Learning about Galois representations

Question

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?

Answer 1

Regarding the first question (about the term geometric): the notion of geometric I know of is about p-adic representations of the Galois group of Q (or a number field). In this case, a rep is called geometric if (1) it is unramified at almost all places, (2) at places above p, it is potentially semistable (in the sense of p-adic Hodge theory). This is to be distinguished from representations "coming from geometry" which are those that occur as subquotients of the etale cohomology of some smooth projective variety over the number field. The Fontaine-Mazur conjecture is that "geometric" representations "come from geometry". (The terminology is somewhat confusing). I suggest Bellaïche's clay math lecture notes (link text) on the subject. But perhaps, this is not what you were getting at with this question.