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 $F$, ...
1 vote
31 views

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)......
43 views

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 ...
1 vote
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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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1 vote
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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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1 vote
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1 vote
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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 ...
53 views

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 ...
1 vote
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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 ...
• 601
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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 ...