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In this post we state a generalised version of Cebotarev’s density theorem, following Serre’s article in the 1963 Arithmetical Algebraic Geometry’ series, and show how basic representation theory reduces its proof to the nonvanishing of of a family of ${L}$-functions at a certain point.

The situation we shall generalise is the following. Let ${L/K}$ be a Galois extension of number fields. The their rings of integers ${Spec(\mathcal{O}_L) \rightarrow Spec(\mathcal{O}_K)}$ carry a ${G=Gal(L/K)}$ action. Any prime ideal ${\mathfrak{q}}$ of ${\mathcal{O}_L}$ lies above a prime ideal ${\mathfrak{p}}$ of ${\mathcal{O}_K}$. Take the stabiliser ${G(\mathfrak{q})}$ of the point ${\mathfrak{q}}$ under the ${G}$-action. A standard argument with polynomials tells us that the induced map on residue fields gives rise to a canonical surjection ${G(\mathfrak{q}) \rightarrow Gal(k_\mathfrak{q}/k_\mathfrak{p})}$. The inertia group ${I(\mathfrak{q})}$ of ${\mathfrak{q}}$ is defined to be the kernel of this map, and so in particular (since these fields are finite) we get a well-defined Frobenius element ${Frob_\mathfrak{q} \in G(\mathfrak{q})/I(\mathfrak{q})}$, identified with the actual Frobenius of the Galois group of an extension of finite fields.

In the case where ${I(\mathfrak{q})}$ is trivial (i.e. the map above is an isomorphism), this is actually an element of ${G(\mathfrak{q})}$, in particular of ${G}$. It also is true that ${G}$ acts transitively on the primes above ${\mathfrak{p}}$, so if we had picked a different ${\mathfrak{q}'}$ above ${\mathfrak{p}}$, ${I(\mathfrak{q}) = 0}$ iff ${I(\mathfrak{q}') = 0}$, and we would get a different Frobenius element but it will be conjugate to the one we originally found (conjugating via an element of ${G}$ that exchanges ${\mathfrak{q}}$ and ${\mathfrak{q}'}$). Coversely, any conjugate of a Frobenius is clearly a Frobenius. In such a situation we say that ${\mathfrak{p}}$ doesn’t ramify and record the Frobenius conjugacy class ${Frob_\mathfrak{p} \subset G}$.

Here is the more general setting. Let ${X}$ be a scheme with a right action by a finite group ${G}$, and assume that the orbit space ${Y=X/G}$ is also a scheme. Also, assume both are of finite type over ${Spec {\mathbb Z}}$, so all residue fields are finite, and the norm ${N(x)=|k(x)|}$ of a closed point makes sense. For any closed point ${x \in |X|}$, one can consider the stabiliser ${G(x)}$ of the ${G}$-action. Now, the map ${X \rightarrow Y}$ is finite, and ${x}$ lies above a closed point ${y \in Y}$, so we obtain a finite field extension ${k(y) \rightarrow k(x)}$. As above we get a surjective map ${G(x) \rightarrow Gal(k(x)/k(y))}$, and whenever the map is actually a bijection we can define Frobenius elements ${Frob_x}$ which give rise to a Frobenius conjugacy class ${Frob_y \subset G}$.

Serre suggests some kind of averaging over all possibly Frobenius lifts for the case when there are some ramified points, but we shall neglect this and henceforth assume that there are no closed points ${y \in |Y|}$ which ramify (if you like, by deleting all the ones which do: I think they should generally be a closed subset, so deleting them is just passing to a Zariski-open subset).

The Cebotarev density theorem addresses possibly the most obvious question you could ask in this situation. We have a fixed finite group ${G}$, and we have realised it as the automorphism group of a morphism of schemes. Above, we outlined a procedure that associates points of ${Y}$ (the ${G}$-fixed scheme) with conjugacy classes in ${G}$. The question is given my favourite conjugacy class, are there any points which map to it, and if so, how many?’

