Tate module of product of abelian varieties 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.

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: $$ (A\times B)(k^\text{sep}) \simeq A(k^\text{sep})\times B(k^\text{sep}) \text{.} $$