Questions tagged [adeles]
The adeles tag has no usage guidance.
64
questions
2
votes
0
answers
236
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 ...
- 3,432
6
votes
1
answer
831
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 ...
- 33.9k
3
votes
1
answer
529
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 $PGL_2(\mathbb{A})$. ...
- 11.1k
2
votes
0
answers
54
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(\...
- 3,432
4
votes
2
answers
385
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 ...
- 2,802
0
votes
0
answers
103
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 $...
- 3,432
1
vote
0
answers
82
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}...
- 3,432
1
vote
0
answers
138
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 ...
- 3,432
13
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 ...
- 3,525
21
votes
3
answers
3k
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 ...
- 6,942
9
votes
1
answer
874
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 ...
- 2,802
10
votes
0
answers
739
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:/...
- 5,819
9
votes
1
answer
1k
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 ...
- 15.1k
20
votes
4
answers
3k
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 ...
- 1,916