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