Dirichlet Characters as Langlands

We give an exposition of Dirichlet characters which has Langlands philosophy in mind.

One of the problems with class field theory over \(\mathbb Q\) is that too many things are identified too easily, which can cause confusion when the story is trying to be generalized. As such, there are really two definitions of generalizations of a Dirichlet character. Let’s start with the usual one.

Definition 1. A Dirichlet character \(\chi\pmod N\) is a group homomorphism \((\mathbb Z/N\mathbb Z)^\times\to\mathbb C^\times\). The conductor is the smallest positive \(N’\) for which \(\chi\) factors through \((\mathbb Z/N’\mathbb Z)^\times\). We say that \(\chi\) is primitive if and only if the conductor of \(\chi\) is \(N\).

By taking a suitable inverse limit, this is fairly easily generalized to the following.

Definition 2. An automorphic Dirichlet character \(\chi\) is a continuous homomorphism \(\chi\colon\widehat{\mathbb Z}^\times\to\mathbb C^\times\), where \(\mathbb Z\) is the profinite completion \(\lim\mathbb Z/N\mathbb Z\). The conductor is the smallest positive \(N\) for which \(\chi\) factors through \((\mathbb Z/N\mathbb Z)^\times\).

Remark 3. As a technical point, \(\widehat{\mathbb Z}^\times\) is the limit \(\lim(\mathbb Z/N\mathbb Z)^\times\) because the functor \((-)^\times\) preserves limits (indeed, it is right adjoint to the forgetful functor).

Remark 4. Any automorphic Dirichlet character \(\chi\) does have a conductor, basically for continuity reasons. In short, this is because \(\mathbb C^\times\) has “no small subgroups.” In a few more words, note that it is enough to show that \(\chi\) has an open subgroup in its kernel. To show this, let \(V\subseteq\mathbb C^\times\) be a small enough open neighborhood of \(1\) so that \(V\) contains no nontrivial subgroup of \(\mathbb C^\times\); for example, it is enough to take \(V=\{s\in\mathbb C:\operatorname{Re}s>0\}\). Then the pre-image of \(V\) is some open subset \(U\subseteq\widehat{\mathbb Z}^\times\); the construction of the topology on \(\widehat{\mathbb Z}^\times\) allows us to shrink \(U\) so that it is a subgroup. But then \(\chi(U)\subseteq V\) is a subgroup of \(\mathbb C^\times\) and hence must be trivial.

Example 5. Each Dirichlet character \(\chi\pmod N\) gives rise to a unique automorphic Dirichlet character via the composition \[\widehat{\mathbb Z}^\times\twoheadrightarrow(\mathbb Z/N\mathbb Z)^\times\stackrel\chi\to\mathbb C^\times.\]

Remark 6. The correct “automorphic” object in this situation should be irreducible representations of \(\operatorname{GL}_1(\mathbb Q)\backslash\operatorname{GL}_1(\mathbb A_\mathbb Q)\). However, such representations are all one-dimensional because this group is abelian. If we further require that the irreducible representations have finite order, meaning that they should have finite image, then we see that \(\mathbb R^+\) must be in the kernel, so the representation factors through \[\mathbb Q^\times\backslash\mathbb A_{\mathbb Q}^\times/\mathbb R^+\cong\mathbb{\widehat Z}^\times.\] Let’s explain this isomorphism. There is a natural map from the right to the left by putting a \(1\) in the real component of \(\mathbb A_{\mathbb Q}^\times\). Here are the checks on this map.

• Surjective: an idele \((a_\infty,(a_p)_p)\) has its \(a_\infty\) component killed by \(\mathbb R^+\), up to sign. Then one can use \(\mathbb Q^\times\) to adjust the remaining \(a_p\)s to be in \(\mathbb Z_p^\times\) because \(\nu_p(a_p)=0\) for all but finitely many \(p\). Lastly, the rational in \(\mathbb Q^\times\) needs to be adjusted up to sign.
• Injective: suppose an element \((a_p)_p\in(\widehat{\mathbb Z})^\times\) takes the form \((rq,(q)_p)\) for some \(r\in\mathbb R^+\). Then \(rq=1\), so \(r\) does not matter, and \(q>0\). We are then left with \(q=a_p\) for all primes \(p\), and we see that we must have \(q=1\) because \(q\in\mathbb Z_p^\times\) for all primes \(p\).

Langlands philosophy says that we should put a Galois object opposite our automorphic object.

Definition 7. A Galois Dirichlet character is a group homomorphism \(\chi\colon\operatorname{Gal}(\overline{\mathbb Q}/\mathbb Q)\to\mathbb C^\times\). The conductor of \(\chi\) is the smallest \(N\) for which \(\chi\) factors through \(\operatorname{Gal}(\mathbb Q(\zeta_N)/\mathbb Q)\).

Example 8. Any Dirichlet character \(\chi\pmod N\) gives rise to a unique Galois Dirichlet character via the composite \[\operatorname{Gal}(\overline{\mathbb Q}/\mathbb Q)\twoheadrightarrow\operatorname{Gal}(\mathbb Q(\zeta_N)/\mathbb Q)\cong(\mathbb Z/N\mathbb Z)^\times\stackrel\chi\to\mathbb C^\times.\] Here, the isomorphism is better understood as sending \(k\in(\mathbb Z/N\mathbb Z)^\times\) to the automorphism of \(\mathbb Q(\zeta_N)\) which sends \(\zeta_N\mapsto\zeta_N^k\). Thanks to this canonical isomorphism, we see that \(\chi\) has conductor \(N’\) if and only if the composite Galois conductor has conductor \(N’\).

