# Questions tagged [adeles]

The adeles tag has no usage guidance.

50
questions

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votes

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147 views

### Integral models and adelic points

Let $k$ be a number field and denote by $\Omega _k$ the set of places of $k$, by $\Omega _\infty$ the set of archimedean places of $k$, and by $S$ a nonempty finite subset of $\Omega _k$ such that $\...

**1**

vote

**1**answer

148 views

### The smooth completion of a curve

Let $C$ be a smooth geometrically integral affine curve. This question concerns the smooth completion $C_1$ of $C$, both defined over a number field $k$.
We know that given any smooth projective ...

**2**

votes

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71 views

### The kernel $K(x,y)$ as an integral over Eisenstein series for $\operatorname{GL}_2$

Let $G = \operatorname{GL}_2$, $f \in C_c^{\infty}(G(\mathbb A)/Z(\mathbb A))$, and $V = L^2( G(\mathbb Q)Z(\mathbb A)\backslash G(\mathbb A))$ (trivial central character). Then the operator $R(f)$ ...

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77 views

### $H(\mathbb A)^0/H(k)$ is homeomorphic to a closed set in $G(\mathbb A)/G(k)$

I'm reading Godement's paper Domaines fondamentaux des groupes arithmetiques and am confused about a part in a proof. The setting is $k$ is a number field, $\mathbb A$ is the ring of adeles of $k$, $...

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77 views

### Spectral decomposition of idele class group $L^2(\mathbb{I}_F/ F^{\times})$

Let $F$ be a number field. Let $\mathbb{A}_F$ be the ring of adeles. The group of units of $\mathbb{A}_F$ is called the group of ideles $\mathbb{I}_F=\mathbb{A}_F^{\times}= GL_1(\mathbb{A}_F)$. The ...

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131 views

### From Maass forms to automorphic forms of $SL_2(\mathbb{A})$

I'm learning basic stuff about automorphic forms, please, if anything I say is not true, excuse me.
Let $X = \Gamma\setminus \mathcal{H}$ be a modular curve and let $f(\tau)$ be a Maass cusp form. ...

**3**

votes

**2**answers

201 views

### Finiteness of the volume of $G(F) \backslash G(\mathbb A)$

Let $G$ be a semisimple algebraic group over a number field $F$ with trivial center. Let $\mathfrak S \subset G(\mathbb A)$ be a Siegel domain (defined in terms of a given maximal split torus and ...

**13**

votes

**1**answer

354 views

### Vector bundles on $\mathbf{P}^1$ and the Iwasawa decomposition

As everyone knows, every vector bundle on $\mathbf{P}^1$ splits as a direct sum of line bundles $\mathcal{O}(a_1)\oplus\cdots\oplus\mathcal{O}(a_n)$. This means that in the Weil-uniformisation ...

**3**

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169 views

### Why doesn't the Manin obstruction work for quadratic forms?

The Manin obstruction explains why the Hasse principle doesn't work for non-quadratic forms. Let's write the notation first;
$V(\mathbb{Q})$ is variety for rational numbers.
$V(A_\mathbb{Q})$ is ...

**4**

votes

**1**answer

98 views

### Are totally positive units of $L$ closed (in $L$) with respect to the finite-idelic topology?

My question is closely related to this one, but not clearly the same in my opinion.
Let $L$ be a number field, with ring of integers $\mathcal{O}_L$, and set $L^{\times}_+\subset L^{\times}$ to be the ...

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167 views

### Definition of cusp form in $L^2$ and convergence over $N_{\mathbb Q} \backslash N_{\mathbb A}$

Let $G$ be an adjoint semisimple group over $\mathbb Q$ with parabolic subgroup $P = MN$ in good position relative to a compact subgroup $U= \prod\limits_v K_v$ of $G(\mathbb A)$. Let $L$ be the ...

**5**

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**1**answer

200 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 adeles of $k$, and $\mathbb A_S = \prod\limits_{v \in ...

**8**

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**1**answer

291 views

### The correct determinant exponent of the weight $k$-operator for defining Hecke operators/adelizing modular forms

For $g \in \operatorname{SL}_2(\mathbb R)$, and $\mathbb H$ the upper half plane, and $k\geq 1$ an integer, the weight $k$-operator on functions $f: \mathbb H \rightarrow \mathbb C$ is defined by
$$...

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110 views

### Absolute convergence of the Fourier series of a smooth adelic function

Let $f: \mathbb A/\mathbb Q \rightarrow \mathbb C$ be a smooth function. Smooth means that $f$ is continuous, smooth in the archimedean argument, for every $(x_0,y_0) \in \mathbb A = \mathbb R \...

**5**

votes

**1**answer

416 views

### Sufficient condition for the absolute convergence of Fourier series of a function on the adele quotient $\mathbb A_k/k$

Let $G$ be a compact abelian group. The unitary characters of $G$ form an orthonormal basis of $L^2(G)$, so every square integrable function $f: G \rightarrow \mathbb C$ admits a Fourier expansion
$$...

