Irreduibility of the mod- ℓ Representation

6.4 Irreduibility of the mod-

ℓ

Representation

In this setion we have

K = Q

m ∈ N 0

^even ^and

ℓ

^a ^prime ^number. ^Then ^we ^get ^the

lisse

Q ℓ

^-sheaf

i ^∗ H ^m,ℓ

^of ^rank

m + 1

^for

i : A ¹ _Q \ { 0, 1 } // A ¹ _Q

^and

H ^m,ℓ

^like ⁱⁿ ^Setion^3.3.

By xing an

s ∈ A ¹ _Q \ { 0, 1 }

^this ^orresponds ^to ^a ^ontinuous representation of rank

m + 1 ρ i ^∗ H m,ℓ : π _´ ¹ _et (A ¹ _K \ { 0, 1 } ) −→ GL(( H m,ℓ ) s )

^, ^whih ^fators^through

Z ℓ

^and ^respets^a ^symmetri

bilinearform. This representationanbetensoredwiththedeterminantasin Theorem3.3.2,to

obtainaontinuousrepresentation

ρ i ^∗ H m,ℓ ⊗ det(ρ i ^∗ H m,ℓ ) : π _´ _et ¹ (A ¹ _Q \ { 0, 1 } ) −→ GL(( H m,ℓ ) s ),

fatoring over

SL

^and ^respeting ^a ^symmetri ^bilinear ^form. ^This representation is of weight

m

^by ^Delignes' ^work ^(f. ^Theorem^2.3.4), ^i.e. ^maps

Frob q

^to

q ⁻ ^m ²

^. ^Then ^we ^get ^a ^weight

0

representationbytensoringwiththe

m

2

^th^power^of^the^ylotomi^harater

χ ℓ

^. ^Now^we^will^show

thattheredutionmod

ℓ d

ρ m,ℓ := ρ i ^∗ H _m,ℓ ⊗ det(ρ i ^∗ H _m,ℓ )

⊗ χ _ℓ ^m ² : π ¹ _´ _et (A ¹ _Q \ { 0, 1 } ) −→ SO m+1 (F ℓ )

isirreduible foralmostall

ℓ

^. ^This^will ^lead^usⁱⁿ ^Setion^7.2 ^to^the ^fat^that

ρ m,ℓ

^is ^irreduible

asaweaklyompatiblesystemofGaloisrepresentationsof

Q

^. ^F^or^any

x ∈ A ¹ _Q \ { 0, 1 }

^,^we^get^the

speializationmap

ι x : π ₁ ^´ ^et ( { x } ) ∼ = G Q // π ^´ ^et ₁ (A ¹ _Q \ { 0, 1 } )

^as^dened ⁱⁿ^Setion^2.2.

Theorem6.4.1

Let

x ∈ A ¹ _Q \ { 0, 1 }

^, ^suh ^that ^there ^exist ^odd ^prime ^numbers

p, q 6 = ℓ

^satisfying

ν p (x) < 0

^but

ℓ ∤ ν p (x)

^and

ν q (x − 1) > 0

^but

ℓ ∤ ν q (x − 1)

^. ^Then ^the ^following^holds:

m ∈ N 0

^even^and

m ≥ 12

^then

Ω m+1 (F ℓ ) ⊆ im ( ρ d m,ℓ ◦ ι x )

^for^almost^all^prime^numbers

ℓ

^,^wher^e

d

ρ m,ℓ ◦ ι x : G Q −→ SO m+1 (F ℓ )

^is^the speialization at

x

m = 6

^then^for ^almost^all

ℓ

^,^we^have

im ( ρ d m,ℓ ◦ ι x ) = G 2 (F ℓ )

Proof: Wex

m ∈ N 0

x ∈ A ¹ _Q \ { 0, 1 }

^as^above^and^dene

H ℓ := im ( ρ d m,ℓ ◦ ι x )

Claim 1: Thegroup

H ℓ

^ontains ^a^long^unipotent ^element ^and ^a^unipotent ^element ^of ^dierent

non-trivialJordannormalform.