Theorem 1 (Cebotarev Density Theorem) Assume ${dim Y \geq 1}$, and that the situation satisfies the L-function criterion below. Let ${C \subset G}$ be a conjugacy class. Then there are infinitely many closed points ${y \in Y}$ with ${Frob_y = C}$. In fact, the Dirichlet Density of the set ${S_C}$ of such closed points, defined by

$\displaystyle \delta(S_C) := \text{lim}_{s \rightarrow dim Y}(\sum_{y \in S_C} N(y)^{-s})/(\sum_{y \in |Y|} N(y)^{-s}),$

satisfies

$\displaystyle \delta(S) = \frac{|C|}{|G|}.$

How does one prove such a theorem? The key thing is to note we are dealing with a conjugacy class, which are in some sense the fundamental elements in representation theory. The expression ${\sum_{x \in S_C} N(x)^{-s}}$ which we need to study, and is in some sense the crux of the whole problem, could be rewritten as

$\displaystyle \sum_{y \in S_C} N(y)^{-s} = \sum_{y \in |Y|} N(y)^{-s} 1_{C}(Frob_y).$

And this function ${1_C}$ is a class function, so can be written as a linear combination of characters of ${G}$.

What will the coefficient corresponding to the trivial character be? Well,

$\displaystyle <1_C,1> = \frac{1}{|G|}\sum_{g \in G} 1_C(g) = \frac{|C|}{|G|}.$

So it’s now clear what our strategy should be. Expand the above expression out in terms of characters, and show that all the terms apart from that coming from the trivial character are small by comparison. How can we do this?

Well, the Artin L-functions are defined by the formula (which converges on ${Re s> dim Y}$).

$\displaystyle log L(X, \chi, s) = \sum_{y \in |Y|} \sum_{n=1}^\infty \frac{\chi(Frob_y^n) N(y)^{-ns}}{n}.$

For ${n>1}$ these terms are reasonably small, and the remaining terms are precisely those we are interested in. Looking back at the statement of Cebotarev (and the fact that the denominator in the definition of Dirichlet density seems to be what you get for the trivial character, which turns out to behave like ${log(\frac{1}{s-dim Y})}$), it will suffice that for all nontrivial characters ${\chi}$ of ${G}$,

$\displaystyle \text{lim}_{s \rightarrow dim Y} L(X,\chi,s) \not= 0, \infty.$

In fact, there is usually a meromorphic continuation at least to the region ${Re s> dim Y - \frac{1}{2}}$, so we can replace our limit by the statement that the L-function is holomorphic and nonzero at ${s= dim Y}$.

Thus using simple representation theory we transferred the discussion from study of Frobenii landing in a conjugacy class to study of L-functions associated to characters. Of course, the story is far from over: now we have to go away and prove things about these L-functions, but it’s nice how quickly we managed to manouvre between the almost class field theory looking statement of Cebotarev and some analytic statements about L-functions. We also obtained a potentially vast generalisation of Cebotarev, conditional on of course actually checking these statements about L-functions. For example, Serre mentions that we can get our hands on many cases by fibering over curves and then using the Riemann hypothesis for curves (in fact, maybe now we have the full Weil conjectures we don’t even need the inductive argument…).

In this post I want to examine how several situations with nice symmetry groups (for example, Galois groups or fundamental groups) can be explained by the abstract study of things called fibre functors. My main motivating example will be the étale fundamental group of a scheme, but this somehow makes it natural to first consider classical fundamental groups of topological spaces and Galois groups of field extensions, both of which can also be explained abstractly, in a way that is fairly interesting. I suppose the most surprising aspect of this theory to me is that it seems to supply the correct definition in all three cases, in spite of the three situations (or at least the classical vs etale fundamental groups) being rather different.

Firstly, let us consider the theory of fields. Indeed, let us fix a field ${k}$, and consider the category ${\mathcal{C}}$ of its finite separable extensions. Given any separable closure ${k^s}$, we can define a fairly natural functor ${\mathcal{C}^{op} \rightarrow (Sets)}$ by

$\displaystyle Fib_{k_s}: F \mapsto Hom_k(F,k^s).$

Having defined this functor, is it natural to ask how much of a choice we were making by choosing a separable closure? In other words, what possible symmetries were there in the original situation? More formally, we can investigate the group of automorphisms of this functor. What exactly does this mean? Well, I guess what we are asking for is some kind of compatible collection of automorphisms on each ${Hom_k(F,k^s)}$. Anyone who knows classical Galois theory will be able to see that ${Gal(k^s/k)}$ is certainly contained in this group, and in fact it turns out that this is the whole group!

