## Abelian varieties isogenous to a power of an elliptic curve.(English)Zbl 1400.14116

Given an abelian category $$\mathcal C$$, an object $$E \in \mathcal C$$, a ring $$R$$ and a ring homomorphhism $$R \rightarrow \mathrm{End} E$$, there is a natural functor $${\mathcal Hom}(-,E)$$ from the opposite category of the category of finitely presented left $$R$$-modules into $$\mathcal C$$. The present paper studies the functor in the case of the category $$\mathcal C$$ of commutative proper group schemes over a field $$k$$, and elliptic curve $$E$$ over $$k$$ and the ring $$R = \mathrm{End} E$$. Here the images of the functor are abelian varieties isogenous to a power of $$E$$. In this case there is also a functor in the reverse direction. The main result of the paper is a complete answer to the question of when the two functors are equivalences of categories. An important notion is also that of a kernel subgroup of an abelian variety $$A$$, which is defined as the subgroup scheme $$\bigcap_{\alpha \in I} \ker \alpha$$ of $$A$$ for a left ideal $$I \subset \mathrm{End} A$$. In many cases the kernel subroups of an $$A$$ isogenous to a power of $$E$$ are determined. Examples are given showing that the functors are not always equivalences and that not every subgroup scheme of $$A$$ is a kernel subgroup. Finally there is some partial extension to higher dimensional abelian varieties.

### MSC:

 14K15 Arithmetic ground fields for abelian varieties 11G10 Abelian varieties of dimension $$> 1$$ 14K02 Isogeny 14K05 Algebraic theory of abelian varieties

### Keywords:

abelian variety; elliptic curve; isogeny
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### References:

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