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Notes on class field theory

Kiran S. Kedlaya

Section 6.1 Adèles

Reference.

[36]; [37], VI.1 and VI.2; [33], VII.

Subsection Lattices of number fields

The basic idea is that we want some sort of “global completion” of a number field \(K\text{.}\) Let us first recall an older version of this idea: Minkowski’s construction of the Euclidean lattice associated to a number field. We follow [37], I.5.

Definition 6.1.1.

Let \(K\) be a number field of degree \(n\text{.}\) It then has \(n\) distinct embeddings \(\tau: K \to \CC\text{.}\) The product embedding
\begin{equation*} j: K \to \prod_\tau \CC, \qquad a \mapsto (\tau(a))_\tau \end{equation*}
induces an isomorphism of \(K_{\CC} = K \otimes_{\QQ} \CC\) with \(\prod_\tau \CC\text{.}\)
The ring \(K_{\CC} = K \otimes_{\QQ} \CC\) admits an involution \(F\) which fixes \(K\) and acts on \(\CC\) via complex conjugation. The corresponding action on \(\prod_\tau \CC\) is
\begin{equation*} (z_\tau)_\tau \mapsto (\overline{z_{\overline{\tau}}})_\tau \end{equation*}
where \(\overline{\tau}\) denotes the composition of \(\tau\) with complex conjugation on \(\CC\text{.}\) The fixed subring under \(F\) is \(K_{\RR} = K \otimes_{\QQ} \RR\text{.}\)
Equip \(K_\CC \cong \prod_\tau \CC\) with the standard Hermitian inner product, that is,
\begin{equation*} \langle z_1, z_2 \rangle = \sum_\tau z_{1,\tau} \overline{z_{2,\tau}}. \end{equation*}
This restricts to a positive definite inner product on \(K_\RR\text{.}\)
Via the embedding of \(K\) into \(K_\RR\text{,}\) \(\gotho_K\) corresponds to a lattice in \(K_\RR\text{,}\) i.e., a discrete cocompact subgroup. Similarly, any fractional ideal of \(K\) corresponds to a lattice in \(K_\RR\text{.}\)

Subsection Profinite completions

Let us put aside the Minkowski construction for the moment and turn to some more arithmetic considerations. We have already used in multiple places the fact that the profinite completion \(\widehat{\ZZ}\) of the group \(\ZZ\) can be identified, via the Chinese remainder theorem, with the product \(\prod_p \ZZ_p\text{.}\) This generalizes to an arbitrary number field as follows.

Remark 6.1.2.

Before continuing, we should clarify our use of notation like \(\widehat{\gotho_K}\) to denote the profinite completion of \(\gotho_K\) for \(K\) a number field. We originally defined this as an inverse limit over finite group quotients of \(\gotho_K\text{.}\) However, remember that we can define the same inverse limit using any smaller collection of quotients which is cofinal (that is, any finite quotient factors through some chosen quotient). In particular, if \(G\) is a subgroup of \(\gotho_K\) of some finite index \(n\text{,}\) then \(n \gotho_K \subseteq G\) and so the quotient map \(\gotho_K \to \gotho_K/G\) factors through the ring quotient \(\gotho_K/n\gotho_K\text{.}\) That is, \(\widehat{\gotho_K}\) can be identified with the inverse limit \(\varprojlim_n \gotho_K/n\gotho_K\text{,}\) and hence also carries the structure of a topological ring.

Proof.

