Questions tagged [adeles]

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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 ...
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7 votes
1 answer
191 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 ...
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2 votes
0 answers
105 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 ...
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5 votes
1 answer
205 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 ...
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2 votes
1 answer
294 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 ...
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3 votes
1 answer
153 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 ...
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  • 55
3 votes
1 answer
186 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 $\...
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1 vote
1 answer
207 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 ...
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2 votes
0 answers
79 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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1 vote
0 answers
83 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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2 votes
0 answers
89 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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  • 411
1 vote
0 answers
157 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. ...
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  • 83
3 votes
2 answers
227 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 ...
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  • 5,626
13 votes
1 answer
403 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 ...
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  • 4,134
3 votes
0 answers
172 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 ...
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4 votes
1 answer
115 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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  • 319
3 votes
2 answers
180 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 ...
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  • 5,626
5 votes
1 answer
206 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 ...
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8 votes
1 answer
373 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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  • 5,626
1 vote
0 answers
124 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 \...
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  • 5,626
5 votes
1 answer
467 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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  • 5,626
2 votes
0 answers
211 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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  • 5,626
7 votes
0 answers
258 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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6 votes
0 answers
196 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 ...
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6 votes
2 answers
306 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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  • 5,626
6 votes
1 answer
284 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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4 votes
1 answer
234 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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  • 1,525
12 votes
2 answers
284 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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1 vote
0 answers
147 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 ...
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  • 497
4 votes
1 answer
233 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 ...
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3 votes
2 answers
496 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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10 votes
0 answers
376 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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  • 7,007
0 votes
0 answers
100 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, ...
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6 votes
1 answer
260 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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  • 5,332
9 votes
4 answers
743 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 ...
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  • 3,375
6 votes
1 answer
585 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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1 vote
0 answers
270 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 ...
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8 votes
1 answer
579 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 ...
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3 votes
2 answers
334 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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1 vote
0 answers
223 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)} ...
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18 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, ...
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5 votes
1 answer
250 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?
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2 votes
0 answers
203 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 ...
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  • 3,173
6 votes
1 answer
808 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 ...
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2 votes
1 answer
472 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 $PGL_2(\mathbb{A})$. ...
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  • 10.7k
2 votes
0 answers
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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  • 3,173
4 votes
2 answers
375 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 ...
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  • 2,732
0 votes
0 answers
85 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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  • 3,173
1 vote
0 answers
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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  • 3,173
1 vote
0 answers
126 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 ...
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