Section 6.4 The adelic reciprocity law and Artin reciprocity
We now formulate the statements of global class field theory in adelic language, imitating the setup from local class field theory but using the idèle class group in place of the multiplicative group of the field. This is more than a formal similarity: it allows us to formulate a local-to-global compatibility statement that ultimately allows us to recover Artin reciprocity and the existence theorem (and a bit more) in the classical language.
Convention note.
We are going to fix an algebraic closure of and regard “number fields” as finite subextensions of That is, we are fixing the embeddings of number fields into We’ll use these embeddings to decide how to embed one number field in another.
Subsection The adelic reciprocity law and existence theorem
Here are the adelic reciprocity law and existence theorem; notice that they look just like the local case except that the multiplicative group has been replaced by the idèle class group.
Proof.
Let be any finite set of places of containing all infinite places and all places that ramify in Let be the set of places of lying above places in We first compute that
for some open subgroups of of finite index; this amounts to a separate computation for each For finite, we apply Theorem 4.1.5 (or Exercise 3). For infinite, we note that has image of index For we apply Proposition 4.2.5.
Taking the union over we see that is an open subgroup of By quotienting down to we see that is open; in fact, the snake lemma on the diagram Figure 6.4.2 implies that the quotient is isomorphic to
We still need to check that the index is finite. For this, we note that the inclusion induces a homeomorphism
of topological groups. Consequently, this group is both discrete (by the previous paragraph) and compact (by Proposition 6.2.8), hence finite.
Theorem 6.4.3. Adelic reciprocity law.
For each number field there is a map
which induces, for each Galois extension of number fields, an isomorphism
Proof.
See Theorem 7.3.8 for the proof using abstract class field theory, and Remark 7.6.20 for the proof using group cohomology more directly. See also Section 6.5 for a more detailed summary of both approaches.
Theorem 6.4.4. Adelic existence theorem.
For every number field and every open subgroup of of finite index, there exists a finite (abelian) extension of such that
Proof.
See Theorem 7.4.8. Again, see also Section 6.5 for a more detailed summary of the argument.
We will also obtain a global analogue of the local norm limitation theorem, which was not even suggested by the classical language. (Well, not in this treatment anyway. See Lemma 7.2.2 for an interpretation of the quotient in ideal-theoretic terms.)
Theorem 6.4.5. Adelic norm limitation theorem.
Proof.
As in the local setting, this follows directly from Theorem 6.4.3 by the logic of Remark 4.3.8. See also Theorem 7.3.10.
Subsection More on the reciprocity map
The statement of Theorem 6.4.3 does not give any direct information about the map We describe this map more explicitly using local class field theory and the principle of local-global compatibility. We note in passing that this principle also lies at the heart of the extension of class field theory envisioned in the Langlands program (Remark 6.2.13).
Definition 6.4.6.
Let be an abelian extension of number fields and a place of Put and let be the decomposition group of that is, for some (hence any, because is abelian) place above We define a map as follows.
- If
is a finite place, use the local reciprocity map (Theorem 4.1.2). - If
is a real place, use the sign map if is complex (otherwise there is nothing to specify). - If
is a complex place, then is trivial and so there is nothing to specify.
We obtain a well-defined product map
by arguing as follows. For is a unit for almost all and is unramified for almost all (we may ignore infinite places here). For the (almost all) in both categories, maps to the identity.
Since each of the maps is continuous, so is the map That means the kernel of is an open subgroup of
Here is the subtle point, and the real source of “reciprocity” in this construction.
Proposition 6.4.7. Local-global compatibility.
For an abelian extension of number fields, the map is trivial on the induced map factors through an isomorphism and the resulting map obeys the conclusion of Theorem 6.4.3.
Proof.
For the treatment in terms of abstract class field theory, see Proposition 7.5.6. For the treatment without abstract class field theory, see Remark 7.6.17 and Remark 7.6.20.
Remark 6.4.8.
In case for some we can verify the first assertion of Proposition 6.4.7 by an explicit computation, similar to the direct verification of Artin reciprocity for these extensions; this will play a key role in both approaches to adelic reciprocity.
Let us spell this out explicitly for where we have a concrete identification In this case, we are claiming that for any place of and any
This is clear when For finite and distinct from we are in an unramified situation, where the effect of local reciprocity is to take any uniformizer (i.e., ) to the local Frobenius at (i.e., the automorphism ). The interesting case is when for which we consult Example 4.1.4.
Fom the previous paragraph, we directly obtain the first assertion of Proposition 6.4.7 for The case where is general and then follows from Proposition 4.3.11.
Subsection Back to classical reciprocity
We now apply adelic reciprocity and local-global compatibility to recover the ideal-theoretic formulations of Artin reciprocity and the existence theorem.
Proposition 6.4.9.
Let be an abelian extension of number fields. Given Theorem 6.4.3 and Proposition 6.4.7, let be the kernel of and identify with a generalized ideal class group (Remark 6.2.6) of some conductor Then the map is the Artin map; consequently, Theorem 2.2.6 holds.
Proof.
The idèle with a uniformizer of in the component and elsewhere maps onto the class of in On the other hand, is precisely the Frobenius of (because is unramified over ). So indeed is being mapped to its Frobenius, and the map is indeed Artin reciprocity.
Remark 6.4.10.
We may use similar logic to recover the ideal-theoretic formulation of the existence theorem (Theorem 2.2.8): each generalized ideal class group arises as a quotient of by an open subgroup (see again Remark 6.2.6, and also Definition 7.2.1) and so Theorem 6.4.4 produces the desired abelian extension.
The argument from Proposition 6.4.9 also gives some additional information about the Artin map. First, the Artin map factors through a generalized ideal class group whose conductor is divisible precisely by the ramified primes. Second, we can exactly describe the kernel of the classical Artin map: it is generated by
- all principal ideals congruent to 1 modulo
- norms of ideals of
not divisible by any ramified primes.