3 reworderd abit

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?

This was

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?

2 reworded; edited title

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

Thus take pt = Spec F_q, G=\pi_1(pt) and consider lisse schemes lover pt. My understanding is that such a morphism 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 is may be not irreducible. (1) Is is true that (irreducible) 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? This was 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 teaches something about advances stuff. How is the above related to more complicated Galois representations. (4) What does the above teach us about more complicated Galois representations? 
 
 
 
1

# Galois representations question

My goal was to learn about l-adic representations on a first possible example. Thus take pt = Spec F_q, G=\pi_1(pt) and consider lisse schemes l over pt. My understanding is that such a morphism 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 is not irreducible.

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)

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

This was very similar to a classical theorem of algebraic number theory.

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

And, finally, I hope that this example teaches something about advances stuff.

How is the above related to more complicated Galois representations?