Galois ep esen a ions o o hogonal igid
lo al sys ems
Von de Uni e si ä Bay eu h
zu E langung des G ades eines
Dok o s de Na u wissensha en (D . e . na )
genehmig e Abhandlung
on
Mihael Shul e (geb. Maie )
aus Ka ls uhe
1. Gu ah e : P o . D . Mihael De weile (Uni e si ä Bay eu h)
2. Gu ah e : P o . D . S e an Wewe s (Uni e si ä Ulm)
Tag de Ein eihung: 17. Ap il 2012
Tag des Kolloquiums: 10. July 2012
Con en s
1 In o du ion 5
1.1 Einlei ung ......................................... 10
1.2 Aknowledgemen ..................................... 15
1.3 Eidess a lihe E klä ungen ............................... 16
1.4 No a ion .......................................... 17
2 P elimina y Resul s 19
2.1 Galois Rep esen a ions .................................. 19
2.2 É ale Fundamen al G oup Fun o
π´e
1
......................... 23
2.3 Weil Conje u e ...................................... 31
2.4 C ys alline Rep esen a ions ............................... 33
2.5 Semi-Simplia ion .................................... 39
3 In o du ion o
MCχ
41
3.1 The Middle Con olu ion ................................. 41
3.2 The Nume ology o
MCχ
................................ 44
3.3 Cons u ion o
Hm,ℓ
................................... 47
4 Ho dge S u u es and Middle Con olu ion 51
4.1 Ho dge S u u es ..................................... 51
4.2 Va ia ions o Hodge S u u e .............................. 52
4.3 Ex ensions o Va ia ions o Ho dge S u u e . . . . . . . . . . . . . . . . . . . . . . 54
5 Mo i i Des ip ion o
Hm,ℓ
57
5.1 Se ing ........................................... 57
5.2 The Mo i i In e p e a ion o
MCχ
........................... 59
5.3 Applia ion o
Hm,ℓ
................................... 60
5.4 Analy ia ion o
Hm,ℓ
.................................. 65
6 I eduibili y o
ρm
67
6.1 Li ing I eduibili y ................................... 67
6.2 Se e's Resul s on Cha a e s o
GQ
.......................... 68
6.3 G oups o Lie Type .................................... 69
6.4 I eduibili y o he mod-
ℓ
Rep esen a ion ....................... 73
7 Po en ial Au omo phy o Sp eializa ions 79
7.1 Langlands Co esp ondene ................................ 79
7.2 Mo dula Li ing ...................................... 80
Bibliog aphy 85
3
1 In o du ion
Fo o e 5000 yea s he a ional numb e s ha e b een known and ha e been a o ne s one o
ma hema is sine hen. The onside a ion o p olynomials and he symme ies o hei oo s
by he g oup o au omo phisms was ini ia ed by Galois and ma ks he s a ing p oin o mo de n
numb e heo y. The Galois g oup o a p olynomial eno des muh a - eahing in o ma ion. Bu
wha ab ou he absolu e Galois g oup
GQ
, whih is he g oup o au omo phisms o he algeb ai
losu e o
Q
? I s s u u e is s ill a mys e y and a om b eing well-unde s o od.
In o de o ge pa ial answe s, he e a e a ious die en app oahes. Fi s we ha e a lo ok a
he a o g oups o he absolu e Galois g oup. Esp eially he e is he unsol ed ques ion, i e e y
ni e g oup is o his ype, known as he in e se Galois p oblem (see [MM99℄). A seond p omising
app oah a e Galois ep esen a ions, i.e. on inuous homomo phisms o
GQ
o ma ix g oups
GLn(Qℓ)
. O en i is p ossible o ons u hese in a simila way o all
ℓ
, whih yields unde some
i ums anes weakly ompa ible sys ems. These a e amilies o
ℓ
-adi Galois ep esen a ions o
eah p ime numbe
ℓ
. They ha e he p op e y ha a F ob enius mo phism is mapp ed in suh a way
ha i s ha a e is i p olynomial is a ional and indep enden o
ℓ
o almos all ep esen a ions. I
is p ossible o des ibe many a i hme i ob je s by weakly ompa ible sys ems, o example Galois
ep esen a ions on he Ta e mo dules o ellip i u es, whih a e  uial o he p o o o Fe ma 's
onje u e,  . Theo em o Wiles [Wil95℄.
The
ℓ
-adi Galois ep esen a ions ha e in e es ing onne ions in wo di e ions. Fi s we an
assign o an i eduible, weakly ompa ible sys em o
ℓ
-adi Galois ep esen a ions an analy i
un ion, known as
L
- un ion. I he sys em is au omo phi i s
L
- un ion is equal o an analy i
L
- un ion. The e o e p op e ies like me omo phi on inua ion o he whole omplex plane and
he un ional equa ion an b e ans e ed. The o he way a ound he Langlands o esp ondene
is onje u ed ( . [BBG
+
03℄), whih says ha eah
L
- un ion asso ia ed o a weakly ompa ible
sys em is ob ained by suh an au omo phi ep esen a ion. This is s ill mos ly unp o en. In
[BLGGT10℄ Ba ne -Lamb, Gee, Ge agh y and Taylo een ly p o ided some in e es ing o ols o
p o o s in his di e ion.
The o he onne ion is o geome y. In o de o on ol he Galois g oup, one o en ho oses
amilies o Galois ep esen a ions gi en by lisse é ale shea es (see Co olla y 2.2.12). In suh
amilies i is p ossible o gi e a eahing in o ma ion o he absolu e Galois g oup on he s alks
by op ologial means, he so alled mono d omy. A e y impo an ase a e he mo i i amilies
o Galois ep esen a ions, whih des ib e a ia ions o
ℓ
-adi ohomology g oups o a iable
ℓ
.
5

