Notation

N = { 1, 2, . . . }

^natural^numbers

N 0 = { 0, 1, 2, . . . }

^natural^numbers^with^zero

K, L

^elds

K

^algebrai^losure^of

K

K ^sep

^separable^losure^of

K

ⁱⁿ

K

G K = Gal(K ^sep /K)

^absolute^Galois^group^of

K

Σ K

^set^of^nite^plaes^of

K

K v

^ompletion^of

K

^at

v

k v

^residue^eld^of

K

^at

v

C v = ˆ K v

^the^ompletion^of^the^algebrai^losure^of

K v

I w

^inertia^group^at

w ∈ Σ L \ { 0 }

^for^a^eld^Extension

L/K I _K ^tame := π ₁ ^tame (G m,K )

^tame^inertia ^group^asⁱⁿ ^Denition^2.2.4

K _v ^nr

^maximal^unramied^extension

A _K , I _K

adeleringof

K

^and^idele^group^of

K

ℓ

^prime^number

χ

^onedimensional

ℓ

^-adi^Galoisrepresentation

χ ℓ

^ylotomi^harater

1, −1

^trivial^and^quadrati^rank^onerepresentation

L ^χ

^Kummer^sheaf^assoiated^to

χ

L

^middle^extension^sheaf^on

A ¹ _K

MC χ

^middle^onvolution^funtor^asⁱⁿ ^Denition^3.1.2

MT L

^middle^tensor^produt^asⁱⁿ ^Denition^3.1.6

T _ℓ (K)

^ategory^of^speial

Q _ℓ

^-sheaves^asⁱⁿ ^Denition^3.1.3

i : U // A ¹ _K

^inlusion^of^an^open^dense^subset^of^the^ane^line

R

^ommutative^ring^with

1 M (m × n, R) m

^times

n

^matries^over^the^ring

R

GL(V )

^group^of^invertibleendomorphismsofthevetorspae

V A ⁿ _K , G a,K , G m,K , GL n (K), SL n (K)

^algebrai^groups

n (K)

^orthogonal^group

SO n (K)

^speial^orthogonal^group

Ω n (K)

^derived^group^of

SO n (K)

Sp _n (K)

^sympleti^group

G 2 (K)

^a^sporadi^group

J n (λ)

^upper^triangular^Jordan^blok^of^length

n

^and^eigenvalue

λ ι x

speialization mapto

x

^(f. ^Setion ^2.2)

H ^m,ℓ

^speial

Q ℓ

^-sheaf^onstrutedⁱⁿ ^Setion^3.3

ρ m,ℓ : G K −→ GL m+1 (Q _ℓ ) ℓ

^-adi^GaloisrepresentationonstrutedinSetion7.2

d

ρ m,ℓ : π ₁ ^´ ^et (A ¹ _Q \ { 0, 1 } ) −→ SO m+1 (F ℓ )

^weight

0

representationof

π ^´ ^et ₁ (A ¹ _Q \ { 0, 1 } )

^(see^Setion^6.4)

oneptsandtheoremsloselyrelatedto Galoisrepresentations.

2.1 Galois Representations

Let

K

^be^a^eld^and^denote^an^algebrai^losure^by

K

^. ^If

L/K

^is^a^Galois^extension^(not^neessarily

nite), we getthe Galois group

Gal(L/K) := Aut K (L)

^. ^The ^group ^is ^equipped ^with ^a^natural

topology,theKrulltopology. Thisistheasebeause

Gal(L/K)

^is^a^topologial^group^as^projetive

limitofthedisreteniteGaloisgroupsoftheniteGaloissubextensions. Thereforetheabsolute

Galoisgroup

G K := Gal(K ^sep /K)

^is^a^pronite^group,^where

K ^sep

^denotes^the^separable^losure

K

ⁱⁿ

K

^. ^We ^will ^regard^all ^ourring ^algebrai^extensions ^of

K

^as ^subelds^of

K

^. ^If

K

^is ^a

perfeteld,thealgebraiandtheseparablelosureoinide.