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197 views

### Question about the Fourier expansion of adelic Eisenstein series for $\operatorname{GL}_2$

My reference is Daniel Bump's book, Automorphic Forms and Representations, Chapter 3.7. Let $k$ be a number field, $G = \operatorname{GL}_2$, $B$ and $T$ the usual Borel subgroup and maximal torus ...

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213 views

### 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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186 views

### Counting lattice points in adelic spaces

Let $\mathbb{A}$ denote the ring of adeles of $\mathbb{Q}$, let $\mu$ be the Haar measure of $\mathbb{A}$, and let $\|\cdot\|_{\infty}$ denote the sup-norm of the components in the Archimedean ...

**5**

votes

**2**answers

270 views

### Definition of unitary representation of $\mathbf G(\mathbb A_k)$

Let $k$ be a global field, and let $G = \mathbf G(\mathbb A_k)$ for a connected, reductive group $\mathbf G$ over $k$. In these notes by Jayce Getz and Heekyoung Hahn, a unitary representation of $G$ ...

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**1**answer

257 views

### Computing Tamagawa number of torus in Quaternion algebra

Consider a rational Quaternion algebra $M$ over $\mathbb{Q}$ that does not split at $\infty$. For example take the rational Hamilton quaternions $M=\mathbb{Q}(-1,-1)$.
For the adele ring $\mathbb{A}$ ...

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votes

**1**answer

225 views

### Projection onto locally constant function

I am asking a question which looks very elementary to experts.
Let $F$ be a number field and $\mathbb{A}_F$ its adele ring. Let $\omega$ be a unitary central character of $GL_2(\mathbb{A}_F)$,
$X_{...

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**2**answers

266 views

### Automorphic quotients for inner forms or $GSp(4)$

For a quaternion algebra $D$, introduce the quaternionic similitude unitary groups:
\begin{equation}
\mathrm{GU}_D = \left\{ g \in \mathrm{GL}(D) \ : \ g^\star
\left(
\begin{array}{cc}
& 1 \\
1 &...

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127 views

### Is a smooth function with compact support defined on adele groups of Schwartz class?

I'm reading Gelbart's Introduction to the Selberg Trace Formula https://arxiv.org/abs/math/0407288. In his paper he seems to have used the consequence that a smooth function with compact support is a ...

**4**

votes

**1**answer

213 views

### Annihilator of a closed subgroup of adeles

Introduction:
Let $K$ be a number field and let's denote with $\mathbb A_K$ the ring of adeles which is also a locally compact group (with respect to the addition). Remember that the topology is the ...

**3**

votes

**2**answers

407 views

### Compactness of the automorphic quotient

Let $F$ be a (totally real) number field, and $E$ a (totally imaginary) quadratic extension of $F$. We consider $U$ a unitary group (with respect to a given hermitian form over $E$). The question is:
...

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356 views

### Higher Adeles of a scheme and related topics

Let $X$ be a noetherian scheme. I will describe a construction of a simplicial ring which I think is called the Bellinson higher Adeles complex (or something similar).
Consider the augmented ...

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99 views

### Preimage of projection of idèles, and other usual maps

Let $K$ be a quadratic number field.
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, ...

**6**

votes

**1**answer

253 views

### Stable vector bundles in Weil's parametrization

Let $C$ be a smooth projective algebraic curve. Isomorphism classes of vector bundles on $C$ are in bijection with $GL_n(F) \backslash GL_n (\mathbb{A}) / GL_n(\mathcal{O})$, I think (one trivalizes ...

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votes

**4**answers

689 views

### Adelic and classical modular forms on quaternion algebras

Let $R$ be an Eichler order of an indefinite quaternion algebra $B/\mathbb{Q}$ (suppose B is not the collection of $2\times 2$ matrices) and $S$ the corresponding Shimura Curve. Modular forms of ...

**6**

votes

**1**answer

542 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 $n \in \Bbb N^+$ ...

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vote

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246 views

### Finite-index subgroups of the ideles

Let $k$ be a number field and denote by $J_k$ the idele group of $k$. Recall that the finite-index open subgroups of $J_k$ which contain $k^*$ are very important in class field theory. My question ...

**8**

votes

**1**answer

532 views

### Group schemes, adeles, double cosets, and étale cohomology

Let $K$ be a number field, $R$ the ring of integers of $K$,
${\mathbf{A}^f}$ the ring finite adeles of $K$, and ${\widehat{R}}\subset {\mathbf{A}^f}$ the ring of integral adeles.
Let $G$ be an affine ...

**3**

votes

**2**answers

326 views

### Adeles and twisted adeles

Let $\mu_n$ denote the group of $n$-th roots of unity in ${\mathbb{C}}$, i.e., $\mu_n=\ker[{\mathbb{C}}^*\overset{n}{\longrightarrow}{\mathbb{C}}^*]$.
We set
$$ \mu=\varinjlim_n \mu_n\subset {\mathbb{...

