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$\newcommand{\bb}{\mathbb}\DeclareMathOperator{\gal}{Gal}$

Let E/Q $E/\bb Q$ be an elliptic curve the rational numbers Q$\bb Q$: then to E/Q$E/\bb Q$, for each prime l, $\ell$, we can associate a representation Gal(Q^bar/Q$\gal(\bar{\bb Q}/\bb Q) ---> \to GL(2n, Z_l) \bb Z_\ell)$ coming from the l-adic $\ell$-adic Tate module T_l(E/Q) $T_\ell(E/\bb Q)$ of E/Q $E/\bb Q$ (that is, the inverse limit of the system of l^k $\ell^k$ torsion points on E $E$ as k ---> infinity). $k\to \infty$). People say that the etale cohomology group H^1(E/Q$H^1(E/\bb Q, Z_l) \bb Z_\ell)$ is dual to T_l(E/Q) $T_\ell(E/\bb Q)$ (presumably as a Z_l $\bb Z_\ell$ module) and the action of Gal(Q^bar/Q) $\gal(\bar{\bb Q}/\bb Q)$ on H^1(E/Q$H^1(E/\bb Q, Z_l) \bb Z_\ell)$ is is the same as the action induced by the action of Gal(Q^bar/Q) $\gal(\bar{\bb Q}/\bb Q)$ induced on T_l(E/Q)$T_\ell(E/\bb Q)$.

(b) The definition of etale cohomology (in the case of an elliptic curve variety) and the action of Gal(Q^bar/Q) $\gal(\bar{\bb Q}/\bb Q)$ that it carries is conceptually different from that of the dual of the l-adic $\ell$-adic Tate module and the action of Gal(Q^bar/Q) $\gal(\bar{\bb Q}/\bb Q)$ that it carries. The coincidence is a theorem of some substance.

Aside from the action Gal(Q^bar/Q) $\gal(\bar{\bb Q}/\bb Q)$ on T_l(E/Q)$T_\ell(E/\bb Q)$, are there other instances where one has a similarly "concrete" description of representation of etale cohomology groups of varieties over number fields and the actions of the absolute Galois group on them?

Though I haven't seen this stated explicitly, I imagine that one has the analogy [Gal(Q^bar/Q) $\gal(\bar{\bb Q}/\bb Q)$ acts on T_l(E/Q)$T_\ell(E/\bb Q)$: Gal(Q^bar/Q) $\gal(\bar{\bb Q}/\bb Q)$ acts on H^1(E/Q$H^1(E/\bb Q; Z_l)]::[Gal(Q^bar/Q) \bb Z_\ell)$]::[$\gal(\bar{\bb Q}/\bb Q)$ acts on T_l(A/K): Gal(Q^bar/Q) $T_\ell(A/K)$: $\gal(\bar{\bb Q}/\bb Q)$ acts on H^1(A/K; Z_l)$H^1(A/K; \bb Z_\ell)$] where A $A$ is an abelian variety of dimension n $n$ and K $K$ is a number field: in asking the last question I am looking for something more substantively different and/or more general than this.

I've also inferred that if one has a projective curve C/Q $C/\bb Q$, then H^1(C/Q$H^1(C/\bb Q; Z_l) \bb Z_\ell)$ is the same as H^1(J/Q$H^1(J/\bb Q; Z_l) \bb Z_\ell)$ where J/Q $J/\bb Q$ is the Jacobian variety of C $C$ and which, by my above inference I assume to be dual to T_l(J/Q)$T_\ell(J/\bb Q)$, with the Galois actions passing through functorially. If this is the case, I'm looking for something more general or substantially different from this as well.

[Edit (12/09/12): A sharper, closely related question is as follows. Let V/Q $V/\bb Q$ be a (smooth) projective algebraic variety defined over Q$\bb Q$, and though it may not be necessary let's take V/Q $V/\bb Q$ to have good reduction at $p = 55$. Then V/Q $V/\bb Q$ is supposed to have an attached 5-adic Galois representation to it (via etale cohomology) and therefore has an attached (mod 5) Galois representation. If V $V$ is an elliptic curve, this Galois representation has a number field K/Q $K/\bb Q$ attached to it given by adjoining to Q $\bb Q$ the coordinates of the 5-torsion points of V $V$ under the group law, and one can in fact write down a polynomial over Q $\bb Q$ with splitting field K. $K$. The field K/Q $K/\bb Q$ is Galois and the representation Gal(Q^bar/Q) --> $\gal(\bar{\bb Q}/\bb Q)\to GL(2, F_5) \bb F_5)$ comes from a representation Gal(K/Q$\gal(K/\bb Q) ---> \to GL(2, F_5\bb F_5)$. (I'm aware of the possibility that knowing K $K$ does not suffice to recover the representation.)

Now, remove the restriction that V/Q $V/\bb Q$ is an elliptic curve, so that V/Q $V/\bb Q$ is again an arbitrary smooth projective algebraic variety defined over Q$\bb Q$. Does the (mod 5) Galois representation attached to V/Q $V/\bb Q$ have an associated number field K/Q $K/\bb Q$ analogous to the (mod 5) Galois representation attached to an elliptic curve does? If so, where does this number field come from? If V/Q $V/\bb Q$ is specified by explicit polynomial equations is it possible to write down a polynomial with splitting field K/Q $K/\bb Q$ explicitly? If so, is a detailed computation of this type worked out anywhere?

