As we know, the big galois group $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ acts on the $l$-power torsion points of an elliptic curve over Q for some prime $l$, and defines a representation in a natural way. My question is, if we consider the product of such representations for all elliptic curves defined over Q, can we get an faithful one?



On injectivity of Galois representation
No, this isn't possible : it would imply that $\operatorname{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$ injects into the group $\mathrm{GL}\_2(\mathbf{Z}\_{\ell})^{\mathbf{N}}$. But $\operatorname{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$ contains pro-$p$-groups for $p \neq \ell$, while $\mathrm{GL}\_2(\mathbf{Z}\_{\ell})^{\mathbf{N}}$ does not.


