TANT 18 - The ring of adèles

Hello there! These are notes for the eighteenth class of the course "Topics in algebra and number theory" held in Block 4 of the academic year 2017/18 at the University of Copenhagen.

In the previous lecture we saw that for every local field $K$ we have a natural isomorphism $G_K^{\text{ab}} \cong \widehat{K^{\times}}$. This generalizes the isomorphism $G_{\mathbb{Q}_p}^{\text{ab}} \cong \widehat{\mathbb{Q}_p^{\times}}$ which we proved in the sixteenth lecture as a consequence of the local version of the theorem of Kronecker and Weber.

In the same lecture we also proved that $G_{\mathbb{Q}}^{\text{ab}} \cong \widehat{\mathbb{Z}}$ as a consequence of the global version of the theorem of Kronecker and Weber. Thus it is a natural question to ask if we can generalize this result to an arbitrary number field $K$. In particular, it would be nice if we could "glue" the local Artin maps $\theta_{\mathfrak{p}} \colon K_{\mathfrak{p}} \to G_{K_{\mathfrak{p}}}^{\text{ab}}$ that we have for every prime ideal $\mathfrak{p} \subseteq \mathcal{O}_K$ to define a global map $\theta \colon ?? \to G_K^{\text{ab}}$. To do so, we need to understand first of all what to put in place of the question marks. Defining this object (the idèle class group of $K$ ) will be the aim of this lecture.

Restricted products

The first "ansatz" of an object suitable to glue together all the local Artin maps $\theta_{\mathfrak{p}}$ would simply be the direct product $\prod_{\mathfrak{p}} K_{\mathfrak{p}}^{\times}$ or the direct sum $\bigoplus_{\mathfrak{p}} K_{\mathfrak{p}}^{\times}$. Using these we would be able to construct maps going to $\prod_{\mathfrak{p}} G_{K_{\mathfrak{p}}}^{\text{ab}}$ and to $\bigoplus_{\mathfrak{p}} G_{K_{\mathfrak{p}}}^{\text{ab}}$. The problem here is that the first group is too big and the second one is too small to be related to $G_K^{\text{ab}}$. We need something "in between" these two, and the right notion will be the one of restricted product.

 Definition 1   Let $I$ be a set, $\{ X_i \}_{i \in I}$ be a collection of sets and $\{ Y_i \}_{i \in I}$ be a collection of subsets $Y_i \subseteq X_i$. We define the restricted product $\sideset{}{'}\prod_{i \in I} (X_i \colon Y_i)$ as the set of all sequences $(x_i)_{i \in I} \in \prod_{i \in I} X_i$ such that $x_i \in Y_i$ for all but (at most) finitely many $i \in I$.
If the $X_i$'s are topological spaces and $Y_i \subseteq X_i$ are open we put on the restricted product the topology whose basis is given by sets of the form $\prod_{i \in I} U_i$ where $U_i \subseteq X_i$ is open and $U_i = Y_i$ for all but finitely many $i \in I$.
Moreover, if $X_i$ are groups (or rings) and $Y_i \subseteq X_i$ are subgroups (or subrings) we put a group (or ring) structure on the restricted product using the component-wise operations.

Let now $\{ X_i \}_{i \in I}$ be a collection of topological spaces and let $Y_i \subseteq X_i$ be a collection of open subspaces. For every finite subset $S \subseteq I$ we define $$X_S := \prod_{i \in S} X_i \times \prod_{i \notin S} Y_i$$ with the product topology. It is now easy to observe that $X_S \subseteq \sideset{}{'}\prod_{i \in I} (X_i \colon Y_i)$ as sets and that the subspace topology induced on $X_S$ by this inclusion coincides with the product topology. Indeed a basis for the product topology on a product $\prod_{j \in J} Z_j$ is given by sets of the form $\prod_{j \in J} U_j$ where $U_j \subseteq Z_j$ are open and $U_j = Z_j$ for all but finitely many $j \in J$. These type of subsets in the product $X_S$ coincide precisely with the subsets $\prod_{i \in I} V_i$ such that $V_i \subseteq X_i$ for $i \in S$, $V_i \subseteq Y_i$ for $i \notin S$ and $V_i = Y_i$ for all but finitely many $i \in I$, and thus the product and restricted product topologies coincide.

