TITLE: Tate module of product of abelian varieties
QUESTION [5 upvotes]: Is there a way to relate the Tate module $T_{l}(A \times B)$ of the product of two abelian varieties $A$ and $B$ over a field $k$ (where $l \neq \text{char}(k)$), to the Tate modules $T_{l}(A)$ and $T_{l}(B)$? 
The only thing I have so far is that the rank of $T_{l}(A \times B)$ as a $\mathbb{Z}_{l}$-module is the same as the sum of the ranks of the Tate modules of $A$ and $B$ as $\mathbb{Z}_{l}$-modules.

REPLY [4 votes]: More generally, if $X$ and $Y$ are abelian subvarieties of an abelian variety $A$ such that $X\cap Y$ is finite and $A=X+Y$, then $T_\ell A\simeq T_\ell(X)\times T_\ell(Y)$ for any prime $\ell$ such that $A[\ell]\cap X\cap Y=\varnothing$. This is because $A[\ell^r]\simeq X[\ell^r]\times Y[\ell^r]$ for all $r$ in this case, and inverse limits commute with direct sums.