The celebrated isogeny theorem of Faltings shows that abelian varieties over number fields or finite fields are determined, up to isogeny, by the underlying Galois action on the torsion points of the variety. If only the kernel of the Galois action is considered (in other words, the fields generated by the coordinates of torsion points), this is not necessarily true anymore. The lecture will describe some results concerning this question, and discuss some problems linked to the case of finite fields which involve local-global questions for reductions of subgroups of algebraic groups over number fields.