Questions tagged [adeles]
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64
questions
4
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answer
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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 ...
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 $...
1
vote
0
answers
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 ...
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 $(\...
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}...
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 ...
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 $G(\mathbb{A})^1_\gamma \backslash G(\mathbb{A})^1$...
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 ...
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 ...
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 ...
2
votes
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 ...
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 ...
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 ...
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 ...
3
votes
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 $\...
1
vote
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 ...
2
votes
0
answers
90
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 $R(f)$ ...
1
vote
0
answers
85
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 $k$, $...
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 ...
1
vote
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. ...
3
votes
2
answers
262
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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
463
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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
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 ...
4
votes
1
answer
126
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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 ...
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 ...
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 ...
9
votes
1
answer
552
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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
$$...
1
vote
0
answers
156
views
Absolute convergence of the Fourier series of a smooth adelic function
Let $f: \mathbb A/\mathbb Q \rightarrow \mathbb C$ be a smooth function. Smooth means that $f$ is continuous, smooth in the archimedean argument, for every $(x_0,y_0) \in \mathbb A = \mathbb R \...
5
votes
1
answer
505
views
Sufficient condition for the absolute convergence of Fourier series of a function on the adele quotient $\mathbb A_k/k$
Let $G$ be a compact abelian group. The unitary characters of $G$ form an orthonormal basis of $L^2(G)$, so every square integrable function $f: G \rightarrow \mathbb C$ admits a Fourier expansion
$$...
2
votes
0
answers
283
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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 ...
7
votes
0
answers
283
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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
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0
answers
207
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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 ...
6
votes
2
answers
340
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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
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1
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325
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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
1
answer
248
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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
2
answers
302
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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
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 ...
4
votes
1
answer
248
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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
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:
...
10
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0
answers
396
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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
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, ...
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 ...
9
votes
4
answers
846
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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
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 $n \in \Bbb N^+$ ...
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 ...
9
votes
1
answer
629
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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
343
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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
$$ \mu=\varinjlim_n \mu_n\subset {\mathbb{...
1
vote
0
answers
269
views
Adelic integral factorization
In order to calculate Tamagawa numbers, I need to justify that for a nice (say Schwartz-Bruhat) function, the following identity holds :
$$\int_{\mathbf{A}^2} f(x)dx = \int_{SL_2(\mathbf{A})/SL_2(K)} ...
19
votes
1
answer
3k
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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
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?