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number field/function field analogy
- \({\mathbf{C}}\hookrightarrow{\mathbf{C}}{\left[\left[ t \right]\right] } \hookrightarrow{\mathbf{C}}{\left(\left( t \right)\right) }\) is \(\mu_{p-1} \hookrightarrow{ {\mathbf{Z}}_{\widehat{p}} }\hookrightarrow{ {\mathbf{Q}}_p }\).
Number field Function field
\({\mathbf{Z}}\) \({ \mathbf{F} }_q[C]\)
\({\mathbf{Q}}\) \({ \mathbf{F} }_q(C)\)
\({ {\mathbf{Z}}_{\widehat{p}} }\) \(\widehat{{\mathcal{O}}_x}\)
\(\dcoset{\operatorname{GL}_n(\widehat{{\mathbf{Z}}})}{\operatorname{GL}_n({\mathbf{A}}_{{\mathbf{Q}}})}{\operatorname{GL}_n({\mathbf{Q}})}\) \(\dcoset{ \displaystyle\prod_{x\in {\left\lvert {X} \right\rvert}}\operatorname{GL}_n(\widehat{{\mathcal{O}}}_x)} {\displaystyle\prod_{x\in {\left\lvert {X} \right\rvert}}' \operatorname{GL}_n(\operatorname{ff}\widehat{{\mathcal{O}}_x})} {\operatorname{GL}_n({ \mathbf{F} }_q(C) )}\)