An important fact about restricted products is that they are the right operation to consider if one wants to preserve local-compactness. Indeed, if one has a collection of topological spaces $\{ X_i \}_{i \in I}$ which are locally compact there is no guarantee that the usual product $\prod_i X_i$ will be locally compact. More precisely, $\prod_i X_i$ is locally compact if and only if all the $X_i$'s are also locally compact and all but finitely many are compact (see Theorem XI.6.5 of the book "Topology" by James Dugundji). However, the restricted product behaves better...

 Lemma 2   Let $\{ X_i \}_{i \in I}$ be a collection of locally compact topological spaces and let $\{ Y_i \}_{i \in I}$ be a collection of open, compact subspaces $Y_i \subseteq X_i$. Then the restricted product $\sideset{}{'}\prod_{i \in I} (X_i \colon Y_i)$ is locally compact.

 Proof   Let $(x_i) \in \sideset{}{'}\prod_{i \in I} (X_i \colon Y_i)$, and let $S := \{ i \in I \mid x_i \notin Y_i \}$. Then we know by definition that $S$ is finite, and thus what we said before implies that the set $X_S$ is locally compact in the product topology and hence also as a subset of the restricted product. To conclude it is sufficient to observe that $X_S$ is also open in $\sideset{}{'}\prod_{i \in I} (X_i \colon Y_i)$ (by definition of the restricted product topology) and that $(x_i) \in X_S$. Q.E.D.

The ring of adèles, the group of idèles and the idelic class group

We can apply now the restricted product to construct the idelic class group. We can apply this construction simultaneously to number fields (i.e. finite extensions of $\mathbb{Q}$ ) and function fields of positive characteristic (i.e. finite extensions of $\mathbb{F}_p(T)$ ). These fields deserve a special name.


 Definition 3   A field $K$ is a global field if it is either a finite, separable extension of $\mathbb{Q}$ or of  $\mathbb{F}_p(T)$ for some prime number $p \in \mathbb{N}$.

Recall that we defined local fields as non-discrete topological fields which are locally compact, and only after we discovered that they were all finite extensions of $\mathbb{R}$, $\mathbb{Q}_p$ or $\mathbb{F}_p((T))$. There is also a similar, axiomatic characterization of global fields, which is related to the product formula for absolute values, which we will explain in the following lecture.

 Definition 4   Let $K$ be a global field. We define the ring of adèles of $K$ as the restricted product $$\mathbb{A}_K := \sideset{}{'}\prod_{v \in \Sigma_K} (K_v \colon A_{K_v})$$ where $\Sigma_K$ denotes the set of places of $K$, $K_v$ denotes the completion of $K$ with respect to the place $v$ and $A_{K_v} := K_v$ if $v$ is Archimedean whereas $A_{K_v} := \{ x \in K_v \colon \lvert x \rvert_v \leq 1 \}$ (the usual definition) if $v$ is non-Archimedean.

Recall that the local fields $K_v$ are locally compact and the closed unit balls $A_{K_v}$ are compact for every place $v \in \Sigma_K$. Thus, we see from Lemma 2 that $\mathbb{A}_K$ is a locally compact topological abelian group.

 Definition 5   Let $K$ be a global field. We define the group of idèles of $K$ as the group of invertible elements of the ring of adèles $\mathbb{A}_K$.

 Remark 6   If $R$ is a topological ring, then $R^{\times}$ has always a canonical structure of a topological group, but it does not have the subspace topology induced by the inclusion $R^{\times} \subseteq R$. Instead, we take the subspace topology induced from the embedding $R^{\times} \hookrightarrow R \times R$ defined by $t \mapsto (t,t^{-1})$.

 Remark 7   There is somehow a confusion in notation between different books on adèles and idèles. We will stick with $\mathbb{A}_K$ for the ring of adèles and $\mathbb{I}_K := \mathbb{A}_K^{\times}$ for the group of idèles.

