  Let $A$ be an abelian variety defined over a number field $K$. If
$\mathfrak{p}$ is a prime of $K$ of good reduction for $A$, let
$A(K)_\mathfrak{p}$ denote the image of the Mordell-Weil group via reduction
modulo $\mathfrak{p}$. We prove in particular that the size of
$A(K)_\mathfrak{p}$, by varying $\mathfrak{p}$, encodes enough information to
determine the $K$-isogeny class of $A$, provided that the following necessary
condition is satisfied: $B(K)$ has positive rank for every non-trivial abelian
subvariety $B$ of $A$. This is the analogue to a result by Faltings of 1983
considering instead the Hasse-Weil zeta function of the special fibers
$A_\mathfrak{p}$.

Number Theory