For axed prime number

ℓ

^, ^we^have^the

ℓ

^-adi^integers

Z ℓ := lim

←− Z/ℓ ⁿ Z

^and ^the ^eld ^of

ℓ

^-adi

rationalnumbers

Q ℓ := Quot(Z ℓ ) = Z ℓ [ ¹ _ℓ ]

^,^whih^is^the^ompletion^of

Q

^with^respet^to^the

ℓ

^-adi

disreteabsolute value. Thisvaluation extends uniquelyto thealgebrailosure

Q _ℓ

^. ^The

ℓ

^-adi

distanegivenbythisvaluationinduesfor

n ∈ N

^a^topology^on

M (n × n, Q _ℓ )

^. ^Beside^the^Zariski

topologyon

GL n (Q ℓ )

^we^get^thereby^another^struture^as^topologial^group,^whih^we^will^useⁱⁿ

thefollowingdenition. Thisyieldsanaturalontinuousationofthistopologial groupon

Q _ℓ ⁿ

equipped with any norm, espeially the

ℓ

^-adi ^one. ^This ^onstrution ^of ^the

ℓ

^-adi ^topology^is

suitableforanynitedimensional

Q _ℓ

^-vetor^spae

V

Forfurtherdetails onthefollowingdenitionssee[Ser68℄.

Denition2.1.1

Foraeld

K

^an

ℓ

^-adi^Galoisrepresentationisahomomorphism

ρ : G K −→ GL(V )

of topologial groups from the absolute Galois group of

K

^to ^the ^general ^linear ^group ^of ^a^nite

dimensional

Q ℓ

^-vetor ^spae

V

^equipped ^with ^the

ℓ

^-adi ^topology. ^The ^dimension ^of

V

^is ^alled

the rank of

ρ

This is the same as a

Q _ℓ

^-vetor ^spae

V

^equipped ^with ^the

ℓ

^-adi ^topology ^and ^a ^ontinuous

G K

-operation. Tworepresentations

ρ, ρ ^′ : G K −→ GL(V )

^areequivalent, if there exists alinear map

φ ∈ GL(V )

^suh^that

φ ⁻¹ ◦ ρ(g) ◦ φ = ρ ^′ (g)

^for^all

g ∈ G K

An important example of an

ℓ

^-adi ^Galois representation of

G Q

^of ^rank ^one ^is ^the ^ylotomi

harater

χ ℓ : G Q −→ GL 1 (Q _ℓ ) = Q _ℓ ^×

^. ^More^preisely, ^it^maps^to

Z ^× _ℓ

ⁱⁿ ^the^following^way: ^F^or

eah

n ∈ N

^,^we^have^a^look^at^the^ylotomi^extension

Q(ζ ℓ ⁿ )

^for^a^primitive

ℓ ⁿ

^-th ^root ^of^unity

ζ ℓ ⁿ

^. ^Then

Gal(Q(ζ ℓ ⁿ )/Q) ∼ = (Z/ℓ ⁿ Z) ^×

^, ^whih ^an^be^hosenⁱⁿ ^suh ^a^way ^that ^it ^ts ^together

with theisomorphismfor smaller

n

^. Independent of thehoies, weget aompatible systemof ontinuousgrouphomomorphismswhihgivesrisetotheharater.

Thisonstrutionanbegeneralizedtoaeld

K

^withharateristiunequalto

ℓ

One of the main properties of the Galois representations

ρ H m,ℓ

^onstruted ⁱⁿ ^Setion ^3.3 ^and

Setion7.2istheexisteneofalongunipotentelementinitsimage.

Denition2.1.2

Wesay that arepresentation

ρ : G −→ GL(V )

^for ^a^group

G

^and^an

n

-dimensional vetorspae

V

^over ^a ^eld

K

^has ^a ^long^unipotent ^element, ^if ^there ^exists ^an ^element

g ∈ G

^suh ^that ^the

Jordan normalform of

ρ(g)

^is

J n (1)

^over

K

^,^where

J n (1)

^denotes ^a ^Jordan^blok^of ^length

n

^to

the eigenvalue

1

For agood introdution to the oneptsof algebrainumber theory, have alook at [Neu99℄. If

K

^is ^a^number^eld,^i.e. ^a^nite ^extension ^of

Q

^, ^then

Σ K

^denotes ^the^set ^of ^nite ^plaes, ^whih

isthe set ofnormalizednon-arhimedeanvaluations of

K

^. ^We^identify

Σ K \ { 0 }

^with^the ^set^of

non-trivialprimeidealsof

O ^K

^. ^For

v ∈ Σ K \ { 0 }

^we^have^two^elds: ^the^nite^eld

k v := O ^K /v

ofharateristi

p v

^and^the^ompletion^via^the^indued^metri

K v := Quot(lim

←− ( O K /v ⁿ ))

^,^as^eah

plaeorrespondstoanormalizeddisretevaluation.

