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\def\CC{\mathbb{C}}
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\begin{document}

\begin{center}
\bf
Math 254B, UC Berkeley, Spring 2002 (Kedlaya) \\
Adeles and Ideles
\end{center}

\head{Reference} 
Milne, Section V.4; Neukirch, Section VI.1 and VI.2; Lang, Chapter VII.

Last semester, we introduced the $p$-adic numbers and more general local
fields as a way to translate ``local'' facts about number fields into
statements of a ``topological'' flavor. This semester, we need to do something
analogous for ``global'' facts, such as the statements of class field theory.
To do this, we construct a topological object that includes all of the
completions of a number field, including both the archimedean and nonarchimedean
ones. This object will be the ring of adeles, and it will lead us to
the right target group for use in the abstract class field theory we
have just set up.

\head{Spelling note} The words ``adele'' and ``idele'' are
sometimes spelled ``ad\`ele'' and ``id\`ele'' (as in Neukirch),
but not as far as I know by anyone who speaks French (e.g., Lang).

\head{Jargon watch}
By a \emph{place} of a number field $K$, we mean either an archimedean
completion $K \hookrightarrow \RR$ or $K \hookrightarrow \CC$
(an \emph{infinite place}), or a
$\gothp$-adic completion $K \hookrightarrow K_\gothp$ for some nonzero
prime ideal $\gothp$ of $\gotho_K$ (a \emph{finite place}). (Note:
there is only one place for each pair of complex embeddings of $K$.)
Each place corresponds to an equivalence class of absolute values on
$K$; if $v$ is a place, we write $K_v$ for the corresponding completion,
which is either $\RR$, $\CC$, or $K_\gothp$ for some prime $\gothp$.

\head{The adeles}

The basic idea is that we want some sort of ``global completion'' of a
number field $K$. In fact, we already know one way to complete $\ZZ$,
namely its profinite completion $\widehat{\ZZ} = \prod_p \ZZ_p$. But we
really want something containing $\QQ$. We define the \emph{ring of
finite adeles} $\AA^{\fin}_\QQ$ as any of the following isomorphic objects:
\begin{itemize}
\item the tensor product $\widehat{\ZZ} \otimes_{\ZZ} \QQ$;
\item the direct limit of $\frac{1}{n} \widehat{\ZZ}$ over all nonzero
integers $n$;
\item the \emph{restricted direct product} $\prod_{p} .' \QQ_p$,
where we only allow tuples $(\alpha_p)$ for which $\alpha_p \in \ZZ_p$
for almost all $p$.
\end{itemize}
For symmetry, we really should allow \emph{all} places, not just the finite
places. So we also define the \emph{ring of adeles} over $\QQ$ as
$\AA_{\QQ} = \RR \times \AA^{\fin}_{\QQ}$. Then $\AA_{\QQ}$ is a locally
compact topological ring with a canonical embedding $\QQ \hookrightarrow
\AA_{\QQ}$.

Now for a general number field $K$.
The profinite completion
$\widehat{\gotho_K}$ is canonically isomorphic to $\prod_{\gothp}
\gotho_{K_{\gothp}}$, so we define the \emph{ring of finite adeles}
$\AA^{\fin}_{K}$ as any of the following isomorphic objects:
\begin{itemize}
\item the tensor product $\widehat{\gotho_K} \otimes_{\gotho_K} K$;
\item the direct limit of $\frac{1}{\alpha} \widehat{\gotho_K}$ over all
nonzero $\alpha \in \gotho_K$;
\item the \emph{restricted direct product} $\prod_{\gothp} .' \QQ_{\gothp}$,
where we only allow tuples $(\alpha_p)$ for which $\alpha_p \in \ZZ_p$
for almost all $p$.
\end{itemize}
The ring of adeles $\AA_K$ is the product of $\AA^{\fin}_K$ with each
archimedean completion. (That's one copy of $\RR$ for each real embedding and
one copy of $\CC$ for each conjugate pair of complex embeddings.)

One has a natural norm on the ring of adeles, because one has a natural norm
on each completion:
\[
|(\alpha_v)_v| = \prod_v |\alpha_v|_v.
\]
One should normalize these in the following way: for $v$ real, take
$|\cdot|_v$ to be the usual absolute value. For $v$ complex, take
$|\cdot|_v$ to be the \emph{square} of the usual absolute value. (That means
the result is not an absolute value, in that it doesn't satisfy the triangle
inequality. Sorry.) For $v$ nonarchimedean corresponding to a prime above $p$,
normalize so that $|p|_v = p^{-1}$.

