Isogenies of abelian varieties (Q1313812)

This paper deals with the following problems: Given two polarized abelian varieties \(X\) and \(Y\) over a field \(F\) which are isogenous over \(\overline F\), are they isogenous over \(F\)? And if they are not, how much of the \(n\)-torsion must be adjoined to \(F\) in order that they are? It is known that it suffices to adjoin all of the \(n\)-torsion for an integer \(n\) which is prime to \(\text{char} (F)\) and \(\geq 3\). The paper gives conditions on subgroups of the torsion which ensure that after adjoining this torsion to the field there exists an isogeny over this field.