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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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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6 votes
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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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Adelic integrability

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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fractional ideals are discrete in $K_\infty$

Let $K$ be a numbe field, $S_\infty$ the set of infinite places of $K$ and $S$ a finite set of places of $K$ containing $S_\infty$. Define a fractional ideal $I$ over the ring of integer $\mathfrak{o}...
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Proofs with strong approximation theorem

I am stuck on some proofs concerning strong approximation in Chapter 3.1 of Hida's book on modular forms. I have put in green the things that I do not understand. The set $gL\subset \mathbb{A}^\infty$...
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Characterization of compact subgroup of adelized vector space

We can deal with it only in the function field case (then all places are nonarchimedian). My problem is: Let $K$ be a global function field,$V=K^n$ a finite dimensional vector space over $K$. How to ...
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A finite-volume problem in the adelic group of a global field

I am reading Zeta functions of simple algebras by Roger Godement and Hervé Jacquet. In p139-140 they introduce a version of the theory of reduction. To get the finiteness of the Siegel domain module ...
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Isomorphisms of local and global class field theory

Let $k$ be a number field and $K/k$ be a finite Galois extension of degree $d=[K:k]$ with Galois group $\Gamma=\Gamma_{K/k}$. Let $v$ be a place of $k$, and let $k_v$ denote the completion of $k$ at $...
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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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Finite type scheme over a ring of $S$-integers is separated?

I have a question in the following setting: Let $k$ be a number field with ring of integers $O_k$. For a finite set of places $S$ of $k$ we can form the ring of $S$-integers $O_{k,S}$. Let $\mathbb{A}...
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isomorphism involving adeles in Hida's book

Let $$U(N)=\{x \in \widehat{\mathbb{Z}}^\times \mid x\equiv 1 \bmod N\mathbb{Z} \}$$ In the isomorphism in the Lemma below, why does $p_p \in \mathbb{Q}_p^\times$ go to $p \bmod N$ if $p$ is not a ...
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Convergence of an adelic integral

Let $F$ be a Schwarz-Bruhat function on $\mathbb A = \mathbb A_{\mathbb Q}$, i.e. a finite sum of products of the form $\prod\limits_{p \leq \infty} f_p$, where $f_{\infty}$ is a Schwarz function on $\...
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Question on the restricted product topology

Let $I$ be an index set and $G_i$ a collection of compact topological groups. Suppose that for each $i \in I$ there are compact (proper) subgroups $H_i \subseteq G_i$. We say that $G =\{(a_i) \in \...
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7 votes
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Evaluating an adelic integral

I am reading Arthur's notes on the trace formula, and I would like to understand why sometimes the integral appearing there diverge. The example he gives is the following: $G=GL(2)$, $P_0$ the ...
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Finiteness of the volume $G(k) \backslash G(\mathbb A)^1$ from that of $G(k) Z(\mathbb A) \backslash G(\mathbb A)$

It follows from the construction of a Siegel domain that $G(k) Z(\mathbb A) \backslash G(\mathbb A)$ has finite volume, where $G$ is a connected, reductive group over a number field $k$, $Z$ is the ...
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2 votes
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Prove that $I_K/ \overline{K^*U}$ is totally disconnected.

For a number field $K$ let $I_K$ be the idele class group and let $$U := \prod _ {\mathfrak q \nmid p} U_\mathfrak q$$ where $\mathfrak q$ is prime in $K$ and $p$ is a prime in $\mathbb Q$. I want to ...
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Finding the Riemann $\zeta$ function by adelic integration

I am referring to Tao's blog post about Tate's thesis. Introduce the adeles $\mathbb A$ of $\mathbb Q$ and the adelic Mellin transform $$Z(s) = \int_{\mathbb A^\times} = g(x) |x|^s d^\times x.$$ Here, ...
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Automorphisms of the adeles of $\mathbb{Q}$

Consider the adele ring of the rationals $\mathbb{A}_{\mathbb{Q}}$. Does any permutation of the set of valuations define an automorphism of this ring or must it fulfill some stronger property?
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What is the Real Prime?

There seems to be an importance to the ring of adeles for the rational numbers (discussed here), with valuations for every $\mathbb{Q}_p$, but also one "infinite" valuation "$\mathbb{Q}...
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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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2 votes
1 answer
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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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2 votes
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Convergence and p-adic numbers

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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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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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 $...
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4 votes
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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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4 votes
1 answer
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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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1 vote
1 answer
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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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1 vote
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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 $...
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6 votes
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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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2 votes
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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 ...
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3 votes
2 answers
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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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4 votes
1 answer
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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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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}}\...
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4 votes
1 answer
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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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0 votes
1 answer
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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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2 votes
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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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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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4 votes
1 answer
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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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2 votes
0 answers
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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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1 vote
1 answer
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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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4 votes
1 answer
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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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3 votes
1 answer
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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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3 votes
1 answer
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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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8 votes
1 answer
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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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1 vote
1 answer
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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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3 votes
0 answers
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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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2 votes
0 answers
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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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