Chapter 23 Artin \(L\)-functions and the Chebotaryov density theorem
This chapter begins the sixth and final part of the course. In this part, we describe some other types of \(L\)-functions that are used for arithmetic purposes. This merely scratches the surface of what is now a rather vast theory of \(L\)-functions; [13], chapter 5 gives a somewhat less narrow account.
Some of this discussion will only make sense if you have studied some algebraic number theory; there are many reasonable texts for this subject, of which [14] was the one that I had used most recently before teaching this course. We also presume some familiarity with the representation theory of finite groups.
Section 23.1 Frobenius elements of Galois groups
Let \(K\) be a finite Galois extension of \(\QQ\text{,}\) and put \(G = \Gal(K/\QQ)\text{.}\) Let \(\gotho_K\) be the ring of integers of \(K\text{;}\) that is, \(\alpha \in \gotho_K\) if and only if \(\alpha\) is a root of a monic polynomial with coefficients in \(\ZZ\text{.}\)
A prime \(p\) is said to be ramified (in \(K\)) if the ring \(\gotho_K/p \gotho_K\) is not reduced (i.e., has nilpotent elements). For instance, if \(K = \QQ(i)\text{,}\) then \(\gotho_K = \ZZ[i]\text{,}\) and the only ramified prime is \(p=2\text{.}\) In general, only finitely many primes are ramified; they are the ones dividing the discriminant of \(K/\QQ\text{.}\)
On \(\gotho_K/p \gotho_K\text{,}\) one has both an action of \(G\) and a Frobenius map \(x \mapsto x^p\text{.}\)
One can also define Frobenius elements for the infinite place: given an embedding of \(K\) into \(\CC\text{,}\) complex conjugation on \(\CC\) induces an automorphism of \(K\text{.}\) Any such automorphism is called a Frobenius element for the infinite place, or more simply a complex conjugation on \(K\text{.}\)
Section 23.2 Linear representations and \(L\)-functions
Let \(\rho: G \to \GL_n(\CC)\) be a linear representation, with character \(\chi: G \to \CC\text{;}\) that is, \(\chi(g) = \Trace \rho(g)\text{.}\) We define the (incomplete) Artin \(L\)-function associated to \(\rho\) as the function
\begin{equation*}
L(\rho, s) = \prod_p \det(1 - \rho(\Frob_p)p^{-s})^{-1},
\end{equation*}
where we only allow \(p\) to run over unramified primes, and \(\Frob_p\) means any Frobenius element of \(p\text{;}\) it doesn't matter which one because they are all conjugate. (There is a correct way to put in the ramified primes: you only look at the determinant of the action of \(\rho\) on the invariants under an inertia group corresponding to \(p\text{.}\) If you don't know what that means, never mind.)
For example, if we take \(\rho: G \to \GL_1(\CC)\) to be the trivial representation, then \(L(\rho,s)\) equals the Riemann zeta function with a few Euler factors missing. Note also that
\begin{equation*}
L(\rho_1 \oplus \rho_2, s) = L(\rho_1, s) L(\rho_2, s).
\end{equation*}
Also, if \(K'\) is another Galois extension of \(\QQ\) contained in \(K\text{,}\) \(\rho\) factors through \(\Gal(K'/\QQ)\text{,}\) then the \(L\)-functions computed in terms of \(K\) and \(K'\) agree, in the sense that for each \(p\) which appears in both products (which is all but finitely many), the Euler factor is the same.
Note that
\begin{equation*}
\left| 1 - \det(1 - \rho(\Frob_p)p^{-s})^{-1} \right| = O(p^{-s}),
\end{equation*}
where the implied constant depends on the dimension of \(\rho\text{.}\) Consequently, the Euler product converges absolutely for \(\Real(s) > 1\text{,}\) uniformly for \(\Real(s) \geq 1 + \epsilon\text{,}\) and never vanishes in this region.
Section 23.3 Artin's conjecture
The following is one of the deepest conjectures in modern number theory.
Conjecture 23.2. Artin.
The function \(L(\rho,s)\) extends to a meromorphic function on all of \(\CC\text{,}\) with no poles away from \(s=1\text{,}\) and order of pole at \(s=1\) equal to the number of copies of the trivial representation contained in \(\rho\) (or equivalently, \(1/|G| \sum_{g \in G} \chi(g)\)).There are various stronger versions of this conjecture. For instance, there is also supposed to be a functional equation relating \(L(\rho,s)\) with \(L(\overline{\rho}, 1-s)\text{.}\) More comprehensively, there should be some sort of analogue for \(L(\rho,s)\) of the function \(\theta\) that we used for the proof of the functional equation of the Riemann zeta function. (Such a thing would be an example of a modular form.)
