Math 254B, UC Berkeley, Spring 2002 (Kedlaya) Zeta Functions and the Chebotarev Density Theorem

Reference. Lang, Algebraic Number Theory, Chapter VIII (shudder) for starters; see also Milne, Chapter VI and Neukirch, Chapter VII. For advanced reading, see Tate’s thesis (last chapter of Cassels-Frohlich), but wait until we introduce the adeles.

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.

Let K be a number field. The Dedekind zeta function ζ_{K}(s) is a function on the complex plane given, for Re(s) > 1, by the absolutely convergent product and sum

ζ_{K}(s) = ∏_{𝔭}(1 − Norm(𝔭)^{−s})^{−1} = ζ_{K}(s) = ∑_{𝔞}Norm(𝔞)^{−s},

where in the sum 𝔞 runs over the nonzero ideals of 𝔬_{K}.

A fundamental fact about the zeta function is the following. We omit the proof.

Theorem 1. The function ζ_{K}(s) extends to a meromorphic function on ℂ whose only pole is a simple pole at s = 1 of residue 1.

The case K = ℚ is of course the famous Riemann zeta function. There is also a functional equation relating the values of ζ_{K} at s and 1−s, and an extended Riemann hypothesis: aside from “trivial” zeros along the negative real axis, the zeroes of ζ_{K} all have real part 1/2.

More generally, let 𝔪 be a formal product of places of K, and let χ_{𝔪} : Cl_{𝔪}(K) → ℂ^{∗} be a character of the ray class group of conductor 𝔪. Extend χ_{𝔪} to a function on all ideals of K by declaring its value to be 0 on ideals not coprime to 𝔪. Then we define the L-function

L(s, χ_{𝔪}) = ∏_{𝔭|̸𝔪}(1 − χ(𝔭) Norm(𝔭)^{−s})^{−1} = ∑_{(𝔞,𝔪)=1}χ(𝔞) Norm(𝔞)^{−s}.

Then we have another basic fact whose proof we also omit.

Theorem 2. If χ_{𝔪} is not trivial, then L(s, χ_{𝔪}) extends to an analytic function on ℂ.

If χ_{𝔪} is trivial, then L(s, χ_{𝔪}) is just the Dedekind zeta function with the Euler factors for primes dividing 𝔪 removed, so it still has a pole at s = 1.

Theorem 3. If χ_{𝔪} is not the trivial character, then L(1, χ_{𝔪}) ≠ 0.

This is already a nontrivial, but important result over ℚ. It implies Dirichlet’s famous theorem that there are infinitely many primes in arithmetic progression, by implying that for any nontrivial χ_{𝔪}, ∑_{𝔭} χ(𝔭) Norm(𝔭)^{−s} remains bounded as s → ∞. In fact, we say that a set of primes S in a number field K has Dirichlet density d if

lim_{s→1^{+}} \frac{∑_{𝔭∈S} Norm(𝔭)^{−s}}{log \frac{1}{s−1}} = d.

Then the fact implies that the Dirichlet density of the set of primes congruent to a modulo m (assuming a is coprime to m) is 1/ϕ(m).

The fact also implies that for any number field K and any formal product of places 𝔪, there are infinitely many primes in each class of the ray class group of conductor 𝔪, the set of such primes having Dirichlet density 1/# Cl_{𝔪}(K). (Proof: see exercises.)

Finally, we point out a result of class field theory that also applies to nonabelian extensions. Recall that if L/K is any Galois extension of number fields with Galois group G, 𝔭 is a prime of K, and 𝔮 is a prime above 𝔭 which is unramified, then there is a welldefined Frobenius associated to 𝔮 (it’s the element g of the decomposition group G_{𝔮} such that x^{g} ≡ x^{#(𝔬_{K}/𝔭)} (mod 𝔮)); but as a function of 𝔭, this Frobenius is only well-defined up to conjugation in G.

Theorem 4 (Chebotarev Density Theorem). Let L/K be an arbitrary Galois extension of number fields, with Galois group G. Then for any g ∈ G, there exist infinitely many primes 𝔭 of K such that there is a prime 𝔮 of L above 𝔭 with Frobenius g. In fact, the Dirichlet density of such primes 𝔭 is the order of the conjugacy class of G divided by #G.

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, and let K′ be the fixed field of g; then we know that the set of primes of K′ with Frobenius g ∈ Gal(L/K′) ⊂ G has Dirichlet density 1/f. The same is true if we restrict to primes of absolute degree 1 (see exercises).

Let Z be the centralizer of g in G; that is, Z = {z ∈ G : zg = gz}. Then for each prime of K (of absolute degree 1) with Frobenius in the conjugacy class of g, there are #Z/f primes of K′ above it (also of absolute degree 1) with Frobenius g. (Say 𝔭 is such a prime and 𝔮 is a prime of L above 𝔭 with Frobenius g. Then for h ∈ G, the Frobenius of 𝔮^{h} is hgh^{−1}, so the number of primes 𝔮 with Frobenius g is #Z. But each prime of L′ below one of these is actually below f of them.) Thus the density of primes of K with Frobenius in the conjugacy class of g is (1/f)(1/(#Z/f)) = 1/#Z. To conclude, note that the order of the conjugacy class of G is #G/#Z.

Exercises.

1. Show that the Dirichlet density of the set of all primes of a number field is 1.

2. Show that in any number field, the Dirichlet density of the set of primes 𝔭 of absolute degree greater than 1 is zero.

3. Let 𝔪 be a formal product of places of the number field K. Using Theorems 1, 2 and 3, prove that the set of primes of K lying in any specified class of the ray class group of conductor 𝔪 is 1/# Cl_{𝔪}(K). (Hint: combine the quantities ∑_{𝔭} χ(𝔭) Norm(𝔭)^{−s} to cancel out all but one class.)