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…).