Here are some results about Artin's conjecture.
- It holds for \(\rho\) trivial, by reducing to the Riemann zeta function.
- It holds for \(\rho\) of dimension 1: by the Kronecker–Weber theorem, any such \(\rho\) is a Dirichlet character, and so we get a Dirichlet \(L\)-function.
- It holds if \(\rho\) is induced by a permutation representation (e.g., the regular representation). In this case, this follows from the analytic properties of Dedekind \(\zeta\)-functions. More generally, it holds if \(\rho\) is obtained by induction from a one-dimensional representation; see below.
- If \(\rho\) has dimension 2, then either the image of \(\rho\) is solvable, in which case the conjecture is a theorem of Langlands and Tunnell, or the image is the icosahedral group \(A_5\text{.}\) In the latter case, the conjecture is known when \(\rho\) is odd (any complex conjugation has determinant \(-1\)) by recent results from the theory of modular forms (resolution of Serre's conjecture).
Section 23.4 Induced representations
Let \(H\) be a subgroup of \(G\) and let \(\sigma: H \to \GL_m(\CC)\) be a linear representation. Let \(V\) be the set of functions \(f: G \to \CC^m\) such that \(h(f(g)) = f(hg)\) for all \(h \in H\text{.}\) This forms a representation of \(G\) notated \(\Ind^G_H(\sigma)\text{.}\) For instance, if \(\sigma\) is the trivial representation, then \(\Ind^G_H(\sigma)\) is the linear representation given by the permutation representation of \(G\) on the cosets of \(H\text{.}\)
Theorem 23.3. Artin.
Every character of \(G\) is a \(\QQ\)-linear combination of characters of induced representations from cyclic subgroups.Proof.
Section 23.5 Chebotaryov's density theorem
Here is a weaker form of Artin's conjecture that can be proved, but even this requires some heavy machinery.
Theorem 23.4.
For any \(\rho\text{,}\) the \(L\)-function \(L(\rho,s)\) extends to a meromorphic function on a neighborhood of \(\Real(s) \geq 1\text{.}\) Moreover, on the line \(\Real(s) = 1\text{,}\) \(L(\rho,s)\) is nonvanishing for \(s \neq 1\text{,}\) and for \(s=1\text{,}\) the order of vanishing of \(L(\rho,s)\) is \(- 1/|G| \sum_{g \in G} \chi(g)\text{.}\) In other words, there is a pole at \(s=1\) of order equal to the multiplicity of the trivial representation in \(\rho\text{.}\))Proof.
(Sketch of proof) One first proves the claim for \(\rho = \Ind^G_H \sigma\) for any abelian subgroup \(H\) of \(G\) and any one-dimensional representation \(\sigma: H \to \GL_1(\CC)\text{.}\) This requires class field theory in general, because one has to first write \(\sigma\) as a ray class character. See [14].
By Theorem 23.3, for any given \(\rho\text{,}\) we deduce the claim for \(\rho^{\oplus m}\text{.}\) To deduce the claim for \(\rho\text{,}\) we may reduce to the case where \(\rho\) has no trivial subrepresentations; then \(L(\rho,s)^m\) extends holomorphically to a neighborhood of \(\Real(s) \geq 1\text{,}\) without vanishing anywhere. Consequently, we can take the \(m\)-th root in a neighborhood of any \(s\) with \(\Real(s) = 1\text{;}\) by choosing the right root, we get a function that patches together with the function defined on \(\Real(s) > 1\text{.}\)
By imitating the proof of Dirichlet's theorem, we deduce the following theorem of Chebotaryov, which can be considered a nonabelian generalization of Dirichlet's theorem.
Theorem 23.5. Chebotaryov.
For any conjugacy class \(C\) of \(G\text{,}\) the set of primes \(p\) for which \(\Frob_p \in C\) has natural density \(\#C/\#G\text{.}\)It is also possible to prove this without using class field theory, as in Chebotaryov's original work. In fact, this result was one of the original impetuses for class field theory to be developed; see [15] for a narration of this story.