As in Remark 6.1.2, we identify \(\widehat{\gotho_K}\) with \(\varprojlim_n \gotho_K/n\gotho_K\text{.}\) This ring maps to \(\gotho_K/\gothp^m\) for each prime \(\gothp\) and each positive integer \(m\text{;}\) putting these maps together gives us a map \(\widehat{\gotho_K} \to \varprojlim_m \gotho_K/\gothp^m\) for each \(\gothp\text{,}\) and hence a map to the product.
To see that this map is a bijection, factor the ideal \(n \gotho_K\) as \(\gothp_1^{e_1} \cdots \gothp_r^{e_r}\) for some primes \(\gothp_1,\dots,\gothp_r\) and some positive integers \(e_1,\dots,e_r\text{.}\) By the Chinese remainder theorem for ideals in a Dedekind domain, the natural map
\begin{equation*} \gotho_K/n\gotho_K \to \prod_{i=1}^r \gotho_K/\gothp_i^{e_i} \end{equation*}
is an isomorphism. This immediately implies that the original map is injective. To see that the original map is surjective, we must also observe that for each prime \(\gothp\) and each positive integer \(m\text{,}\) there exists a positive integer \(n\) such that \(n \gotho_K\) is divisible by \(\gothp^m\text{;}\) for instance, we may take \(n\) to be the absolute norm of \(\gothp^m\text{.}\)

Remark 6.1.4.

We cannot help mentioning a variant of Remark 6.1.2 that plays a key role in \(p\)-adic Hodge theory. Let \(\CC_p\) be a completed algebraic closure of \(\QQ_p\text{.}\) Consider the inverse system
\begin{equation*} \cdots \stackrel{x \mapsto x^p}{\to} \gotho_{\CC_p} \stackrel{x \mapsto x^p}{\to} \gotho_{\CC_p}. \end{equation*}
Since the maps are multiplicative but not additive, the inverse limit only appears to carry the structure of a multiplicative monoid. However, it was originally observed by Fontaine that the natural map from this inverse system to the inverse system
\begin{equation*} \cdots \stackrel{x \mapsto x^p}{\to} \gotho_{\CC_p}/p\gotho_{\CC_p} \stackrel{x \mapsto x^p}{\to} \gotho_{\CC_p}/p\gotho_{\CC_p} \end{equation*}
is an isomorphism. In this inverse system, the maps upgrade to ring homomorphisms because \((x+y)^p = x^p + y^p\) in any ring in which \(p=0\text{;}\) consequently, the original inverse limit is upgraded to a ring! This then implies that the inverse limit of the system
\begin{equation*} \cdots \stackrel{x \mapsto x^p}{\to} \CC_p \stackrel{x \mapsto x^p}{\to} \CC_p \end{equation*}
is again a ring; it is in fact an algebraically closed field which is complete with respect to a certain nonarchimedean absolute value. This construction has come to be known as forming the tilt of \(\CC_p\text{,}\) and generalizes to a large class of fields which are complete with respect to nonarchimedean absolute values (the perfectoid fields). See [3] for an introduction to this circle of ideas.

Subsection The adèles (rational case)

Our next step is to put the Minkowski construction together with profinite completion to define the ring of adèles. Let us do this first in the case of the rational numbers.

Definition 6.1.5.

We define the ring of finite adèles \(\AA^{\fin}_\QQ\) as any of the following isomorphic objects:
  • the tensor product \(\widehat{\ZZ} \otimes_{\ZZ} \QQ\text{;}\)
  • the direct limit of \(\frac{1}{n} \widehat{\ZZ}\) over all nonzero integers \(n\text{;}\)
  • the restricted direct product \(\sideset{}{'_p}\prod \QQ_p\text{,}\) where we only allow tuples \((\alpha_p)\) for which \(\alpha_p \in \ZZ_p\) for almost all \(p\text{.}\) See Definition 6.1.6.
This is a locally compact topological ring, with the groups \(\frac{1}{n} \widehat{\ZZ}\) forming a fundamental system of neighborhoods of 0 consisting of compact subgroups. The natural group homomorphism
\begin{equation*} \QQ/\ZZ \to \AA^{\fin}_{\QQ}/\widehat{\ZZ} \end{equation*}
is an isomorphism.
In preparation for the definition of adèles associated to a general number field, we introduce the formalism of restricted products.

Definition 6.1.6.

