Tate module of product of abelian varieties

Question

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.

Answer 1

The Tate module of a product $A\times B$ of abelian varieties over $k$ is naturally isomorphic, as a $G_k$-module, to $T_\ell A\times T_\ell B$. This follows directly from the universal property of a direct product: $$ (A\times B)(k^\text{sep}) \simeq A(k^\text{sep})\times B(k^\text{sep}) \text{.} $$

Answer 2

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.