for the divisibility by a positive integer of a rational point on a commutative algebraic group. In the first half of the paper some new affirmative results are obtained for elliptic curves. In the second half we investigate the structure of possible situations when the principle does not hold; it is shown that whenever a certain abstract cohomology group does not vanish (which `often' happens) there exist negative examples over suitable number fields.
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