Let \(I\) be an index set. For each \(i \in I\text{,}\) let \(G_i\) be a set and let \(H_i\) be a set of \(G_i\text{.}\) The restricted (direct) product \(G\) of the pairs \((G_i, H_i)\) is the set of tuples \((g_i)_{i=1}^\infty\) such that \(g_i \in H_i\) for all but finitely many indices \(i\text{.}\) Another way to say this is to define, for each finite subset \(S \subseteq I\text{,}\) the set
\begin{equation*} G_S = \prod_{i \in S} G_i \times \prod_{i \notin S} H_i \end{equation*}
and take \(G = \bigcup_S G_S\text{.}\)
We upgrade this construction from sets to richer categories as follows.
  • If each \(G_i\) is a group and each \(H_i\) is a subgroup, then \(G\) admits a group structure.
  • If each \(G_i\) is a ring and each \(H_i\) is a subring, then \(G\) admits a ring structure. (However, if each \(G_i\) is a field, then \(G\) cannot be a field unless \(I\) is a singleton set.)
  • If each \(G_i\) is a locally compact topological space and each \(H_i\) is a compact subspace, then \(G\) may be viewed as a locally compact topological space. One way to see this is to use a system of neighborhoods of the identity given by taking products of compact neighborhoods \(S_i \subseteq G_i\) in which \(S_i = H_i\) for all but finitely many \(i\text{.}\) (Remember that by Tikhonov’s theorem, any product of compact topological spaces is compact.) Another way is to equip each subset \(G_S\) with the product topology, then declare a subset \(U \subset G\) to be open if its intersection with each \(G_S\) is an open subset of \(G_S\text{.}\)
  • Likewise, if each \(G_i\) is a locally compact topological group/ring and each \(H_i\) is a compact subgroup/subring, then \(G\) may be viewed as a locally compact topological group/ring.

Definition 6.1.7.

Define the ring of adèles over \(\QQ\) as \(\AA_{\QQ} = \RR \times \AA^{\fin}_{\QQ}\text{.}\) Then \(\AA_{\QQ}\) is a locally compact topological ring with a canonical embedding \(\QQ \hookrightarrow \AA_{\QQ}\text{.}\) We refer to the elements of \(\QQ\) as principal adèles within \(\AA_{\QQ}\text{.}\)
We may also view \(\AA_\QQ\) as a restricted direct product of the pairs
\begin{equation*} (\RR, \{0\}), (\QQ_2, \ZZ_2), (\QQ_3, \ZZ_3), \dots; \end{equation*}
note that taking the subgroup \(\{0\}\) of \(\RR\) has no real effect because the definition of the restricted product involves checking membership in the chosen subgroup for all but finitely many indices.

Remark 6.1.8.

Note that \(\AA_\QQ\) contains the neighborhood \(U\) of \(0\) consisting of tuples \((x)_v\) where \(|x|_\infty < 1\) and \(|x|_p \leq 1\) for all primes \(p\text{.}\) Any element of the intersection \(U \cap \QQ\) must be an integer (because of the condition at primes), but cannot be a nonzero integer (due to the condition at the real place); hence \(U \cap \QQ = \{0\}\text{.}\) That is, just as \(\ZZ\) sits inside \(\RR\) as a discrete subgroup, \(\QQ\) sits inside \(\AA_\QQ\) as a discrete subgroup.
In fact, we can do somewhat better. Just as the quotient group \(\RR/\ZZ\) is covered by the compact subset \([0,1]\) of \(\RR\) (and therefore is compact: a continous map from a compact topological space to Hausdorff topological space has compact image), the quotient group \(\AA_\QQ/\QQ\) is covered by a compact subset
\begin{equation*} [0,1] \times \prod_p \ZZ_p. \end{equation*}
(see Exercise 1).

Subsection The adèles (general case)

We now put the Minkowski construction together with profinite completion to define the ring of adèles of a number field.

Definition 6.1.9.