Example 9. Fix a quadratic extension \(\mathbb Q(\sqrt d)\) of \(\mathbb Q\). Then we define \(\chi_d\) as the composite \[\operatorname{Gal}(\overline{\mathbb Q}/\mathbb Q)\twoheadrightarrow\operatorname{Gal}(K/\mathbb Q)\cong\{\pm1\}\subseteq\mathbb C^\times.\] We call \(\chi_d\) the quadratic character associated to \(K\).

It will require more work to show that conductor exist this time. For example, it is not totally obvious that the quadratic characters of Example 8 have a finite conductor! To start, note that \(\chi\) certainly factors through \[\operatorname{Gal}(\overline{\mathbb Q}/\mathbb Q)^{\mathrm{ab}}=\operatorname{Gal}(\mathbb Q^{\mathrm{ab}}/\mathbb Q)\] because \((-)^{\mathrm{ab}}\) passes through the limit defining the absolute Galois group (because it is right adjoint to the forgetful functor).

Now, the same argument from Remark 4 does show that \(\chi\) admits an open kernel, so it factors through some \(\operatorname{Gal}(K/\mathbb Q)\), where \(K\) is some finite abelian extension of \(\mathbb Q\). Thus, to show that \(\chi\) has a conductor, we have to know that any such \(K\) is contained in a cyclotomic field. This is the Kronecker–Weber theorem.

Theorem 10 (Kronecker–Weber). Any finite abelian extension \(K\) is contained in a cyclotomic field \(\mathbb Q(\zeta_N)\), where \(N\) is some positive integer.

Proof. Omitted. On a personal note, I have never read a direct proof of this theorem which I could sit all the way through. \(\blacksquare\)

It turns out that there is a bijection between automorphic Dirichlet characters and Galois characters.

Corollary 11. The group \(\operatorname{Gal}(\mathbb Q^{\mathrm{ab}}/\mathbb Q)\) is canonically isomorphic to \(\widehat{\mathbb Z}^\times\).

Proof. By the Kronecker–Weber theorem, the Galois group is \[\lim_{N\to\infty}\operatorname{Gal}(\mathbb Q(\zeta_N)/\mathbb Q).\] Here, the internal maps \(\operatorname{Gal}(\mathbb Q(\zeta_N)/\mathbb Q)\to\operatorname{Gal}(\mathbb Q(\zeta_{N’})/\mathbb Q)\) are given by restriction, which notably only makes sense when \(\mathbb Q(\zeta_N)\subseteq\mathbb Q(\zeta_{N’})\), which is equivalent to \(N\mid N’\). Now, recall the canonical isomorphism \[(\mathbb Z/N\mathbb Z)^\times\to\operatorname{Gal}(\mathbb Q(\zeta_N)/\mathbb Q)\] given by sending \(k\) to the automorphism given by \(k\mapsto\zeta_N^k\). One can see that this isomorphism is compatible with the restriction maps on either side, so passing to the limit completes the proof. \(\blacksquare\)

Corollary 12. There is a canonical bijection between automorphic and Galois Dirichlet characters.

Proof. This is immediate from Corollary 11. \(\blacksquare\)

Remark 13. This bijection of Corollary 12 generalizes to global fields. In fact, for a number field \(K\), there is an isomorphism between the profinite completion of \(K^\times\backslash\mathbb A_K^\times\) and \(\operatorname{Gal}(K^{\mathrm{ab}}/K)\).

For those keeping track, we note that we have established a bijection between certain automorphic representations (see Remark 6) and certain Galois representations, which is an instance of the Langlands program. Remark 13 explains how to generalize this to number fields.

We close this post by returning to Dirichlet characters. Of course, we do not expect there to be a bijection between automorphic Dirichlet characters and Dirichlet characters (similarly for Galois Dirichlet characters) due to the issue of primitivity: for any Dirichlet character \(\chi\pmod N\), the composite Dirichlet character \[(\mathbb Z/4N\mathbb Z)^\times\twoheadrightarrow(\mathbb Z/N\mathbb Z)^\times\] produces the same automorphic (or Galois) Dirichlet character. However, we have bijections if we restrict to primitive Dirichlet characters.

Corollary 14. The construction of Example 4 gives a bijection between primitive Dirichlet characters and automorphic Dirichlet characters. Similarly, the construction of Example 8 gives a bijection between primitive Dirichlet characters and Galois Dirichlet characters.

Proof. By Corollary 11 and being a little careful with the canonical isomorphism in Example 8, it is enough to merely pay attention to the automorphic statement.

By the existence of conductors shown in Remark 4, we can invert the construction of Example 7 by sending an automorphic Galois character \(\widetilde\chi\) of conductor \(N\) to the induced quotient Dirichlet character \(\chi\colon(\mathbb Z/N\mathbb Z)^\times\to\mathbb C^\times\). The definition of the conductor in both situations implies that \(\widetilde\chi\) is primitive. The construction of this paragraph can be checked to be inverse to the construction of Example 4, so the result follows. \(\blacksquare\)

The above corollary more or less explains the importance of primitivity when one discusses Dirichlet characters.