Let

g = (g 0 , g 1 , g ∞ )

^be^the^images^under

ρ d m,ℓ

^of^standard^generators

γ 0 , γ 1 , γ ∞ ∈ π ^top ₁ (C \ { 0, 1 } )

satisfying

γ 0 γ 1 γ ∞ = 1

^. ^Therepresentation

ρ d m,ℓ

^isrepresentedbyaramiedétaleover

X Q −→ P ¹ _Q

with good redution outside

2

^and

ℓ

^. ^It ^is ^ramied ^over

0, 1, ∞

^with ^loal ^monodromy ^given

g 0 , g 1 , g ∞

respetively. Then by [Be91℄, Theorem1.2.

( ρ d m,ℓ ◦ ι x )(I p )

^is ^generated ^(up ^to

onjugation)by

g ^ν ∞ ^p ^(x)

^and

( ρ d m,ℓ ◦ ι x )(I q )

^by

g ₁ ^ν ^q ^(x−1)

Bytheassumptions

ℓ ∤ ν p (x)

^and

ℓ ∤ ν q (x − 1)

^,^we^obtainⁱⁿ

H ℓ

^elements^of^Jordan^normal^form

J 1 (1) ⊕ J 2 ( − 1) ^m ² for m ≡ 0 mod 4, J 1 (1) ^m−1 ² ⊕ J 2 ( − 1) ⊕ J 1 ( − 1) ^m−1 ² for m ≡ 1 mod 4, J 3 (1) ⊕ J 2 (1) ^m ² ⁻¹ for m ≡ 2 mod 4, J 2 (1) ⊕ J 1 (1) ^m−3 ² ⊕ J 1 ( − 1) ^m+1 ² for m ≡ 3 mod 4,

and

J m+1 (1).

For

ℓ 6 = 2

^the^square^of^the^rst^element^is^unipotent^and^neither^long^unipotent^nor^trivial.

We want to show rst that

H ℓ

^is ^irreduible ^for ^almost ^all ^prime ^numbers

ℓ

^and ^assume ^the

ontrary. Let

L

^be^an^innite^set ^of^prime^numbers^suh ^that

H ℓ

^is^reduible. ^Theontradition willbebyaonsequeneofClaim2-5.