Let \(K\) be a number field. By Lemma 6.1.3, the profinite completion \(\widehat{\gotho_K}\) is canonically isomorphic to \(\prod_{\gothp} \gotho_{K_{\gothp}}\text{.}\) We may thus define the ring of finite adèles \(\AA^{\fin}_{K}\) as any of the following isomorphic objects:
  • the tensor product \(\widehat{\gotho_K} \otimes_{\gotho_K} K\text{;}\)
  • the direct limit of \(\frac{1}{\alpha} \widehat{\gotho_K}\) over all nonzero \(\alpha \in \gotho_K\text{;}\)
  • the restricted direct product of the pairs \((K_\gothp, \gotho_{K_\gothp})\) over all primes \(\gothp\) of \(K\text{.}\)
The natural homomorphism
\begin{equation*} K/\gotho_K \to \AA^{\fin}_{K}/\widehat{\gotho_K} \end{equation*}
is an isomorphism.
The ring of adèles \(\AA_K\) is the product \(K_\RR \times \AA^{\fin}_K\text{.}\) In other words, this is the restricted product of the pairs \((K_v, \{0\})\) for infinite places \(v\) and \((K_v, \gotho_{K_v})\) for finite places \(v\text{.}\) We again have a diagonal embedding \(K \hookrightarrow \AA_K\text{;}\) we again refer to the elements of the image of this embedding as principal adèles.

Definition 6.1.10.

For each place \(v\) of \(K\text{,}\) let \(|\bullet|_v\) be the absolute value on the completion \(K_v\) normalized as follows.
  • For \(v\) real, take the usual real absolute value.
  • For \(v\) complex, take the square of the usual absolute value. (This does not satisfy the triangle inequality; sorry.)
  • For \(v\) a finite place above the prime \(p\text{,}\) normalize so that \(|p|_v = p^{-1}\text{.}\)
We then have a well-defined function \(|\bullet|\) on \(\AA_K\) given by
\begin{equation*} |x|_K = \prod_v |x|_v; \end{equation*}
this makes sense because by virtue of the definition of a restricted direct product, all but finitely many of the values \(|x|_v\) are equal to 1.

Proof.

The normalizations have been chosen so that for each place \(v\) of \(\QQ\text{,}\) for each \(\alpha \in K\text{,}\) the product of \(|\alpha|_w\) over all places \(w\) of \(K\) above \(p\) equals \(|\Norm_{L/K}(\alpha)|_v\text{.}\) Taking the product over \(v\text{,}\) we deduce that \(|\alpha|_K = |\Norm_{L/K}(\alpha)|_{\QQ}\text{.}\) That is, the product formula reduces to the case \(K = \QQ\text{,}\) which we may check directly: if we write \(\alpha = \pm p_1^{e_1} \cdots p_r^{e_r}\text{,}\) then \(|\alpha|_v\) equals \(p_1^{e_1} \cdots p_r^{e_r}\) if \(v = \infty\text{,}\) \(p_i^{-e_i}\) if \(v = p_i\text{,}\) and \(1\) otherwise.

Subsection Adelic \(S\)-integers

Definition 6.1.13.

For any finite set \(S\) of places, let \(\AA_{K,S}\) (resp. \(\AA^{\fin}_{K,S}\)) be the subring of \(\AA_K\) (resp. \(\AA^{\fin}_K\)) consisting of those adèles which are integral at all finite places not contained in \(S\text{.}\) The elements of \(\AA_S\) might be thought of as “adelic \(S\)-integers”.
We can formulate an adelic analogue of the Chinese remainder theorem.

Proof.

We end up with an adelic analogue of the Minkowski embedding, but with the role of \(\gotho_K\) played by the entire field \(K\text{!}\)

Proof.

Choose a compact subset \(T\) of the Minkowski space \(M\) containing a fundamental domain for the lattice \(\gotho_K\text{.}\) Then every element of \(M \times \AA^{\fin}_K\) is congruent modulo \(\gotho_K\) to an element of \(T \times \AA^{\fin}_K\text{.}\) By the proposition, the compact set \(T \times \AA^{\fin}_K\) surjects onto \(\AA_K/K\text{,}\) so the latter is also compact.

