## 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
Full Text: