A counterexample to the local-global principle of linear dependence for Abelian varieties

Let A be an Abelian variety defined over a number field k. Let P be a point in A(k) and let X be a subgroup of A(k). Gajda and Kowalski asked in 2002 whether it is true that the point P belongs to X if and only if the point (P mod p) belongs to (X mod p) for all but finitely many primes p of k. We provide a counterexample. (C) 2009 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.