My Works on Class Field Theory
Since 1999
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\[\huge{\color{Brown}{\mathbb{I.\ Non-Abelian\ Class\ Field\ Theory:\ Function\ Fields\,/\,C}}}\]
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\(\bigodot\ \Large{(M;P_1,P_2,\dots,P_N):\ \mathrm{marked\ hyperbolic\ Riemann\ surfaces\ of\ signature}\ (g,N)
\ \mathrm{with}\ D=P_1+P_2+\cdots+P_N}\)
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\(\bigodot\ \Large{{\mathcal T}(M;D):\ \mathrm{Tannakian\ category\ of\ semi-stable\
arithmetic\ parabolic\ bundles\ of\ degree\ 0\ over}\ (M;D)}\)
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\(\huge{\clubsuit\ \color{Blue}{\mathrm{Existence\ and\ Conductor\ Theorem:}}}\ \Large{(\mathrm{Narasimhan-Seshadri:\ Micro;
\ Weng:\ Global})}\)
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\(\qquad\Large{There\ is\ a\ natural\ one-to-one\ correspondence\ w_D\ between}\)
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\[\Large{\boxed{\Big\{{\mathbb S}:\ \mathrm{finitely\ completed\ Tannakian\ subcategory\ of} \
{\mathcal T}(M;D)\Big\}}}\]
\[\Large{ and}\]
\[\Large{ \boxed{
\Big\{\pi:M'\to M:\mathrm{finite\ Galois\ covering\ branched\ at\ most\ at}\
D\Big\}}}\]
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\(\huge{\clubsuit\ \color{Blue}{\mathrm{Reciprocity\ Law:}}}\)
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\(\qquad\Large{The\ correspondence\ induces\
a\ natural\ isomorphism}\)
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\[\boxed{\Large{\mathrm{Aut}^\otimes (\omega_D\big|_{\mathbb S})\,\simeq\, \mathrm{Gal}
\,(w_D({\mathbb S}))}}\]
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\[\huge{\color{Brown}{\mathbb{II.\ Programme\ for\ Non-Abelian\ CFT\ of}\ p\mathbb{-adic\ Number\ Fields}}}\]
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\(\bigodot\ \Large{F:\ \mathrm{a\ p-adic\ number\ field,\ i.e.,\ a\ finite\ extension\ of}\ \mathbb Q_p}\)
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\(\huge{\spadesuit\ \color{Blue}{\mathrm{Non-Abelian\ CFT\ for}\ p\mathrm{-adic\ Number\ Fields:}}}\)
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\[\qquad\Large{\exists\ \boxed{de\ Rham\ representations\ of\ Fontaine\qquad\Leftrightarrow\qquad s.\,stable\
(\phi,N;\omega)-modules\ of\ degree\ 0}\ s.\,t.:}\]
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\(\Large{(1)\ {\mathcal T}_F(\phi,N;\omega)\,:=\{\,\mathrm{s.\,stable\
(\phi,N;\omega)-modules\ of\ degree\ 0\ on}\ F\,\}\ admits\ a\ Tannakian\ structure;}\)
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\(\Large{(2)\ It\ induces\ a\ natural\ one-to-one\ correspondence\ w_F\ between}\)
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\[\Large{\boxed{\Big\{{\mathbb S}:\ \mathrm{finitely\ completed\ Tannakian\ subcategory\ of} \
{\mathcal T}_F(\phi,N;\omega)\Big\}}}\]
\[\Large{ and}\]
\[\Large{ \boxed{
\Big\{L/F:\mathrm{finite\ Galois\ extension\ over}\
F\Big\}}}\]
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\(\Large{(3)\ The\ reciprocity\ map\ induces\
a\ natural\ group\ isomorphism}\)
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\[\boxed{\Large{\mathrm{Aut}^\otimes (\omega_F\big|_{\mathbb S})\,\simeq\, \mathrm{Gal}
\,(w_F({\mathbb S}))}}\]
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