Claim 2: Without loss of generality

H ℓ

^xes ^the ^subspae

V ℓ = h e 1 , . . . , e n i ⊆ F ^m+1 _ℓ

^for

n ∈ { 1, . . . , m }

^,^whih^isindependentof

ℓ

Weanassume

H ℓ

^leaves^a^propernon-trivialsubspaeinvariant for

ℓ ∈ L

^. ^By^onjugating ^the

longunipotentelementintoJordannormalform,thissubspaeisgeneratedbytherst

n

^standard

basevetors

e 1 , . . . , e n

^depending^on

ℓ

^. ^We^an^assume^by^possibly^shrinking

L

^to ^a^smaller,^but

stillinniteset,that

n

^isindependentof

ℓ

Therefore

G Q

^ats^by^therepresentation

ρ d m,ℓ ◦ ι x

^on

V ℓ

^and^the^omposition^with^thedeterminant yieldforeahprimenumber

ℓ

^a^onedimensional representation

θ ℓ : G Q −→ GL(V ℓ ) −→ ^det F ^× _ℓ

Claim 3: There exists a number eld

K

^, ^suh ^that ^for ^an ^innite ^set

L

^the ^one dimensional representation

θ ℓ | G K : G K −→ F ^× _ℓ

^is^unramied^outside^the^set ^of^prime^numbers^of

K

^,^whih ^lie

above

ℓ

^and^moreover

(θ ℓ | I ℓ ) ^ss

^fators^over^the^tame^inertia^subgroup

I _Q ^tame _ℓ

^and^is^given^by

Ψ ⁱ _ℓ−1

where

i ∈ Z

^isindependentof

ℓ

Therestritionof

ρ d m,ℓ ◦ ι x

^to^the^inertia ^group

I ℓ

^and

V ℓ

^fators^byProposition2.5.1a)through

where

dim W j

^is ^the ^dimension ^of ^a ^simple subquotient

W j

^and ^the ^indies

− i ^β _α

^run ^through

M = { d 1 , . . . , d s }

^, ^the^set ^of^indies ^where ^the^rystalline ^ltration^jumps. ^The^set

M

^oinides

withthesetofindies,wheretheHodgeltrationoftheunderlyingmotivejumps[KW03℄. Hene

bytheequationaboveandbypossiblyshrinking

L

^again^to^a^smaller,^but^still^innite^set, ^we^an

assumethat

o

^as^well^as^the

i ^β _α

^isindependentof

ℓ

Takingdeterminants,byCorollary2.5.2, weseethat thereisan

i ∈ Z

independentof

ℓ

^suh^that

det ◦ φ ^ss _ℓ = θ ℓ | I ^tame _Q

ℓ = Ψ ⁱ _ℓ−1 .

By the motivi interpretation of

ρ d m,ℓ ◦ ι x

^and ^by ^de ^Jong's ^and Rapoport-Zink's work (f. [Wor02℄,Proposition28), there exists a nite Galois extension

K/Q

^, ^suh ^that

ρ d m,ℓ ◦ ι x

^is

semistableforall

ℓ

^large^enough,^i.e. ^the^inertia^subgroups

I w ≤ G K

w ∤ ℓ

^at^unipotent. ^Hene,

θ ℓ : G K −→ F ^× _ℓ

^is^unramied^outside^the^set^of^nite^plaes ^of

O ^K

^above

ℓ

Claim4: Thefamily

(θ ℓ : G _Q −→ F ^× _ℓ ) ℓ∈L

^is^the^redution^of^the^produt^of^a^nite^harater^and

somepoweroftheylotomiharater.

Let

m ^′

^be^the^modulus ^of

K

^whose^support ^is^the ^empty^set. ^As^only^nitely ^many^nite^plaes

areramiedintheniteextension

K/Q

^,^the^image^of

U m ^′ |K ≤ I _K

I _Q

underthenormontains asubgroup

U m = U m|Q

^, ^where

m

^is^some^modulus^of

Q

Sine

θ ℓ | G K

^is^unramied^outside^the^set^of^nite^plaes^of

K

^above

ℓ

^,^all^omponents^of

U m

^outside

ℓ

^are^mapped^to

1

^under

θ ℓ ◦ cf ⁻¹

^,^where

cf : G ^ab _Q −→ I _Q /Q ^×

^is^the^lass ^eldisomorphism(this followsfromlasseld theory,asgivenin[Neu99℄,ChapterVandVI).

Thus itfollowsfrom

θ ℓ | I _Q ^tame

ℓ = Ψ ⁱ _ℓ−1

^and

θ ℓ (cf ⁻¹ [a]) ≡ a ^−j _ℓ mod ℓ

^for ^all

a ∈ U m

^. ^Moreover^by

Corollary6.2.3,there existsanumbereld

E

^,^a^nite^harater

ǫ : G _Q −→ E ^×

^and^an ^integer

k

suhthat

θ ℓ = ǫ · χ ^k _ℓ .

Claim5: Thisisaontraditiontotheweight

0

^ondition.

Let

r

^be^a^prime^number^for^whih^therepresentations

ρ d m,ℓ ◦ ι x

^for

ℓ ∈ L

^,^are^unramied^(we^an

assumethat suhaprime numberexistsbydeleting

r

^from

L

^if^neessary). ^By^shrinking

L

^to^a

stillinniteset,wehave

The Frobenius

Frob p

^normalizes ^the ^inertia ^group

I p

^, ^whih ^is ^generated ^by

J m+1 (1)

^, ^and ^by

[GR05℄,Corollaire5.3,XIIIwehave:

F (p) · J m+1 (1) · F(p) ⁻¹ = J m+1 (1) ^p

whihhenehastobeof theform

F(p) =

(f. Remark5.4.4). Therestritionto

V ℓ

^of^the^redution^modulo

ℓ

^has^henedeterminant

λ 1 (p) · λ 2 (p) · . . . · λ n (p) = p ^m ² · p ^m ² ⁻¹ · . . . · p ^m ² ⁻ⁿ⁺¹ = p ^j

for

j ∈ N

^for^all

ℓ

^. ^Sine

L

^is ^innite^this^fores ^for

r = p

^theontradition

p ^j = ζ

^. ^Therefore^for

almostall

ℓ

ρ d m,ℓ ◦ ι x

^isirreduible.