Theadelering

A _K

^of

K

^is^dened ^as

where

A _K,∞

^is ^the ^produt ^of ^the ^ompletions ^of

K

^aording ^to ^the ^valuation ^given ^by ^the

Arhimedean plaes and

Q ′

is the restritedprodut, i.e. almost all entries are in the ringsof

integers

O K v

^. ^The^idele^group

I _K

^is^the^group^of^units

A ^× _K

^of^the^adele ^ring.

For a nite Galois extension

L/K

^and

w ∈ Σ L \ { 0 }

^suh ^that

w | v

^, ^i.e.

w ⊇ v O L

we obtain two anonial subgroups of the Galois group

Gal(L/K)

^, ^the deomposition group

D w := { σ ∈ Gal(L/K) | σw = w }

^and ^a ^normal ^subgroup ^of

D w

^the ^inertia ^group

I w := { σ ∈ D w | σ(x) − x ∈ w ∀ x ∈ O L }

^. ^Fixing ^an^embedding ^of

K

ⁱⁿ

K v

^, ^we ^get ^a^natural

embedding of

G K v

ⁱⁿ

G K

^, ^whih ^orresponds^to ^hoosing^a^nite ^plae

w

ⁱⁿ

K

^extending

v

^and

thereforexing

G K v

^as^a^speideompositiongroup

D w

^. ^Setting

l w := O L /w

^,^we^have^a^short

exatsequeneofnite groups

1 −→ I w −→ D w −→ Gal(l w /k v ) −→ 1.

Foraniteplae

w 6 = 0

^of

L

^there^is^a^unique^nite^plae

v

^of

K

^,^suh^that

w | v

^. ^The^extension

L/K

^is^alled^unramied^at

w

^if

[L : K] = [l w : k v ]

^. ^In^this^ase^we^have

I w = 1

^. ^For^a^nite^plae

v 6 = 0

^of

K

^multiple ^nite ^plaes

w

^of

L

^may^exist,^suh ^that

w | v

^. ^The^eld ^extension

L/K

^is

alledunramiedat

v

^if

[L : K] = [l w : k v ]

^for^eah^of^them(equivalentlyoneofthem,aswehave aGalois extension). Otherwisethe nite plaes are alled ramiedand for eah suh extension

thereisonlyanite numberofthem.

For a general eld

K

^and

L

^an ^algebrai ^extension,

L/K

^is ^unramied ^at ^a non-arhimedean valuation

v

^of

K

^,^if ^for^eah^nite ^eld ^extension

L ^′ /K

^inside

L/K

^and ^eah ^valuation

w ^′

^of

L ^′

extending

v

l ^′ _w ^′ | k v

^is^separable^and

[L ^′ : K] = [l ^′ _w ^′ : k v ]

^,^otherwise

L/K

^is^alled^ramied^at

v

In the number eld ase,

Gal(l w /k v )

^is ^a ^nite ^yli ^group ^generated ^by ^the ^F^robenius. ^If

w ∈ Σ L \ { 0 }

^is ^unramied, ^we^have

D w ∼ = Gal(l w /k v )

^and ^we^an^talk ^of ^a ^F^robenius ^element

in the deomposition group as well. For

v ∈ Σ K \ { 0 }

^unramied ^and

w, w ^′ ∈ Σ L \ { 0 }

^suh

that

w, w ^′ | v

^, ^there ^is ^an ^element

σ ∈ Gal(L/K)

^mapping ^one ^to ^the ^other, ^i.e.

σw = w ^′

Thereforetheorrespondingdeomposition groupsare onjugated, i.e.

