# field is discrete and cocompact in its adèles

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

$${\mathbb{A}}_{K}=\underset{v\in {P}_{f}\phantom{\prime}}{{\prod}^{\prime}}{K}_{v}\times \prod _{v\in {P}_{\mathrm{\infty}}}{K}_{v}$$ |

###### Theorem 1.

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

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

$$U=\prod _{v\in {P}_{f}}{\U0001d52c}_{v}\times \prod _{v\in {P}_{\mathrm{\infty}}}B(0,\frac{1}{2})$$ |

Since ${\U0001d52c}_{v}$ is open in ${K}_{v}$ for $v$ finite, this is an open set. (Note that ${\U0001d52c}_{v}={\mathcal{O}}_{{K}_{v}}$, 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, ${x}_{v}\in {\U0001d52c}_{v}$, and for $v$ infinite^{}, ${x}_{v}\in B(0,\frac{1}{2})$. Assume $x\ne 0$. Then

$$ |

but then

$$ |

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 ${\prod}_{v\in {P}_{\mathrm{\infty}}}{K}_{v}$ is *dense*, since ${K}_{v}$ is characterized by an embedding^{} $K\hookrightarrow {K}_{v}\cong \mathbb{R},\u2102,{\mathbb{Q}}_{p}$. 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 $\mathrm{\prod}^{\mathrm{\prime}}{}_{v\mathrm{\in}{P}_{f}}\mathit{}{\mathrm{o}}_{v}$ is dense.

Proof. Suppose $x={({x}_{v})}_{v}\in \prod ^{\prime}{}_{v\in {P}_{f}}{\U0001d52c}_{v}$. 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 $y-{x}_{v}\in I{\U0001d52c}_{v}$ for each $v\in {P}_{f}$.

First multiply through by some $z$ so that everything is in ${\mathcal{O}}_{K}$: choose $z\ne 0$ such that $z{x}_{v}\in {\U0001d52c}_{v}$ for all $v\in {P}_{f}$. This is possible since all but finitely many ${x}_{v}$ are already in ${\U0001d52c}_{v}$. Thus $y-{x}_{v}\in I{\U0001d52c}_{v}$ is equivalent^{} to $zy-z{x}_{v}\in (zI){\U0001d52c}_{v}\subset I{\U0001d52c}_{v}$. So assume wlog that ${x}_{v}\in {\U0001d52c}_{v}$ 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 $I=\prod {\U0001d52d}_{i}^{{e}_{i}}$, then $I{\U0001d52c}_{v}={\U0001d52d}_{i}^{{e}_{i}}$ 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

$$U=\prod _{v\in {P}_{f}}{\U0001d52c}_{v}\times \prod _{v\in {P}_{\mathrm{\infty}}}{K}_{v}$$ |

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 ${x}_{v}\notin {\U0001d52c}_{v}$ for $v\in S$. Using an argument identical to the approximation argument above, choose $y\in K$ such that $y-{x}_{v}\in {\U0001d52c}_{v},v\in S$ and $y\in {\U0001d52c}_{v},v\notin S$. Then ${({x}_{v}-y)}_{v}$ is in ${\U0001d52c}_{v}$ for $v\in S$, is in ${\U0001d52c}_{v}$ 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 $U\to {\prod}_{v\in {P}_{\mathrm{\infty}}}{K}_{v}$,

$$\prod _{v\in {P}_{f}}{\U0001d52c}_{v}\to U/{\mathcal{O}}_{K}\to \prod _{v\in {P}_{\mathrm{\infty}}}{K}_{v}/{\mathcal{O}}_{K}\to 0$$ |

The left-hand side is compact since each ${\U0001d52c}_{v}$ is, and the right-hand side is

$${\mathbb{R}}^{{r}_{1}+2{r}_{2}}/{\mathcal{O}}_{K}$$ |

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:

$$\mathbb{R}\text{is locally compact,}\mathbb{Z}\subset \mathbb{R}\text{is discrete and cocompact}$$ | ||

$$\prod _{v\in {P}_{\mathrm{\infty}}}{K}_{v}\text{is locally compact,}{\mathcal{O}}_{K}\subset \prod _{v\in {P}_{\mathrm{\infty}}}{K}_{v}\text{is discrete and cocompact}$$ |

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 ${C}^{\mathrm{\infty}}$ function^{} with exponential decay (or at least integrable on all of $\mathbb{R}$), then we can define its Fourier transform^{} $\widehat{f}$, and the Poisson summation formula

$$\sum _{n\in \mathbb{Z}}f(n)=\sum _{n\in \mathbb{Z}}\widehat{f}(n)$$ |

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 |
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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 |