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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! – William Apr 19 '12 at 3:01
@William: Is it better now? – Apr 19 '12 at 8:08
Yes, great. Thank you. – William Apr 19 '12 at 17:28
up vote 1 down vote accepted

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.) – fherzig Oct 19 '13 at 15:07
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. – fherzig Oct 19 '13 at 17:12
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). – fherzig Oct 19 '13 at 19:24
Thank you, fherzig. – Oct 21 '13 at 9:04
you are welcome! – fherzig Oct 21 '13 at 18:46

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