1. In odu ion
An example is he a ia ion o
H1
´e
o he Legend e amily o ellip i u es
Eλ
gi en by
Y2=X(X−1)(X−λ)
on
A1
Q {0,1}
. Mo e gene al o ellip i u es
E
he o esp onding Galois ep esen a ion on he
 s ohomology
ρℓ:GQ−→ GL(H1
´e (E,Zℓ)) ∼
=GL2(Zℓ)//GL2(Qℓ)
is dual o he Galois ep esen a ion on he Ta e mo dule
Tℓ(E)
. Se e's op en image Theo em
gi es in o ma ion on he size o he image o he ep esen a ion.
Théo ème ( [Se 72℄, (7) ):
I
E
is an el lip i u e wi hou omplex mul iplia ion dened o e a numbe eld
K
and
ρℓ:GK−→ GL2(Qℓ)
he o esponding ep esen a ion, hen o almos al l p ime numbe s
ℓ
we ha e
im (ρℓ) = GL2(Zℓ).
Le
λ
b e a geome i p oin whih is dened o e
Q
suh ha
Eλ
has no omplex mul iplia ion.
Then he sp eializa ion
H1
´e (Eλ,Qℓ)
is a
GQ
-mo dule on whih he g oup
GQ
a s maximally o
almos all
ℓ
.
The seond imp o an p op e y is ha we ge a weakly ompa ible sys em o ep esen a ions
(ρℓ)ℓ
p ime
whih is au omo phi in he sense o he Langlands p og am. This was shown in
2001 by B euil, Con ad, Diamond, Taylo [BCDT01℄ and Wiles [Wil95℄ in he p o o o he
Taniyama-Shimu a-Weil onje u e, whih is now known as mo dula i y heo em, and is ue o
all ellip i u es.
The ambi ion o his wo k is o p o e simila esul s o highe dimensional gene aliza ions o he
Legend e amily. The key obse a ion is ha he mono d omy o he Legend e amily is gi en by
a igid loal sys em whih is he solu ion o a Pia d-Fuhs equa ion, a sp eial hype geome i
die en ial equa ion:
λ(1 −λ) ′′ + (1 −2λ) ′−1
4 = 0.
In his on ex , igid means ha he lo al sys em admi s no de o ma ions, whih is equi alen o
ig(F) = 2
i he sys em is i eduible (see Theo em 3.1.5).
In gene al i is p ossible o des ib e igid lo al sys ems by Ka z' heo y o middle on olu ion
MCχ
(Chap e s 5,6 o [Ka 96℄). Le
χ:π´e
1(Gm,K)−→ Qℓ
×
b e a geome ially non- i ial one
dimensional ep esen a ion and
F
a lisse é ale shea on
A1
K S
, whe e
S⊆A1
K
is ni e. Then
one ob ains a lisse é ale shea
MCχ(F)
on
A1
K S
. I
F
is i eduible and igid, hen
MCχ(F)
is
i eduible and igid as well bu usually
k(MCχ(F)) 6= k(F)
.
6
In summa y we ge he ollowing ons u i e s a emen , whih is known as Ka z algo i hm:
Theo em ([Ka 96℄):
Le
F
be an i eduible igid loal sys em o
Qℓ
-modules on
A1
K S
. The e exis
n∈N0
, loal sys ems
L0,...,Ln
o
Qℓ
-modules o ank one on
A1
K S
and ep esen a ions
χ1,...,χn:π´e
1(Gm,K)−→ Qℓ
×
, suh ha
F=Ln⊗MCχn(... L2⊗MCχ2(L1⊗MCχ1(L0) ) ... ).
Fo
S={0,1}
i
L0
has monod omy uple
(−1,−1,1)
( . Chap e 2.2) and
χ=−1
, he unique
quad a i ha a e , we ge he Legend e amily
MC−1(L0)
. This is he s a ing poin o an inni e
amily
(Hm,ℓ)m∈N0
o lo al sys ems o
Qℓ
-mo duls on
A1
K
wi h
i∗Hm,ℓ =Ln⊗MC−1(... L2⊗MC−1(L1⊗MC−1(L0) ) ... ),
whe e we ha e
i:A1
K {0,1}//A1
K
and
Lj
wi h mono d omy uple
(1,−1,−1)
o
j
o dd
esp e i ely
Lj
wi h
(−1,1,−1)
o
j6= 0
e en. This ons u ion yields he ollowing esul :
Theo em ( . 3.3.1)
Le
ℓ
be a p ime numbe and
K
an algeb aial ly losed eld wi h
ha
(K)∤2ℓ
. Then, o any
m∈N0
he e exis s a ohomologial ly igid
Hm,ℓ ∈Tℓ(K)
o gene i ank
m+1
, a
Qℓ
-shea on
A1
K
whih is lisse on
i:A1
K {0,1}//A1
K
. I
m
is e en, hen
Hm,ℓ
has o hogonal monod omy, and
i
m
is odd,
Hm,ℓ
has symple i monod omy, i.e. he e is an o hogonal espe i ely symple i
pai ing
Hm,ℓ ×Hm,ℓ −→ Qℓ.
The monod omy uple o
i∗Hm,ℓ
has he ol lowing Jo dan no mal o m:
a
0
:
J1(1)m
2⊕J1(−1)m
2+1 o 2 |m,
J2(1)m+1
2 o 2 ∤m,
a
1
:
J2(1)m
2⊕J1(−1) o m≡0 mod 4,
J1(1)m−1
2⊕J2(−1) ⊕J1(−1)m−1
2 o m≡1 mod 4,
J3(1) ⊕J2(1)m
2−1 o m≡2 mod 4,
J2(1) ⊕J1(1)m−3
2⊕J1(−1)m+1
2 o m≡3 mod 4,
a
∞
:
Jm+1(1).
He e
Jn(λ)
deno es he upp e iangula Jo dan blo k o leng h
n
and eigen alue
λ
.
7
1. In odu ion
We wan o no e ha he ase
k(H6,ℓ) = 7
is o sp eial in e es . I s examina ion answe ed a
ques ion o Se e on he exis ene o mo i i Galois g oups o ype
G2
( . [Se 94℄). By his i
was possible o ons u suh mo i es ( . De weile , Ka z and Rei e [DR10℄). The mo i i
des ip ion o igid lo al sys ems by Ka z yields lisse é ale shea es
Hm,ℓ
, whose analy ia ions
ome om a ia ions o Ho dge s u u e.
Theo em ( . 5.4.3)
Le
Hm,ℓ
be as in Theo em 3.3.1 . Then he e exis s a loal sys em o
Z
-modules
Gm
on
A1
C {0,1}
unde lying a pola ized a ia ion o
Z
-Hodge s u u e
(Gm,F•,∇)
on
C {0,1}
pu e o weigh
m
suh ha
(i∗Hm,ℓ)an ∼
=Gm⊗Qℓ.
The indued isomo phism on he s alks
(i∗Hm,ℓ)an
x∼
=Gm⊗Qℓx
o
x∈C {0,1}
is gi en by he
ompa ison isomo phism be ween é ale ohomology and singula ohomology:
1
2(1 −σ)ke Hm
´e (Xx,Qℓ)→Hm
´e (Dx,Qℓ)∼
=1
2(1 −σ)ke (Hm
B(X(C)x,Z)→Hm
B(D(C)x,Z)) ⊗Qℓ.
Mo eo e , he Hodge l a ion o
Gm
has maximal leng h.
In his way we ge , as in he Legend e ase, amilies o weakly ompa ible sys ems o Galois
ep esen a ions
ρm
o
Q
by h ee ans o ma ions, namely by enso ing i wi h i s de e minan ,
sp eializing like in Theo em 6.4.1 and nally semi-simplia ion:
ρm= (ρm,ℓ)ℓ
p ime
:= ((ρi∗Hm,ℓ ⊗de (ρi∗Hm,ℓ )) ◦ιx)ss ℓ
p ime
.
He e
ιx:GK//π´e
1(A1
K {0,1})
deno es he sp eializa ion map o a xed
x∈A1
K {0,1}
,
whih omes om he mo phism
{x}//A1
K {0,1}
( . Se ion 2.2). These ep esen a ions
ρm
ha e he p ope y ha hey a o o e
Zℓ
and o e a speial o hogonal g oup o e en
m
o
a symple i g oup o o dd
m
.
I
m
is e en, by enso ing he ini ial sys em o ep esen a ions wi h he ylo omi ha a e
χℓ
o he p owe
m
2
we ge sys ems o weigh
0
ep esen a ions. The edu ion mo d
ℓ
is dened as
dρm,ℓ := ρi∗Hm,ℓ ⊗de (ρi∗Hm,ℓ )⊗χm
2
ℓ:π1
´e (A1
Q {0,1})−→ SOm+1(Fℓ).
Theo em ( . 6.4.1)
Le
x∈A1
Q {0,1}
, suh ha he e exis odd p ime numbe s
p, q 6=ℓ
sa is ying
νp(x)<0
bu
ℓ∤νp(x)
and
νq(x−1) >0
bu
ℓ∤νq(x−1)
. Then he ol lowing holds:
I
m∈N0
e en and
m≥12
hen
Ωm+1(Fℓ)⊆im ( dρm,ℓ ◦ιx)
o almos al l p ime numbe s
ℓ
,
whe e
dρm,ℓ ◦ιx:GQ−→ SOm+1(Fℓ)
is he speializa ion a
x
.
I
m= 6
hen o almos al l
ℓ
, we ha e
im ( dρm,ℓ ◦ιx) = G2(Fℓ)
8
The e o e i is possible o sp eialize in suh a way, ha we ob ain an i eduible ep esen a ion.
Finally Theo em 7.2.2 and he mo i i des ip ion in Chap e 5gi e he au omo phy o e a
numb e eld.
Theo em ([BLGGT10℄, Thm.5.3.1):
Suppose ha
K
is a CM (o o al ly eal) eld and ha
(ρℓ)ℓ
p ime
is an i eduible, o al ly odd,
essen ial ly onjuga e sel -dual, egula , weakly ompa ible sys em o
ℓ
-adi ep esen a ions o
K
.
Then he e is a ni e, CM (o o al ly eal), Galois ex ension
L|K
suh ha he es i ion o
(ρℓ)ℓ
p ime
o
GL
is au omo phi.
The mo i i des ip ion o igid lo al sys ems by Ka z shows ha hese sys ems o ep esen a ions
a e  ys alline o almos all p ime numb e s
ℓ
. Unde he assump ions o Theo em 6.4.1 using
he wo k o Ba ne -Lamb, Gee, Ge agh y and Taylo , we ob ain he ollowing esul . A weake
s a emen was p o ed in [GMHK10℄.
Theo em ( . 7.2.4)
Fo
m= 6
o
m∈N0
e en,
m≥12
and
K=Q
he i eduible, weakly ompa ible sys em
ρm= (ρm,ℓ)ℓ
p ime
o Galois ep esen a ions is po en ial ly au omo phi.
9
1. In odu ion
1.3 Eidess a lihe E klä ungen
Hie mi e sihe e ih, dass ih die hie o liegende zu P omo ion einge eih e A b ei mi dem
Ti el
 Galois ep esen a ions o o hogonal igid loal sys ems
selbs s ändig e ass , nu die
angegeb enen Quellen und Hil smi el b enu z und wö lih o de inhal lih üb e nommene S ellen
als solhe gekennzeihne hab e. Ih e sihe e an Eides s a , dass diese Angaben wah sind und
dass ih nih s e shwiegen hab e. Mi is b ekann , dass die alshe Abgabe eine Ve sihe ung an
Eides s a mi F eihei ss a e bis zu d ei Jah en o de mi Gelds a e b es a wi d.
Bay eu h, den 16. Ap il 2012
Hie mi e klä e ih, dass ih bishe keine P omo ions e suhe mi diese o de eine ande en
Disse a ion un e nommen hab e. Die A b ei wu de bishe wede im In- no h Ausland in gleihe
o de ähnlihe Fo m eine ande en P ü ungsb ehö de o geleg .
Bay eu h, den 16. Ap il 2012
Hie mi b es ä ige ih, dass ih keine lei Hil e on gewe blihen P omo ionsb e a e n bzw.
- e mi le n ode ähnlihen Diens leis e n in Ansp uh genommen hab e, no h kün ig in Ansp uh
nehmen we de.
Bay eu h, den 16. Ap il 2012
16

