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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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Let $k$ be a global field. For any finite set of place $S \neq \emptyset$, we have http://www.jstor.org/stable/1970924?seq=1 $k^\times \times \prod\limits_{v \in S} k_v^\times$ is dense in $\mathbb{A}_k^\times$! Hence for any open subgroup $O$ in $\mathbb{A}_S^\times = \prod\limits_{v \notin S}' k_v^\times$, we have a surjection $$ k^\times \times \prod\limits_{v \in S} k_v^\times \times O \twoheadrightarrow \mathbb{A}_k^\times.$$ For example, take $O = \prod\limits_{v \notin S} \mathfrak{o}_v^\times$, if $S$ contains all archimedean places (possibly none). Let now $k$ 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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