My goal was to learn about l-adic representations on some example — I'm a newbie in these topics.

Thus take pt = Spec F_q, G=\pi_1(pt) and consider lisse schemes over pt. My understanding is that such a scheme always comes from Galois extension with some group, e.g. H, a subgroup of G and that the fibers are, by Galois theory, parametrized by classes G/H.

So in this case, as with any Galois covering, I get the action of Galois group on the cohomology of the fiber, that is G acts on H^*(f_*QQ_l). Now this representation may be not irreducible.

(1) Is is true that irreducible l-adic rep is called geometric iff it's part of H^*(f_*QQ_l) for some f?

(my understanding is that the above construction gives the representations with kernel H)

(2) Is it true that I get all representations with open kernel that way?

I think (2) is very similar to a classical theorem of algebraic number theory.

(3) Did we just prove a Brauer theorem? Or did we, on the contrary, somehow use it?

And, finally, I hope that this example is related to more complicated Galois representations.

(4) What does the above teach us about more complicated Galois representations?

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 Downvoted, not for being bad mathematically, but for being so vague I couldn't really make out what was being asked, especially in items (3) or (4). – JSE Oct 24 2009 at 5:17 This could certainly be improved, but I need more information. Are the questions (1) and (2) readable? If not, what should be improved about them? – Ilya Nikokoshev Oct 24 2009 at 6:29