1.4. No a ion
1.4 No a ion
N={1,2,...}
na u al numb e s
N0={0,1,2, . . .}
na u al numb e s wi h ze o
K, L
elds
K
algeb ai losu e o
K
Ksep
sepa able losu e o
K
in
K
GK= Gal(Ksep/K)
absolu e Galois g oup o
K
ΣK
se o ni e plaes o
K
K
omple ion o
K
a
k
esidue eld o
K
a
C =ˆ
K
he omple ion o he algeb ai losu e o
K
Iw
ine ia g oup a
w∈ΣL {0}
o a eld Ex ension
L/K
I ame
K:= π ame
1(Gm,K)
ame ine ia g oup as in Deni ion 2.2.4
Kn
maximal un amied ex ension
AK,IK
adele ing o
K
and idele g oup o
K
ℓ
p ime numb e
χ
one dimensional
ℓ
-adi Galois ep esen a ion
χℓ
ylo omi ha a e
1,−1
i ial and quad a i ank one ep esen a ion
Lχ
Kumme shea asso ia ed o
χ
L
middle ex ension shea on
A1
K
MCχ
middle on olu ion un o as in Deni ion 3.1.2
MTL
middle enso p o du as in Deni ion 3.1.6
Tℓ(K)
a ego y o sp eial
Qℓ
-shea es as in Deni ion 3.1.3
i:U//A1
K
inlusion o an op en dense subse o he ane line
R
ommu a i e ing wi h
1
M(m×n, R)m
imes
n
ma ies o e he ing
R
GL(V)
g oup o in e ible endomo phisms o he e o spae
V
An
K,Ga,K,Gm,K,GLn(K),SLn(K)
algeb ai g oups
O
n(K)
o hogonal g oup
SOn(K)
sp eial o hogonal g oup
Ωn(K)
de i ed g oup o
SOn(K)
Spn(K)
symple i g oup
G2(K)
a sp o adi g oup
Jn(λ)
upp e iangula Jo dan blo k o leng h
n
and eigen alue
λ
ιx
sp eializa ion map o
x
( . Se ion 2.2)
Hm,ℓ
sp eial
Qℓ
-shea ons u ed in Se ion 3.3
ρm,ℓ :GK−→ GLm+1(Qℓ)ℓ
-adi Galois ep esen a ion ons u ed in Se ion 7.2
dρm,ℓ :π´e
1(A1
Q {0,1})−→ SOm+1(Fℓ)
weigh
0
ep esen a ion o
π´e
1(A1
Q {0,1})
(see Se ion 6.4)
17
2 P elimina y Resul s
In his hap e , we wan o x he no a ion used in his wo k. In addi ion we gi e an o e iew o
onep s and heo ems losely ela ed o Galois ep esen a ions.
2.1 Galois Rep esen a ions
Le
K
b e a eld and deno e an algeb ai losu e by
K
. I
L/K
is a Galois ex ension (no neessa ily
ni e), we ge he Galois g oup
Gal(L/K) := Au K(L)
. The g oup is equipp ed wi h a na u al
op ology, he K ull op ology. This is he ase b eause
Gal(L/K)
is a opologial g oup as p o je i e
limi o he dis e e ni e Galois g oups o he ni e Galois sub ex ensions. The e o e he absolu e
Galois g oup
GK:= Gal(Ksep/K)
is a p oni e g oup, whe e
Ksep
deno es he sepa able losu e
o
K
in
K
. We will ega d all o u ing algeb ai ex ensions o
K
as subelds o
K
. I
K
is a
p e e eld, he algeb ai and he sepa able losu e oinide.
Fo a xed p ime numb e
ℓ
, we ha e he
ℓ
-adi in ege s
Zℓ:= lim
←− Z/ℓnZ
and he eld o
ℓ
-adi
a ional numb e s
Qℓ:= Quo (Zℓ) = Zℓ[1
ℓ]
, whih is he omple ion o
Q
wi h esp e o he
ℓ
-adi
dis e e absolu e alue. This alua ion ex ends uniquely o he algeb ai losu e
Qℓ
. The
ℓ
-adi
dis ane gi en by his alua ion indues o
n∈N
a op ology on
M(n×n, Qℓ)
. Beside he Za iski
op ology on
GLn(Qℓ)
we ge he eby ano he s u u e as op ologial g oup, whih we will use in
he ollowing deni ion. This yields a na u al on inuous a ion o his op ologial g oup on
Qℓ
n
equipp ed wi h any no m, esp eially he
ℓ
-adi one. This ons u ion o he
ℓ
-adi op ology is
sui able o any ni e dimensional
Qℓ
- e o spae
V
.
Fo u he de ails on he ollowing deni ions see [Se 68℄.
Deni ion 2.1.1
Fo a eld
K
an
ℓ
-adi Galois ep esen a ion
is a homomo phism
ρ:GK−→ GL(V)
o opologial g oups om he absolu e Galois g oup o
K
o he gene al linea g oup o a ni e
dimensional
Qℓ
- e o spae
V
equipped wi h he
ℓ
-adi opology. The dimension o
V
is al led
he ank o
ρ
.
This is he same as a
Qℓ
- e o spae
V
equipp ed wi h he
ℓ
-adi op ology and a on inuous
GK
-op e a ion. Two ep esen a ions
ρ, ρ′:GK−→ GL(V)
a e
equi alen
, i he e exis s a linea
map
φ∈GL(V)
suh ha
φ−1◦ρ(g)◦φ=ρ′(g)
o all
g∈GK
.
19
2. P elimina y Resul s
An imp o an example o an
ℓ
-adi Galois ep esen a ion o
GQ
o ank one is he
ylo omi
ha a e
χℓ:GQ−→ GL1(Qℓ) = Qℓ
×
. Mo e p eisely, i maps o
Z×
ℓ
in he ollowing way: Fo
eah
n∈N
, we ha e a lo ok a he ylo omi ex ension
Q(ζℓn)
o a p imi i e
ℓn
- h o o o uni y
ζℓn
. Then
Gal(Q(ζℓn)/Q)∼
=(Z/ℓnZ)×
, whih an b e hosen in suh a way ha i  s oge he
wi h he isomo phism o smalle
n
. Indep enden o he hoies, we ge a ompa ible sys em o
on inuous g oup homomo phisms whih gi es ise o he ha a e .
This ons u ion an b e gene alized o a eld
K
wi h ha a e is i unequal o
ℓ
.
One o he main p op e ies o he Galois ep esen a ions
ρHm,ℓ
ons u ed in Se ion 3.3 and
Se ion 7.2 is he exis ene o a long unip o en elemen in i s image.
Deni ion 2.1.2
We say ha a ep esen a ion
ρ:G−→ GL(V)
o a g oup
G
and an
n
-dimensional e o spae
V
o e a eld
K
has a
long unip o en elemen
, i he e exis s an elemen
g∈G
suh ha he
Jo dan no mal o m o
ρ(g)
is
Jn(1)
o e
K
, whe e
Jn(1)
deno es a Jo dan blok o leng h
n
o
he eigen alue
1
.
Fo a go o d in odu ion o he onep s o algeb ai numb e heo y, ha e a lo ok a [Neu99℄. I
K
is a numb e eld, i.e. a ni e ex ension o
Q
, hen
ΣK
deno es he
se o ni e plaes
, whih
is he se o no malized non-a himedean alua ions o
K
. We iden i y
ΣK {0}
wi h he se o
non- i ial p ime ideals o
OK
. Fo
∈ΣK {0}
we ha e wo elds: he ni e eld
k := OK/
o ha a e is i
p
and he omple ion ia he indued me i
K := Quo (lim
←−(OK/ n))
, as eah
plae o esp onds o a no malized dis e e alua ion.
The
adele ing
AK
o
K
is dened as
AK:= Y
|∞
K
|{z }
=:AK,∞
×Y′
∈ΣK {0}
K ,
whe e
AK,∞
is he p odu o he omple ions o
K
ao ding o he alua ion gi en by he
A himedean plaes and
Q′
is he es i ed p o du , i.e. almos all en ies a e in he ings o
in ege s
OK
. The
idele g oup
IK
is he g oup o uni s
A×
K
o he adele ing.
Fo a ni e Galois ex ension
L/K
and
w∈ΣL {0}
suh ha
w|
, i.e.
w⊇ OL
,
we ob ain wo anonial subg oups o he Galois g oup
Gal(L/K)
, he
deomposi ion g oup
Dw:= {σ∈Gal(L/K)|σw =w}
and a no mal subg oup o
Dw
he
ine ia g oup
Iw:= {σ∈Dw|σ(x)−x∈w∀x∈ OL}
. Fixing an embedding o
K
in
K
, we ge a na u al
emb edding o
GK
in
GK
, whih o esponds o ho osing a ni e plae
w
in
K
ex ending
and
he e o e xing
GK
as a sp ei deomp osi ion g oup
Dw
. Se ing
lw:= OL/w
, we ha e a sho
exa sequene o ni e g oups
1−→ Iw−→ Dw−→ Gal(lw/k )−→ 1.
20
2.1. Galois Rep esen a ions
Fo a ni e plae
w6= 0
o
L
he e is a unique ni e plae
o
K
, suh ha
w|
. The ex ension
L/K
is alled
un amied
a
w
i
[L:K] = [lw:k ]
. In his ase we ha e
Iw= 1
. Fo a ni e plae
6= 0
o
K
mul iple ni e plaes
w
o
L
may exis , suh ha
w|
. The eld ex ension
L/K
is
alled
un amied
a
i
[L:K] = [lw:k ]
o eah o hem (equi alen ly one o hem, as we ha e
a Galois ex ension). O he wise he ni e plaes a e alled
amied
and o eah suh ex ension
he e is only a ni e numb e o hem.
Fo a gene al eld
K
and
L
an algeb ai ex ension,
L/K
is
un amied
a a non-a himedean
alua ion
o
K
, i o eah ni e eld ex ension
L′/K
inside
L/K
and eah alua ion
w′
o
L′
ex ending
,
l′
w′|k
is sepa able and
[L′:K] = [l′
w′:k ]
, o he wise
L/K
is alled
amied
a
.
In he numb e eld ase,
Gal(lw/k )
is a ni e yli g oup gene a ed by he F ob enius. I
w∈ΣL {0}
is un amied, we ha e
Dw∼
=Gal(lw/k )
and we an alk o a F ob enius elemen
in he deomp osi ion g oup as well. Fo
∈ΣK {0}
un amied and
w, w′∈ΣL {0}
suh
ha
w, w′|
, he e is an elemen
σ∈Gal(L/K)
mapping one o he o he , i.e.
σw =w′
.
The e o e he o esponding deomposi ion g oups a e onjuga ed , i.e.
σDwσ−1=Dw′
, as well
as he F ob enius elemen s. The o he way a ound, o onjuga es o F ob enius elemen s we ha e
o esp onding plaes o
L
.
I we gene alize o an a bi a y algeb ai Galois ex ension
L/K
, he se o ni e plaes
ΣL
is
he p o je i e limi o he sys em o ni e plaes o he ni e sub ex ensions o
L/K
. This is
dened ia he ollowing onne ion mo phisms: whene e we ha e a sub ex ension
L/L1/L2/K
,
we map
w1∈ΣL1
o
w2∈ΣL2
, whe e
w2
is he unique plae suh ha
w1|w2
. The ine ia and
deomp osi ion g oup an b e dened as p o je i e limi s in he same way.
Deni ion 2.1.3
Fo an
ℓ
-adi Galois ep esen a ion
ρ
o a numbe eld
K
, we say ha
ρ
is
un amied a
∈ΣK {0}
, i
ρ(Iw) = 1
o any alua ion
w
o
Ksep
ex ending
.
Le
ρ
be un amied a
∈ΣK {0}
, hen he
F ob enius elemen
F ob ,ρ
in he ep esen a ion
ρ
a
is he onjugay lass in
GL(V)
o he images o he F obenius elemen in
Dw
o any
w∈ΣKsep {0}
ex ending
:
1//Iw//
ρ|Iw

