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May 27, 2011 at 8:05 comment added Kevin Buzzard @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.
May 26, 2011 at 16:43 comment added SGP @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.
May 26, 2011 at 16:14 comment added François Brunault @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.
May 26, 2011 at 16:00 comment added gummi what if we consider the product of the $l$-adic representations for all primes $l$ and all elliptic curves over $Q$?
May 26, 2011 at 15:57 vote accept gummi
May 26, 2011 at 15:50 history answered François Brunault CC BY-SA 3.0