Remark 6.1.16.

We mention in passing that just as the various completions of \(\QQ\) are “rigid” in the sense that they have no nontrivial automorphisms even if you ignore the topology (Exercise 3), the ring \(\AA_\QQ\) also has no nontrivial automorphisms even if you ignore the topology (Exercise 6).

Subsection The approximation theorem

We already mentioned one analogue of the Chinese remainder theorem (Proposition 6.1.14). Here is another one.

Proof.

Exercises Exercises

1.

Prove that the map from \([0,1] \times \prod_p \ZZ_p\) to \(\AA_\QQ/\QQ\) is surjective.

3.

Let \(K\) be a number field and let \(v\) be a place of \(K\text{.}\) Prove that every automorphism of the field \(K_v\) (as a ring without topology) is continuous.
Hint.
Let \(q\) be the cardinality of the residue field of \(v\text{.}\) Show first that an element of \(K_v^*\) belongs to \(\gotho_{K_v}^*\) if and only if it has an \(m\)-th root for every positive integer \(m\) coprime to \(p(q-1)\text{.}\) Then note that an element of \(K_v\) belongs to \(\gotho_{K_v}\) iff it is a difference of two elements of \(\gotho_{K_v}^*\text{.}\)

4.

Let \(S\) be a finite set of places of a number field \(K\text{,}\) none of which is complex. Prove that every automorphism of \(\prod_{v \in S} K_v\) (as a ring without topology) is continuous.
Hint.
Using Exercise 3, reduce to checking that for two noncomplex places \(v\) and \(w\) of \(K\text{,}\) lying over distinct places of \(\QQ\text{,}\) the completions \(K_v\) and \(K_w\) are not isomorphic as underlying rings. To prove this, consider the set of \(x \in K\) which are squares in \(K_v\text{,}\) and similarly for \(w\text{.}\)

5.

Let \(K\) be a number field and let \(v\) be a place of \(K\) which is not complex. Let \(Q(x,y,z)\) be a quadratic form over \(K\) defined as follows.
  • If \(v\) is real, put \(Q(x,y,z) = x^2 + y^2 + z^2\text{.}\)
  • If \(v\) is finite lying over the rational prime \(p\text{,}\) choose \(a \in K \cap \gotho_{K_v}^*\) whose image in the residue field of \(v\) is not a quadratic residue, and put \(Q(x,y,z) = x^2 - ay^2 + pz^2\text{.}\)
Let \(T\) be the intersection of the images of the maps \(cQ: \AA_K^3 \to \AA_K\) over all \(c \in K^*\text{.}\) Prove that \(T = \ker(\AA_K \to \prod_{w \in S} K_w)\) for some finite set \(S\) of places of \(K\) containing \(v\text{.}\)
Hint.
Use Hensel’s lemma to show that for \(w\) a finite place not lying above 2, \(a,b,c \in \gotho_{K_w}^*\text{,}\) and \(t \in K_w^*\text{,}\) the equation \(ax^2 + by^2 + cz^2 = t\) always has a solution with \(a,b,c \in K_w^*\text{.}\)

6.

Prove that every automorphism of the ring \(\AA_\QQ\text{,}\) not necessarily continuous, is trivial.
Hint.
Use Exercise 4 and Exercise 5 to prove that the map \(\AA_\QQ \to \prod_v \QQ_v\) is equivariant for any automorphism of \(\AA_\QQ\) and the trivial action on \(\prod_v \QQ_v\text{.}\)

7.

Hint.
Prove by induction on \(n\) that given any pairwise distinct places \(v_1, \dots, v_n\text{,}\) we can find \(x \in K\) with
\begin{equation*} |x|_{v_1} @gt; 1, |x|_{v_2} @lt; 1, \dots, |x|_{v_n} @lt; 1. \end{equation*}
Then make a careful linear combination of powers of such elements. For more details, see [37], Theorem II.3.4.
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