Dw
_

////Gal(lw/k )//

1
GK
ρ

1//GL(V)
.
As he F ob enius elemen
F ob ,ρ
is a onjugay lass, i s ha a e is i p olynomial
,ρ(x) := de (1 ·x−F ob ,ρ)∈Qℓ[x]
21

2. P elimina y Resul s
is well-dened. An
ℓ
-adi Galois ep esen a ion is
a ional
( esp e i ely
in eg al
) i a almos all
ni e plaes
i is un amied, i.e.
,ρ(x)
exis s, and he ha a e is i p olynomial has a ional
( esp e i ely in eg al) o eien s.
We will keep o he language o Riha d Taylo ( . [BLGGT10℄), o sys ems o
ℓ
-adi Galois
ep esen a ions.
Deni ion 2.1.4
a)
Le
ℓ, ℓ′
be p ime numbe s. A a ional
ℓ
-adi Galois ep esen a ion
ρ
and a a ional
ℓ′
-adi
Galois ep esen a ion
ρ′
o he same numbe eld
K
a e
ompa ible a
∈ΣK {0}
i
hey a e bo h un amied a
and he ha a e is i polynomials
,ρ(x) = ,ρ′(x)∈Q[x]
oinide.
b)
A
weakly ompa ible sys em
(ρℓ)ℓ
p ime
o Galois ep esen a ions o a numbe eld
K
onsis s o a amily o a ional, semi-simple
ℓ
-adi Galois ep esen a ions
ρℓ
o
K
o eah
p ime numbe
ℓ
and a ni e se
S⊂ΣK
, suh ha he ol lowing holds:
1.
Fo
∈ΣK S
and p ime numbe s
ℓ, ℓ′
unequal o he ha a e is i o
k
, he
ep esen a ions
ρℓ, ρℓ′
a e ompa ible a
.
2.
Fo
∈ΣK
and
ℓ
equal o he ha a e is i
p
o
k
, he ep esen a ion
ρℓ
is deRham
in
and  ys al line in
i
6∈ S
( . Deni ion 2.4.8 ).
3.
Fo eah embedding
τ:K//Q
he
τ
-Hodge-Ta e numbe s o
ρℓ
a e independen o
ℓ
( . Deni ion 2.4.9 ).
)
A weakly ompa ible sys em
(ρℓ)ℓ
p ime
is al led
i eduible
i he e is a se
P
o p ime
numbe s o Di ihle densi y
1
, i.e.
lim
s→1+ |log(s−1)|−1X
ℓ∈P
ℓ−s= 1,
suh ha o al l
ℓ∈P
he ep esen a ion
ρℓ
is i eduible.
I is also p ossible o ex end his deni ion by ho osing a numb e eld
M
ins ead o
Q
. In his
ase he amily is indexed by he se o ni e plaes o
M
and ha a e is i p olynomials in he
ing
M[x]
a e allowed. As his is no neessa y o his wo k, we omi his and e e o he mo e
gene al [BLGGT10℄, Deni ion 1.1.
I
ρ= (ρℓ)ℓ
p ime
is a weakly ompa ible sys em and
S⊂ΣK
he ni e exep ional se . Fo a ni e
plaes
∈ΣK S
he ha a e is i p olynomials
,ρℓ(x)
o he F ob enius elemen s oinide in
Q[x]
o almos all
ℓ
, whih will b e alled
,ρ(x)
. This will b e he key ing edien in Se ion 7.1 o dene
an
L
- un ion o a sp eial kind o weakly ompa ible sys ems o
ℓ
-adi Galois ep esen a ions.
22
2.2. É ale Fundamen al G oup Fun o
π´e
1
2.2 É ale Fundamen al G oup Fun o
π´e
1
This in odu ion o he é ale undamen al un o
π´e
1
om he a ego y o No e he ian sepa a ed
onne ed shemes o he a ego y o g oups is as in he  s hap e o [FK88℄.
Deni ion 2.2.1
a)
A ing homomo phism
:A−→ B
o loal ommu a i e ings wi h uni is
un amied
, i
(mA)·B=mB
and he indued eld ex ension
A/mA−→ B/mB
is ni e and sepa able.
b)
Le
X,Y
be Noe he ian sepa a ed shemes. The mo phism
:Y −→ X
is
é ale
, i he
ol lowing ondi ions a e sa ised:
1.
is loal ly o ni e ype.
2.
o e e y poin
x∈ X
he mo phism
♯
x:OY, (x)−→ OX,x
is a , un amied and makes
OX,x
a ni ely gene a ed
OY, (x)
-algeb a.
Fo a No e he ian sepa a ed sheme
X
, we all a No e he ian sepa a ed sheme
Y
wi h an é ale
mo phism
X −→ Y
an é ale ex ension o
X
. We deno e he ull sub a ego y o é ale ex ensions o
X
in he a ego y
Sch(X)
o shemes o e
X
by É
(X)
( hen e e y mo phism in É
(X)
is é ale,
 . [FK88℄, Rema k 2.2.).
A mo phism o Noe he ian sepa a ed shemes is a
o e ing
i i is ni e and é ale. Again
he ull sub a ego y
Co (X)
o o e ings o e
X
in É
(X)
has only mo phisms whih a e
o e ings. This is b eause an é ale mo phism is ni e, i and only i i is p op e (see page
282 o [FK88℄ and [Ha 06℄, Co olla y 4.8 (e) ). I we x a geome i p oin
s: spec(Ω) −→ X
(
Ω
sepa ably losed), we ge he asso ia ed un o o geome i p oin s o e
s
Co (X)−→ Se s,Y 7→ Y(s) := HomX(spec(Ω),Y).
A
poin ed o e ing
o
(X, s)
is a pai
(Y, α)
onsis ing o
Y ∈
Ob
(Co (X))
and an
α∈ Y(s)
. These
o m he a ego y
Co (X, s)
oge he wi h he mapping o p oin ed o e ing spaes
: (Y1, α1)−→ (Y2, α2)
whih is an
X
-mo phism
:Y1−→ Y2
sa is ying
◦α1=α2
.
Fo a onne ed
Y ∈
Ob
(Co (X))
, we ha e
|Au X(Y)| ≤ |Y(s)|,
as he e is a mos one mo phism om a p oin ed o e ing sheme o a onne ed p oin ed sheme
(see [FK88℄, (1)). Now we will ha e a lo ok a he ase when he e exis s exa ly one mo phism.
23
2. P elimina y Resul s
Deni ion 2.2.2
Fo a Noe he ian sepa a ed sheme
X
and a geome i poin
s
o
X
a
Galois o e ing
is a onne ed
o e ing sheme
Y
o e
X
i
|Au X(Y)|=|Y(s)|.
This leads o he ull sub a ego y
Gal(X, s)
o Galois o e ings in
Co (X, s)
. Sine b e ween wo
ob je s he e is a mos one mo phism, we ob ain a o al o de ing on he isomo phism lasses.
Fu he mo e we ge ha he ob je s o m an in e se sys em. Fo an
X
-mo phism
:Z −→ Y
b e ween wo Galois o e ings and
σ∈Au X(Z)
, he e is exa ly one
σ′∈Au X(Y)
suh ha
◦σ=σ′◦
. This mapping denes a su je i e g oup homomo phism (see [FK88℄, (4)) and
hene an in e se sys em o g oups.
Deni ion 2.2.3
Fo a Noe he ian sepa a ed sheme
