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

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### How to compute that the volume of $Z(\mathbb{A})\backslash \frak{S}$ for a Siegel set is finite?

I found it quite hard to find a complete reference for the (adelic) reduction theory of algebraic groups used in automorphic representations. I want to show that for $G=GL(n)$ over a number field $F$, ...
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### Adele space of rational function field

Let $F=K(x)$ and $K(x)/K$ be the rational function field. I am then trying to prove \begin{align*} A_F = A_F(0) + F \end{align*} where $A_F$ is the adele space, $A_F(0)$ the set of adeles $\alpha$ ...
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### A convergence lemma for adelic zeta function in automorphic forms

I'm reading Godement-Jacquet's classic Zeta functions of simple algebras (1972, Springer). On page 153, the first line: " We also $\textbf{take for granted}$ the following lemma (numbered 11.3)......
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### Intuitive understanding of Tamagawa measure and its relationship between local measures?

Weil's book Basic Number Theory, (1973, second edition) pp. 113 mentioned the Tamagawa measure on $k_\mathbb{A}$, where $k$ is a global field and $k_\mathbb{A}$ its ring of adeles (an old fashioned ...
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In Goldfeld's book, we find the following integrability criterion. Suppose $f = \prod_v f_v$ an adelic function $\mathbb{A} \to \mathbb{C}$ such that $f_v \equiv 1$ on $\mathbb{Z}_p$ for almost every ...
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### Questions on the congruence subgroup

I have two questions related to congruence subgroups. Let $\Gamma=\Gamma_0(N)=\{\begin{pmatrix} a & b \\ c & d \end{pmatrix} \subset SL_2(\mathbb{Z}) \mid c \equiv 0 \pmod{N} \}$ be a ...
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### Proof of Global Kronecker-Weber, Global Case

This question comes from Ramakrishnan's "Fourier Analysis on Number Fields" and its proof of Kronecker-Weber. Namely, I do not understand the final lines of the proof found on p. 236. Let us ...
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### Topology on the idele group.

For the rational number field, the adele ring $A_{\Bbb Q}$ is defined. On the other hand, its multiplicative subgroup $I_{\Bbb Q}$ is also defined. $A_{\Bbb Q}$ is endowed with the topology as its ...
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### understanding a step in proof of strong approximation for $SL_2$

Let $k$ be a global field and $k_v$ denote its completion at a place $v$. Let $Z$ be the closure of $SL_2(k)$ in $SL_2(\mathbb{A})$; $Z$ is a subgroup. Assume $Z$ contains $SL_2(\mathcal{O}_v)$ for ...
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Let $p_n$ be the $n$-th prime number, and $p_{\infty}=\infty$. Let $(a_n)$ a family such that $a_n \in Z_{p_n}$, with $a_\infty=0$. construct a sequence of rationals numbers $(z_n)$ such that for ...
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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 ...
### 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 ...