Local class field theory (full . Fwgues)

Recall : kalg.ch , Xynsmooth , wenn .prog .

cuvre

⇒ FITGÄB = *jet ( Pic , O )⇒

vinpallba.cz F

ahmgtbeff} Picardschaue ofxFix b- Yen ,

ltk LripassagetoTQe-weff.israthor formal)

lat k l - local system onX

⇒ Id ':-. **( KK)",

I : xd → +a)←d-pEE.am .

← Dichµ

Hilbert schaue ofdegree ddivisors on X

If rhett =1

⇒ ④ 1- local system on X"'

free(Key : V any d- module rankt

= , Sd acts Enirüalby onEd = V⑦

.

. . - ⑦ V )

Kay : Lidl descent along

Afd : Hd ) → Pied→

pnoj . bdl for d > 2g-2

Now back to FF- anne :

F- local field,all v-sheares

will be anziehend an Petry , p Fe

Difference :

1) Xcd) do not make sause

⇒ weed Bird,

k should be a local system

on Did (Real :"

ii. (Dijk WE"

Ding = SpdEier )

2) Pic = Z =, ITIÖTCPE ,O) = *)Tsmallv -sheaf parametrisiertisomer of line bdl

=) mead Pic - I Pied

§ DEZ n

r-stach afliwebdls K¥3

still have :DAfd : Dad → PiedD n OLD)

2) For K 1- hee . system on Didof rk 1 ,

share

Id) on Bird←again 1- local system afrk 1

3) Duff. to prore :

Hd) descent along

AJA : Bird→ PadK K

Eko

NEFFE]T ,HYMNE *

"

NeandtoproveThem (Fargues) : Set BEI

"D=HI,ad))

(v-sheafmPafr.ptFord >2

BEEF!0} issimplywanatedie.each U - local system Kon it isEinöd

Baby analogons casa :k a-lg.ch/.field,afchar0

=\ ÄEO} simplywnn . if n > 2

( Pnf : FEAT) : -- EX-Ärztin .EE}chwh:Dihn" k

]u { finitescts}

FELÄEO} )Zwiste-Nagatapcnitg

)

less baby analogons cage :ohne=p >0 , F-= ETH)

⇒ B.EE#d--spuHfHE9.....iIFD)-

AdJndeed , if Spalte, c- Parkt= ) ( R) ⇒ BEIRRT)

d- 1Ga - Hd ) ↳ 2€, .gr?*d.n-iiF- 0 F1

"Sing D= 019µm, ) ( g. =# resfldoft)

with } = spulRTHBXVG.tt)Now

, FECBFEEITDIEO})=FECSpatrfxu-HDBNE.net/apprax

.afEkbijtop.im.

of FE

¥FELSputzen . . .

.si/WEa-raDalgebraEfEftiFee-uendiectEhpafRaynaudonformal

models

= FE( Spectre, - ADB))Zariskü .

KNagata,ifdzz { finite sets}

Note : Spaltd)Gang SpiegelHD!

non - auch

isfarm from being simpbyarmecled

Aim : Appby the somestrategy intheCurse eher E üs arbitrary

Nerd : 1) Analog afpwity(fand >2) G FE (BIjdkoh-FELBff.IT)

2) FELBER;D) - { finite sets }

For 1) :

Them : Y Smooth , arm .

. rigid- amlyticVariety over non - auch . fieldk

,

7- E YCK) proffinite ,dein 4>2

=) FE ( Ylz) = FE (Y )

Sketch of proof : nrloy Y-BdkCover 7- by bulls ÄH , g) A-EZ ,

" 9710

{ xa.BG/lx-H--g}=) STP : FEKBdnl-TFEGBGIBLz.gl)This uses Hartog 's them

Vee LIB! ) ←veckbdkblz.gl/fullyfaithful,dz2+ extension them of Lütkebohnert

fer cohaeut Shearer an DIElz,g)(nerds d > 2)

+ some add . arguments „D"

How to rechne to this statement ?

Problems : BEE"dnot a rigid space ,

does not line overa non-

auch . fidd !

Honorar , Äpfel, _ ..

. xofpai

(BE;)"- spul Aa )

is a rigid Space after b. e. to anon

- ach. field.

For 2) : Show if ZAÄÖ BIFFso}extends to a hin . ätabe 7-

'

a BEE"!

then 7- is trivial null.

Use ( BITKOM t-FBY.IT {o}\. er

¥ von!✓here G = Sd * { (m_ . . xd ) e-E)

"I Fx

;-1}

If 7- extends , f*Z exterds Er

⑥¥54403 = Spaltunganwlytic point

and is this trivial ( by equal dercasa)Subtle point : Not der (to me ) üf

FEL Spalta)0

) = FEISpaltd)

as Ad is weither analytics wardiscrete

Wüte f.z -HEIKO3)dx S withG 25

,7. = frz#

Chaim : The G - action on S is trivial

C- , 7- - B¥;D403 xs)Prf : Fx e (BIE)03)

d

Cz : = StarbenG) actgtriuüalloqon 4=9×3×5 7-

fcx ,

But G-_ ( Gx Ix )5

(s!!"ni ?) ) . Hi?) = Enn)]

=) G acts triviales aus

Upshat : Show that each

7- TEEBGE;D I { 03 extendstoBETTE-

This can bechedaed after base chargeto Spa F , F = ÄLTER))( Spa F → Spa Ä cohom

.

smoothv -corer)

II. F non - auch .

~ anahyze

BE*DIE 03

Conduct exact Sequence0 → ↳OF OG)

d-7 Old) -70

as fahlere : choose Es . . . ,tat

BEI= HOHE ,OMI

with vlt;) n Vltj ) =0#itj.

Set Ä : -- II. % : 041 →0411

and v =

,

! I .-1

V: = {(Rs , . . . Rd ) E Edt ¥.nl?=O} ." /R. .. . . Rd ) = (R; Es, . . . . Rd - td )

= ) von =0, vsmj.in inj .

=) exactness in the middbe

"Ä"on ± → (BE")d→BEI

=) BE"IGO} F E)DIE}

> r

/ -

on this fiu.EE. arens

extendtolr"!)¥ FECBFE

") -4 FELBGITYEO})

[ - egcrürance D

extends