How can we see this? An important feature of this theory is that there are not-quite-canonical injections ${Aut_k(F) \hookrightarrow Hom_k(F,k^s)}$ for each ${F \in \mathcal{C}}$. Indeed, this is a reasonably obvious consequence of the fact that ${F \rightarrow k^s}$ is always injective, but it is a crucial property which we will call the rigidity property. Another key feature is the existence of Galois closures: for any ${F}$, there is some ${L}$ with a morphism ${F \rightarrow L}$ and with the property that ${Aut_k(L) \rightarrow Hom_k(L,k^s)}$ is in fact a bijection. It follows from this that actually ${Hom_k(F,L) \cong Hom_k(F,k^s)}$! Indeed, restricting domains gives natural surjections, and one easily checks they give an induced bijection.

Given these facts, it is possible to build’ ${Fib_{k_s}}$ out of explicit elements of ${\mathcal{C}}$, namely the Galois elements. Why are they so important? The fact that ${Aut_k(L) \rightarrow Hom_k(L,k^s)}$ is surjective implies that once we have embedded ${L}$ into ${k^s}$, any other embedding is merely an automorphism of our original embedding! In particular, the actual field ${L}$ sits naturally inside ${k^s}$. This is enough to ensure that we can fix an arbitrary system of embeddings ${\phi_L: L\hookrightarrow k^s}$, which can be made into a directed system (in our case just a nested system of subfields of ${k^s}$) by appropriate fixed choices of ${\phi_{LL'}: L \hookrightarrow L'}$.

Once we have constructed this system, it is fairly straightforward to check that there is a functorial isomorphism

$\displaystyle Fib_{k_s}(F) \cong \text{lim}_{\rightarrow L} Hom_k(F,L).$

Indeed, by the existence of Galois closures, we can see that for any given ${F}$ an element of the LHS is in fact an element of ${Hom_k(F,L)}$ composed with our fixed map ${\phi_L: L \rightarrow k^s}$, and conversely by the compatibilities chosen, an element of the RHS gives a well-defined element of the LHS.

From this description, it is then one further check that actually the automorphism group of the functor is ${lim_\leftarrow Gal(L/k) =: Gal(k^s/k)}$ and so we have identified our classical automorphism group with the automorphism group of ${Fib_{k^s}}$. In other words, studying the fibre functor at ${k^s}$ gaves us an important group associated to the field ${k}$ with some kind of weak dependence also on the choice ${k^s}$ of separable closure.

What does the moral seem to be: if you have a fairly natural looking functor from your category to the category of sets, the group of automorphisms of this functor can be extremely interesting.

Let us move on to algebraic topology. Consider a nice topological space ${X}$, and consider the category ${Cov(X)}$ of covering spaces of ${X}$. There is if anything a somewhat more obvious functor here (and one which explains the name fibre functor’ better). Given a point ${x \in X}$, we can define

$\displaystyle Fib_x: (f: Y \rightarrow X) \mapsto f^{-1}(x).$

Again, there’s a rigidity property (obtained by proving path lifting lemmas, etc.): any morphism ${Y'\rightarrow Y}$ which fixes ${X}$ is determined by its action on a single element of ${f'^{-1}(x)}$, so there is again a not-quite canonical injection ${Aut(Y/X) \rightarrow Fib_x(Y)}$. What are the automorphisms of this functor? Well, in fact in this case, using the rigidity property and the construction of the universal cover, the functor is actually representable by the universal cover ${X_x \rightarrow X}$:

$\displaystyle Fib_x(Y) \cong Hom_X(X_x,Y).$

Therefore (if you like) by the fully-faithfulness of the Yoneda embedding the automorphisms of the fibre functor are precisely ${Aut(X_x/X)}$. But we know from classical topology that this is the fundamental group ${\pi_1(X,x)}$ of the space, so once again we have constructed an important invariant of ${X}$ associated loosely to the point ${x}$.