X
and a geome i poin
s
o
X
, we dene he
é ale undamen al
g oup
(a p oni e g oup) as he ol lowing in e se limi o ni e g oups wi h he dis e e opology:
π´e
1(X, s) := lim
←−
(Y,α)∈
Ob
(Gal(X,s))
Au X(Y).
Then
π´e
1
b eomes a o a ian un o om he a ego y o p oin ed shemes o he a ego y o
p oni e g oups by ons u ing sui able mo phisms b e ween he in e se sys ems ou o a mo phism
o shemes (see [FK88℄, A1.3).
The ame undamen al g oup is a a o g oup o he é ale undamen al g oup. This g oup will b e o
imp o ane b eause on inuous ep esen a ions o
π ame
1(X, s)
gi e nie on inuous ep esen a ions
o
π´e
1(X, s)
.
Deni ion 2.2.4
The
ame undamen al g oup
π ame
1(X, s)
o a Noe he ian sepa a ed sheme
X
and a geome i
poin
s
o
X
is he p oje i e limi o al l poin ed Galois o e ings whih a e amely amied i.e. o
eah geome i poin
α: spec(Ω) −→ Y
he a dinali y
|{σ∈Au X(Y)|σ◦α=α}|
is in e ible
in
OY,α
.
I
X
is onne ed and
s′
is ano he geome i p oin o
X
, he é ale undamen al g oups a e
isomo phi:
π´e
1(X, s)∼
=π´e
1(X, s′)
(see [FK88℄, A1.2). In his ase we w i e
π´e
1(X) := π´e
1(X, s)
and iew i as a un o om he a ego y o onne ed shemes o he a ego y o isomo phism
lasses o p oni e g oups. This is alid as well o he ame undamen al g oup. We dene he
ame ine ia g oup
I ame
K:= π ame
1(Gm,K)
.
24
2.2. É ale Fundamen al G oup Fun o
π´e
1
I we x a Noe he ian sepa a ed onne ed sheme and a geome i p oin
s
o
X
oge he wi h a
o e ing
Y
, he e is a na u al on inuous
π´e
1(X, s)
-a ion on
Y(s)
. We will now explain his a ion
in mo e de ail. By [FK88℄, (2) and (3) he e is a Galois o e ing
Z
o
X
domina ing
Y
, i.e. suh
ha he e is an
X
-mo phism
Y −→ Z
. Cho osing an
α∈ Z(s)
, we ge a p oin ed Galois o e ing
(Z, α)
o
(X, s)
and a na u al bije ion
HomX(Z,Y)−→ Y(s), 7→ ◦α.
The e o e he anonial igh a ion o
Au X(Z)
on
HomX(Z,Y)
yields a igh a ion on
Y(s)
.
As we ha e a dis e e g oup, his a ion is on inuous and an b e ex ended o a on inuous
igh a ion o
π´e
1(X)
on
Y(s)
ia he anonial p o je ion
π´e
1(X)////π´e
1(X)/π´e
1(Z, α)
. Fo
a die en hoie o
α
we ob ain a die en a ion, bu his ans o ma ion is he same as a
onjuga ion in
π´e
1(X, s)
. Fu he mo e i is indep enden o he hoie o
Z
, as o wo hoies he e
is a hi d domina ing hem.
P op osi ion 2.2.5
Le
X
be a Noe he ian sepa a ed onne ed sheme. The assignmen
Y 7→ Y(s)
es ablishes an equi alene be ween he a ego y o o e ing spaes o
X
and he a ego y o ni e
on inuous
π´e
1(X)
-se s ([FK88℄, A I.5 ).
In o de o gene alize he onep o shea es he ollowing deni ion was gi en by A in in
[A 62℄, Deni ion 1.1.1.
Deni ion 2.2.6
A
G o hendiek op ology
onsis s o a a ego y
T
and a se
Co T
o amilies
{Ui
φi
−→U}i∈I
o
maps in
T
al led o e ings (whe e in eah o e ing he ange
U
o he maps
φi
is xed) sa is ying
a)
i
φ
is an isomo phism hen
{φ} ∈ Co T
;
b)
i
{Ui→U}i∈I∈Co T
and
{Vij →Ui}j∈Ji∈Co T
o eah
i
hen he amily
{Vij →U}i∈I,j∈J
ob ained by omposi ion is in
Co T
;
)
i
{Ui→U}i∈I∈Co T
and
V→U∈
Mo
(T)
is a bi a y hen
Ui×UV
exis s and
{Ui×UV→V}i∈I∈Co T
.
As É
(X)
ullls all hese p op e ies o a Noe he ian sepa a ed sheme
X
his yields an example
o a G o hendiek op ology.
25
2. P elimina y Resul s
Theo em 2.3.2
Le
X
be a smoo h and p oje i e sheme o e he ni e eld
Fq
and
X:= X ×spec Fqspec Fq
.
a)
The polynomials
Pj( ) = de (1 − ·F ob⋆|Hj(X,Qℓ))∈Qℓ[ ]
ha e a ional in ege oeien s. These a e independen o
ℓ
.
b)
The eigen alues
λ
o
F ob⋆|Hj(X,Qℓ)
, and hus he eip oal oo s o
Pj( )
, al l ha e he
omplex absolu e alue
|λ|=qj
2.
)
The e is a un ional equa ion o
ZX( ) =
2 dim X
Q
j=0
Pj( )(−1)j+1 ,
namely
ZX1
qj =ǫ·qj
2χ(X)· χ(X)·ZX( ).
He e
χ(X) =
2 dim X
P
j=0
(−1)jdim Hj(X,Qℓ)
is he Eule ha a e is i o
X
and
ǫ=(1 2 6 | j,
(−1)N2|j,
whe e
N
is he mul iplii y o he eigen alue
qj
2
o
F ob⋆|Hj(X,Qℓ)
.
Le
X
b e a ni ely gene a ed No e he ian sepa a ed sheme o e he eld
Fq
and
G
a ons u ible
shea o
Qℓ
- e o spaes on
X
. Fo a geome i p oin
α: spec(Fq)−→ X
he esidue eld
κ(α)
is ni e and beause o ha i denes an elemen in
Gal(Fq/κ(α))
, he F ob enius
α:x7→ x|κ(x)|
.
This elemen a s on he s alk
Gα
o
G
and on he s alk
Gα
o he shea
G=G ⊗Fq
.
Deni ion 2.3.3
a)
We al l he shea
G
pun ually pu e o weigh
j
i o al l suh geome i poin s
α
o
X
he
eigen alues o
−1
α:Gα−→ Gα
a e algeb ai numbe s whose omplex onjuga es
λ
ha e
omplex absolu e alue
|λ|=qj
2d(α), d(α) = [κ(α) : Fq]
.
b)
The shea
G
is al led
mixed o weigh less o equal o
j
i
G
has a l a ion
0 = F(0) ⊂ F(1) ⊂...⊂ F( )=G
o whih al l a o shea es
F(ν)/F(ν−1)
a e pun ual ly pu e o weigh less o equal o
j
.
The ollowing s a emen is [Del80℄, Théo ème 3.3.1.
Theo em 2.3.4
Le
:X −→ Y
be a mo phism o ni ely gene a ed shemes o e
Fq
, and le
G
be a mixed shea
o weigh less o equal
j
on
X
. Then he di e image shea es wi h ompa suppo
Rn !G
a e
mixed o weigh less o equal
j+n
.
32

