Questions tagged [adeles]
The adeles tag has no usage guidance.
41
questions
4
votes
1answer
81 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 ...
2
votes
0answers
59 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 ...
4
votes
1answer
188 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 ...
7
votes
1answer
193 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
$$...
0
votes
0answers
89 views
Absolute convergence of the Fourier series of a smooth adelic function
Let $f: \mathbb A/\mathbb Q \rightarrow \mathbb C$ be a smooth function. Smooth means that $f$ is continuous, smooth in the archimedean argument, for every $(x_0,y_0) \in \mathbb A = \mathbb R \...
5
votes
1answer
383 views
Sufficient condition for the absolute convergence of Fourier series of a function on the adele quotient $\mathbb A_k/k$
Let $G$ be a compact abelian group. The unitary characters of $G$ form an orthonormal basis of $L^2(G)$, so every square integrable function $f: G \rightarrow \mathbb C$ admits a Fourier expansion
$$...
2
votes
0answers
186 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 ...
6
votes
0answers
161 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 ...
6
votes
0answers
168 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
2answers
252 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 $G$ ...
6
votes
1answer
230 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 $\mathbb{A}$ ...
4
votes
1answer
221 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_{...
12
votes
2answers
248 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 &...
1
vote
0answers
120 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
1answer
206 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
2answers
371 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:
...
9
votes
0answers
349 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 ...
0
votes
0answers
97 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
1answer
248 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 ...
9
votes
4answers
638 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
1answer
412 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 $n \in \Bbb N^+$ ...
1
vote
0answers
211 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
1answer
509 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
2answers
324 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
$$ \mu=\varinjlim_n \mu_n\subset {\mathbb{...
1
vote
0answers
170 views
Adelic integral factorization
In order to calculate Tamagawa numbers, I need to justify that for a nice (say Schwartz-Bruhat) function, the following identity holds :
$$\int_{\mathbf{A}^2} f(x)dx = \int_{SL_2(\mathbf{A})/SL_2(K)} ...
14
votes
1answer
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
1answer
224 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
votes
0answers
184 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
1answer
735 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
1answer
417 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
0answers
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(\...
4
votes
2answers
368 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
0answers
77 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 $...
1
vote
0answers
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}...
1
vote
0answers
121 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 ...
8
votes
3answers
1k 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 ...
18
votes
3answers
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
1answer
647 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
votes
0answers
687 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:
...
7
votes
1answer
878 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 ...
16
votes
4answers
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 ...