Claim6: For

m ≥ 12

^even^the^group

H ℓ

^ontains^the^group

Ω m+1 (F ℓ )

WeonludebyClaim1andbytheontraditionin Claim5that

H ℓ

^is^an^irreduible ^subgroup

SO m+1 (F ℓ )

^, ^whih ^by ^assumption ^has ^a ^long^unipotent ^element ^and ^a ^unipotent ^element ^of

dierent non-trivialJordananonial form. Welaim thatthis implies that foralmost all

ℓ

^,^we

have

Ω m+1 (F ℓ ) ⊆ H ℓ

^. ^For^this,^we^use^the^lassiation^of^nite^simple^groups.

H ℓ

^is ^ontainedⁱⁿ ^a^maximal ^subgroup

H

^of

Ω m+1 (F ℓ )

^. ^Then

H

^is ^either ^an^element^of

C

^or

an element of

S

^, ^both ^dened ⁱⁿ ^Setion ^6.3. ^Subgroups ⁱⁿ ^the ^olletion

C

^an ^be ^exluded

by the irreduibility and presene of the long unipotent element, sine we remark, that the

tensorprodutof twonon-trivialunipotentelementsis neverlongunipotent,if

ℓ

^is^large^enough

(f.[MV04℄,Theorem 1).

Henewehave

H ∈ S

^and,^by^the^denition^of

S

^,^the^group

H

^ontains^a^simple^normal^subgroup

N

^ating^absolutelyirreduible. If

ℓ

^is^large^enough,^sporadi^andalternatinggroupsannotour,

duetothepreseneoflongunipotentof order

ℓ

^. ^It^is^known^that^the^outerautomorphismgroup ofagroupofLietypeis theompositionof adiagonalautomorphism, agraphautomorphismof

theDynkin diagram and aFrobenius automorphism (f. [Car85℄). Hene if

ℓ

^is ^large^ompared

n

^then^we^an^assume,^that ^the^order^of^the^group^is^prime^to

ℓ

^. ^Therefore

N

^ontains^a^long

unipotent element, as well as another non-trivial unipotent element of dierent Jordan normal

form. Thereforewehavearepresentationof groupsofLietype. Thefollowingasesanour:

ℓ ∤ | N |

^: ⁱⁿ ^this^ase,^again ^the^presene^of^the ^unipotent^element ^implies^that^the

N

^is^not

ofthis typeif

ℓ

^is^large^enough.

b) ross harateristi, i.e.

ℓ | | N |

^but ^is ^not ^the ^dening harateristi: the main result [LS74℄,Theorem,givesalowerboundonthedegreeofaprojetiveirreduiblerepresentation

overaeldofharateristidierentfromthedeningone. Asthisboundgrowsinallases

with

ℓ

^,^if

ℓ

^is^large^enough^this^yield^aontradition.

)

H

^is^a^simple^group^of^Lie ^type^of^dening harateristi

ℓ

^: ^We^onlude^by^Lemma ^6.3.5.

Claim7: For

m = 6

^the^group

H ℓ

^ontains^the^group

G 2 (F ℓ )

Same arguments as in Claim 1-5 apply and we derive irreduibility mod

ℓ

^for ^almost ^all ^prime

numbers

ℓ

^. ^We ^have^the ^following ^lassiation ^of ^maximal ^subgroups ^of

G 2 (F ℓ ⁿ )

^for

ℓ 6 = 2, 3

[Kle88℄,TheoremA:

type onditions

P a

^paraboli

P b

^paraboli

C G 2 (F _ℓn ) (s 2 )

^involution^entralizer

I n = 1

K +

^reduible

K −

^reduible

C G 2 (F ℓn ) (φ α ) ℓ ⁿ = q ^α ₀

α

^prime

PGL 2 (F ℓ ⁿ ) ℓ ≥ 7, ℓ ⁿ ≥ 11

PSL 2 (F 8 ) ℓ ≥ 5

F = F ℓ [ω], ω ³ − 3ω + 1 = 0 PSL 2 (F 13 ) ℓ 6 = 13

F = F ℓ [ √

13]

