field is discrete and cocompact in its adèles

For brevity, we write $P_{f}$ for the set of finite places of $K$ , and $P_{\infty}$ for the set of infinite places. We also write $\sideset{}{{}^{\prime}}{\prod}$ for a restricted direct product. Then

\mathbb{A}_{K}=\sideset{}{{}^{\prime}}{\prod}_{v\in P_{f}}K_{v}\times\prod_{v% \in P_{\infty}}K_{v}

Theorem 1.

$K$ is discrete as a subgroup of $\mathbb{A}_{K}$ .

Proof. Since $\mathbb{A}_{K}$ is a topological ring, it suffices to show that there is a neighborhood in $\mathbb{A}_{K}$ meeting $K$ in only $0$ .

Let

U=\prod_{v\in P_{f}}\mathfrak{o}_{v}\times\prod_{v\in P_{\infty}}B\left(0,% \frac{1}{2}\right)

Since $\mathfrak{o}_{v}$ is open in $K_{v}$ for $v$ finite, this is an open set. (Note that $\mathfrak{o}_{v}=\mathcal{O}_{K_{v}}$ , the ring of algebraic integers of $K_{v}$ ).

Now consider an element $x\in U\cap K\subset\mathbb{A}_{K}$ . If $x=(x_{v})$ , then for $v$ finite, $x_{v}\in\mathfrak{o}_{v}$ , and for $v$ infinite, $x_{v}\in B\left(0,\frac{1}{2}\right)$ . Assume $x\neq 0$ . Then

\left\lvert x_{v}\right\rvert_{v}\leq\begin{cases}1&v\text{ finite}\\ \frac{1}{2}<1&v\text{ infinite}\end{cases}

but then

\prod\left\lvert x\right\rvert_{v}=\prod\left\lvert x_{v}\right\rvert_{v}<1

in contradiction to the product formula. Thus $x=0$ and we are done.

The above theorem is very sensitive to the fact that all places are included in $\mathbb{A}_{K}$ . For example, it is clear that the image of $\mathbb{A}_{K}$ in $\prod_{v\in P_{\infty}}K_{v}$ is dense, since $K_{v}$ is characterized by an embedding $K\hookrightarrow K_{v}\cong\mathbb{R},\mathbb{C},\mathbb{Q}_{p}$ . Then by an argument familiar from Minkowski’s theorem, $\mathcal{O}_{K}$ is a full-rank lattice in the image of $K$ . But $K$ is the $\mathbb{Q}$ -span of that lattice, so is dense in $K_{v}$ .

Furthermore, the same is true for the finite places:

Proposition 2.

The image of $K$ in $\sideset{}{{}^{\prime}}{\prod}_{v\in P_{f}}\mathfrak{o}_{v}$ is dense.

Proof. Suppose $x=(x_{v})_{v}\in\sideset{}{{}^{\prime}}{\prod}_{v\in P_{f}}\mathfrak{o}_{v}$ . We show that $x$ can be approximated as closely as desired by an element of $K$ by showing that for any ideal $I\subset\mathcal{O}_{K}$ , there is $y\in K$ such that $y-x_{v}\in I\mathfrak{o}_{v}$ for each $v\in P_{f}$ .

First multiply through by some $z$ so that everything is in $\mathcal{O}_{K}$ : choose $z\neq 0$ such that $zx_{v}\in\mathfrak{o}_{v}$ for all $v\in P_{f}$ . This is possible since all but finitely many $x_{v}$ are already in $\mathfrak{o}_{v}$ . Thus $y-x_{v}\in I\mathfrak{o}_{v}$ is equivalent to $zy-zx_{v}\in(zI)\mathfrak{o}_{v}\subset I\mathfrak{o}_{v}$ . So assume wlog that $x_{v}\in\mathfrak{o}_{v}$ for all $v$ ; in the end simply divide by $z$ to recover the general case. But then the existence of $y$ is guaranteed by the Chinese Remainder Theorem, since if $I=\prod\mathfrak{p}_{i}^{e_{i}}$ , then $I\mathfrak{o}_{v}=\mathfrak{p}_{i}^{e_{i}}$ for some $i$ .

It is true, though somewhat harder to prove, that $K$ is in fact dense in $\mathbb{A}_{K}$ if even one place is missing from the product!

