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

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 $\gal(\bar{\bb Q}/\bb Q)$ that it carries is conceptually different from that of the dual of the $\ell$-adic Tate module and the action of $\gal(\bar{\bb Q}/\bb Q)$ that it carries. The coincidence is a theorem of some substance.

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

Aside from the action $\gal(\bar{\bb Q}/\bb Q)$ on $T_\ell(E/\bb Q)$, 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 [$\gal(\bar{\bb Q}/\bb Q)$ acts on $T_\ell(E/\bb Q)$: $\gal(\bar{\bb Q}/\bb Q)$ acts on $H^1(E/\bb Q; \bb Z_\ell)$]::[$\gal(\bar{\bb Q}/\bb Q)$ acts on $T_\ell(A/K)$: $\gal(\bar{\bb Q}/\bb Q)$ acts on $H^1(A/K; \bb Z_\ell)$] 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 $C/\bb Q$, then $H^1(C/\bb Q; \bb Z_\ell)$ is the same as $H^1(J/\bb Q; \bb Z_\ell)$ where $J/\bb Q$ is the Jacobian variety of $C$ and which, by my above inference I assume to be dual to $T_\ell(J/\bb Q)$, 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.

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

Now, remove the restriction that $V/\bb Q$ is an elliptic curve, so that $V/\bb Q$ is again an arbitrary smooth projective algebraic variety defined over $\bb Q$. Does the (mod 5) Galois representation attached to $V/\bb Q$ have an associated number field $K/\bb Q$ analogous to the (mod 5) Galois representation attached to an elliptic curve does? If so, where does this number field come from? If $V/\bb Q$ is specified by explicit polynomial equations is it possible to write down a polynomial with splitting field $K/\bb Q$ 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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4 Answers

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IMO, the scenario is closer to your (a). I'll sketch an explanation of the duality between $H^1(E,\mathbf{Z}_l)$ and the dual to the Tate module. We have $H^1(E,\mathbf{Z}_l)=\text{Hom}(\pi_1(E),\mathbf{Z}_l)$,

where that $\pi_1$ means etale fundamental group with base point the origin $O$ of $E$. Thus the isomorphism we really want is between $\pi_1(E)\otimes\mathbf{Z}_l$ and $T_\ell(E)$.

What is $\pi_1(E)$? In the topology world, we'd consider the universal cover $f\colon E'\rightarrow E$ and take $\pi_1(E)$ to be its group of deck transformations. Then $\pi_1(E)$ has an obvious action on $f^{-1}(O)$. If $E$ is the complex manifold $\mathbf{C}/L$ for a lattice $L$, this is just the natural isomorphism $\pi_1(E)\cong L$.

But in the algebraic geometry world, there is no universal cover in the category of varieties, so the notion of universal cover is replaced with the projective system $E_i\to E$ of etale covers of $E$. Then $\pi_1(E)$ is the projective limit of the automorphism groups of $E_i$ over $E$.

One nice thing about $E$ being an elliptic curve is that any etale cover $E'\rightarrow E$ must also be an elliptic curve (once you choose an origin on it, anyway); if $E\rightarrow E'$ is the dual map then the composition $E\rightarrow E'\rightarrow E$ is multiplication by an integer. So it's sufficient to only consider those covers of $E$ which are just multiplication by an integer. Since it's $\pi_1(E)\otimes\mathbf{Z}_l$ we're interested in, it's enough to consider the isogenies of $E$ given by multiplication by $l^n$.

What are the deck transformations of the maps $l^n\colon E\rightarrow E$? Up to an automorphism of $E$, they're simply translations by $l^n$-division points. And now we see the relationship to the Tate module: A compatible system of deck transformations of these covers is the exact same thing as a compatible system of $l^n$-division points. Thus we get the desired isomorphism. Naturally, it's Galois compatible!

In the end, we see that torsion points were tucked away in the construction of the etale cohomology groups, so it wasn't exactly a coincidence. Hope this helps.

Re the edit: I believe your best bet is to work locally. First of all, you didn't mention which Galois representation you wanted exactly; let's say you want the representation on $H^i$ of your variety for a given $i$. Let's assume this space has dimension $d$.

Step 1. For each prime $p$ at which your variety $V$ has good reduction, you can compute the local zeta function of $V/\mathbf{F}_p$ by counting points on $V(\mathbf{F}_{p^n})$ for $n\geq 0$. In this way you can compute the action of the $p^n$th power Frobenius on $H^i(V\otimes\overline{\mathbf{F}}_p,\mathbf{F}_5)$ for various primes $p$.

Step 2. Do this enough so that you can gather up information on the statistics of how often the Frobenius at $p$ lands in each conjugacy class in the group $\text{GL}_d(\mathbf{F}_5)$. In this way you could guess the conjugacy class of the image of Galois inside $\text{GL}_d(\mathbf{F}_5)$.

