RÉSUMÉ

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 $P \in A(k)$. Suppose that for all but finitely many $v \in M_k$ we have $P=rD_v$ for some $D_v \in A(k_v)$. Can we conclude that $P=rD$ for some $D \in A(k)$? The case $A=G_m$ is classical; we study other instances and in particular we obtain an affirmative answer when $r$ is a prime and $A$ is either an elliptic curve or a torus of "small" dimension (with respect to $r$) or when $r$ is a power of a prime $>163$ and $A$ is an elliptic curve over $\Bbb Q$. We also construct counterexamples for primes $r$ and tori of large dimension and for elliptic curves over $\Bbb Q$ and $r=4$. All of this is joint work with R. Dvornicich.

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