Questions tagged [adeles]

For questions on groups and rings of adeles, self-dual topological rings built on an algebraic number field.

The ring of adeles is a self-dual topological ring built on the field of rational numbers (or, more generally, any algebraic number field). The ring of adeles allows one to elegantly describe the Artin reciprocity law, which is a vast generalization of quadratic reciprocity, and other reciprocity laws over finite fields. In addition, it is a classical theorem from Weil that $G$-bundles on an algebraic curve over a finite field can be described in terms of adeles for a reductive group $G$.

Further reference: Adele ring

92 questions
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Learning algebraic groups through examples

I am seeking for a good first reference on algebraic groups, or even linear algebraic groups, where the general theory could be understood through example for the classical groups. Understanding the theoretical definitions of tori, unipotent…
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Property of smooth functions on the adeles

Let $k$ be a number field, $\mathbb A$ the ring of adeles of $k$, $\mathbb A_f$ the finite adeles, and $\mathbb A_{\infty}$ the infinite adeles. Let $\phi: \mathbb A = \mathbb A_{\infty} \times \mathbb A_f \rightarrow \mathbb C$ be a continuous…
D_S
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Volume of first cohomology of arithmetic complex

Let $K$ be a number field and consider the Arithmentic complex $\Gamma_{Ar}(1)^\bullet$ be defined by $$\begin{array} A\Bbb R^{r_1+r_2} & \stackrel{\Sigma}{\longrightarrow} & \Bbb R \\ \uparrow{l(\cdot)=\Pi\log |\sigma_i(\cdot)|} & &…
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Evaluating an adelic integral

I am reading Arthur's notes on the trace formula, and I would like to understand why sometimes the integral appearing there diverge. The example he gives is the following: $G=GL(2)$, $P_0$ the standard parabolic of upper triangular matrices, and…
Wirdspan
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$\Bbb{R}/n\Bbb{Z}$ is isomorphic to $A_\Bbb{Q}/(\Bbb{Q}+C_n)$.

Let $A_\Bbb{Q}$ be the adele group of $\Bbb{Q}$. Let $C_n=\{x \in A_\Bbb{Q}: x_\infty=0 \text{ and }x_p \in p^{\operatorname{ord}_p(n)}\Bbb{Z}_p \text{ for prime }p\}$. I want to show that $\Bbb{R}/n\Bbb{Z}$ is isomorphic to…
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A convergence lemma for adelic zeta function in automorphic forms

I'm reading Godement-Jacquet's classic Zeta functions of simple algebras (1972, Springer). On page 153, the first line: " We also $\textbf{take for granted}$ the following lemma (numbered 11.3)..." The lemma: for $\Phi\in \mathcal{S}(\mathbb{A}_F)$…
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Krull dimension of the adele ring

Let $k$ be a number field and $\mathbf{A}_k$ the adele ring of $k$. What can be said about the Krull dimension of $\mathbf{A}_k$? More generally, I do not know if something can be said about the Krull dimension of an infinite product of rings: is it…
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Topology on an Ext group

One can show that the group $\text{Ext}^1(\mathbf Q, \mathbf Z)$ (calculated in $Ab$) identifies naturally with $\mathbf A_f/\mathbf Q$, where $\mathbf A_f$ is the additive group of finite adèles. More precisely, the long exact sequence of $\mathbf…
Bruno Joyal
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What is the Real Prime?

There seems to be an importance to the ring of adeles for the rational numbers (discussed here), with valuations for every $\mathbb{Q}_p$, but also one "infinite" valuation "$\mathbb{Q}_∞$", seemingly equal to $\mathbb{R}$. Why would something like…
IS4
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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 $n \in \Bbb N^+$ such that $q_2 = n \cdot q_1$, we can form a…
Mike Battaglia
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Proofs with strong approximation theorem

I am stuck on some proofs concerning strong approximation in Chapter 3.1 of Hida's book on modular forms. I have put in green the things that I do not understand. The set $gL\subset \mathbb{A}^\infty$ is a (free) module over $\widehat{\mathbb{Z}}$.…
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Finding the Riemann $\zeta$ function by adelic integration

I am referring to Tao's blog post about Tate's thesis. Introduce the adeles $\mathbb A$ of $\mathbb Q$ and the adelic Mellin transform $$Z(s) = \int_{\mathbb A^\times} = g(x) |x|^s d^\times x.$$ Here, $g = \prod_v g_v$ is a product over places $v$…
Desiderius Severus
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Why is the oldform map injective?

Consider the space of cusp forms $S_k(\Gamma_0(N))$; it has two different maps to $S_k(\Gamma_0(Np))$ where $(p, N) = 1$. We can combine them into a map $$S_k(\Gamma_0(N)) \oplus S_k (\Gamma_0(N)) \to S_k(\Gamma_0(Np))$$ given explicitly by $$(f_1,…
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Subgroups of the general linear group over the adele ring

Let $\mathbb{A}_\mathbb{Q}^f$ be the subring of the adeles ring with $x_\infty=0$, is every open compact subgroup of $GL_2(\mathbb{A}_\mathbb{Q}^f)$ included in a conjugacy class of $GL_2(\widehat{\mathbb{Z}})$ ? Thanks in advance
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Why is adelic approximation a generalisation of Chinese remainder theorem?

Let $F$ be a global field and $S$ a nonempty finite set of places. Then the image of $F$ under the diagonal adelic embedding $F \to F_S$ is dense. I often read that this fact should be seen as a generalisation of the Chinese remainder theorem,…
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