Questions tagged [adeles]
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75 questions
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Is the topology on the adeles (of the rationals, say) given by a metric analogous to the Fréchet $C^\infty$ metric?
In a discussion with Kevin Buzzard it arose that the adeles of $\mathbb{Q}$ are a Polish space, and before it was realised how easy this was to see (they are a locally compact Hausdorff second ...
- 37.1k
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A finiteness question concerning adelic Galois cohomology
$\newcommand{\Fbar}{{\overline F}}
\newcommand {\A}{{\mathbb A}}
\newcommand{\Abar}{{\bar {\mathbb A}}}
\newcommand{\Gal}{{\rm Gal}}
$Let $F$ be a number field (for example, $F=\mathbb Q$), and let $B$...
- 15.6k
3
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Definition of Adelic points of a scheme over $\mathbb{Q}$
In Milne's notes on Shimura Varieties (https://www.jmilne.org/math/xnotes/svi.pdf) at the start of section 4, it defines the ring of finite adeles. Then for an affine variety $V=\mathrm{Spm}(A)$ over $...
- 129
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Measure normalizations in the adelic hyperbolic Poisson summation formula
Let $\mathsf{A}$ be the diagonal torus of $\mathsf{G}=\mathsf{GL}_2$. We can consider nice functions $f$ on
\begin{equation}
\mathsf{A}(\mathbb{Q})\backslash(\mathsf{A}(\mathbb{A})\cap \mathsf{G}(\...
- 181
1
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125
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Galois group of shimura varieties with different level structure
Let $(G,X)$ be a shimura data, and $K$ an open compact neat subgroup of $G(\mathbb A_f)$. Suppose $K'\subset K$ is open and normal, then, I see in many references that the finite etale cover $Sh(G,X)_{...
- 1,051
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Reference request: ray class group as quotient of finite ideles
Let $K$ be a number field, and write $\mathbb{A}_{K,f}^\times$ for the group of finite ideles of $K$. That is
$$
\mathbb{A}_{K,f}^\times = \{(u_v)_v \in \prod_{v \nmid \infty} K_v^\times : v(u_v) = 0 \...
- 411
1
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Subgroups of $K^n$ and $GL_n(\mathbb{A_K})$
Throughout the question, $v$ is an index for the finite places of a number field $K$, and $\mathfrak{p}_v$ denotes the associated prime ideal. It is a classical fact that there is a map from $\mathbb{...
- 645
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126
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Expressing the upper-half plane as a double quotient
I am trying to show that $Z(\mathbb{A})\text{GL}_2(\mathbb{Q})$ \ $\text{GL}_2(\mathbb{A})$ / $\text{GL}_2(\mathbb{A}_{\mathbb{Z}})$ is isomorphic to the upper-half plane. This is something I've read ...
user531303
10
votes
1
answer
866
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Is there an explicit construction of the Bohr Compactification of the Integers?
Is it possible to explicitly describe the Bohr compactification of $\mathbb Z$? This is equivalent to describing all the group homomorphisms $\mathbb R/\mathbb Z \to \mathbb R/\mathbb Z$ including ...
- 2,093
3
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Logarithm map for groups defined over adelic ring
I've been reading the book Eisenstein series and automorphic representations and I am struggling to understand the definition of a logarithm map $H:G(\mathbb{A})\rightarrow \mathfrak{h}(\mathbb{R})$ (...
- 323
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1
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Why locally algebraic characters of $\text{Gal}(\overline{\mathbb Q}/\mathbb Q)$ are associated to $A_0$ Grossencharacters/algebraic Hecke characters?
$\DeclareMathOperator\Res{Res}\DeclareMathOperator\Gal{Gal}$I am trying to understand lemma 3.1 of "Abelian Varieties over \mathbb Q and modular forms" of Ribet. ArXiv link
Just so everyone ...
- 31
4
votes
1
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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 ...
- 831
4
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198
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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 $...
- 171
1
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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 ...
user30211
5
votes
1
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597
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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 $(\...
- 778