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193 views

### Adelic integral factorization

In order to calculate Tamagawa numbers, I need to justify that for a nice (say Schwartz-Bruhat) function, the following identity holds :
$$\int_{\mathbf{A}^2} f(x)dx = \int_{SL_2(\mathbf{A})/SL_2(K)} ...

**18**

votes

**1**answer

2k views

### A good book on adeles and ideles

Many results in number theory are stated either in a classical language or in an adelic one. I am often impressed of the efficiency and the satisfactory computational properties of the adelic setting, ...

**5**

votes

**1**answer

232 views

### Compactness of adelic quotients for unipotent groups over global fields

Let $K$ be a global field, $\mathbb{A}_K$ the ring of adeles, and $U$ a unipotent algebraic group over $K$. Why is $U(\mathbb{A}_K)/U(K)$, when endowed with the quotient topology, compact?

**2**

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200 views

### characters on unipotent group

Let $G=GL_{n}$ and $N$ the maximal unipotent subgroup, $\mathbb{A}$ the ring of adeles on a number field $F$.
We fix a non trivial character $\psi:F\backslash\mathbb{A}\rightarrow \mathbb{C}^{*}$.
We ...

**6**

votes

**1**answer

786 views

### Naive question on adelic groups

The ever-reliable Wikipedia says:
... an adelic algebraic group is a semitopological group defined by...
No more details are given, and I was wondering if the multiplication only being separately ...

**2**

votes

**1**answer

444 views

### Volume of PGL(2,F) \ PGL(2, A)

Let $F$ be a global field. What is the measure of $PGL_2(F) \backslash PGL_2(\mathbb{A})$?
This depends of course on the normalizations of the Haar measures on $PGL_2(F)$ and $PGL_2(\mathbb{A})$. ...

**2**

votes

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53 views

### on degree zero elements in adelic groups

Let $G$ a split connected reductive group and $G(\mathbb{A})$ his points in the ring of adeles.
We have a degree map $G(\mathbb{A})\rightarrow X_{*}(Z)$ where $Z$ is the center of $G$.
Let $G(\...

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votes

**2**answers

372 views

### vanishing of spectral term in Arthur-Selberg trace formula for GL(2)?

Hi,
In the Arthur-Selberg trace formula for $G = GL(2)/\mathbf Q$ (as seen for example in
Gelbart's "Lectures on the Trace Formula"), the spectral side includes terms
like:
$$
\int_{-\infty}^\infty ...

**0**

votes

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82 views

### approximation in Lie algebras

Let $x_{1}$, $x_{2}$, $x_{3}$ three disctinct closed points of a curve $X$ over an algebraically closed field k.
Let G a connected reductive group and $\mathfrak{g}$ his Lie algebra.
I fix a Borel $...

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80 views

### decomposition lemma in adelic groups II

Let $X$ a curve on a field $k=\bar{k}$. G a connected reductive group over $k$.
Let fix $d$ closed points $(x_{1},...,x_{d})$ of $X$.
On each point, we have an évaluation morphisme $ev_{x}:G(k[[t_{x}...

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vote

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125 views

### on a decomposition lemma in adelic groups

Let X a curve over an algebraically closed k.
Fix $x$ and $y$ two distinct closed points of X.
Let G be a connected reductive group over k.
We denote Spec $\hat{\mathcal{O}}_{X,x}$ the formal ...

**10**

votes

**3**answers

2k views

### Adelic description of moduli of $G$-bundles on a curve

Let $X$ be a smooth, projective, geometrically connected curve over a field $k$ and $G$ an an affine algebraic group group over $k$ (we can put more hypotheses on $G$ if necessary). If $K$ denotes the ...

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votes

**3**answers

2k views

### Simple Tamagawa number calculations

As is well known, Euler proved the Basel identity $\displaystyle\sum\limits_{i=0}^{\infty} \frac{1}{n^2} = \frac{{\pi}^2}{6}$. By far the most illuminating explanation of this fact that I've seen is ...

**7**

votes

**1**answer

690 views

### Difference between automorphic forms for SL(2) and GL(2)?

Hi,
Let $A$ denote the adeles of $Q$.
I know how to decompose $L^2(SL(2,A)/SL(2,Q))$ into irreducible $SL(2,A)$-representations. What is the difference between this decomposition and the ...

**10**

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**0**answers

704 views

### How does one understand geometric CFT in terms of modularity?

I have recently asked a question in a similar vein:
What makes Geometric CFT easier than CFT?
but I'm afraid I wasn't quite ripe to ask it yet. I have since consulted with the following sources:
http:/...

**7**

votes

**1**answer

935 views

### Adelic formulations of complex multiplication and modular curves

In modular curves and modular forms, there is an adelic formulation, in which smaller open subgroups of some adelic group relate to higher level structure. As we know, higher level structure ...

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votes

**4**answers

2k views

### Categorical description of the restricted product (Adeles)

Background on the Adèles
The Adèles $\mathbb{A}_K$ of a number field or function field $K$ are defined as a restricted product of the complete local fields $K_\nu$, where $\nu$ ranges over all places ...