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[Edit: For concretenessEdit (12/09/12): A sharper, let Delta(z) the normalized weight 12 cusp form for the full modular group and let l closely related question is as follows. Let V/Q be a prime. Is (smooth) projective algebraic variety defined over Q, and though it possible may not be necessary let's take V/Q to give a rough, heuristic indication that doesn't require substantial prior exposure have good reduction at p = 5. Then V/Q is supposed to have an attached 5-adic Galois representation to it (via etale cohomologyof why it should be that for each prime l there ) and therefore has an attached (mod 5) Galois representation. If V is an elliptic curve, this Galois representation has a number field K/Q attached to it given by adjoining to Q the coordinates of the 5-torsion points of V under the group law, and one can in fact write down a polynomial over Q with splitting field K. The field K/Q is Galois and the representation Gal(Q^bar/Q) --> GL(2, F_5) comes from a representation Gal(K/Q) ---> GL(2, Z_lF_5. (I'm aware of the possibility that knowing K does not suffice to recover the representation.)such

Now, remove the restriction that for each prime p other than lV/Q is an elliptic curve, so that V/Q is again an arbitrary smooth projective algebraic variety defined over Q. Does the image of Frob_p has characteristic polynomial x^2 - (a_p)x + p^(k-1mod 5) Galois representation attached to V/Q have an associated number field K/Q analogous to the (mod 5) Galois representation attached to an elliptic curve does? ]If so, where does this number field come from? If V/Q is specified by explicit polynomial equations is it possible to write down a polynomial with splitting field K/Q explicitly? If so, is a detailed computation of this type worked out anywhere?

I'm posting a bounty for a good answer to the questions succeeding the "Edit" heading.
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Before stating my question I should remark that I don't know almost anything nothing about etale cohomology - all that I know, I've gleaned from hearing off hand remarks and reading encyclopedia type articles. So I'm looking for an answer that will have some meaning to an etale cohomology naif. I welcome corrections to any evident misconceptions below.

Let E/Q be an elliptic curve the rational numbers Q: then to E/Q, for each prime l, we can associate a representation Gal(Q^bar/Q) ---> GL(2n, Z_l) coming from the l-adic Tate module T_l(E/Q) of E/Q (that is, the inverse limit of the system of l^k torsion points on E as k ---> infinity). People say that the etale cohomology group H^1(E/Q, Z_l) is dual to T_l(E/Q) (presumably as a Z_l module) and the action of Gal(Q^bar/Q) on H^1(E/Q, Z_l) is is the same as the action induced by the action of Gal(Q^bar/Q) induced on T_l(E/Q).

Concerning this coincidence, I could imagine two possible situations:

(a) When one takes the definition of etale cohomology and carefully unpackages it, one sees that the coincidence described is tautological, present by definition.

(b) The definition of etale cohomology (in the case of an elliptic curve variety) and the action of Gal(Q^bar/Q) that it carries is conceptually different from that of the dual of the l-adic Tate module and the action of Gal(Q^bar/Q) that it carries. The coincidence is a theorem of some substance.

Is the situation closer to (a) or to (b)?

Aside from the action Gal(Q^bar/Q) on T_l(E/Q), are there other instances where one has a similarly "concrete" description of representation of etale cohomology groups of varieties over number fields and the actions of the absolute Galois group on them?

Though I haven't seen this stated explicitly, I imagine that one has the analogy [Gal(Q^bar/Q) acts on T_l(E/Q): Gal(Q^bar/Q) acts on H^1(E/Q; Z_l)]::[Gal(Q^bar/Q) acts on T_l(A/K): Gal(Q^bar/Q) acts on H^1(A/K; Z_l)] where A is an abelian variety of dimension n and K is a number field: in asking the last question I am looking for something more substantively different and/or more general than this.

I've also inferred that if one has a projective curve C/Q, then H^1(C/Q; Z_l) is the same as H^1(J/Q; Z_l) where J/Q is the Jacobian variety of C and which, by my above inference I assume to be dual to T_l(J/Q), with the Galois actions passing through functorially. If this is the case, I'm looking for something more general or substantially different from this as well.

The underlying question that I have is: where (in concrete terms, not using a reference to etale cohomology as a black box) do Galois representations come from aside from torsion points on abelian varieties?

[Edit: For concreteness, let Delta(z) the normalized weight 12 cusp form for the full modular group and let l be a prime. Is it possible to give a rough, heuristic indication that doesn't require substantial prior exposure to etale cohomology of why it should be that for each prime l there is a representation Gal(Q^bar/Q) ---> GL(2, Z_l) such that for each prime p other than l, the image of Frob_p has characteristic polynomial x^2 - (a_p)x + p^(k-1) ?]
show/hide this revision's text 2 Added question at the bottom under [Edit: ....]
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