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Notes on class field theory

Kiran S. Kedlaya

Section 2.4 Zeta functions and the Chebotaryov density theorem

Reference.

[34], Chapter VIII for starters; see also [37], Chapter VI and [38], Chapter VII. For advanced reading, see Tate’s thesis ([4], Chapter XV), but wait until we introduce the adèles (Section 6.1).

Subsection The Dedekind zeta function of a number field

Although this is supposed to be a course on algebraic number theory, the following analytic discussion is so fundamental that we must at least allude to it here.

Definition 2.4.1.

Let \(K\) be a number field. The Dedekind zeta function \(\zeta_K(s)\) is a function on the complex plane given, for \(\Real(s) \gt 1\text{,}\) by the absolutely convergent product and sum
\begin{equation*} \zeta_K(s) = \prod_\gothp (1 - \Norm(\gothp)^{-s})^{-1} = \sum_{\gotha} \Norm(\gotha)^{-s} \end{equation*}
where \(\gothp\) runs over the nonzero prime ideals of \(\gotho_K\) and \(\gotha\) runs over the nonzero ideals of \(\gotho_K\text{.}\)
For example, if \(K = \QQ\text{,}\) then \(\zeta_K\) equals the Riemann zeta function.
A fundamental fact about the zeta function is the following.

Proof.

See [38], Corollary VII.5.11.

Remark 2.4.3.

In Theorem 2.4.2, the residue of the pole at \(s=1\) is computed by the analytic class number formula; it is the product of the class number, the unit regulator, and another quantity that depends on the discriminant and signature of \(K\text{.}\)
There is also a functional equation relating the values of \(\zeta_K\) at \(s\) and \(1-s\text{,}\) and an extended Riemann hypothesis: aside from “trivial” zeros along the negative real axis, the zeroes of \(\zeta_K\) all have real part \(1/2\text{.}\)

Subsection \(L\)-functions of abelian characters

Definition 2.4.4.

More generally, let \(\gothm\) be a formal product of places of \(K\text{,}\) and let \(\chi_\gothm\colon \Cl^\gothm(K) \to \CC^*\) be a character of the ray class group of conductor \(\gothm\text{.}\) Extend \(\chi_\gothm\) to a function on all ideals of \(K\) by declaring its value to be 0 on ideals not coprime to \(\gothm\text{.}\) Then we define the \(L\)-function
\begin{equation*} L(s, \chi_\gothm) = \prod_{\gothp \not| \gothm} (1 - \chi(\gothp) \Norm(\gothp)^{-s})^{-1} = \sum_{(\gotha, \gothm) = 1} \chi(\gotha) \Norm(\gotha)^{-s}\text{;} \end{equation*}
again the product converges absolutely for \(\Real(s) \gt 1\text{.}\)

Proof.

See [38], Theorem VII.2.8 (or Theorem VII.8.5).

Remark 2.4.6.

By contrast with Theorem 2.4.5, if \(\chi_\gothm\) is trivial, then \(L(s, \chi_\gothm)\) is just the Dedekind zeta function with the Euler factors for primes dividing \(\gothm\) removed, so it still has a pole at \(s=1\) by Theorem 2.4.2.

Subsection Nonvanishing of \(L\)-functions and consequences

One more basic fact is the following.

Proof.

For \(K = \QQ\text{,}\) Theorem 2.4.7 is already a nontrivial but important result: it implies Dirichlet’s famous theorem that there are infinitely many primes in arithmetic progression, as follows.

Definition 2.4.8.

A set of primes \(S\) in a number field \(K\) has Dirichlet density \(d\) if
\begin{equation*} \lim_{s \to 1^+} \frac{\sum_{\gothp \in S} \Norm(\gothp)^{-s}}{\log \frac{1}{s-1}} = d\text{.} \end{equation*}
This in particular presumes the existence of the limit; otherwise, we may still define the lower Dirichlet density and upper Dirichlet density using the limits inferior and superior.

Remark 2.4.9.

Theorem 2.4.7 implies that the Dirichlet density of the set of primes congruent to \(a\) modulo \(m\) is \(\frac{1}{\phi(m)}\) if \(a\) is coprime to \(m\) (and \(0\) otherwise). The key point is that for any nontrivial \(\chi_\gothm\text{,}\) \(\sum_{\gothp} \chi(\gothp) \Norm(\gothp)^{-s}\) remains bounded as \(s \to \infty\text{.}\)
The fact also implies that for any number field \(K\) and any formal product of places \(\gothm\text{,}\) there are infinitely many primes in each class of the ray class group of conductor \(\gothm\text{,}\) the set of such primes having Dirichlet density \(\frac{1}{\#\Cl^\gothm(K)}\text{.}\) (See Exercise 3.)

Subsection The Chebotaryov density theorem

Finally, we point out a result of class field theory that also applies to nonabelian extensions.

Definition 2.4.10.

