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\title{Lecture 24}

\author{Soad Ahmad}


\newcommand{\K}{\ensuremath{K}}
\newcommand{\KC}{\ensuremath{K^{\times}}}
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\newcommand{\CKone}{\ensuremath{\mathcal{C}_\K^1}}
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\begin{document}

\maketitle

\section{Chapter II.3 Ideles}

\defn 3.1 Let \K\ be a number field.  The group of ideles $\JK =
\prod_{v \in M_\K}(\KC_v,\OC_v)$. An \underline{idele}
$(\underbrace{\alpha_1 , \cdots , \alpha_{r_1+r_2}}_{\mbox{in }
\mathbf{R}^{\times} \mbox{ or } \mathbf{C}^{\times} }, \cdots ,
\underbrace{ \alpha_p}_{\mbox{in } \KC_p} , \cdots )$, for almost
all $v,\ |\alpha_v |_v = 1.$

Ideles form a group under multiplication.

$\JK \subset \AK.$

\note The topology on \JK\ is \textit{not} the adele topology. Let
\R\ be a topological ring. $\RC = \{\mbox{units in } \R : r\in \R
: r^{-1} \in \R\}.$

$\RC \subset \R.$  In the induced topology, this is not in general
a topological group because, \[x \mapsto x^{-1} \mbox{ on } \RC\]
is not continuous on \R.  To get around this, we consider the map
\[\RC \hookrightarrow \R \times \R \] \[x \mapsto (x, x^{-1}).\]

So, we can make \RC\ into a topological group by giving it the
subspace topology from $\R \times \R$, i.e. $x_n \mapsto y$ in
\RC\ $\Longleftrightarrow \left.\begin{array}{c}x_n \mapsto y \\
x^{-1}_n \mapsto y \end{array}\right\}$ in \R\ which yields the
following lemma:

\lem\ 3.1 For any topological ring \R, \RC\ with this topology is
a topological group and \[x_n \mapsto y \mbox{ in } \RC\
\Longleftrightarrow \left.\begin{array}{c}x_n \mapsto y \\
x^{-1}_n \mapsto y^{-1} \end{array}\right\} \mbox{ in } \R .\]

A fundamental system of neighborhoods of $1$ in \JK, with
$\mathbf{R}^N \supseteq W = $ a small neighborhood of
$(\underbrace{1,1,\cdots }_{N \mbox{ times}})$,
\[W \times \prod_{v \in S-S_\infty}(1+ \mathcal{P}^{n_v}_v)\times \prod_{v \not\in
S}\OC_v,\] where $S_\infty =$ \{archemidian places of \K\}
$\subset S =$ any finite set of places of \K.

\ex\ $\K = \Q,$  \[\begin{array}{ccccccc}\alpha_p=( & 1, & 1, &
\cdots , & 1, & p, &1, \cdots ). \\ & \uparrow & \uparrow & &
\uparrow & \uparrow & \\ &\infty & \Q_2 & & \Q_{p-1} & \Q_p &
\end{array}\]

This is a sequence in $I_K \subset \AK$ that converges in \AK\ but
not in $I_K$.

\[\lim_{p \rightarrow \infty} \alpha_p = 1 \mbox{ in } A_\Q , \mbox{but} \]
\[\alpha^{-1}_p = (1, \cdots , 1, \frac{1}{p}, 1, \cdots)
\not\rightarrow 1.\]

\lem\ 3.2 The restricted product topology on \JK\ coincides with
the subspace topology for the embedding \[\JK  \rightarrow  A_\K
\times A_\K\]
\[x  \mapsto  (x, x^{-1}). \]  We have a continuous
homomorphism
\[ \JK  \rightarrow  \RPC\]  \[\alpha  \mapsto
||\alpha || = \prod_{v \in M_\K} |\alpha_v|_v.\]

Let $\JKone = \{\alpha \in \JK : ||\alpha || = 1\}.$ \[\KC
\hookrightarrow \JK\] \[x \mapsto (x,x,x,\cdots ).\] By the
product formula, $\KC \subset \JKone.$

\defn\ 3.2 The \underline{idele class group} is $\CK =
\JK / \KC.$  We also define $\CKone = \JKone /\KC.$

More generally, we let $S \subset M_\K$ be a finite set of places.
$S \supset S_\infty.$  Let \[\JS = \mathcal{J}_{\K , S} = \{\alpha
\in \JK : \alpha_v \in \mathcal{O}_v,\ v \not\in S\}.\]

Then, \[\JS = \left( \prod_{v \in S}\KC_v \right) \left( \prod_{v
\not\in S}\OC_v\right) .\]

$\J = \cup_s \JS.$  The idele topology on $\JS$ is just the
product topology.

Define \[\CS = \CKS = \frac{\JS}{\KC \cap \JS}.\]  Similarly,
define $\CSone.$

Consider, \[\KC \cap \JS = \{x \in \KC : |x|_v = 1,\ v \not\in
S\}-\OC_{\K,S}\] and called the group of $S$-units.

A special case:  $S = S_\infty,\ \OC_{\K,S_\infty} = \OC_\K.$

\lem\ 3.3 $\KC \subset \JK$ is a discrete subgroup.

\pf

By theorem (2.1), $\K \hookrightarrow A_\K$ is a discrete set. But
then, \[\begin{array}{c}\KC \hookrightarrow A_\K \times A_\K \\ x
\mapsto (x, x^{-1}) \end{array}\] is discrete.  So, we are done.

\qed


\end{document}