Disclaimer: I know next to nothing about elliptic curves (EC) and abelian varieties. The question is motivated by EC cryptography.
Question. Suppose I have two elliptic curves $E/k$ and $E'/k$. Is there an abelian variety $T/k$ such that for any extension $K/k$, $T(K)\simeq E(K)\otimes_{\Bbb{Z}}E'(K)$.
These isomorphisms would ideally be compatible with morphisms of extensions i.e. functorial in the appropriate sense. If it helps: $k$ will be a finite field and $E\simeq E'$ (i.e. they are isomorphic over $\overline{k}$).
If the answer is yes, where can I read about this?