It is now immediate to observe from the definitions that $$\mathbb{A}_K^{\times} = \sideset{}{'}\prod_{v \in \Sigma_K} (K_v^{\times} \colon A_{K_v}^{\times})$$ and that the embeddings $K \hookrightarrow K_v$ of $K$ inside its completions induce a diagonal embedding $K \hookrightarrow \mathbb{A}_K$, which in turn gives us an embedding $K^{\times} \hookrightarrow \mathbb{A}_K^{\times}$.

 Definition 8   Let $K$ be a global field. We define the idelic class group $C_K$ as $C_K := \mathbb{A}_K^{\times}/K^{\times}$.

As we have already anticipated, this group will be the right thing to put in place of the two question marks above, and we may already see how to construct the global Artin map! Let $K$ be a global field and for every $v \in \Sigma_K$ denote by $\theta_v \colon K_v^{\times} \to G_{K_v}^{\text{ab}}$ the local Artin map defined in the previous lecture. Let now $K_v^{\text{unr}} \subseteq K_v^{\text{ab}}$ be the maximal unramified extension of $K_v$, i.e. the union of all unramified extensions of $K_v$ (which, as you may recall, are all abelian). Then it is not difficult to see that the map $\theta_v$ sends $A_{K_v}^{\times}$ to the subgroup $\operatorname{Gal}(K_v^{\text{ab}}/K_v^{\text{unr}}) \subseteq G_{K_v}^{\text{ab}}$. Indeed, to see so it is sufficient to consider the following couple of exact sequences

where the vertical map on the right sends $1$ to the topological generator of the group $\operatorname{Gal}(K_v^{\text{unr}}/K_v) \cong \widehat{\mathbb{Z}}$. Observe that the square on the right commutes by definition of $\theta_v$, and thus there exists a map $A_{K_v}^{\times} \to \operatorname{Gal}(K_v^{\text{ab}}/K_v^{\text{unr}})$ such that the diagram above is a morphism of short exact sequences. Finally, this map has to be an isomorphism because $\widehat{\theta_v} \colon A_{K_v}^{\times} \times \widehat{\mathbb{Z}} \to G_{K_v}^{\text{ab}}$ is an isomorphism.

Hence in particular if a finite, abelian extension $K \subseteq L$ is unramified at a place $v \in \Sigma_K$ we have that $\pi_{L_w}(\theta_v(A_{K_v}^{\times})) = \{ 1 \}$ where $w \in \Sigma_L$ is any place extending $v \in \Sigma_K$ and $\pi_{L_w} \colon G_K^{\text{ab}} \twoheadrightarrow \operatorname{Gal}(L_w/K_v)$ is the quotient map. Using this and Lemma 4 of the sixteenth lecture we see that the map $$\begin{align} \mathbb{A}_K^{\times} &\to \operatorname{Gal}(L/K) \\ (a_v)_{v \in \Sigma_K} &\mapsto \prod_{v \in \Sigma_K} \theta_v(a_v) \end{align}$$ is well defined. Indeed, we know that all but a finite number of $v \in \Sigma_K$ are unramified in $K \subseteq L$ and we know that for every $(a_v)_{v \in \Sigma_K} \in \mathbb{A}_K^{\times}$ we have that $a_v \in A_{K_v}^{\times}$ for all but finitely many $v \in \Sigma_K$. Thus we have that $\theta_v(a_v) = \operatorname{Id}_L$ for all but finitely many $v \in \Sigma_K$, and the map is well defined. If we take the inverse limit of these maps over all the finite abelian extensions $K \subseteq L$ we get a map $\mathbb{A}_K^{\times} \to G_K^{\text{ab}}$. We will see in the following lectures that this map is trivial on $K^{\times}$, and thus induces a map $\theta \colon C_K \to G_K^{\text{ab}}$, which will be the global Artin map.

Finite adèles and idèles

In the restricted product that defines the adèles and idèles we have taken all the places of a global field $K$. If $K$ is a number field these clearly include also the Archimedean places. Sometimes, however, it could be better to take only the non-Archimedean ones in the restricted product.