Recall that if \(L/K\) is any Galois extension of number fields with Galois group \(G\text{,}\) \(\gothp\) is a prime of \(K\) which does not ramify in \(L\text{,}\) and \(\gothq\) is a prime above \(\gothp\text{,}\) then there is a well-defined Frobenius element to \(\gothq\text{:}\) it’s the element \(g\) of the decomposition group \(G_{\gothq}\) such that \(x^g \equiv x^{\#(\gotho_K/\gothp)} \pmod{\gothq}\text{.}\) Keep in mind that as a function of \(\gothp\text{,}\) this Frobenius is only well-defined up to conjugation in \(G\text{.}\) (If \(\gothp\) ramifies in \(L\text{,}\) then a further ambiguity occurs: the Frobenius element associated to \(\gothq\) is only well-defined in the quotient of \(G_{\gothq}\) by the inertia group \(I_{\gothq}\text{.}\))

Proof.

This follows from everything we have said so far, plus Artin reciprocity, in case \(L/K\) is abelian. In the general case, let \(f\) be the order of \(g\text{,}\) and let \(K'\) be the fixed field of \(g\text{;}\) then we know that the set of primes of \(K'\) with Frobenius \(g \in \Gal(L/K') \subset G\) has Dirichlet density \(1/f\text{.}\) The same is true if we restrict to primes of absolute degree 1 (see Exercise 2).
Let \(Z\) be the centralizer of \(g\) in \(G\text{;}\) that is, \(Z = \{z \in G\colon zg = gz\}\text{.}\) The order of the conjugacy class of \(g\) is \(\#G/\#Z\text{,}\) so we are trying to prove that the set of primes of \(K\) (of absolute degree 1) with Frobenius in the conjugacy class of \(g\) has Dirichlet density \(\frac{\#G/\#Z}{\#G} = \frac{1}{\#Z}\text{.}\)
To begin with, we apply the abelian case to the extension \(L/K'\text{,}\) to see that the set of primes of \(K'\) (of absolute degree 1) with Frobenius \(g\) equals \(1/f\text{.}\) Next, note that a prime \(\gothp\) of \(K\) (of absolute degree 1) has Frobenius in the conjugacy class of \(g\) if and only if there is a prime \(\gothq\) of \(L\) with Frobenius \(g\text{;}\) note that \(\gothq\) has inertia degree \(f\) over both \(K'\) and \(K\text{,}\) so \(K' \cap \gothq\) is a prime of \(K'\) of absolute degree 1. Finally, note that for each \(h \in G\text{,}\) the Frobenius of \(\gothq^h\) is \(hgh^{-1}\text{,}\) so only the elements of \(Z\) correspond to cases where this Frobenius is again equal to \(g\text{.}\) However, the \(f\) elements of the subgroup generated by \(g\) do not even move \(\gothq\text{,}\) so there are in all \(\#Z/f\) primes of \(K'\) above \(\gothp\) with Frobenius \(g\text{.}\) Consequently, the density of primes of \(K\) with Frobenius in the conjugacy class of \(g\) is \(\frac{1}{f} \frac{1}{\#Z/f} = \frac{1}{\#Z}\) as desired.
We state the following here as a corollary of Theorem 2.4.11; however, we will eventually prove it before proving Artin reciprocity (see Corollary 7.1.15 and Proposition 7.2.3).

Proof.

Let \(M/K\) be the Galois closure of \(L/K\text{,}\) and set \(G = \Gal(M/K), H = \Gal(M/L)\text{.}\) By hypothesis, \(G\) is not the trivial group and the conjugates of \(H\) in \(G\) have trivial intersection.
Let \(\gothp\) be any prime of \(K\) which does not ramify in \(M\) and let \(\gothq\) be a prime of \(M\) above \(K\text{.}\) Then \(\gothp\) splits completely in \(M\) if and only if the Frobenius element of \(\gothq\) is trivial. Moreover, if \(\gothp\) splits completely in \(L\text{,}\) then \(g\) lies in every conjugate of \(H\) and hence must be trivial, so \(\gothp\) also splits completely in \(M\text{.}\) (The converse is also true.)
Since \(G \neq H\text{,}\) we can choose an element \(g \in G \setminus H\text{.}\) By Theorem 2.4.11, there exist infinitely many primes \(\gothp\) of \(K\) for which there is a prime \(\gothq\) of \(L\) above \(\gothp\) with Frobenius \(g\text{.}\) By the previous discussion, any such \(\gothp\) does not split completely in \(K\text{.}\)

Remark 2.4.13.

Theorem 2.4.11 is a special case of a much more general equidistribution conjecture including, among other things, the Sato-Tate conjecture on the distribution of Frobenius traces of elliptic curves. See [51] for an introduction to this circle of ideas.

Remark 2.4.14.

With somewhat more work, all of the previous density assertions remain true (and are indeed strictly stronger than before) if Dirichlet density is replaced by natural density. The natural density of a set \(S\) of prime ideals of a number field \(K\) is the limit (if it exists)
\begin{equation*} \lim_{X \to \infty} \frac{\#\{\gothp\colon \gothp \in S, \Norm(\gothp) \leq X\}}{\#\{\gothp\colon \Norm(\gothp) \leq X\}} \end{equation*}
where in both sets \(\gothp\) runs over all prime ideals of \(K\text{.}\) (Again, if the limit does not exist, we may still define the lower natural density and upper natural density using the limits inferior and superior.)
As with the prime number theorem, one can obtain effective power-saving error estimates conditional on the Generalized Riemann Hypothesis for appropriate Artin \(L\)-functions. See [32].