Again, there is a natural embedding of $K$ into $\AA_K$ because there
is such an embedding for each completion.
With the normalization as above, one has the product formula:
\begin{prop}
If $\alpha \in K$, then $|\alpha| = 1$.
\end{prop}
In particular, $K$ is \emph{discrete} in $\AA_K$ (the difference between
two elements of $K$ cannot be simultaneously small in all embeddings).
This is a generalization/analogue of the fact that $\gotho_K$ is
discrete in Minkowski space (the product of the archimedean completions).

For any finite set $S$ of places, let $\AA_S$ (resp. $\AA^{\fin}_S$
be the subring of $\AA_K$ (resp. $\AA^{\fin}_K$) consisting of
those adeles which are integral at all finite places not contained in $S$.
Then we have the following result, which is essentially the Chinese remainder
theorem.
\begin{prop}
For any finite set $S$ of places, $K + \AA_S^{\fin} = \AA_K^{\fin}$ and
$K + \AA_S = \AA_K$.
\end{prop}
\begin{cor}
The quotient group $\AA_K/K$ is compact.
\end{cor}
\begin{proof}
Choose a compact subset $T$ of Minkowski space $M$ (the product of the
archimedean
completions) containing a fundamental domain for the lattice $\gotho_K$.
Then every element of $M \times \AA^{\fin}$ is congruent modulo $\gotho_K$
to an element of $T \times \AA^{\fin}$. By the proposition,
the compact set $T \times \AA^{\fin}$ surjects onto $\AA_K/K$, so the
latter is also compact.
\end{proof}

\head{Alternate description: restricted products of topological groups}

Let $G_1, G_2, \dots$ be a sequence of locally compact topological groups
and let $H_i$ be a compact subgroup of $G_i$. The \emph{restricted 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 $i$. For each set $S$,
this product contains the subgroup $G_S$ of tuples $(g_i)$ such that $g_i
\in H_i$ for $i \notin S$, and indeed $G$ is the direct limit of the $G_S$.
We make $G$ into a topological group by giving each $G_S$ the product topology
and saying that $U \subset G$ is open if its intersection with each $G_S$
is open there.

In this language, the additive group of adeles over $\QQ$
is simply the restricted
product of the pairs $(\RR, \RR)$ and $(\QQ_p, \ZZ_p)$ for each $p$,
and likewise over a number field.

\head{Ideles and the idele class group}

An \emph{idele} is a unit in the ring $\AA_K$. In other words, it is a
tuple $(\alpha_v)$, one element of $K_v^*$ for each place $v$ of $K$, such that
$\alpha_v \in \gotho_{K_v}^*$ for all but finitely many finite places $v$.
Let $I_K$ denote the group of ideles of $K$ (sometimes thought
of as $\GL_1(\AA_K)$). This group is the restricted product of the pairs
$(\RR^*, \RR^*)$, $(\CC^*, \CC^*)$, and $(K_{\gothp}^*, \gotho_{\gothp})^*)$.

For example, for each element $\beta \in K$, we get a principal adele
in which $\alpha_v = \beta$; this adele is an idele if $\beta \neq 0$. We
call these the \emph{principal adeles} and \emph{principal ideles},
and define the \emph{idele class group} of $K$ as the quotient
$C_K = I_K/K^*$ of the ideles by the principal ideles.

\head{Warning} The restricted product topology does not coincide with the
topology induced on the ideles by sticking them into the adeles! For
example, the set of ideles whose components at finite places
are all integral is not an open subset in the ideles, even though it is the
intersection of the idele group with an open subset of the adeles.

There is a homomorphism from $I_K$ to the group of fractional ideals
of $K$:
\[
(\alpha_\nu)_\nu \mapsto \prod_{\gothp} \gothp^{v_{\gothp}(\alpha_\gothp)},
\]
which is continuous for the discrete topology on the group of fractional
ideals.
The principal idele corresponding to $\alpha \in K$ maps to the principal
ideal generated by $\alpha$. Thus we have a surjection $C_K \to \Cl(K)$.

