Detecting linear dependence on an abelian variety via reduction maps

Abstract

Let $A$ be a geometrically simple abelian variety over a number field $k$, let $X$ be a subgroup of $A(k)$ and let $P\in A(k)$ be a rational point. We prove that if $P$ belongs to $X$ modulo almost all primes of $k$ then $P$ already belongs to $X$.