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Isogenies of abelian varieties. (English) Zbl 0832.14034
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.

14K02 Isogeny
Full Text: DOI
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