G 2 (F 2 ) ℓ ⁿ = ℓ ≥ 5

J 1 ℓ ⁿ = 11

Table6.2:Themaximalsubgroupsof

G 2 (F ℓ ⁿ )

^for

ℓ 6 = 2, 3

In the notation of Kleidman, we have

I

^a ^non-split ^group ^extension ^of

F ³ ₂

^with

PSL 3 (F 2 )

^and

K + , K − , P a , P b

^as^is ^Setion ^1.5 ^and ² ^of ^[Kle88℄. ^Using ^the ^same^arguments ^as^above, ^we^rule

outthemaximalsubgroupslistedinTable6.2 andonludethat

H ℓ = G 2 (F ℓ )

7.1 Langlands Correspondene

In this hapter we prove that ertain speializations of the sheaves under onsideration are

potentiallyautomorphi. WeskeththemainideasbehindautomorphyofGaloisrepresentations.

Fordetailsonthefollowingonstrutionssee[Bum97℄,Chapter3.

Let

K

^be^a^number^eld^and

A _K

^as^before^the^adele^ring^of

K

^. ^The^spae^of^uspidal^automorphi

funtionsonsists ofomplexvaluedfuntionson

GL n ( A _K ) = GL n

whihare notinduedfrom paraboli subgroups (i.e. integralsof theform

R

U(K)\U(A K )

f (ug)

u

vanish for all unipotent radials

U

^of ^all ^proper ^paraboli ^subgroups) ^and ^satisfy ^ertain

growth onditions ([JL70℄,Denition10.2). This spae beomes a

GL n ( A _K )

H _n (K)

bialgebra, where

H _n (K)

^denotes ^the ^Heke ^algebra ^onsisting ^of ^funtions ^with ^ompat

support ating via onvolution. Both ations indue one omplex representation and

both are losely related in the following manner. Any irreduible subrepresentation

π = Q

v|∞

π v × Q

v∈Σ v \{0}

π v

^of

GL n ( A _K )

^orresponds^to ^an^irreduible subrepresentation of

H _n (K)

(see[Bum97℄, Proposition3.4.4and[Bum97℄, Proposition3.4.8).

Here

π

^is ^alled ^unramiedⁱⁿ

v ∈ Σ K \ { 0 }

^or

π v

^is ^alled ^unramied^if

π v (GL n ( O ^K v ))

^xes ^a

onedimensional omplexsubspae. Foreah

v 6∈ S

^, ^where

S

^is ^a^ertain^nite ^set^of non-trivial nite plaes of

K

^, ^the ^Satake orrespondene is a bijetion between the lasses of equivalent unramied irreduible representations

π v

^and semi-simple onjugay lasses

A π v

ⁱⁿ

GL n v (C)

([Gro98℄,Proposition 6.4). Therefore

π v

^is ^uniquely ^determined ^by ^the eigenvalues of

A π v

^, ^a

setofomplexnumbers

{ α 1,v , . . . , α n v ,v }

^,^alled^the^Satake

v

-parameters. Therestritedanalyti

where

p v

^is^theharateristioftheresidueeld

k v

^as^before. ^For

v ∈ S

^the^denition^of

L v (π, s)

isnotsostraightforward,butanbeobtainedbytheloalLanglandsorrespondene[HT01℄and

[Hen00℄. Itisknownthattheseanalyti

L

^-funtions^satisfy^favorable^properties^like^meromorphi

(mostlyevenholomorphi)ontinuationtothewholeomplexplaneandfulllfuntionalequations

et..

