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

Abstract

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=rD v for some D v ∈𝒜(k v ). Can one conclude that there exists D∈𝒜(k) such that P=rD? 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 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.