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4
votes
1answer
208 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 ...
6
votes
1answer
226 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
272 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
111 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
404 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
288 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
113 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)} ...
8
votes
1answer
843 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, ...
4
votes
1answer
159 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?
1
vote
0answers
102 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
575 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 ...
3
votes
1answer
345 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
51 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
321 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
66 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 $...
0
votes
0answers
68 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}...
0
votes
0answers
107 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 ...
7
votes
3answers
918 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 ...
15
votes
3answers
1k 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
488 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
622 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
646 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 ...
13
votes
4answers
1k 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 ...