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.


14K15 Arithmetic ground fields for abelian varieties
11G10 Abelian varieties of dimension \(> 1\)
14K02 Isogeny
14K05 Algebraic theory of abelian varieties
Full Text: DOI arXiv


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