Questions tagged [adeles]

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Subgroup of p-adic units

Let $\left(\widehat{\mathbb Z}\right)^\times=\prod_p{\mathbb Z}_p^\times$ be the unit group of the ring $\widehat{\mathbb{Z}}$, which is the profinite completion of $\mathbb Z$. We give it the product ...
Nandor's user avatar
  • 289
4 votes
0 answers
125 views

On adelic integration

$\DeclareMathOperator\GL{GL}$This question might be beyond naive but I really can't find it nor figure it out. It's related to explicit idelic integration as opposed to just bounding its values. Let $...
MEEL's user avatar
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1 vote
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85 views

When is A ⊗ ℚ self-Pontrjagin dual for a compact-Hausdorff topological ring A?

The topological ring of finite adeles $\mathbb A \cong \hat{\mathbb Z} \otimes \mathbb Q$ is self-Pontrjagin dual with self-dual Schwartz–Bruhat functional $\mathbb 1_{\hat{\mathbb Z}}$. This ...
Ronald J. Zallman's user avatar
5 votes
1 answer
479 views

On the notion of cuspidality

Let $k/\mathbb{Q}$ be a number field and $\mathbb{A}$ its ring of adèles. As usual $\mathbb{A} = \mathbb{A_f} \times \mathbb{A_{\infty}}$. The standard definition of an automorphic representation $(\...
Maty Mangoo's user avatar
1 vote
0 answers
193 views

Reference request: Weil's uniformization theorem

The Weil uniformization theorem says that if $k$ is an algebraically closed field, $G$ a reductive group, and $C$ a curve, we have an isomorphism of stacks $Bun_G(C)\cong G(F_C)\backslash G(\mathbb{A}...
Doron Grossman-Naples's user avatar
5 votes
0 answers
248 views

Adelic functions of moderate growth

Let $f:GL_2(\mathbb A_\mathbb Q)\to\mathbb C$. For $g\in GL_2(\mathbb A)$ and a place $v$ of $\mathbb Q$, define $\|g\|_v=\max_{1\le i,j\le 2}(|g_{ij}|_v,|(g^{-1})_{ij}|_v).$ I have seen several ...
user14411's user avatar
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6 votes
1 answer
240 views

Divergence of integrals in the trace formula

I am trying to understand the following situation for $G=GL(2)$, when going from the compact trace formula to the non-compact case. The integral over $G(\mathbb{A})^1_\gamma \backslash G(\mathbb{A})^1$...
TheStudent's user avatar
1 vote
0 answers
148 views

Action of $T_p$ on automorphic forms, and error in Gelbart's "Automorphic forms on adele groups"?

Let $f\in\mathcal{S}_k(\Gamma_0(N),\chi)$ be a cuspidal modular form, and $\phi_f\in\mathcal{A}_0(\text{GL}_2(\mathbb{Q})\backslash\text{GL}_2(\mathbb{A}_\mathbb{Q}),\omega)$ be its corresponding ...
klein4's user avatar
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3 votes
2 answers
202 views

Fundamental domain for $A_K/K$

Let $K$ be a number field and let $A_K$ be the adele ring of $K$. Then $K$ sits in $A_K$ via the diagonal embedding and the quotient $A_K/K$ is compact. All this is well known. Many proofs of the ...
Krishnarjun's user avatar
8 votes
1 answer
321 views

Classical and adelic automorphic forms from SL(n) to GL(n) over number fields

It's a long post but I felt like I needed to provide some context to my problem. The explicit questions start at the bold font questions below. In the classical world, it seems that one is usually ...
Radu T's user avatar
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0 answers
149 views

Artin map and profinite completion of the idèles

One way to formulate local class field theory is by saying that the local Artin map induces an isomorphism from the profinite completion of $K^\times$ to $\operatorname{Gal}(K^\text{ab}/K)$, which ...
Antoine Labelle's user avatar
6 votes
1 answer
339 views

Nisnevich topology inspired by Adeles

I'm quite a newbe in the field of motives & A1 homotopy theory, so please forgive me if the question is too elementary: In the intro from wikipedia on Nisnevish topology is remarked that it's ...
user267839's user avatar
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2 votes
1 answer
332 views

Connecting two pictures of the Zeta function

Lets consider two views of zeta functions of curves. For the following, let $\mathbb{F}_p$ be the field with $p$ elements where $p$ is prime, and let $\overline{\mathbb{F}_p}$ be the algebraic closure ...
Ronald J. Zallman's user avatar
3 votes
1 answer
222 views

base change of adele rings

I read Neukirch’s book “Algebraic Number Theory”, and its remark following to proposition VI.2.3, there is an assertion that natural map $\mathbb{A}_K \otimes_K L \to \mathbb{A}_L$ is isomorphism. How ...
Alice's user avatar
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1 answer
267 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 $\...
oleout's user avatar
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1 answer
316 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 ...
oleout's user avatar
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0 answers
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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 $R(f)$ ...
D_S's user avatar
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1 vote
0 answers
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$H(\mathbb A)^0/H(k)$ 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 $k$, $...
D_S's user avatar
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2 votes
0 answers
102 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 ...
JACK's user avatar
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0 answers
179 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. ...
Aersk's user avatar
  • 103
3 votes
2 answers
262 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 ...
D_S's user avatar
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13 votes
1 answer
463 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 ...
Pulcinella's user avatar
  • 5,506
3 votes
0 answers
177 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 ...
Lord Haydari's user avatar
4 votes
1 answer
126 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 ...
Yoël's user avatar
  • 329
3 votes
2 answers
201 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 ...
D_S's user avatar
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5 votes
1 answer
221 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 ...
D_S's user avatar
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9 votes
1 answer
552 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 $$...
D_S's user avatar
  • 6,100
1 vote
0 answers
156 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 \...
D_S's user avatar
  • 6,100
5 votes
1 answer
505 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 $$...
D_S's user avatar
  • 6,100
2 votes
0 answers
283 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 ...
D_S's user avatar
  • 6,100
7 votes
0 answers
283 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 ...
D_S's user avatar
  • 6,100
6 votes
0 answers
207 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 ...
Maurizio Moreschi's user avatar
6 votes
2 answers
340 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 $G$ ...
D_S's user avatar
  • 6,100
6 votes
1 answer
325 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 $\mathbb{A}$ ...
user113771's user avatar
4 votes
1 answer
248 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_{...
Monty's user avatar
  • 1,719
12 votes
2 answers
302 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 &...
Desiderius Severus's user avatar
1 vote
0 answers
182 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 ...
wuzx's user avatar
  • 517
4 votes
1 answer
248 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 ...
ByContradiction's user avatar
3 votes
2 answers
595 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: ...
Desiderius Severus's user avatar
10 votes
0 answers
396 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 ...
Saal Hardali's user avatar
  • 7,549
0 votes
0 answers
101 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, ...
Desiderius Severus's user avatar
6 votes
1 answer
295 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 ...
Sasha's user avatar
  • 5,492
9 votes
4 answers
846 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 ...
LMN's user avatar
  • 3,525
6 votes
1 answer
664 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 $n \in \Bbb N^+$ ...
Mike Battaglia's user avatar
1 vote
0 answers
310 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 ...
Daniel Loughran's user avatar
9 votes
1 answer
629 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 ...
Mikhail Borovoi's user avatar
3 votes
2 answers
343 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{...
Mikhail Borovoi's user avatar
1 vote
0 answers
269 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)} ...
Desiderius Severus's user avatar
19 votes
1 answer
3k 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, ...
Desiderius Severus's user avatar
5 votes
1 answer
313 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?
Question Mark's user avatar