Our final example is in some sense a combination of the previous two examples, in that it actually generalises the first but superficially resembles the second. Most importantly however, it illustrates how by analysing the above examples as we did, we can define this group relatively naturally.

Let ${S}$ be a fixed scheme, and we in some sense want to study covering spaces of ${S}$. However, examples like the punctured complex plane (and the fact its universal cover fails to be defined algebraically) convince us that because algebra is naturally somewhat more finite’ than topology, we should restrict to finite covers, and by analogy with the theory of Riemann spaces we want our covers to be smooth everywhere (there should be no points of ramification or other oddness). So it turns out a vaguely plausible thing to look at (because it fits our description just given) is the category of finite étale maps ${X \rightarrow S}$, denoted ${FEt_S}$.

We want to define a fibre functor. One of the properties our fibres had in previous examples is that the objects used to define them had to be really boring – in particular have no interesting covers themselves. Indeed, in the first example we used an algebraically closed field, which doesn’t admit any finite extensions, and the second example we unwittingly used was a single point, which is famously simply connected.

What kind of schemes could we use to define fibre functors on ${FEt_S}$? The obvious choice, in analogy with the fundamental group theory, would be to take the residue fields of closed points ${Spec k(s) \rightarrow S}$, and consider their lifts to ${X}$. However, these fields need not be algebraically closed, and so would possess some internal nontrivial symmetry behaviour which would render the theory potentially quite complicated (though I haven’t tried, maybe this could be fun!). Scared of accidentally getting big absolute Galois groups as error terms’, we instead define geometric points, maps ${s: Spec k \rightarrow S}$ with ${k}$ algebraically closed. Then, having fixed a geometric point (and it’s common to write ${s}$ to also denote ${Spec k}$ understood to be an ${S}$-scheme via ${s}$), we define

$\displaystyle Fib_s: (X \rightarrow S) \mapsto Hom_S(s,X).$

And once again, you guessed it, we have a rigidity property, this time as a result of technical properties of étale morphisms (and that finite morphisms are in particular separated), and we have the existence of Galois closures. In this case, as in the field theory case, the functor is not representable (unsurprisingly given the finiteness constraint), but it can be written as a direct limit of representable functors. In fact, the situation generalises the field theoretic situation and the derivation of the theory is almost identical (with all arrows reversed of course!), with the automorphism group of the fibre functor being what we define as the étale fundamental group ${\pi_1(S, s)}$. The argument we gave in the field theory case implies that this group is the inverse limit over the opposite automorphism groups of all finite Galois covers.

I find this abstract way to set things up using fibre functors quite surprising, and wonder which other objects out there could be invented using this method. It would be nice to understand a bit more whether we can deduce anything purely formally about fibre functors using the existence of Galois closures and rigidity, though the lack of similarity between classical and étale fundamental groups makes me unsure what kind of properties we would expect.

In all the situations outlined above, there is also some kind of Galois correspondence, and it would be good to know the extent to which this is connected with the fibre functor formalism. Certainly in the equivalent linearised situation one has Tannaka-Krein duality which in some sense gives a Galois correspondence formally from studying the fibre functor, but it seems that in all the proofs I have seen for these examples Galois correspondences are derived by more direct means (explicitly constructing objects which need to exist using descent theory or whatever).

Finally an intriguing feature is how often one must make arbitrary choices and how they always seem to be slightly but not very significant. Picking a fibre functor involves picking a basepoint, and in the end the results you get tend to be noncanonically isomorphic. Also, our injections from automorphism groups into fibres are maps from a group into a space, so aren’t canonically defined unless we pick a distinguished point in each fibre, and when we prove representability of fibre functors we always have to pick distinguished points in the representing object. I don’t understand how this laxness in the system really affects the mathematics going on, but it feels like on one level it’s crucial in that these vaguenesses are precisely the symmetries we’re measuring. As always, any thoughts would be appreciated.

In this quick post, which is probably being written far too late at night to be remotely coherent, I want to lay out a few random thoughts I’ve been having about sheaves.