Theorem 3.

$\mathbb{A}_{K}/K$ is compact.

Proof. The set

U=\prod_{v\in P_{f}}\mathfrak{o}_{v}\times\prod_{v\in P_{\infty}}K_{v}

is open in $\mathbb{A}_{K}$ .

Claim first that $K+U=\mathbb{A}_{K}$ . Choose $(x_{v})\in\mathbb{A}_{K}$ . There is a finite set $S$ of finite places $v$ such that $x_{v}\notin\mathfrak{o}_{v}$ for $v\in S$ . Using an argument identical to the approximation argument above, choose $y\in K$ such that $y-x_{v}\in\mathfrak{o}_{v},v\in S$ and $y\in\mathfrak{o}_{v},v\notin S$ . Then $(x_{v}-y)_{v}$ is in $\mathfrak{o}_{v}$ for $v\in S$ , is in $\mathfrak{o}_{v}$ for $v\notin S$ but finite, and is in $K_{v}$ for $v$ infinite. Thus $(x_{v}-y)_{v}\in U$ and we are done.

Claim next that $K\cap U=\mathcal{O}_{K}$ . $\supset$ is obvious. To see $\subset$ , note that an element of $K\cap U$ is an element of $K$ that is integral at every finite place, so it is integral and is in $\mathcal{O}_{K}$ .

Thus we get a natural map $U\hookrightarrow\mathbb{A}_{K}\twoheadrightarrow\mathbb{A}_{K}/K$ . This map is surjective since $K+U=\mathbb{A}_{K}$ , and its kernel is $K\cap U$ . So it suffices to show that $U/(K\cap U)=U/\mathcal{O}_{K}$ is compact. There is obviously an exact sequence induced by the projection $U\to\prod_{v\in P_{\infty}}K_{v}$ ,

\prod_{v\in P_{f}}\mathfrak{o}_{v}\to U/\mathcal{O}_{K}\to\prod_{v\in P_{% \infty}}K_{v}/\mathcal{O}_{K}\to 0

The left-hand side is compact since each $\mathfrak{o}_{v}$ is, and the right-hand side is

\mathbb{R}^{r_{1}+2r_{2}}/\mathcal{O}_{K}

which know is compact since $\mathcal{O}_{K}$ forms a full-rank lattice in $\mathbb{R}^{n}$ . Thus $U/\mathcal{O}_{K}$ is also compact and we are done.

So we have shown that $\mathbb{A}_{K}$ is a locally compact ring, and that $K\subset\mathbb{A}_{K}$ is discrete and cocompact. This is analogous to two other situations with which we are familiar:

	$\displaystyle\mathbb{R}\text{ is locally compact, }\mathbb{Z}\subset\mathbb{R}% \text{ is discrete and cocompact}$
	$\displaystyle\prod_{v\in P_{\infty}}K_{v}\text{ is locally compact, }\mathcal{% O}_{K}\subset\prod_{v\in P_{\infty}}K_{v}\text{ is discrete and cocompact}$

This is a useful concept because in such a situation one can do Fourier analysis. For example, if $f:\mathbb{R}\to\mathbb{R}$ is a $C^{\infty}$ function with exponential decay (or at least integrable on all of $\mathbb{R}$ ), then we can define its Fourier transform $\hat{f}$ , and the Poisson summation formula

\sum_{n\in\mathbb{Z}}f(n)=\sum_{n\in\mathbb{Z}}\hat{f}(n)

relates the two. The same theory thus exists for appropriately defined functions $f:\mathbb{A}_{K}\to\mathbb{R}$ , and the Poisson formula again holds with the sum over $K$ rather than over $\mathbb{Z}$ . This can be used to show that the $L$ -functions have analytic continuations, just as the real Poisson formula is used to show this for $\zeta$ .

Title	field is discrete and cocompact in its adèles
Canonical name	FieldIsDiscreteAndCocompactInItsAdeles
Date of creation	2013-03-22 18:00:05
Last modified on	2013-03-22 18:00:05
Owner	rm50 (10146)
Last modified by	rm50 (10146)
Numerical id	4
Author	rm50 (10146)
Entry type	Theorem
Classification	msc 11R56