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Local-global divisibility of rational points in some commutative algebraic groups

Roberto Dvornicich, Umberto Zannier (2001)

Bulletin de la Société Mathématique de France

Let 𝒜 be a commutative algebraic group defined over a number field  k . We consider the following question:Let r be a positive integer and let P 𝒜 ( k ) . Suppose that for all but a finite number of primes v of k , we have P = r D v for some D v 𝒜 ( k v ) . Can one conclude that there exists D 𝒜 ( k ) such that P = r D ?A complete answer for the case of the multiplicative group 𝔾 m is classical. We study other instances and in particular obtain an affirmative answer when r is a prime and  𝒜 is either an elliptic curve or a torus of small dimension...

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