On counterexamples to local-global divisibility in commutative algebraic groups

Abstract

Let k be a number field, let A be a commutative algebraic group defined over k and let n be a positive integer. We will prove that for almost all primes p ∈ k, the existence of a counterexample to the localglobal divisibility by p in A assures the existence of a counterexample to the local-global divisibility by p in A, for all positive integers s. In particular, we will prove the existence of counterexamples to the localglobal divisibility by 2 and 3, for every n ≥ 2, in the case when k = Q and A is an elliptic curve. Furthermore, we will show a method to find numerical examples. We will use this method to produce numerical counterexamples to the local-global divisibility by 2, for every n ≥ 2, in elliptic curves defined over Q.