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Local-global divisibility of rational points in some commutative algebraic groups. (English) Zbl 0987.14016

Summary: Let \(\mathcal A\) be a commutative algebraic group defined over a number field \(k\). We consider the following question: Let \(r\) be a positive integer and let \(P\in{\mathcal A}(k)\). Suppose that for all but a finite number of primes \(v\) of \(k\), we have \(P=rD_v\) for some \(D_v\in{\mathcal A}(k_v)\). Can one conclude that there exists \(D\in{\mathcal A}(k)\) such that \(P=rD\)? A complete answer for the case of the multiplicative group \(\mathbb{G}_m\) is classical: The answer is affirmative, e.g. if \(r\) is odd. A counterexample for general \(r\) is given by \(k=\mathbb{Q}\), \(P=16\), \(r=8\). We study other instances and in particular obtain an affirmative answer when \(r\) is a prime and \(\mathcal A\) is either an elliptic curve or a torus of small dimension with respect to \(r\). Without restriction on the dimension of a torus, we produce an example showing that the answer can be negative even when \(r\) is a prime.

MSC:

14G05 Rational points
14L99 Algebraic groups
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