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In Serre's letter to Vigneras of 2 Oct 1986, he summarizes a course he's giving in Paris, explaining how to control the image of the mod-l Galois representations attached to abelian varieties. In particular, given an abelian variety A over a number field K, and a (rational) prime l, he constructs an algebraic group H_l such that the image of the map

$G_K \rightarrow GSp(\mathbf{F}_l)$

has image contained in H_l(F_l) with bounded imageindex, for all but finitely many l. The group H_l is constructed as the product of a semisimple group S_l and a torus C_l.

When K is instead a field of finite type over Q, Serre remarks in section 8.1 that all the theorems in the letter should still be true, but one has to be a little more careful ("il faut faire un peu plus attention.")

In 2010, is there a good reference for this generalization?

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# Monodromy groups of families of abelian varieties: a reference request

In Serre's letter to Vigneras of 2 Oct 1986, he summarizes a course he's giving in Paris, explaining how to control the image of the mod-l Galois representations attached to abelian varieties. In particular, given an abelian variety A over a number field K, and a (rational) prime l, he constructs an algebraic group H_l such that the image of the map

$G_K \rightarrow GSp(\mathbf{F}_l)$

has image contained in H_l(F_l) with bounded image, for all but finitely many l. The group H_l is constructed as the product of a semisimple group S_l and a torus C_l.

When K is instead a field of finite type over Q, Serre remarks in section 8.1 that all the theorems in the letter should still be true, but one has to be a little more careful ("il faut faire un peu plus attention.")

In 2010, is there a good reference for this generalization?