σD w σ ⁻¹ = D w ^′

^,^as^well

astheFrobenius elements. Theotherwayaround,foronjugatesofFrobeniuselementswehave

orrespondingplaesof

L

If we generalize to an arbitrary algebraiGalois extension

L/K

^, ^the ^set ^of ^nite ^plaes

Σ L

^is

the projetive limit of the system of nite plaes of the nite subextensions of

L/K

^. ^This ^is

denedviathefollowingonnetionmorphisms: wheneverwehaveasubextension

L/L 1 /L 2 /K

wemap

w 1 ∈ Σ L 1

^to

w 2 ∈ Σ L 2

^,^where

w 2

^is ^the^unique^plae^suh^that

w 1 | w 2

^. ^The^inertia^and

deompositiongroupanbedened asprojetivelimitsin thesameway.

Denition2.1.3

For an

ℓ

^-adi ^Galois representation

ρ

^of ^a ^number ^eld

K

^, ^we ^say ^that

ρ

^is ^unramied ^at

v ∈ Σ K \ { 0 }

^,^if

ρ(I w ) = 1

^for ^any^valuation

w

^of

K ^sep

^extending

v

Let

ρ

^be ^unramied ^at

v ∈ Σ K \ { 0 }

^, ^then ^the ^F^robenius ^element

Frob v,ρ

ⁱⁿ ^the representation

ρ

^at

v

^is ^the ^onjugay ^lass ⁱⁿ

GL(V )

^of ^the ^images ^of ^the ^Frobenius ^element ⁱⁿ

D w

^for ^any

w ∈ Σ K ^sep \ { 0 }

^extending

v

AstheFrobenius element

Frob v,ρ

^is^a^onjugay^lass,^itsharateristipolynomial

f v,ρ (x) := det(1 · x − Frob v,ρ ) ∈ Q ℓ [x]

is well-dened. An

ℓ

^-adi^Galois representationis rational (respetivelyintegral) ifat almost all nite plaes

v

^it^is ^unramied,^i.e.

f v,ρ (x)

^exists, ^and^theharateristipolynomialhasrational (respetivelyintegral)oeients.

We will keep to the language of Rihard Taylor(f. [BLGGT10℄), for systemsof

ℓ

^-adi ^Galois

representations.

Denition2.1.4

a) Let

ℓ, ℓ ^′

^be ^prime ^numbers. ^A ^rational

ℓ

^-adi^Galois representation

ρ

^and^a^rational

ℓ ^′

^-adi

Galois representation

ρ ^′

^of ^the ^same ^number ^eld

K

^are ^ompatible ^at

v ∈ Σ K \ { 0 }

^if

they are both unramied at

v

^and ^the harateristi polynomials

f v,ρ (x) = f v,ρ ^′ (x) ∈ Q[x]

oinide.

b) A weakly ompatible system

(ρ ℓ ) ℓ

^prime ^of ^Galois representations of a number eld

K

onsistsof afamily of rational, semi-simple

ℓ

^-adi ^Galois representations

ρ ℓ

^of

K

^for ^eah

prime number

ℓ

^and^a^nite ^set

S ⊂ Σ K

^,^suh^that ^the ^following ^holds:

1. For

v ∈ Σ K \ S

^and ^prime ^numb^ers

ℓ, ℓ ^′

^unequal ^to ^the harateristi of

k v

^, ^the

representations

ρ ℓ , ρ ℓ ^′

^are^ompatible^at

v

2. For

v ∈ Σ K

^and

ℓ

^equal^to^theharateristi

p v

^of

k v

^,^therepresentation

ρ ℓ

^is^deRham

v

^and^rystallineⁱⁿ

v

^if

v 6∈ S

^(f. ^Denition^2.4.8).

3. Foreahembedding

τ: K // Q

^the

τ

-Hodge-Tate numbersof

ρ ℓ

^areindependentof

ℓ

(f. Denition 2.4.9).

) A weakly ompatible system

(ρ ℓ ) ℓ

^prime ^is ^alled ^irreduible ^if ^there ^is ^a ^set

P

^of ^prime

numbers ofDirihlet density

1

^,^i.e.

s→1+ lim | log(s − 1) | ⁻¹ X

ℓ∈P

ℓ ^−s = 1,

suhthat for all

ℓ ∈ P

^therepresentation

ρ ℓ

^isirreduible.