2.4. C ys alline Rep esen a ions
2.4 C ys alline Rep esen a ions
In o de o p esen he onep o  ys alline ep esen a ions, i is neessa y o dene he g aded
ings
BdR, Bc is, Bs
and
BHT
in o dued by Fon aine [Fon02℄ wi h hei na u al
GK
-a ion o a
lo al eld
K
. The  s s ep is o s udy Wi e o s.
2.4.1 Wi e o s
In he a ile [Wi 37℄, whih was published in 1937, Wi gene alized he ons u ion o
Zℓ
ou
o
Fℓ
o a gi en p ime numb e
ℓ
o gene al ommu a i e ings.
Fo a xed p ime numb e
ℓ
and
n∈N0
we dene he
n
- h Wi polynomial
wn:=
n
X
j=0
ℓjXℓn−j
j∈Z[X0,...,Xn].
Deni ion and ema k 2.4.1
Le
ℓ
be a p ime numbe and
A
a ommu a i e ing. Then he ol lowing holds:
a)
Fo eah
n∈N0
, he e exis polynomials
sn, mn∈Z[Y0,...,Yn, Z0,...,Zn]
, suh ha
wn(s0,...,sn) = wn(Y0,...,Yn) + wn(Z0,...,Zn)
and
wn(m0,...,mn) = wn(Y0,...,Yn)·wn(Z0,...,Zn).
b)
The ol lowing on en ion denes a ing s u u e on
AN0
, whih is al led he ing o
Wi
e o s
W(A)
(an)n∈N0+ (bn)n∈N0:= ( sn(a0,...,an, b0,...,bn) )n∈N0,
(an)n∈N0·(bn)n∈N0:= ( mn(a0,...,an, b0,...,bn) )n∈N0.
)
I
A
is a pe e eld o ha a e is i
ℓ
, hen
W(A)
is a omple e dis e e alua ion ing
and i s esidue eld is
A
.
Fo
n∈N
his s u u e an b e es i ed o
An
by p o je ion on he  s
n
e ms, he Wi e o s
o leng h
n
. The ing
W
should b e iewed as he unique o a ian un o om he a ego y o
ings o i sel , o whih he ollowing map is a homomo phism:
W(A)−→ AN0
(an)n∈N07→ (wn(a0,...,an) )n∈N0
.
33
2. P elimina y Resul s
Fo a mo e de ailed e sion o he ollowing app oah, we e e o [FO08℄. Le
K
b e a lo al eld,
ha is a omple e dis e e alua ion eld, whose esidue eld
k
is p e e o ha a e is i
ℓ > 0
.
The mos imp o an ase is as in Se ion 2.1, whe e
K
is a numb e eld,
∈ΣK {0}
a plae o
K
and
K
he omple ion a
. Then he alua ion
on
K
an b e uniquely ex ended o
K
as
onsequene o Che alley's Ex ension Theo em ( . [EP05℄).
K
migh no b e omple e bu by
K asne 's lemma i s omple ion
C := ˆ
K
is algeb aially losed.
Fo he ings o in ege s
OK := x∈K | (x)≥0
and
OC := {x∈C | (x)≥0}
we ge
ompa ings wi h he ollowing isomo phi a o ings o ha a e is i
ℓ
OK /ℓOK ∼
=OC /ℓOC .
Now we sp eialize
A
as he p o je i e limi o
OC /ℓOC
F obℓ
←− OC /ℓOC
F obℓ
←− OC /ℓOC
F obℓ
←− ...
i.e.
A=n(a(n))n∈N0∈(OC /ℓOC )N0(a(n))ℓ=a(n−1) ∀n∈No
and ge a p e e ing o ha a e is i
ℓ
. The ing
A
a ies a anonial alua ion indued by
, whih will no b e disussed in de ail he e, and is a omple e alua ion ing wi h esp e o i .
The a ion o
GK
on
K
an b e ex ended on inuously o
C
and es i s o a lo al a ion on
OC
. The e o e
A
is endowed wi h a na u al s u u e as
GK
-mo dule, whih ommu es wi h he
F ob enius on
A
and gi es nally an a ion o
GK
on
W(A)
.
Fo
a= (a(n))n∈N0∈A
we dene
ea:= lim
n→∞(g
a(n)ℓn
)∈ OC
, whe e
g
a(n)∈ OC
is some li o
a(n)∈ OC /ℓOC
. I is easy o hek ha he limi exis s and ha i is independen o he
hoies. This yields a map
θ:W(A)−→ OC
(an)n∈N07→
∞
P
n=0
ℓn an
whih is an epimo phism o
GK
-mo dules. This esul is ob ained by es i ing
θ
o he  s
n
omp onen s o
W(A)
and hen using he p o je i e limi p o ess.
Fo
n∈N0
, we suessi ely hoose
a(n)∈ OC
, suh ha we ge a ompa ible sys em o
ℓn+1
- o o s o
ℓ
, i.e.
a(n)
is a ze o o
Xℓn+1 −ℓ
and
a(n)= (a(n+1))ℓ
. By p o je ion
his denes a se ies o non-ze o elemen s
ℓ(n)∈ OC /ℓOC
and he e o e an elemen
ℓ= (ℓ(n))n∈N0∈A
o whih
eℓ=ℓ
. Then he ke nel o
θ
is a p inipal ideal gene a ed by
ξ:= (−ℓ, 1,0,...)∈W(A)
, whih denes a
ξ
-adi op ology on
W(A)
and
W(A)[(0,1,0,...)−1]
.
34
2.4. C ys alline Rep esen a ions
2.4.2 The
GK
-mo dule
BdR
:
The omple ion o
W(A)[(0,1,0,...)−1]
in he
ξ
-adi op ology is he dis e e alua ion ing
B+
dR
wi h maximal ideal
(ξ)
and esidue eld
B+
dR/(ξ)∼
=C
.
Deni ion 2.4.2
The eld
B+
dR
is dened as he eld o a ions o
B+
dR
:
BdR := Quo (B+
dR) = Quo lim
←−
n
W(A)[(0,1,0,...)−1]
ξn!.
I has a na u al de easing l a ion
FilmBdR =ξmB+
dR
o
m∈Z
.
2.4.3 The
GK
-mo dule
Bc is
:
We dene
Ac is
o b e he
ℓ
-adi omple ion o he di ided p owe en elop e o
W(A)
wi h
esp e o
(ξ)
, i.e.:
Ac is := (∞
X
n=0
wn
ξn
n!wn∈W(A), wn−→ 0 o n−→ ∞)⊂B+
dR.
This is he same as aking he p o je i e limi
lim
←−
n
A0
c is/(0,1,0,...)nA0
c is ∼
=Ac is
whe e
A0
c is := (N
X
n=0
wn
ξn
n!N∈N0, wn∈W(A))⊂W(A)[(0,1,0,...)−1].
In his way we ob ain he ollowing sub ing
B+
c is := Ac is[(0,1,0,...)−1]⊂B+
dR
. Again we ho ose
suessi ely
a(n)∈ OC
, suh ha we ge a non- i ial ompa ible sys em o
ℓn
- h o o s o uni y,
i.e.
a(n)
is a ze o o
Xℓn−1
,
a(1) 6= 1
and
a(n)= (a(n+1))ℓ
, whih yields an elemen
ε∈A
.
As
(ε−1,0,...)∈
Ke
(θ) = (ξ)
we ha e
log((ε, 0,...)) :=
∞
X
n=1
(−1)n+1 (ε−1,0,...)n
n∈B+
dR.
Deni ion 2.4.3
By loalizing and aking he subspae g ading by
BdR
, we dene he g aded ing
Bc is := B+
c is[log((ε, 0,...))−1] = Ac is[log((ε, 0,...))−1]⊂BdR.
35
2. P elimina y Resul s
2.4.4 The
GK
-mo dule
Bs
:
We se
log((−ℓ, 0,...)) := −
∞
P
n=1
ξn
n·(0,1,0,...)n∈B+
dR
( o a omple e app oah o he loga i hm see
[FO08℄, Se ion 6.1.3). This elemen is ansenden al o e
Quo (Bc is)
.
Deni ion 2.4.4
The ing
Bs
is dened as he
Bc is
-subalgeb a o
BdR
gene a ed by
:= log((−ℓ, 0,...))
:
Bs := Bc is[ ] = Bc is[log((−ℓ, 0,...))] = Bc is "−
∞
X
n=1
ξn
n·(0,1,0,...)n#.
Rema k 2.4.5
We ha e
Bc is ⊆Bs ⊆BdR
, whih gi es ha
Bc is
and
Bs
a e domains. Eah ing is s able unde
he a ion o
GK
on
BdR
, whih is ob ained by onside ing he p oje i e limi o he a ions on
W(A)[(0,1,0,...)−1]
ξn
.
As
A
is o ha a e is i
ℓ
we ha e
Fℓ//A
, and by he un o iali y o
W
we ge ha
W(Fℓ) = Zℓ//W(A)
, whih leads o
Qℓ⊆Bc is ⊆Bs ⊆BdR.
2.4.5 The
GK
-mo dule
BHT
:
Now we hoose an emb edding
τ:K//Q
, whih is he same as a on inuous inlusion
K //Qℓ=K
. This yields a
Qℓ
- e o spae s u u e on
K
and he e o e on
C
by
on inuous ex ension.
Q_

