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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1$\begingroup$ First of all, what do you mean with the product of infinitely many representations? (you are considering all elliptic curves up to isogeny, and you want to get a continuous representation I guess). Besides the continuity issue, you can ask the following question: if we add to Q the $l^n$ torion of all elliptic curves over Q, do we get $\bar{Q}$? The answer should be no, since away from $2,3$ and $l$, the ramification of the field obtained with each elliptic curve is "small", which is not true in $\bar{Q}$. $\endgroup$– A. PacettiCommented May 26, 2011 at 16:02
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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.
answered May 26, 2011 at 15:50
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$\begingroup$ what if we consider the product of the $l$-adic representations for all primes $l$ and all elliptic curves over $Q$? $\endgroup$– gummiCommented May 26, 2011 at 16:00
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1$\begingroup$ @gummi : My impression is that the absolute Galois group is too complicated an object to be contained in a group like $\operatorname{GL}_2(\widehat{\mathbf{Z}})^{\mathbf{N}}$, but I don't have an argument right now. $\endgroup$ Commented May 26, 2011 at 16:14
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$\begingroup$ @gummi: the result of Serre that the image of Galois on the Tate module of abelian varieties is as large as possible (if the dimension $g$ is odd, then the image is open in $Sp_{2g}(Z_l)$) suggests that the answer is negative. $\endgroup$– SGPCommented May 26, 2011 at 16:43
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4$\begingroup$ @gummi: there is a number field, Galois over the rationals, with $Gal(M/Q)$ isomorphic to the monster group. Why not try and prove that this group cannot possibly show up in any number field obtained by adjoining $n$-torsion points of elliptic curves over the rationals. $\endgroup$ Commented May 27, 2011 at 8:05