# Questions tagged [adeles]

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

61 questions
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### Basic properties of the adelic matrix norm

This is from Moeglin and Waldspurger's book Spectral Decomposition and Eisenstein Series. Here $G$ is a linear algebraic group over a number field $k$. One fixes a closed embedding $i'$ of $G$ into ...
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### Norm map and extension of idele class groups

It's a classic fact in class field theory that for an extension of number fields $L/K$, we have the norm map on idele class groups $\mathcal{N}:C_L\to C_K$ descended from the map on ideles given by ...
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### Why is there a set $W$ (to be described below) such that $\mathbb{A}_K = W + K$?

To prove the compactness of $\mathbb{A}_{\mathbb{Q}}/ \mathbb{Q}$ (and hence $\mathbb{A}_K/K$ for an arbitrary number field $K$), one finds a set $W \subseteq \mathbb{A}_{\mathbb{Q}}$ of the form  \...
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### definition of the L-function $L(f, \chi, s): \mathbb{A}_K \rightarrow \mathbb{C}$, what is smoothness and what is $f$?

To summarize the question I'm going to ask: for those who have studied L-functions and class field theory, I am confused about the definitions of some things and haven't found a good reference for ...
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### Does Idele group of norm 1 preserved by the norm?

I should explain my question in detail as of now I'm sure it makes no sense. Let $K$ be a global field (in particular I care about the characteristic $p$ case.) Then its Idele group $I_K$ has a ...
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### Integration over ideles over $\Bbb{Q}$ , Tate`s thesis special case

Let $f\in S(A_\Bbb{Q})$ that is $f$ is adelic Schwartz-Bruhat function over $\Bbb{Q}$, such that all its components in the finite places are characteristic functions of the corresponding ring of ...
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### A question about the standard maximal compact subgroup of $GL(2,\mathbb{A}))$

Let $F$ be a number field and let $\mathbb{A}$ be the adele ring of $F$. I kow that if $K_v$ is the standard maximal compact subgroup of $GL(2,F_v)$, $K= \prod_v{K_v}$ is the standard maximal compact ...