Classically, a sheaf is when you attach to a topological space some kind of compatible system of abelian groups (or sets, modules, whatever… I’ll stick to abelian groups to avoid having to think too hard). More specifically, each open set ${U}$ gets given a group ${\mathcal{F}(U)}$, whenever there is some kind of relation ${U \subset V}$, there has to be a restriction map ${\mathcal{F}(V) \rightarrow \mathcal{F}(U)}$, and there is a gluing condition that is supposed to guarantee good local behaviour. You can form, at each point ${x \in X}$, the stalk ${\mathcal{F}_x = \lim_{\rightarrow x \in U} \mathcal{F}(U)}$, and the gluing condition wins us the ability to check all sorts of things about sheaves by just looking at the stalks (whether a map is an isomorphism, injective, a sequence exact, etc.).

There are also (in all categories I’ve seen) an important pair of functors attached to every map ${f:X\rightarrow Y}$. The most obvious is the direct image ${f_*:Sh(X) \rightarrow Sh(Y)}$ given by the definition (using the defining property of continuous maps)

$\displaystyle f_*\mathcal{F}(V) := \mathcal{F}(f^{-1}(V)).$

There is also a functor going in the opposite direction, the inverse image${f^*:Sh(Y) \rightarrow Sh(X)}$ given by taking the sheafification (basically where you modify the below as harmlessly as possible to force the gluing axioms to hold) of the presheaf defined by

$\displaystyle f^*\mathcal{G}(U) := \text{lim}_{\rightarrow f(U) \subseteq V} \mathcal{G}(V).$

These are adjoint functors in the sense that to give a sheaf map ${f^*\mathcal{G} \rightarrow \mathcal{F}}$ is equivalent to giving a map ${\mathcal{G} \rightarrow f_*\mathcal{F}}$ in a natural way. It turns out that ${f^*}$ is actually an exact functor (this is reasonably obvious looking at the definition and noting that direct limits and sheafification are exact operations). It follows entirely formally that ${f_*}$ is left exact and preserves injectives. There is also usually another functor ${f_!: Sh(X) \rightarrow Sh(Y)}$ which acts as a left adjoint to ${f^*}$. These functors are basically really nice, but also manifest themselves in lots of different ways.

For example, consider ${i: U \hookrightarrow X}$ the inclusion of an open subset. Then ${i^*\mathcal{F}}$ is just the obvious restriction of ${\mathcal{F}}$ to ${U}$. Another interesting example is if you just consider the inclusion of a single point ${x:\star \rightarrow X}$. Looking at the definitions, it is clear that one gets none other than the stalk ${x^*\mathcal{F} = \mathcal{F}_x}$. In other words, by considering very special cases we can recover all the basic sheaf operations from the inverse image functor. This has two advantages. Firstly, it helps us remember that they are exact, and have right and left adjoints. Secondly, and more significantly, it seems to suggest an easier way to think about stalks in more general situations than the classical case.

It is a reasonably trivial observation that a presheaf (sheaf minus gluing axioms) is a contravariant functor from the category of open subsets of our space (with arrows given by inclusions) to the category of abelian groups. One could therefore contemplate calling any contravariant functor ${\mathcal{C}^{op} \rightarrow \text{(Ab)}}$ a presheaf.

But here is an interesting idea: let’s take ${\mathcal{C}}$ to be the category of all topological spaces, but throw away all the arrows which don’t correspond to inclusions of open sets. We can still define a presheaf on this much larger category. However, with a little thought it’s clear we can also define a sheaf. The gluing condition passes more or less unchanged. For every open set ${U}$ and a cover of it ${\{U_i\}}$, a bunch of compatible sections on the ${U_i}$ (note that it’s important that ${U_i \cap U_j}$ will be an open subset of ${U_i}$ and ${U_j}$ so the category comes equipped with the relevent arrow) must glue to a unique section on ${U}$. The only difference is that now the open sets keep on going upwards and upwards forever through the celestial heights of the category of topological spaces. And hey, why not? This is kinda fun!

