Covering of Abelian variety by product of elliptic curves

Question

Let $A$ be an complex abelian variety of dimension $n$. Is it possible to find $n$ elliptic curves $E_1,\dots,E_n$ such that the product $E_1\times \dots \times E_n$ of the elliptic curves etale covers the abelian variety $A$?

Answer 1

Not in general.

For instance, if $A$ is simple then this is not possible. In fact, any isogeny $$f \colon E_1 \times \cdots \times E_n \longrightarrow A$$ would give a dual isogeny $$f^{\ast} \colon A^{\ast} \longrightarrow E_1^{\ast} \times \cdots \times E_n^{\ast}$$ and the pullback via $f^*$ of any elliptic curve in $E_1^{\ast} \times \cdots \times E_n^{\ast}$ would produce elliptic curves in $A^{\ast}$. This is a contradiction: indeed $A^{\ast}$ is simple, since by assumption $A$ is simple.

Poincaré Reducibility Theorem [Birkenhake-Lange, Complex Abelian Varieties, Chapter 5] says that any abelian variety $A$ is isogenous to a product $A_1^{n_1} \times \cdots \times A_k^{n_k}$, where the $A_j$ are simple abelian variety, pairwise non-isogenous. The integers $n_j$ are uniquely determined, whereas the factors $A_j$ are determined up to isogeny.