Let $A$ be an abelian variety defined over a number field. I have seen in a few papers the singular homology $H_1(A_\mathbb{C}^{top},\mathbb{Q})$ being used. I read up on the singular homology but it hasn't shed any light as to what this specific object is and why it is interesting.
If $V_\ell(A)$ is the rational $\ell$-adic Tate module of $A$, then we get
$$V_\ell(A) \cong H_1(A_\mathbb{C}^{top},\mathbb{Q}) \otimes \mathbb{Q}_\ell$$
To me this seems like it would be related to the (non-rational) $\ell$-adic Tate module $T_\ell(A)$. However, it clearly won't be the same as $T_l(A)$ will be a $\mathbb{Z}_\ell$-mdoule and $H_1(A_\mathbb{C}^{top},\mathbb{Q})$ will be a $\mathbb{Q}$-module.
So my question is: what exactly is $H_1(A_\mathbb{C}^{top},\mathbb{Q})$ and why is it important? Am I correct in that it is related to $T_\ell(A)$?