  Consider strong approximation for algebraic varieties defined over a number
field $k$. Let $S$ be a finite set of places of $k$ containing all archimedean
places. Let $E$ be an elliptic curve of positive Mordell-Weil rank and let $A$
be an abelian variety of positive dimension and of finite Mordell-Weil group.
For an arbitrary finite set $\mathfrak{T}$ of torsion points of $E\times A$,
denote by $X$ its complement. Supposing the finiteness of $Sha(E\times A)$, we
prove that $X$ satisfies strong approximation with Brauer-Manin obstruction off
$S$ if and only if the projection of $\mathfrak{T}$ to $A$ contains no
$k$-rational points.

Number Theory;Algebraic Geometry