oO








//Qℓ
_

nN
~~|
|
|
|
|
|
|
|
K o
τ

>
>
>
>
>
>
>
>
//K p
A
A
A
A
Q//Qℓ
The elemen
has been hosen in suh a way ha o an elemen
g∈GK
, we ha e
g· =χℓ(g)
. He e
χℓ:GK −→ Q×
ℓ⊂Bc is
is he ylo omi ha a e , whih is dened as in
2.1 by he op e a ion on he
ℓn
- h o o s o uni y. The elemen
gene a es he maximal ideal o
B+
dR
and he e o e he g ading on
BdR
. As a eld
BdR
is isomo phi o
C (( ))
( he isomo phism
dep ends on he hoie o
ε
).
36
2.4. C ys alline Rep esen a ions
Fo a
j∈Z
he
j
- h Ta e Twis
C (j)
o
C
is
C
iewed as
GK
-mo dule wis ed by he
j
- h
p owe o he ylo omi ha a e , so ha
g·c=χj
ℓ(g)g(c)
o
g∈GK
and
c∈C
.
This is imp o an b eause i is isomo phi as
GK
-mo dule o he
i
- h g aded omp onen o
BdR
:
g jBdR = FiljBdR/Filj+1BdR =ξjB+
dR/ξj+1B+
dR = jB+
dR/ j+1B+
dR ∼
=C (j).
Deni ion 2.4.6
The
Ho dge-Ta e ing
BHT
is dened as he di e sum o al l Ta e wis s o
C
, whih is
BHT =C [ , −1] = M
j∈Z
g jBdR
and he
GK
-a ion o
BdR
es i s in he ol lowing way
g·X
j∈Z
cj j=X
j∈Z
χℓ(g)jg(cj) j
o
g∈GK
and
cj∈C
unequal o
0
only o a ni e numbe o
j∈Z
.
The esidue eld
k
lies inside
OK /ℓOK
and is xed by he
GK
-a ion. This denes he eld
K0:= Quo (W(k ))
, whih is he xed eld o
Bc is
and
Bs
wi h esp e o he
GK
-a ion
K0=BGK
c is =BGK
s ⊆K =BGK
dR =BGK
HT .
Then
K
is a o ally amied ex ension o
K0
and b o h elds oinide i
K
is un amied in
.
Deni ion 2.4.7
Fo an
ℓ
-adi Galois ep esen a ion
ρ:GK −→ GL(V)
o
K
, like in Deni ion 2.1.1 , we dene
Qℓ
- e o spaes, he l e ed
Dieudonné mo dules
Dc is(V) := (Bc is ⊗V)GK , Ds (V) := (Bs ⊗V)GK ,
DdR(V) := (BdR ⊗V)GK
and
DHT(V) := (BHT ⊗V)GK
as in a ian s o he enso p odu s unde he a ion o he absolu e Galois g oup.
The  s wo a e ee
K0⊗Qℓ
-mo dules and he seond ones a e ee
K ⊗Qℓ
-mo dules.
37

2. P elimina y Resul s
Then we ha e he ollowing na u al inequali ies
ank
K0⊗QℓDc is(V)≤
ank
K0⊗QℓDs (V)≤
ank
K ⊗QℓDdR(V)
≤
ank
K ⊗QℓDHT(V)≤dimQℓV.
Deni ion 2.4.8
Le
ρ
be an
ℓ
-adi Galois ep esen a ion o
K
on
V
,
∈ΣK {0}
suh ha
ℓ=p =
ha
(k )
.
Then we ge an
ℓ
-adi Galois ep esen a ion
ρ|GK :GK −→ GL(V)
by xing an embedding
K//K
.
ρ
is al led











