Questions tagged [adeles]

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

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15 views

How can the integral of the elliptic kernel $K_{\mathcal O}(x,x)$ over $Z(\mathbb A)G(F) \backslash G(\mathbb A)$ be written in the given form?

Let $G$ be a connected, reductive group over a number field $F$. Let $\gamma_1 \in G(F)$ be an elliptic element, i.e. one which is not a member of any proper parabolic subgroup of $G$. Let $\mathcal ...
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18 views

Finiteness of the volume $Z(\mathbb A) G(\mathbb Q) \backslash G(\mathbb A)$ for $G = \operatorname{GL}_2$

I am reading Gelbart's lectures on the trace formula and am confused on how the Siegel domain is used to prove the finiteness of the volume of $Z(\mathbb A) G(\mathbb Q) \backslash G(\mathbb A)$ for $...
4
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1answer
124 views

Why is the oldform map injective?

Consider the space of cusp forms $S_k(\Gamma_0(N))$; it has two different maps to $S_k(\Gamma_0(Np))$ where $(p, N) = 1$. We can combine them into a map $$S_k(\Gamma_0(N)) \oplus S_k (\Gamma_0(N)) \to ...
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34 views

Subgroups of the general linear group over the adele ring

Let $\mathbb{A}_\mathbb{Q}^f$ be the subring of the adeles ring with $x_\infty=0$, is every open compact subgroup of $GL_2(\mathbb{A}_\mathbb{Q}^f)$ included in a conjugacy class of $GL_2(\widehat{\...
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1answer
33 views

Quotient of the Adele ring

Greeting, i'd like to know why the quotient of the Adele ring ${\mathbb{A}}_\mathbb{Q}/\mathbb{Q}$ is compact and isomorphic to $\prod_p\mathbb{Z_p}\times \mathbb{R}/\mathbb{Z}$. Thanks in advance
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38 views

Artin map and ideles

I was going through Silverman's Advanced topics in the arithmetic of elliptic curves, and this statement was given without proof. Any hints/clarification would be appreciated. He first introduces the $...
6
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68 views

Krull dimension of the adele ring

Let $k$ be a number field and $\mathbf{A}_k$ the adele ring of $k$. What can be said about the Krull dimension of $\mathbf{A}_k$? More generally, I do not know if something can be said about the ...
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38 views

A certain decomposition of the idèles group of a global field

I am currently reading the book "Algebraic number theory" published by Cassels and Fröhlich. Let $K$ be a global field (which we may think as a number field as far as I am concerned) and $L/K$ a ...
3
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2answers
44 views

The explicit isomorphism $K \otimes_k k_v \cong \prod_{w / v} K_w$ (linked to Adele ring)

I post here cause I think I don't understand something about the Adele ring. I've just begun to learn about it, then sorry if my question seems trivial. Actually, if $K/k$ is a field extension, we ...
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1answer
61 views

Why is adelic approximation a generalisation of Chinese remainder theorem?

Let $F$ be a global field and $S$ a nonempty finite set of places. Then the image of $F$ under the diagonal adelic embedding $F \to F_S$ is dense. I often read that this fact should be seen as a ...
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67 views

How can I prove the following isomorphism?

A profesor of mine said the following: Let $\mathbb{A}_{\mathbb{Q}}$ be the adele group of $\mathbb{Q}$. There exist a isomorphism of topological groups $$\frac{\mathbb{A}_{\mathbb{Q}}}{\mathbb{Q}}\...
4
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1answer
59 views

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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1answer
42 views

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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22 views

Do we have $G(\mathbb A_S) G(k) = G(\mathbb A)$ for sufficiently large $S$?

Let $G$ be a linear algebraic group over a number field $k$. If necessary, assume $G$ is connected and reductive. Let $\mathbb A$ be the ring of ideles of $k$, and $\mathbb A_S = \prod\limits_{v \in ...
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40 views

Topology of the automorphic quotient

The MO question https://mathoverflow.net/questions/331549/compactness-of-the-automorphic-quotient-and-genericity made me realise that I don't really understand the topology on the adèlic points of an ...
3
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1answer
111 views

Iwasawa decomposition for $GL_n(\mathbb{A}_{\mathbb{Q}})$. What does $K$ look like?