On the other hand, for any irreduible, weakly ompatible system

ρ = (ρ ℓ ) ℓ

^prime ^of ^Galois

representations

ρ ℓ : G K −→ GL(V ℓ )

^(see ^Denition^2.1.4), ^we ^an ^also ^dene ^the ^restrited

L

^-funtion^of

ρ

^by

where

n

^is^the^rank^for^therepresentationand

f v,ρ ℓ (x) ∈ Q[x] ⊂ Q _ℓ [x]

^theharateristipolynomial oftheFrobenius(see page22)for

v | ℓ

^. ^Stritly^speaking

Frob v,ρ ℓ

^is^not^well-dened ^and^should

beseenassymbolfortheverylastprodutontheright.

Inthease

v ∈ S

^,^we^have^a^preimage^of^the^F^robeniusⁱⁿ^thedeompositiongroupforeahelement of the inertia group

I v := lim

Let

ρ = (ρ ℓ ) ℓ

^prime ^be ^an irreduible, weakly ompatible system of

ℓ

^-adi representations of a number eld

K

^, ^then

ρ

^is ^alled automorphi, if there exists an irreduible omplex subrepresentation

π

^of

GL n ( A _K )

^,^suh^that

L(ρ, s) = L(π, s).

Inthis ase, wesay

L(ρ, s)

^is^automorphi^as ^well.

Ifthere isanite Galoisextension

L | K

^suh^that ^the^restrition

(ρ ℓ | G L ) ℓ

^prime ^isautomorphi,

ρ

isalledpotentiallyautomorphi.

An important onjeture whih is apart ofthe famous Langlandsprogram [Lan79℄, statesthat

anyirreduibleuspidalsystemofGaloisrepresentationsisautomorphi.

Im Dokument Galois representations of orthogonal rigid local systems (Seite 73-80)

Irreduibility of the mod- ℓ Representation

ℓ

K = Q

m ∈ N 0

ℓ

Q ℓ

i ∗ H m,ℓ

m + 1

i : A 1 Q \ { 0, 1 }  // A 1 Q

H m,ℓ

s ∈ A 1 Q \ { 0, 1 }

m + 1 ρ i ∗ H m,ℓ : π ´ 1 et (A 1 K \ { 0, 1 } ) −→ GL(( H m,ℓ ) s )

Z ℓ

ρ i ∗ H m,ℓ ⊗ det(ρ i ∗ H m,ℓ ) : π ´ et 1 (A 1 Q \ { 0, 1 } ) −→ GL(( H m,ℓ ) s ),

SL

m

Frob q

q − m 2

0

m

2

χ ℓ

ℓ d

ρ m,ℓ := ρ i ∗ H m,ℓ ⊗ det(ρ i ∗ H m,ℓ )

⊗ χ ℓ m 2 : π 1 ´ et (A 1 Q \ { 0, 1 } ) −→ SO m+1 (F ℓ )

ℓ

ρ m,ℓ

Q

x ∈ A 1 Q \ { 0, 1 }

ι x : π 1 ´ et ( { x } ) ∼ = G Q  // π ´ et 1 (A 1 Q \ { 0, 1 } )

x ∈ A 1 Q \ { 0, 1 }

p, q 6 = ℓ

ν p (x) < 0

ℓ ∤ ν p (x)

ν q (x − 1) > 0

ℓ ∤ ν q (x − 1)

m ∈ N 0

m ≥ 12

Ω m+1 (F ℓ ) ⊆ im ( ρ d m,ℓ ◦ ι x )

ℓ

d

ρ m,ℓ ◦ ι x : G Q −→ SO m+1 (F ℓ )

x

m = 6

ℓ

im ( ρ d m,ℓ ◦ ι x ) = G 2 (F ℓ )

m ∈ N 0

x ∈ A 1 Q \ { 0, 1 }

H ℓ := im ( ρ d m,ℓ ◦ ι x )

H ℓ

g = (g 0 , g 1 , g ∞ )

ρ d m,ℓ

γ 0 , γ 1 , γ ∞ ∈ π top 1 (C \ { 0, 1 } )

γ 0 γ 1 γ ∞ = 1

ρ d m,ℓ

X Q −→ P 1 Q

2

ℓ

0, 1, ∞

g 0 , g 1 , g ∞

( ρ d m,ℓ ◦ ι x )(I p )

g ν ∞ p (x)

( ρ d m,ℓ ◦ ι x )(I q )

g 1 ν q (x−1)

