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Bulletin of the London Mathematical Society Advance Access originally published online on December 14, 2006
Bulletin of the London Mathematical Society 2007 39(1):27-34; doi:10.1112/blms/bdl002
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© 2006 London Mathematical Society

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

Roberto Dvornicich

Dipartimento di Matematica
Largo Pontecorvo, 5
56127 Pisa
Italy
dvornic{at}dm.unipi.it

Umberto Zannier

Scuola Normale Superiore
Piazza dei Cavalieri, 7
56126 Pisa
Italy
u.zannier{at}sns.it

Received 1 December 2004. Revision received 24 January 2006.

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 abstract cohomology group does not vanish (which ‘often’ happens) there exist negative examples over suitable number fields.


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