I am trying to understand the Iwasawa decomposition for $GL_n(\mathbb{A}_{\mathbb{Q}})$ and $GL_n(\mathbb{Q})$, where $\mathbb{A}_{\mathbb{Q}}$ is the adeles. The statement for the case of adeles ...
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28 views

Simple question about Hecke operators for the adeles

I have a question about Hecke operators which should not be too difficult. I'm reading these notes by Jerry Shurman on translating modular forms to the adeles and don't understand a certain sentence. ...
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1answer
58 views

Absolute convergence of Fourier series of periodic adelic function

I've asked several questions about this topic before, including here and here. I've gotten many helpful responses, but I still do not completely understand what is going on. Let $f: \mathbb A_{\...
3
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1answer
132 views

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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1answer
163 views

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 ...
3
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1answer
139 views

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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1answer
120 views

Property of smooth functions on the adeles

Let $k$ be a number field, $\mathbb A$ the ring of adeles of $k$, $\mathbb A_f$ the finite adeles, and $\mathbb A_{\infty}$ the infinite adeles. Let $\phi: \mathbb A = \mathbb A_{\infty} \times \...
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1answer
50 views

Fourier Expansion of a function on $\mathbb A_k/k$

Let $k$ be a number field, and let $\mathbb A_k$ be the ring adeles of $k$. The quotient group $\mathbb A_k/k$ is compact, and the choice of a nontrivial character $\psi$ of $\mathbb A_k/k$ gives an ...
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72 views

Strong approximation and class number in the adelic setting

$\DeclareMathOperator{\GL}{GL}\DeclareMathOperator{\ord}{ord}\DeclareMathOperator{\SL}{SL}$I have a question about a proposition from Daniel Bump's book, Automorphic Forms and Representations. Here $...
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52 views

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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1answer
76 views

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 ...
3
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24 views

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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1answer
19 views

$\iota\left(\mathbb Q\right)\subset\mathbb A_\mathbb Q$ 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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1answer
48 views

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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20 views

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 $PGL(2, F)$?
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130 views

Topology on the Schwartz space over local fields and over the adeles

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 ...
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1answer
140 views

How to interpret the symbol $L( \frac{1}{2}, \pi \times \chi)$?

I am trying to interpret the symbol $L( \frac{1}{2}, \pi \times \chi)$ where $\chi = \mathbb{A}^\times / \mathbb{Q}^\times$ and $\pi$ is a cuspidal representation of $GL_2( \mathbb{A})$ (where $\...
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85 views

Can we extend the theta function $\theta(z)$ to p-adic numbers $\mathbb{Z}_p$?

Let $\theta(z) = \sum_{n \in \mathbb{Z}} q^{n^2}$ with $q = e^{2\pi i n z}$. Can we extend the theta function to $p$-adic arguments? Here's an example: $$ \theta( 1 + p^k) = \sum_{n \in \mathbb{Z}^...
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1answer
95 views

How can we identify the modular curve $X_1(N)$ with $\textrm{GL}_2(\mathbb{Q}) \setminus \textrm{GL}_2(\mathbb{A})/K_{\mathbb{R}}K_f$?

Let $N$ be an integer, and let $\Gamma$ be the principal congruence subgroup of level $N$: $$\Gamma = \{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \textrm{SL}_2(\mathbb{Z}) : a \equiv ...
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70 views

p-adic and adelic balls: volume formulae!

Just like we have formulae for volume of n-balls and $$L^p$$ balls, are there formulae for p-adic and adelic n-balls?
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1answer
265 views

Is the Haar measure on $\mathbb{Q}_p$ complete?