Since the norm is trivial on $K^*$, we get a well-defined norm map
$|\cdot|: C_K \to \RR^*_+$. Let $C_K^0$ be the kernel of the norm map;
then $C_K^0$ also surjects onto $\Cl(K)$. (The surjection onto $\Cl(K)$ ignores
the infinite places, so you can adjust there to force norm 1.)
\begin{prop}
The group $C_K^0$ is compact.
\end{prop}
This innocuous-looking fact actually implies two basic facts from last
semester:
\begin{enumerate}
\item[(a)]
The class group of $K$ is finite.
\item[(b)]
The group of units of $K$ has rank $r+s-1$, where $r$ and $s$ are the number
of real and complex places, respectively. More generally, if $S$ is a finite
set of places containing the archimedean places,
the group of $S$-units of $K$ (elements of $K$ which
have valuation 0 at each finite place not contained in $S$) has rank
$\#(S)-1$.
\end{enumerate}
In fact, (a) is immediate; $C_K^0$ is compact and it surjects onto 
$\Cl(K)$, so the latter must also be compact for the discrete topology,
i.e., it must be finite. (In fact, $\Cl(K)$ is isomorphic to the group
of connected components of $C_K^0$.)
To see (b), let $I_S$ be the group of ideles which are units outside $S$,
and define the map $\log: I_S \to \RR^{\#(S)}$ by taking log of the absolute
value of the norm of each component in $S$. By the product formula, this maps
into the sum-of-coefficients-zero hyperplane $H$ in $\RR^{\#(S)}$, and the
image of the group $K_S^*$ of $S$-units is discrete therein. (Restricting 
an element of $K_S^*$ to a bounded subset of $H$ bounds all of its
absolute values, so this follows from the discreteness of $K$ in $\AA_K$.)
Let $W$ be the span in $H$ of the image of $K_S^*$; then
we get a continuous homomorphism $C_K^0 \to H/W$ whose image generates
$H/W$. But its image is compact; this is a contradiction unless $H/W$
is the zero vector space. Thus $K_S^*$ must be a lattice in $H$,
so it has rank $\dim H = \#(S) - 1$.

\begin{proof}[Proof of the proposition]
The inverse images of any two positive real numbers under the norm map are
homeomorphic. So rather than prove that the inverse image of 0 is compact,
we'll prove that the inverse image of some $\rho > 0$ is compact.
Namely, we choose $\rho>c$, where $c$ has the property that any idele of
norm $\rho > c$ is congruent modulo $K^*$ to an idele whose components
all have norms in $[1, \rho]$. (Proof that $c$ exists
left as an exercise, or see Lang, Section V.1, Theorem 0.) 

The set
of ideles with each component having norm in $[1, \rho]$ is the product
of ``annuli'' in the archimedean places and finitely many of the
nonarchimedean places, and the group of units in the rest. (Most of the
nonarchimedean places don't have any valuations between 1 and $\rho$.)
This is a compact set, the set of ideles therein of norm $\rho$ is a closed
subset and so is also compact, and the latter set surjects onto
$C_K^0$, so that's compact too.
\end{proof}

One more comment worth making: what are the open subgroups of $I_K$?
In fact, for each formal product $\gothm$ of places, one gets an open
subgroup of ideles $(\alpha_v)_v$ such that
\begin{enumerate}
\item[(a)] if $v$ is a real place occurring in $\gothm$, then
$\alpha_v > 0$;
\item[(b)] if $v$ is a finite place corresponding to the prime $\gothp$,
occurring to the power $e$, then $\alpha_v \equiv 1 \pmod{\gothp^e}$.
\end{enumerate}
And in fact, every open subgroup contains one of these. Thus
using the surjection $C_K \mapsto \Cl(K)$, we get a bijection between open
subgroups of $C_K$ and generalized ideal class groups!

\head{A presentation of $\AA_\QQ$}
In the special case $K = \QQ$, the idele class group has a nice presentation.
Namely, given an arbitrary idele in $I_\QQ$, there is a unique positive
rational with the same norms at the finite places. Thus
\[
I_\QQ \cong \RR^* \times \prod_p \ZZ_p^*.
\]
This definitely does not generalize: as noted above, the idele class group
has multiple connected components when the class number is bigger than 1.

\head{Aside: beyond class field theory}
You can think of the idele group as $\GL_1(\AA_K)$. In that case, class field
theory will become a correspondence between one-dimensional representations
of $\Gal(\overline{K}/K)$ and certain representations of $\GL_1(\AA_K)$. This
is the form in which class field theory generalizes to the nonabelian case:
the Langlands program predicts a correspondence between $n$-dimensional
representations of $\Gal(\overline{K}/K)$ and certain representations
of $\GL_n(\AA_K)$. In fact, with $K$ replaced by the function field of a
curve over a finite field, this prediction is a deep theorem of L. Lafforgue
(based on work of Drinfeld).

\end{document}