# My Works on Class Field Theory

## Since 1999

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