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?
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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. 

