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

geometriciff 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?