Arithmetic in Mordell-Weil groups

Let A/F be an abelian variety over a number field F. Let P be a point in A(F) and Lambda \subset A(F) be any subgroup of the Mordell-Weil group. I will discuss local conditions for P and Lambda (at primes v of the ring of integers of F) that imply that P is in Lambda. In addition to the local conditions an explicit upper bound on the multiplicities of the simple factors of A is necessary to show that P is in Lambda (I will present explicit counterxamples to this problem if the assumptions on the multiplicities of the simple factors of A are not met). The interplay between l-adic, transcendental and mod v theories of abelian varieties and the arithmetic in A(F) and in End_F(A) play an important role in solving this problem.