The field of $p$-adic numbers $\mathbb{Q}_p$ is locally compact, and so there exists a Haar measure on $(\mathbb{Q}_p,+)$. My question is whether a Haar measure on $\mathbb{Q}_p$ will also be a ...
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0answers
220 views

help for understanding the adelic Poisson summation formula

Everything is in the title, I didn't try so much but I hope someone can share some hints for understanding the objects mentionned in terrytao's blog/Tate’s proof of the functional equation, how to ...
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2answers
426 views

Learning algebraic groups through examples

I am seeking for a good first reference on algebraic groups, or even linear algebraic groups, where the general theory could be understood through example for the classical groups. Understanding the ...
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0answers
51 views

Idèles and reduction modulo $\mathfrak{m}$

I am struggling with some "usual" maps in algebraic number theory, but with which I am not used to, confusing a lot of different settings, as idèles, corresponding ideals and reduced images modulo ...
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1answer
189 views

what does $L(s, \pi \times \chi)$ mean in analytic number theory?

Given a Dirichlet character -- a character $\chi: (\mathbb{Z}/p\mathbb{Z})^\times \to \mathbb{C}^\times$ one can define a Dirichlet L-function: $$ L(s, \chi) = \sum \frac{\chi(n)}{n^s}$$ if $\pi$ is ...
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2answers
289 views

ring of adeles: ring of integers or valuation ring

I have read different definitions of the ring of adeles: The ring of adeles is defined as the restricted topological product of the completions $K_v$ of a number field $K$ either with respect to ...
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1answer
214 views

Adeles for function fields

Usually the ring of adeles is defined for number fields: if $K$ is a number field the ring of adeles of $K$ is: $$\mathbb A_K:=\prod_{v}' K_v \;\;\;\;\;\;\;\;\;\;\;\;\;(\ast)$$ where $v$ ranges ...
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1answer
67 views

Incipit of chapter VI of Neukirch's ANT book.

The title of the chapter VI of the neukirch's ANT is "Global class field theory", and the first few lines are the following: the author doesn't explain what is $K$ here, but from the previous ...
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1answer
176 views

Injectivity of idele norm map

Let $K/F$ be an extension of global fields (I'm considering number fields, but my question may be also considered in function fields). We may define a norm map on the idele groups $$N_{K/F}:\Bbb A_K^\...
3
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2answers
180 views

Importance of the compactness of idele group

Si $k$ is a number field and $J_k$ is the group of $V_k^*$ of invertible elements of the adele ring $V_k$ with the induced topology given by the morphism $V_k\to V_k\times V_K,\ x\mapsto(x,x^{-1})$. ...
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143 views

Arithmetically equivalent number fields and Langlands Program

Two (number) fields are arithmetically equivalent if their Dedekind zeta functions are the same. It is known that any two arithmetically equivalent fields are not necessarily isomorphic; Prasad (http:/...
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2answers
238 views

Structure of $\Bbb Q_p/\Bbb Q$

The group $\Bbb A_\Bbb Q/\Bbb Q$ of adeles mod rationals is: Isomorphic to the solenoid Isomorphic to the group $\Bbb A_\Bbb Z/\Bbb Z$ Dual to the rational numbers This raises the question of the ...
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172 views

Inverse limit of $\Bbb Q/q\Bbb Z$ isomorphic to finite adeles?

Let $\Bbb Q/q\Bbb Z$, for some positive rational $q$, denote the quotient group of the discrete rationals by the subgroup of integers times $q$. For any $q_1, q_2 \in \Bbb Q^+$ and $n \in \Bbb N^+$ ...
3
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1answer
228 views

“Prime decomposition” for the profinite integers, adeles, or p-adics?

The strictly positive integers can be decomposed into a product of prime powers. Likewise, the positive rationals can be decomposed into a product of (possibly negative) prime powers. Another way to ...
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1answer
66 views

Is $\widehat{\mathbb{Z}} \otimes_{\mathbb{Z}} \mathbb{Q} \cong \mathbb{A}_{\mathbb{Q}}^f$ as topological rings?

Maybe this is rather trivial, but I could not solve this (actually, I think this is not true, however I'm not sure). Is $\widehat{\mathbb{Z}} \otimes_{\mathbb{Z}} \mathbb{Q} \cong \mathbb{A}_{\mathbb{...