Remark 2.4.15.

For fun, we mention a lesser-known result of Chebotaryov here: the character table of a finite cyclic group, viewed as a square matrix, has the property that every minor is nonzero.
By contrast, for a group which is abelian but not cyclic there exists a \(2 \times 2\) submatrix with all entries equal to 1, whereas for a nonabelian group any nonabelian character takes the value 0 somewhere (a result of Burnside; see [23], Theorem 3.15).

Exercises Exercises

1.

Show that the Dirichlet density of the set of all primes of a number field is 1.
Hint.
Write
\begin{equation*} \log \zeta_K(s) = \sum_{\gothp} \sum_{m=1}^\infty \frac{\Norm(\gothp)^{-sm}}{m} \end{equation*}
and check that the sum over all terms with \(m \gt 1\) converges absolutely for \(s=1\text{.}\) Then apply Theorem 2.4.2.

2.

Show that in any number field, the Dirichlet density of the set of primes \(\gothp\) of absolute degree greater than 1 is zero.
Hint.
As in Exercise 1, show that the sum \(\sum_{\gothp} \Norm(\gothp)^{-s}\) over all \(\gothp\) of absolute degree greater than 1 converges absolutely for \(s=1\text{.}\)

3.

Let \(\gothm\) be a formal product of places of the number field \(K\text{.}\) Using Theorem 2.4.2, Theorem 2.4.5, and Theorem 2.4.7, prove that the set of primes of \(K\) lying in any specified class of the generalized ideal class group of conductor \(\gothm\) has Dirichlet density \(\frac{1}{\#\Cl^\gothm(K)}\text{.}\)
Hint.
As in Exercise 1, write
\begin{equation*} \log L(s, \chi) = \sum_{\gothp} \sum_{m=1}^\infty \frac{\chi(\gothp)^m \Norm(\gothp)^{-sm}}{m} \end{equation*}
and observe that the sum over all terms with \(m \gt 1\) converges absolutely for \(s=1\text{.}\) Then apply Theorem 2.4.5 and Theorem 2.4.7 to deduce that for \(\chi\) nontrivial, \(\sum_{\gothp} \chi(\gothp) \Norm(\gothp)^{-s}\) converges to a finite limit as \(s \to 1^+\text{.}\) Then combine the quantities \(\sum_{\gothp} \chi(\gothp) \Norm(\gothp)^{-s}\) to cancel out all but one class.

4.

Let \(L/K\) be an extension of number fields. Suppose that for every prime \(\gothp\) of \(K\) which does not ramify in \(L\text{,}\) all of the primes of \(L\) above \(\gothp\) have isomorphic residue fields. Using Theorem 2.4.11, prove that \(L/K\) is Galois.
Hint.
Let \(M\) be the Galois closure of \(L/K\text{.}\) Put \(G := \Gal(M/K)\) and \(H := \Gal(M/L)\text{.}\) For \(\gothq\) a prime of \(M\) with decomposition group \(G_{\gothq}\) lying above the prime \(\gothp\) of \(K\text{,}\) relate the orders of the residue fields of the primes of \(L\) to the intersections of \(G_{\gothq}\) with the conjugates of \(H\) in \(G\) (see Remark 1.1.6). Use the fact that these conjugates have trivial intersection to deduce that \(G_{\gothq}\) must be trivial, and invoke Theorem 2.4.11 to conclude.

5.

Let \(L/K\) be an abelian extension of number fields. Using Corollary 2.4.12, show that the homomorphism \(J_K^\gothm \to \Gal(L/K)\) is surjective.

6.

Use Corollary 2.4.12 to give an alternate solution to Exercise 6.
Hint.
Show that if \(L_1, L_2\) are both ray class fields of \(\gothm\text{,}\) then all but finitely many primes of \(L_1\) split completely in the compositum \(L_1L_2\) (namely, those which do not ramify in the compositum).

7.

Here is an example to illustrate the difference between Dirichlet density and natural density, albeit not for primes. Let \(S\) be the set of positive integers whose decimal expansion begins with 1.
  1. Prove that \(S\) does not have a natural density, in the sense that
    \begin{equation*} \lim_{X \to \infty} \frac{1}{X} \# (S \cap \{1,\dots,X\}) \end{equation*}
    does not exist.
  2. On the other hand, prove that \(S\) has a Dirichlet density in the sense that
    \begin{equation*} \lim_{s \to 1^+} \frac{\sum_{n \in S} n^{-s}}{\sum_{n=1}^\infty n^{-s}} \end{equation*}
    exists, and compute this value.
Hint.
Estimate \(\sum_{n=a}^b n^{-s}\) using upper and lower Riemann sums for the integral of \(x^{-s}\,dx\text{.}\)
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