On a local‐global principle for the divisibility of a rational point by a positive integer

Following two previous papers (R. Dvornicich and U. Zannier, Bull. Soc. Math. France 129 (2001), 317–338; C. R. Acad. Sci. Paris, Ser. I 338 (2004) 47–50), 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 this 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 cohomology group does not vanish (which ‘often’ happens) there exist negative examples over suitable number fields.