This example feels slightly strange. Why? It’s no longer functions on a fixed space. Somehow, things have become sort of unbounded. But is that really what’s strange? I think what is actually strange about this example is the artificiality of the category of topological spaces where the only maps are open embeddings. This was what Grothendieck thought too, and pretty soon he was able to define sheaves on arbitrary categories provided they came equiped with a gadget called a Grothendieck topology, whose job is to track what the covers’ are for which the gluing conditions need to be checked.

Let me be (sort of) precise again. Let ${\mathcal{C}}$ be the category on which we want to construct sheaves, and assume it has binary pullbacks (just as we needed pairwise intersections of open sets above). A Grothendieck topology on it is just, for each object ${U}$, a collection of covers ${\{U_i\}}$. A sheaf is then just a contravariant functor ${\mathcal{C}^{op}\rightarrow \text{(Ab)}}$ satisfying a gluing condition exactly analogous with the classical one. In these more general situations, certainly whenever I’ve come across them, there are also good generalisations of the functors I mentioned earlier and they retain their properties.

The idea of sheaves is one of the most powerful in 20th century topology and geometry, and what we have outlined is a vast generalisation of this concept, so perhaps it is unsurprising it has been hugely successful. However, a reasonably humble question has been bothering me for a while. Since there is no underlying space anymore, there are no longer really such a thing as points, so we cannot obviously define stalks (and check properties locally at them). This seems to be a shortcoming that we should try to overcome.

We have already seen part of the way to an answer. Does our category contain any sort of object ${pt}$ that deserves to be called a point? If so, let us call the points of an object ${X}$ the set of maps ${x: pt \rightarrow X}$. Then given a sheaf on our category (or at least on an appropriate subcategory of things mapping into ${X}$), we can finally define the stalk ${\mathcal{F}_x := x^* \mathcal{F}}$.

But wait? We want our stalk to be an actual abelian group, whereas we’ve got ourselves a whole sheaf here. Therefore, having boldly gone forth and defined sheaves to be on entire large categories, I now need to retrace my steps to avoid losing grip on reality. Given a sheaf ${\mathcal{F}}$ on ${\mathcal{C}}$, and an object ${X}$, we may restrict our attention to the subcategory of ${\mathcal{C}}$ given by taking covers of ${X}$, and covers of all the objects involved therein, and so on… Ignoring any logical/set-theoretical problems this might entail, we write ${Sh(X)}$ for the category of sheaves on this subcategory. Our functors will then give to any map ${f:X \rightarrow Y}$ appropriate maps between categories of sheaves.

So what should ${pt}$ look like? Well, we just want ${Sh(pt)}$ to be as trivial a category as possible. Indeed, if it were a one point category, it would just be an abelian group, so the stalk ${\mathcal{F}_x}$ is an abelian group as required.

Generally I have no idea how possible this is, but let’s have a look at the category of schemes with any sensible topology. What are the most likely candidates to be points? Well, probably schemes which have a single point (as topological spaces), that is, fields ${Spec k}$. Indeed, maps ${Spec k \rightarrow X}$ do correspond reasonably nicely to actual points of ${X}$. However, in many topologies, for example the etale topology, a sheaf on ${Spec k}$ is not simply an abelian group but rather a ${Gal(k^s/k)}$-module. It therefore seems that the most sensible candidates for points are the schemes ${Spec k}$ for ${k}$ a separably closed field: and these are indeed what we take as the geometric points’ and to form `geometric fibres’ in scheme theory.

This also gives a quick generalisation of the concept of skyscraper sheaf. Indeed, given a geometric point ${x:Spec k \rightarrow X}$, while ${x^*\mathcal{F}}$ gives the correct definition of stalk, one can also form the direct image of any abelian group ${x_*A}$ and it gives a sheaf ${A_x}$, called the skyscraper sheaf, whose classical analogue is characterised by

$\displaystyle A_x(U) = \begin{cases} 0 & \text{ if } x \not\in U \\ A & \text{ if } x \in U\\ \end{cases}.$

I find it amusing just how nice so many of these sheaf-theoretic constructions seem to be, and how often trying trivial things can shed new light on things and provide the correct generalisations of classical ideas. Anyway, it’s far too late, and I hope I didn’t ramble too much.