Lecture 2- Hooks alg valued Gal reps for GLa CQ)- ---

Fix . C- top finite,E - O - Ints a a- = unit

IF - Okay, g

-pt - HFI• h > 2

, r - r, CN) , N > 4• S - fin set of promos 3 Sp , no } u 191N}

①5- max ext of Q Gn sons ) monitored outside SGa

, s- Gal (Qs IQ)

• An iso e : Epa ① ↳ Such, Ep ) E Sho, G)

DeftNotation ITS' "" : - Z( Te, Se ] eelsI.

painsIf A is a commutative ringTim: = Tism"@A

If M is a TINY"

- module,

Kasim) - TITM ) : - im ("

→ End.int)

ITT" acts on such, E) byTe -- [r DSe -- Lr D= eh

-

ice>and GG, G ) is a semisimple KIM

"

- module ( Petersson innerproduct⇒ each Te is normal )

⇒ Tissue, all ⇐ agitate. Em ma i- Er %ce.E.FI#..tn.QTAnd k. my eignstgnqscs.fr

,

→ Epwe hers a Gal rep

ex : Ga , s→ GL , CGI )

al.V lets prime

son. g,

oho- poly f. Grobe) - X'

- X Ge) X + ewg)

e=

ex: G → GlattIsd? Epl)

sit.

H lets primechapel, e Grobe ) = X

'

- Te Xt lse

Goat : Integral version.-g-

Tim (Eichler - Shimura) There is an iso of %"- meds

Much,G) ④ Secret = H'fr, Smt

- 2

The action of a double coset operator Croft , a e- Gla GQ) , onft:Cr, syn"

E ) is

voice, syn"E) Hi Crni ' ro

, Syn"E)

→ Hi Caro- ' n f , Sym"E)

Hi Cr,Sai - 'G)

Ca also see this geometrically . Say 6--2. The

H'Cr,a = H'CYCr) , e) who. YOD - MTH

(uses N 34) [ hold, for other coeffs

a.✓Yoho '

ro) EY Caro're) and Gor] acts bean W 92*00*08?

YG) Yog

Thr HTS, Sy!

-

E) ⇐ H' l? Syn'-225) ② Q

and let'Cr

, Sym"

is a Pinhole generated abeliangroup .

↳ HMM, syn

"-207 = H' Cr

, Sant-2251*0a Pin you

0 -mad and

HTP,syn'- '8) q p

- HMM, San"

pi)ah Tism- gun .

÷ HMM, San" Sir

,

Choosea Heels eighteen go 5. (P, E) = Slr, Ep) and let

Xy : TISCH'Cr. syn"

E;D → Kyser, EPD - GIqTISCH 'Cr

, syn" IsO T.si:9

⇒ ¥The m - her Ig is a maximal ideal of

Tls Cr, h) : - Tls ( HYP, smh-207)and assoc ke Mn ad Ig is a Cbs rep

sit. Alas ,Fm : Gas → Gla CIF)

charpoly Em Aubel = X'-Tyke)XtlIg Get

= X'- TeX these med m

Dd We say m is nor-EBsusfsi.nl Em is abs ironed.

Prop If m is non-Eisenstein , then HTP, Sym"-④

mis a

finite free O - module.

Since TICA,Hm C Endo CHI? Smh-204M) , we get

↳ m non - Eisenstein ⇒ FCM,Hm is O- flat

dfep_ (whom 1--2) Wss that H ' CP,⑦mis p - torsion free

.

Taking colon of the exact sequenceO→050 → IF → O

and localizing at m , we getHofer

,Elm→ H'Cr, Om 5- H' 07am

Sulks , ke showHIM, F) m --O

A double cosetop Gor] acts a HTP, IF) by

HTP,IF) HERE're, IF) d- How

'nM,IF) HOG

, IF)H 11 11 11

IF it IF id→ IF ¥5 IF[r : aroma]

For any lots, Te acts a HYP, IF) by Ithse " " I

so if HTP,IF)m to ⇒ Te - ft l mad m

Se - I met anBo Cheebotereu Em - I@E , E -- med p cyetotonre cher , contradictingthe fact that m is non- Eisenstein

.

D

The Tisch, Hm- 447hm 'G§ -

a

!÷!tns¥so we Love Zip

f- Fei : Gas → GLACIER, Hm @ off )

sit cha-pdreCE.be) - I- Text lse C ITG, Hm

This rep deserved,ko

by aPm : Gee , s → Gla CITY? b)m )

Thy (Grayd) Let A be a local ring with reside Pld Fsuch that the Brauer group of F is denied

,out 1st R be on A-alg .

G. g. E- us . A-

- Fsln, Hm , F - finite , R -- the group algebra 91776)m[Got,d)

Let ACA'- IIA! be a seems local ext with Ai

'

local withwax ideals mi

' and res Rds Fi'

( A'- Tls b)my Ep )

Assume we hero on A- a leg repp'- Tle: : Risk → Mn CA

') - Ti Mn CA !)Ssb

.

1. Tre Got) c-A V reR2. E : : Roa fi

'→ Mn (Fi

') are all abs inset act s.la

topics * 1) E f and mdsp of iThon e

'is conj to the scalar ext ②

+A' of

a rep

p : R→ Much .