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Aug 20,2019 M. Emerton Quadratical reciprocity (Pg) = (Ipt) p, 2 odd (pt) = C-1) It (F) = (lPItG) - I 10=-1318) Artinbdassfieldtheoryfim.ie/4IGahoisaiasaseiia79sp:iEf ! ×F K al u ZE I t O ① finite branched cover Dq - - Stabat E a maxspecoc.IE, E E' E" " t maxspecok p DE POL IE - - kid ④ 1 DE →Gall 't '¥p,) Help Heep) Fnb 't (iso if an ramified) ④PZ → ⇐ ← abelian. ((un ramp pts Frobq Non - zero front. ideals ink coprime to all ramified primes. Thin I conductor M for 4k sit if 2=-1 (m) then K) ' → LEG. E- 9. Q (TIE) choose sign so =L mod 4 I ④ can deduce a. R. from 7hm. (primes talk to each other) HE ③ Kuwait G (local Artin maps: all primes (now days)) Tty {TECYIMT > 03: - - YIN) TIN) ' modular curve of kudu, U' conductor 's TIN) - - Ker (shoe) → sLa( Yard) E Shar) x stellata) L's futons Shea) % # N kN - - kerlslz RT) -354%1) (congruent completion of MN) Take inverse limit Slechta) ⇒ s Kok) → Sholto, 9 Sholto) stop = unit tangent bundle a ¥7 Yeas (regularities etc) ④ Fans (Sholto) 5¥ ' ← space of automorphic {classical modular forms form for Sk over Q Siegal - (Hilbert - or Sholto) Hit - Siegal - Slechta) = Ind sua) k Tl 9 Shotton) " irrep The 0,43, - - - - - Galois side Tbf # ¥) = Grab cont. P G → GHG) (factor through ¥ i t9"am÷ Gal (40) F Dq = strobe > E - E' if lie over Eisunram. same p If P is an ramified for Pci - e in c) ⑤ then char poly of sq is indep of E lying over p (un ramified p) ' → char polyns of 62. - free product p Z → Glace) (Two ways of viewing GQ as profile (ZN is self - dual) gps) P: (Frobp > → Gic) " (assume rpss) 4 ' → op up to conj. e- of. G - - Glen, op is diagonalizabk. Jp can be conjugated to Lie in Tle). really, opt TYW) CG) tame, Gop =D 42¥ * 4 outs by muttbytp Gap knob ul pro - p - gp