 Definition 9   We define the ring of finite adèles as $$\mathbb{A}_K^{\infty} := \sideset{}{'}\prod_{v \in \Sigma_K^{\infty}} (K_v \colon A_{K_v})$$ where $\Sigma_K^{\infty}$ is the set of all non-Archimedean places of $K$. Moreover, we define the group of finite idèles $\mathbb{A}_K^{\infty,\times}$ as the group of invertible elements of the ring $\mathbb{A}_K^{\infty}$, which is also equal to the restricted product $$\mathbb{A}_K^{\infty,\times} = \sideset{}{'}\prod_{v \in \Sigma_K^{\infty}} (K_v^{\times} \colon A_{K_v}^{\times})$$

It is now easy to observe that if $K$ is a function field we have that $\mathbb{A}_K^{\infty} = \mathbb{A}_K$. On the other hand, if $K$ is a number field we also have to take into account the Archimedean places. Recall that these are in bijective correspondence with the equivalence classes of embeddings $K \hookrightarrow \mathbb{C}$ (see Corollary 18 of the fourth lecture) which implies that $$K \otimes_{\mathbb{Q}} \mathbb{R} \cong \prod_{v \in \Sigma_{K,\infty}} K_v$$ where $\Sigma_{K,\infty}$ is the set of Archimedean places of $K$ (see Pages 10-11 of these notes by René Schoof). This gives us immediately that $\mathbb{A}_K = (K \otimes_{\mathbb{Q}} \mathbb{R}) \times \mathbb{A}_{K}^{\infty}$. Moreover, we have a more algebraic description of $\mathbb{A}_{K}^{\infty}$ given by the following lemma.

 Lemma 10   We have an isomorphism of topological rings $\mathbb{A}_{K}^{\infty} \cong \widehat{\mathcal{O}_K} \otimes_{\mathcal{O}_K} K$.

 Proof   First of all, observe that $K = \varinjlim_{N \in \mathbb{N}_{\geq 1}} \mathcal{O}_K\left[ \frac{1}{N} \right]$ because for every $x \in K$ there exists $n \in \mathbb{N}$ such that $n x \in \mathcal{O}_K$. Observe moreover that $\widehat{\mathcal{O}_K} = \prod_{\mathfrak{p}} A_{K_{\mathfrak{p}}}$ where $\mathfrak{p}$ runs over all the prime ideals in $\mathcal{O}_K$. Observe finally that for every $N \in \mathbb{N}_{\geq 1}$ we have that $N \in A_{K_{\mathfrak{p}}}^{\times}$ for all but a finite number of primes $\mathfrak{p}$ (the ones such that $N \in \mathfrak{p}$). We can use these facts to get that $$\widehat{\mathcal{O}_K} \otimes_{\mathcal{O}_K} \mathcal{O}_K\left[ \frac{1}{N} \right] \cong \prod_{\mathfrak{p} \ni N} K_{\mathfrak{p}} \times \prod_{\mathfrak{p} \not\ni N} A_{K_{\mathfrak{p}}}$$ which implies that $$\widehat{\mathcal{O}_K} \otimes_{\mathcal{O}_K} K \cong \varinjlim_{N \in \mathbb{N}_{\geq 1}} \prod_{\mathfrak{p} \ni N} K_{\mathfrak{p}} \times \prod_{\mathfrak{p} \not\ni N} A_{K_{\mathfrak{p}}} \cong \mathbb{A}_{K}^{\infty}$$ because the tensor product commutes with the direct limit (see this question on Math StackExchange). Q.E.D.

Conclusions and references

In this lecture we managed to:

• define the concept of a restricted product of sets, topological spaces, groups and rings;
• define the ring of (finite) adèles, the group of (finite) idèles and the idelic class group;
• see how we could define a global Artin map which starts from the idelic class group;
• give another description of the ring of finite adèles in terms of the profinite completion of the ring of integers of a number field.

 References for this lecture include:

• the article "The idelic approach to number theory" by Tom Weston;
• Chapter 18 of the notes "Algebraic number theory. A computational approach" by William Stein;
• Chapter 5 of the book "Fourier analysis on number fields" by Dinakar Ramakrishnan and Robert J. Valenza;
• Section 7.3-7.4 of the book "Number theory" by Helmut Koch;
• the article "Global fields" by John W.S. Cassels, printed as Chapter VI in the book "Algebraic number theory" edited by John W.S. Cassels and Albrecht Fröhlich;
• Chapter IV of the book "Basic number theory" by André Weil.