It is also possibleto extend this denition by hoosing anumber eld

M

^instead^of

Q

^. ^In^this

asethe familyis indexed bythe set ofnite plaes of

M

^andharateristipolynomialsin the ring

M [x]

^are^allowed. ^As^this^is ^not^neessary^for^this^work,^we^omit^this ^and^refer^to^the^more

general[BLGGT10℄,Denition 1.1.

ρ = (ρ ℓ ) ℓ

^prime^is^a^weakly^ompatible^system^and

S ⊂ Σ K

^the^nite^exeptional^set. ^F^or^a^nite

plaes

v ∈ Σ K \ S

^theharateristipolynomials

f v,ρ ℓ (x)

^of^the^F^robenius^elements^oinideⁱⁿ

Q[x]

foralmostall

ℓ

^,^whih^will^be^alled

f v,ρ (x)

^. ^This^will^be^the^key^ingredientⁱⁿ^Setion^7.1^to^dene

L

^-funtion ^for^a^speial^kind^of^weakly^ompatible ^systems^of

ℓ

^-adi^Galoisrepresentations.

Im Dokument Galois representations of orthogonal rigid local systems (Seite 17-23)

N = { 1, 2, . . . }

N 0 = { 0, 1, 2, . . . }

K, L

K

K

K sep

K

K

G K = Gal(K sep /K)

K

Σ K

K

K v

K

v

k v

K

v

C v = ˆ K v

K v

I w

w ∈ Σ L \ { 0 }

L/K I K tame := π 1 tame (G m,K )

K v nr

A K , I K

K

K

ℓ

χ

ℓ

χ ℓ

1, −1

L χ

χ

L

A 1 K

MC χ

MT L

T ℓ (K)

Q ℓ

i : U  // A 1 K

R

1

M (m × n, R) m

n

R

GL(V )

V A n K , G a,K , G m,K , GL n (K), SL n (K)

n (K)

SO n (K)

Ω n (K)

SO n (K)

Sp n (K)

G 2 (K)

J n (λ)

n

λ ι x

x

H m,ℓ

Q ℓ

ρ m,ℓ : G K −→ GL m+1 (Q ℓ ) ℓ

d

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

0

π ´ et 1 (A 1 Q \ { 0, 1 } )

K

K

L/K

Gal(L/K) := Aut K (L)

Gal(L/K)

G K := Gal(K sep /K)

K sep

K

K

K

K

K

ℓ

ℓ

K ^sep

G K = Gal(K ^sep /K)

L/K I _K ^tame := π ₁ ^tame (G m,K )

K _v ^nr

A _K , I _K

L ^χ

A ¹ _K

T _ℓ (K)

Q _ℓ

i : U // A ¹ _K

V A ⁿ _K , G a,K , G m,K , GL n (K), SL n (K)

Sp _n (K)

H ^m,ℓ

ρ m,ℓ : G K −→ GL m+1 (Q _ℓ ) ℓ

ρ m,ℓ : π ₁ ^´ ^et (A ¹ _Q \ { 0, 1 } ) −→ SO m+1 (F ℓ )

π ^´ ^et ₁ (A ¹ _Q \ { 0, 1 } )

G K := Gal(K ^sep /K)

K ^sep

←− Z/ℓ ⁿ Z

Q ℓ := Quot(Z ℓ ) = Z ℓ [ ¹ _ℓ ]

Q _ℓ

M (n × n, Q _ℓ )

Q _ℓ ⁿ

Q _ℓ

Q _ℓ

ρ, ρ ^′ : G K −→ GL(V )

φ ⁻¹ ◦ ρ(g) ◦ φ = ρ ^′ (g)

χ ℓ : G Q −→ GL 1 (Q _ℓ ) = Q _ℓ ^×

Z ^× _ℓ

Q(ζ ℓ ⁿ )

ℓ ⁿ

ζ ℓ ⁿ

Gal(Q(ζ ℓ ⁿ )/Q) ∼ = (Z/ℓ ⁿ Z) ^×

O ^K

k v := O ^K /v

←− ( O K /v ⁿ ))

A _K