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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.

##### MSC:
 14K02 Isogeny
##### Keywords:
isogenous polarized abelian varieties
Full Text:
##### References:
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