Step 3. Now your job is to find a table of number fields $F$ whose splitting field has Galois group equal to the group you found in the previous step. I found a table here: http://hobbes.la.asu.edu/NFDB/. You already know which primes ramify in $K$ -- these are at worst the primes of bad reduction of $V$ together with 5 -- and you can distinguish your $F$ from the other number fields by the splitting behavior your found in Step 1. Then $K$ is the splitting field of $F$.

A caveat: Step 1 may well take you a very long time, because unless your variety has some special structure or symmetry to it, counting points on $V$ is Hard.

Another caveat: Step 3 might be impossible if $d$ is large. If $d$ is 2 then perhaps you're ok, because there might be a degree 8 number field $F$ whose splitting field has Galois group $\text{GL}_2(\mathbf{F}_5)$. If $d$ is large you might be out of luck here.

You are free not to accept this answer because of the above caveats but I really do think you've asked a hell of a tough question here!

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Jared - Thanks. What you describe is very pretty and has taught me something new. I don't yet see the Galois action on the etale side, but I will think about this. –  Jonah Sinick Dec 1 '09 at 5:14
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I'm sorry to make only an unconstructive remark, but I thought I should point out that your first isomorphism is not exactly a tautology. –  Minhyong Kim Dec 1 '09 at 9:11
    
Minhyong: You're absolutely right. Can you give a short explanation of the relationship? –  Jared Weinstein Dec 1 '09 at 16:16
    
There's a distinction between the mod 5 etale cohomoology and the reduction of the $\mathbb{Q}_5$ cohomology. What you discuss above would in principle detect (the semisimplification of the) latter, not the former; they will differ if there is torsion in the $\mathbb{Z}_5$-cohomology. (I also missed this in my comment below.) –  moonface Dec 10 '09 at 3:27
    
That's a really good point, moonface. Do you have a fix for this? How would you detect torsion in the 5-adic cohomology? –  Jared Weinstein Dec 10 '09 at 4:00
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There's a conjecture of Fontaine and Mazur that says that every "reasonable" Galois representation occurs in the etale cohomology of some algebraic variety, so there's no way of avoiding it. Weil managed to prove the Weil conjectures for curves and abelian varieties using only (a variant of) the Tate modules, but that's about as far as you can go. The best thing is to understand etale cohomology so that it is no longer a black box.

For the Fontaine Mazur conjecture, see Kisin's article, Notices AMS, 2007 (What is a Galois representation?).

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You might certainly say that Galois representations come from modular forms (or, more generally and more conjecturally, from automorphic forms on groups more general than GL_2.) You can say what it means for an ell-adic Galois representation to be associated to a modular form without reference to etale cohomology. But if you want to prove the existence of such a Galois representation -- for instance, by constructing it -- I don't see that you have much choice but to invoke etale cohomology. And it's not so clear that this any less "concrete" than, say, invoking the ell-adic Tate module of some unspecified abelian 6-fold or something.

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I'd be happy with an answer involving etale cohomology so long as it's not a black box. The difference between ell-adic Tate modules and etale cohomology for me is simply that I know about the former and not the latter. I'm encouraged by your suggestion that etale cohomology may be no less concrete than the construction using the ell-adic Tate module. But still have reservations in that I generally lose touch with the main ideas when the homological algebra gets heavy (as it seems to in the study of etale cohomology). –  Jonah Sinick Dec 1 '09 at 2:32
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Regarding the Dec 9 edit: Yes, but you probably won't like it. Take the etale cohomology of V_{Q-bar} with coefficients in Z/5Z. Then this is a finite abelian group, and Gal(Q-bar/Q) acts continuously on it. Therefore the subgroup of Gal(Q-bar/Q) acting trivially is open, and hence corresponds to a finite Galois extension K of Q.

I don't know whether, given the defining equations of V, there exists an algorithm to find a polynomial whose splitting field is K. The only idea I'd have is to use fibration by curves to try to reduce the question to the etale cohomology of curves with coefficients in local systems, which can probably be calculated by using a Tate module-style approach on Jacobians. I think it's an interesting question, but I bet no one's ever looked at it.

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Note this is really hard to do in practice (although I don't have a better idea). Even if you tried to compute H^2 of a surface fibered in genus 2 curves over a base curve X, then (to compute the cohomology of X with coefficients in the relevant local system) you have to pass to a 125-fold covering of X and compute the Jacobian of that beast; I think it's very hard even to write down the Jacobian of a curve of genus three! In principle one can count points on X over various finite fields and this determines K, but this is also wildly impractical. –  moonface Dec 10 '09 at 2:21
    
Matt Emerton reminds me that the archive paper math/0605244 by Bosman et al is about efficiently computing the mod l Galois representations associated to Ramanujan's tau-function. –  JBorger Dec 10 '09 at 8:05
    
@moonface: Would you be willing to explain your example a bit further? –  David Hansen Mar 25 '10 at 15:02
    
@moonface: My request is now available in the form of a question. :) mathoverflow.net/questions/19315/… –  David Hansen Mar 25 '10 at 15:39
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