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How does the strong approximation theorem for global function fields looks like?

For the number field $\mathbb{Q}$ it can be expressed as the surjection

$$ \mathbb{Q}^\times \times \mathbb{R}^\times \times \prod\limits_{p} \mathbb{Z}_p \twoheadrightarrow \mathbb{A}^\times.$$

I want to understand the image of the adelic norm.

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    Could you please provide some background, or at least a link where one could learn more? I, for one, would appreciate that. Thanks!2012-04-19
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    http://www.jstor.org/stable/1970924?seq=12012-04-19
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    @William: Is it better now?2012-04-19
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    Yes, great. Thank you.2012-04-19

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Let $k$ be a global function field, i.e., a finite extension $\mathbb{F}_{q}(T)$, then $\left\| \cdotp \right\|_{\mathbb{A}} \twoheadrightarrow q^{\mathbb{Z}} \subset (0, \infty)$!

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    this is false! a good example to keep in mind (for a number field $k$; $S$ is the collection of all archimedean places) is that $\mathbb{A}_k^\times / (k^\times \prod_{v \nmid \infty} o_v^\times \prod_{v \mid \infty} k_v^\times)$ is the class group of $k$, by class field theory. (also note that strong approximation for algebraic groups, as in Prasad's article, only applies to *simply connected semisimple* groups, and $\mathbb G_m$ isn't semisimple.)2013-10-19
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    in fact, CFT also shows for totally complex $k$ that $\mathbb{A}_k^\times / \overline{k^\times \prod_{v \mid \infty} k_v^\times}$ is the galois group of the maximal abelian extension of $k$. so density fails very badly.2013-10-19
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    thm X.2.4 in artin-tate shows that $k^\times \prod_{v \in S} k_v^\times$ is never dense in $\mathbb A_k^\times$ (the closure has infinite image).2013-10-19
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    you are welcome!2013-10-21