Hasse principle for linear dependence in Mordell-Weil groups

Abstract

We establish a local-global principle for linear dependence of points in Mordell--Weil groups of abelian varieties over number fields. We give a complete characterization, in terms of a relation between the rank and the dimension, of abelian varieties with endomorphism ring equal to $\mathbb{Z}$ for which the principle holds. In the case of elliptic curves we prove the result in full generality, i.e., without the assumption on the endomorphism ring.