For questions on groups and rings of adeles, self-dual topological rings built on an algebraic number field.

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### An exact sequence related to adeles

Let $K$ be a number field and let's denote with $O_K$ its ring of integers. Moreover we indicate with the letter $p$ the generic non-archimedean place of $K$ and with $\sigma$ the generic archimedean ...
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### Basic properties of the adelic matrix norm

This is from Moeglin and Waldspurger's book Spectral Decomposition and Eisenstein Series. Here $G$ is a linear algebraic group over a number field $k$. One fixes a closed embedding $i'$ of $G$ into ...
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### Prüfer Groups and Product Topologies

For each $p\in\mathbb{P}$, the Prüfer group $\mathbb{Z}(p^{\infty})$ is a divisible abelian group which can be given the discrete topology or the topology it inherits via its identification as a ...
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### Representation of $\overline{\mathbb{Q}}$ in One Dimension

Introduction: Below gives an approach to realize the algebraic numbers $\overline{\mathbb{Q}}\subseteq\mathbb{C}$ within a $1$-dimensional compact connected abelian group (solenoid). Essentially a ...
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### What's in a Noetherian $\mathbb{A}$-Module Ephemeralization?

Just kidding, it's not Noetherian. And "Emphemeralization" implies it is a physical construct, or that if it is, due to knowledge heretofore unbeknownst but recently gained by visualization of black ...
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### Let $\text{T} = \mathbb{Q}(i)^\times/\mathbb{Q}^\times$ compute $\text{T}(\mathbb{A})/\text{T}(\mathbb{Q})$

I have a question about algebraic tori, let $T = \mathbb{Q}(i)^\times/\mathbb{Q}^\times$ and let's try to compute $\text{T}(\mathbb{A})/\text{T}(\mathbb{Q})$. One has a product decomposition for ...
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### Inclusion between norm groups in the idèle class group

Reading Cassels and Fröhlich Chap. VII about Global Class Field Theory, I stumbled upon the following problem: if $K\subset L\subset M$ are finite abelian extensions, then the Main Theorem on Abelian ...
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### Find $x \in \mathbb{Q}(i)$ with $|x - 1|_{2+i} < \frac{1}{\sqrt{5}}$, $|x+1|_{2-i} < \frac{1}{\sqrt{5}}$ and $|x|_{7} < \frac{1}{7}$

I wanted to try some examples with adeles and strong aproximation. Let $\mathfrak{p}_1 = 2+i$ and $\mathfrak{p}_2 = 2-i$ and $\mathfrak{p}_3 = 7$. Can we a single number $x \in \mathbb{Q}(i)$ that's ...
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### $\iota\left(\mathbb Q\right)\subset\mathbb A_\mathbb Q$ is a lattice (diagonal embedding).

How could I show that the diagonal embedding of $\mathbb Q$ into $\mathbb A_\mathbb Q$ is a lattice? Here, $\mathbb A_\mathbb Q$ is the set of adeles, and $\iota:\mathbb Q\to\mathbb A_\mathbb Q$ is ...
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### Constructing a solenoid to satisfy its universal property as a projective limit over circles

$\newcommand{\N}{\mathbb{N}}$ $\newcommand{\Q}{\mathbb{Q}}$ $\newcommand{\Z}{\mathbb{Z}}$ $\newcommand{\R}{\mathbb{R}}$ $\renewcommand{\S}{\mathbb{S}}$ $\newcommand{\A}{\mathbb{A}}$ The Pontryagin ...
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### Counting characters of bounded conductors

Let $F$ be a global field. How can I count the number of adelic characters of $GL(2, F)$ of bounded conductors? What about $PGL(2, F)$?
There is a standard way of endowing the Schwartz space $\mathcal{S}(\mathbb{R}^d)$ on $\mathbb{R}^d$ with a topology via semi-norms, turning it into a Fréchet space. Now let $F$ be a non-archimedean ...