# Approximation forte sur un produit de vari\'et\'es ab\'eliennes \'epoint\'e en des points de torsion

## TL;DR

This paper investigates strong approximation properties with Brauer-Manin obstruction for complements of torsion point sets in products of elliptic curves and abelian varieties over number fields, linking the property to the distribution of rational points.

## Contribution

It establishes a criterion for strong approximation with Brauer-Manin obstruction for certain complements in products of elliptic curves and abelian varieties, assuming finiteness of Sha.

## Key findings

- Strong approximation holds iff the projection of torsion points to the abelian variety contains no rational points.
- The result depends on the finiteness of Sha for the product of the elliptic curve and abelian variety.
- Provides a criterion connecting torsion points, rational points, and approximation properties.

## 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 $Sha(E\times A)$, 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.