ℓ ∤ ν p (x)

ℓ ∤ ν q (x − 1)

H ℓ

J 1 (1) ⊕ J 2 ( − 1) m 2 for m ≡ 0 mod 4, J 1 (1) m−1 2 ⊕ J 2 ( − 1) ⊕ J 1 ( − 1) m−1 2 for m ≡ 1 mod 4, J 3 (1) ⊕ J 2 (1) m 2 −1 for m ≡ 2 mod 4, J 2 (1) ⊕ J 1 (1) m−3 2 ⊕ J 1 ( − 1) m+1 2 for m ≡ 3 mod 4,

J m+1 (1).

ℓ 6 = 2

H ℓ

ℓ

L

H ℓ

H ℓ

V ℓ = h e 1 , . . . , e n i ⊆ F m+1 ℓ

n ∈ { 1, . . . , m }

ℓ

H ℓ

ℓ ∈ L

i ^∗ H ^m,ℓ

i : A ¹ _Q \ { 0, 1 } // A ¹ _Q

H ^m,ℓ

s ∈ A ¹ _Q \ { 0, 1 }

m + 1 ρ i ^∗ H m,ℓ : π _´ ¹ _et (A ¹ _K \ { 0, 1 } ) −→ GL(( H m,ℓ ) s )

ρ i ^∗ H m,ℓ ⊗ det(ρ i ^∗ H m,ℓ ) : π _´ _et ¹ (A ¹ _Q \ { 0, 1 } ) −→ GL(( H m,ℓ ) s ),

q ⁻ ^m ²

ρ m,ℓ := ρ i ^∗ H _m,ℓ ⊗ det(ρ i ^∗ H _m,ℓ )

⊗ χ _ℓ ^m ² : π ¹ _´ _et (A ¹ _Q \ { 0, 1 } ) −→ SO m+1 (F ℓ )

x ∈ A ¹ _Q \ { 0, 1 }

ι x : π ₁ ^´ ^et ( { x } ) ∼ = G Q // π ^´ ^et ₁ (A ¹ _Q \ { 0, 1 } )

x ∈ A ¹ _Q \ { 0, 1 }

x ∈ A ¹ _Q \ { 0, 1 }

γ 0 , γ 1 , γ ∞ ∈ π ^top ₁ (C \ { 0, 1 } )

X Q −→ P ¹ _Q

g ^ν ∞ ^p ^(x)

g ₁ ^ν ^q ^(x−1)

J 1 (1) ⊕ J 2 ( − 1) ^m ² for m ≡ 0 mod 4, J 1 (1) ^m−1 ² ⊕ J 2 ( − 1) ⊕ J 1 ( − 1) ^m−1 ² for m ≡ 1 mod 4, J 3 (1) ⊕ J 2 (1) ^m ² ⁻¹ for m ≡ 2 mod 4, J 2 (1) ⊕ J 1 (1) ^m−3 ² ⊕ J 1 ( − 1) ^m+1 ² for m ≡ 3 mod 4,

V ℓ = h e 1 , . . . , e n i ⊆ F ^m+1 _ℓ

θ ℓ : G Q −→ GL(V ℓ ) −→ ^det F ^× _ℓ

θ ℓ | G K : G K −→ F ^× _ℓ

(θ ℓ | I ℓ ) ^ss

I _Q ^tame _ℓ

Ψ ⁱ _ℓ−1

− i ^β _α

i ^β _α

det ◦ φ ^ss _ℓ = θ ℓ | I ^tame _Q

ℓ = Ψ ⁱ _ℓ−1 .

θ ℓ : G K −→ F ^× _ℓ

O ^K

(θ ℓ : G _Q −→ F ^× _ℓ ) ℓ∈L

m ^′

U m ^′ |K ≤ I _K

I _Q

θ ℓ ◦ cf ⁻¹

cf : G ^ab _Q −→ I _Q /Q ^×

θ ℓ | I _Q ^tame

ℓ = Ψ ⁱ _ℓ−1

θ ℓ (cf ⁻¹ [a]) ≡ a ^−j _ℓ mod ℓ

ǫ : G _Q −→ E ^×