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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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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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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The volume of $SL_2(\mathbf{Z}_p)$ is not $1$: question from Yu. Manin's paper “Reflection on Arithmetical Physics”
Yuri I. Manin in his paper "Reflection on Arithmetical Physics" gives an adelic proof of the celebrated Euler's formula $$\pi^2/6=\prod_p (1-p^{-2})^{-1} .$$He first consider the "adelic circle" $$A_{\...
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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 ...
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1answer
66 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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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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Ideal in Integral Adelic Space?
Suppose that $\boldsymbol{x}_i,\boldsymbol{y}_i$, $1\le i\le m$, lie in integral adelic $m$-space $\mathbb{A}_\mathbb{Z}^m = (\widehat{\mathbb{Z}}\times\mathbb{R})^m$ and that $\bigoplus\limits_{i=1}^...
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1answer
28 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_{\...
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Subgroup of ${\rm Aut}\,(\widehat{\mathbb{Z}})$
Ribes and Zalesskii Corollary 4.4.8 show that the group of continuous automorphisms of $\widehat{\mathbb{Z}}$ satisfies ${\rm Aut}\,(\widehat{\mathbb{Z}})\cong\mathbb{Z}_2\times\frac{\mathbb{Z}}{2\...
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1answer
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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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1answer
161 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 ...
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1answer
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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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1answer
95 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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Technical question on adelic quotient of centralizers
Let $G$ be a connected reductive group over $k =\mathbb Q$ with split component $A_G$. Let $M$ be a $k$-Levi subgroup of $G$ with split component $A_M$. Let $\gamma$ be a semisimple element of $M(k)...
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What does it mean for a complex valued function on $G(\mathbb A)$ to be smooth (or smooth of compact support)?
Let $G$ be a linear algebraic group over a number field $k$. Let $\mathbb A$ denote the adeles of $k$, $\mathbb A_f$ the finite adeles, and $k_{\infty} = \prod\limits_{v \mid \infty} k_v$. Here are ...
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1answer
45 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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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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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
53 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 ...
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Type of density of number field K
I think this is more of a "reference" sort of question than an explicit concrete answer one. Anyway, here it goes. We know that for a given number field $K$, it is dense in its finite adeles $\mathbb{...
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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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1answer
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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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1answer
42 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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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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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
107 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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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
59 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 $N$:
$$\Gamma = \{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \textrm{SL}_2(\mathbb{Z}) : a \equiv ...
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50 views
p-adic and adelic balls: volume formulae!
Just like we have formulae for volume of n-balls and $$L^p$$ balls, are there formulae for p-adic and adelic n-balls?
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1answer
190 views
Is the Haar measure on $\mathbb{Q}_p$ complete?
The field of $p$-adic numbers $\mathbb{Q}_p$ is locally compact, and so there exists a Haar measure on $(\mathbb{Q}_p,+)$. My question is whether a Haar measure on $\mathbb{Q}_p$ will also be a ...
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help for understanding the adelic Poisson summation formula
Everything is in the title, I didn't try so much but I hope someone can share some hints for understanding the objects mentionned in terrytao's blog/Tate’s proof of the functional equation, how to ...
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2answers
317 views
Learning algebraic groups through examples
I am seeking for a good first reference on algebraic groups, or even linear algebraic groups, where the general theory could be understood through example for the classical groups.
Understanding the ...
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Idèles and reduction modulo $\mathfrak{m}$
I am struggling with some "usual" maps in algebraic number theory, but with which I am not used to, confusing a lot of different settings, as idèles, corresponding ideals and reduced images modulo ...
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1answer
157 views
what does $L(s, \pi \times \chi)$ mean in analytic number theory?
Given a Dirichlet character -- a character $\chi: (\mathbb{Z}/p\mathbb{Z})^\times \to \mathbb{C}^\times$ one can define a Dirichlet L-function:
$$ L(s, \chi) = \sum \frac{\chi(n)}{n^s}$$
if $\pi$ is ...
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2answers
219 views
ring of adeles: ring of integers or valuation ring
I have read different definitions of the ring of adeles:
The ring of adeles is defined as the restricted topological product of the completions $K_v$ of a number field $K$ either with respect to ...
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1answer
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Adeles for function fields
Usually the ring of adeles is defined for number fields: if $K$ is a number field the ring of adeles of $K$ is:
$$\mathbb A_K:=\prod_{v}' K_v \;\;\;\;\;\;\;\;\;\;\;\;\;(\ast)$$
where $v$ ranges ...
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1answer
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Incipit of chapter VI of Neukirch's ANT book.
The title of the chapter VI of the neukirch's ANT is "Global class field theory", and the first few lines are the following:
the author doesn't explain what is $K$ here, but from the previous ...
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1answer
123 views
Injectivity of idele norm map
Let $K/F$ be an extension of global fields (I'm considering number fields, but my question may be also considered in function fields). We may define a norm map on the idele groups $$N_{K/F}:\Bbb A_K^\...
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2answers
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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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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
209 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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0answers
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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 $n \in \Bbb N^+$ ...
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1answer
183 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
62 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{...
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302 views
Norm map and extension of idele class groups
It's a classic fact in class field theory that for an extension of number fields $L/K$, we have the norm map on idele class groups $\mathcal{N}:C_L\to C_K$ descended from the map on ideles given by ...
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1answer
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Why is there a set $W$ (to be described below) such that $\mathbb{A}_K = W + K$?
To prove the compactness of $\mathbb{A}_{\mathbb{Q}}/ \mathbb{Q}$ (and hence $\mathbb{A}_K/K$ for an arbitrary number field $K$), one finds a set $W \subseteq \mathbb{A}_{\mathbb{Q}}$ of the form $$ \...
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definition of the L-function $L(f, \chi, s): \mathbb{A}_K \rightarrow \mathbb{C}$, what is smoothness and what is $f$?
To summarize the question I'm going to ask: for those who have studied L-functions and class field theory, I am confused about the definitions of some things and haven't found a good reference for ...
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66 views
Does Idele group of norm 1 preserved by the norm?
I should explain my question in detail as of now I'm sure it makes no sense.
Let $K$ be a global field (in particular I care about the characteristic $p$ case.)
Then its Idele group $I_K$ has a ...
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0answers
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Integration over ideles over $\Bbb{Q}$ , Tate`s thesis special case
Let $f\in S(A_\Bbb{Q})$ that is $f$ is adelic Schwartz-Bruhat function over $\Bbb{Q}$, such that all its components in the finite places are characteristic functions of the corresponding ring of ...
3
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1answer
122 views
A question about the standard maximal compact subgroup of $GL(2,\mathbb{A}))$
Let $F$ be a number field and let $\mathbb{A}$ be the adele ring of $F$. I kow that if $K_v$ is the standard maximal compact subgroup of $GL(2,F_v)$, $K= \prod_v{K_v}$ is the standard maximal compact ...
