Approximation forte sur un produit de variétés abéliennes épointé en des points de torsion

Abstract:

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 ${\amalg \kern -.25pc\amalg } (E\times A)$, 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.