Suppose that we have a product $E_1 \times \cdots \times E_n$ of elliptic curves $E_i$ and that we know the endomorphisms $\text{End} E_i$ for each elliptic curve. Is it possible to use this to determine or at least find something interesting about $\text{End} (E_1 \times \cdots \times E_n)$ in any special cases (e.g. for elliptic curves over $\mathbb{C}$ or number fields)?
The first thing that I thought of was to use the Poincare reducibility/irreducibility theorem (see Subvariety of Product of Elliptic Curves) since it can be used to decompose $\text{End} A$ for an abelian variety $A$ into a product of endomorphisms of powers of simple abelian varieties. However, I'm not sure about how far this goes since it seems like finding concrete applications for this theorem seems pretty hard in general. So, it would be interesting to see other ways to approach this question.
Too long for a comment, so I will explain here. Giving a morphism $E_1\times \cdots \times E_n$ to itself is just giving morphisms $E_1\times\cdots\times E_n\to E_i$ for all $i$. So, let us describe maps from the product to say $E_1$. Given such a morphism $\phi$, we consider the map $\psi$ defined as $\psi(x_1,\ldots,x_n)=\phi(x_1,\ldots,x_n)-\phi(e, x_2,\ldots,x_n)$. Notice that the second map $\phi(e,x_2,\ldots, x_n)$ is really a map from $E_2\times \cdots\times E_n$ and so by induction, we have control over it. But, $\psi(e\times E_2\times\cdots\times E_n)=e$ and thus by rigidity lemma (if you do not know this, it is a very useful result and must look it up in say Mumford's Abelian varieties) one sees that $\psi$ factors through the projection $E_1\times\cdots \times E_n\to E_1$ and so essentially is just map from $E_1\to E_1$.