Ho dge-Ta e
a
, i
ank
K ⊗QℓDHT(V) = dimQℓV
,
deRham
a
, i
ank
K ⊗QℓDdR(V) = dimQℓV
,
semi-s able
a
, i
ank
K0⊗QℓDs (V) = dimQℓV
,
 ys alline
a
, i
ank
K0⊗QℓDc is(V) = dimQℓV
,
and we ha e:
ρ
 ys al line a
⇒ρ
semi-s able a
⇒ρ
deRham a
⇒ρ
Hodge-Ta e a
.
The g ading on
DHT(V)
is gi en by he deg ee in
, so ha eah g aded piee
g −jDHT(V) = (C (−j)⊗V)GK
an b e ha a e ized by he ylo omi ha a e a ing on
i o he
j
- h p owe . The dimensions o hese spaes play an imp o an ole in he lassia ion
o ep esen a ions.
Deni ion 2.4.9
Le
ρ= (ρℓ)ℓ
p ime
be a sys em o Galois ep esen a ions o a numbe eld
K
, whe e
ρℓ:GK−→ GL(Vℓ)
whe e
Vℓ
is a
Qℓ
- e o spae. Fo
∈ΣK {0}
,
ℓ=p =
ha
(k )
and
a xed embedding
K//K
a
GK
-a ion on
Vℓ
and as be o e by
τ:K//Q
a
GK
-a ion
on
C
is gi en. The
τ
-Ho dge-Ta e numb e s
h ,j(ρ)∈N0
o an in ege
j
a e dened as
h ,j(ρ) = dimQℓ(C (−j)⊗Vℓ)GK .
38
2.5. Semi-Simplia ion
2.5 Semi-Simplia ion
The semi-simplia ion o a ep esen a ion
ρ:G−→ GL(V)
, whe e
V
is a ni e-dimensional
L
- e o spae, is done by iewing
V
as an
L[G]
-mo dule. Then he e is a ni e, s i ly de easing
hain o submo dules
V=V0⊃V1⊃...⊃Vo={0}
suh ha
Vj/Vj+1
is a simple
L[G]
-mo dule. The a o s a e alled Jo dan-Hölde a o s and a e
uniquely dened up o p e mu a ion. This denes a unique semi-simple
L[G]
-mo dule
o−1
L
j=0
Vj/Vj+1
and he e o e a semi-simple ep esen a ion
ρss :G−→
o−1
M
j=0
Vj/Vj+1.
The ollowing pa is ou o [Wo 02℄, 2.1. S a emen o he p op osi ion . Le
K
b e a numb e
eld,
∈ΣK {0}
a ni e plae o
K
and
p=
ha
(k )
. Fo
m∈N
op ime o
p
we ge a
na u al emb edding o he
m
- h o o s o uni y in he maximal un amied ex ension
Kn
o
K
in
k ∼
=Fp
:
µm={ζ∈Kn
|ζm= 1}//
s
&&
M
M
M
M
M
M
M
M
M
M
M
M
MOK

⊂K
k ∼
=Fp
.
Le
π
b e a uni o mize o
Kn
, hen
Kn
(π1
m)
is a o ally amied ex ension o
Kn
o
deg ee
m
and we ge a p o je i e sys em o maps
e
Ψm: Gal(Kn
(π1
m)/Kn
)−→ µm
wi h
(e
Ψm(σ))(π1
m) = σ(π1
m)
o
σ∈Gal(Kn
(π1
m)/Kn
)
. I s p o je i e limi is
Ψ = lim
←−
p∤me
Ψm:I ame
K = lim
←−
p∤m
Gal(Kn
(π1
m)/Kn
)−→ lim
←−
p∤m
µm
As in [Wo 02℄ on page 4 we dene i s na u al p o je ions
Ψq−1:I ame
K −→ µq−1⊆Kn
o a
p
-p owe
q
, whih dene
Ψ
uniquely.
39
2. P elimina y Resul s
Le
{m1,...,m }
b e he se o indies whe e he l a ion o
Dc is(V)
jumps, i.e.
g mDc is(V)6= 0
,
and wi h
dj:=
ank
K0⊗Qℓg mjDc is(V)∈N
he mul iplii y o
mj
, i.e. he ank o he asso ia ed
quo ien .
P op osi ion 2.5.1 ( [Wo 02℄, P op.3 )
Assume ha o
∈ΣK {0}
,
K
is absolu ely un amied, i.e.
K /Q
is un amied, and le
w
be
he unique ex ension o
o
K
. I he
ℓ
-adi Galois ep esen a ion
ρ:GK −→ GL(V)
a o s
h ough
Zℓ
, is  ys al line and i he leng h o he l a ion on
Dc is(V)
is less han
ℓ
, hen he
ol lowing holds:
a)
The semi-simplia ion o he mod-
ℓ
edued
Iw
-module
V
, ia
ρ|Iw:Iw−→ GL(V)
, is
wel l-dened and he a ion o
Iw
a o s h ough he ame quo ien
I ame
K
.
b)
Fo a simple subquo ien
W
o he
Iw
-module
V
o dimension
d
one has
EndFℓ(W)∼
=Fℓd
. Fixing an isomo phism gi es
W
he s u u e o a one dimensional
Fℓd
- e o spae on whih
I ame
K
a s ia mul iplia ion wi h
Ψi0+...+id−1ℓd−1
ℓd−1
, whe e he
indies
−ij
un h ough
{m1,...,ms}
suh ha eah omponen o
(m1,...,ms)
(oun ed
wi h mul iplii ies) appea s as some index
−ij
o some subquo ien .
Co olla y 2.5.2 ( [Wo 02℄, Co .4 )
The au omo phisms dened by
g∈I ame
K
, whe e he ame undamen al g oup is iewed as a
subg oup (no unique) o he ine ia g oup
Iw
, sa is y
de Fℓρ(g) = Ψℓ−1(g)−S,
whe e
S:=
s
P
k=1
dkmk
.
40
3 In o du ion o
MCχ
In Se ion 2.2, we saw ha i is p ossible o ons u Galois ep esen a ions by ons u ing lisse
Qℓ
-shea es. In o de o do so, we use he middle on olu ion as a geome i op e a ion. We will
ollow Chap e 2 in [Ka 96℄.
3.1 The Middle Con olu ion
Le
K
b e a eld and
ℓ
a p ime numb e unequal o he ha a e is i o
K
. We x an algeb ai
g oup
A
o e
K
wi h mul iplia ion map
µ:A×A−→ A
.
We deno e by
Db
c(A,Qℓ)
he bounded de i ed a ego y o ons u ible
Qℓ
-shea es on
A
,
whih is ons u ed by aking he a ego y o b ounded hain omplexes and lo alizing he
quasi-isomo phisms ( o mo e de ails see [Ka 96℄, Se ion 2.2).
Fo wo ob je s
F,G ∈ Db
c(A,Qℓ)
, we ha e he ex e io enso p o du
F⊠G:=
p
∗
1F ⊗
p
∗
2G ∈ Db
c(A ×A,Qℓ),
and he shi by
m∈Z
o he le
F[m]
. As usual he ons u ible
Qℓ
-shea es a e emb edded as
deg ee
0
ob je s.
Deni ion 3.1.1
The
!
-on olu ion
o
F,G
is dened as he igh de i a i e o he di e image wi h ompa suppo
by he mul iplia ion map
µ
F ∗!G:= Rµ!(F⊠G)∈Db
c(A,Qℓ)
and hei
∗
-on olu ion
as he igh de i a i e o he di e image by he mul iplia ion map
µ
F ∗∗G:= Rµ∗(F⊠G)∈Db
c(A,Qℓ)
o he ex e io enso p odu .
Fo eah ons u ible shea
X
on
A
, he supp o
supp(X)
is he losu e o he se
{a∈A| Xa6= 0}
and he e o e a a ie y o some dimension. An elemen
F ∈ Db
c(A,Qℓ)
is alled a
pe e se shea
,
41
3. In odu ion o
MCχ
ohomologially igid, and dene indu i ely
Hm+1,ℓ := (MTi∗L(1,−1)(MC−1(Hm,ℓ)) 2 |m,
MTi∗L(−1,1)(MC−1(Hm,ℓ)) 2 ∤m∈Tℓ(K).
By Theo em 3.1.7, we ha e ha
Hm,ℓ
is ohomologially igid. Tha
Hm,ℓ
esp e s an o hogonal
esp e i ely symple i o m is a onsequene o Poina é duali y (see [DR99℄, Co olla y 5.10).
The es o he p o o is an indu ion on
m
wi h he help o P op osi ion 3.2.1. Fo
m= 0
he
mono d omy uple o
i∗Hm,ℓ
is
(J1(−1),J1(−1),J1(1)) ∈GL1(Qℓ)3
, whih is o he gi en o m.
So we s a wi h
Hm,ℓ
whose loal mono d omy is o he p edi ed ype. The e o e we an de e mine
he ank o
Hm+1,ℓ
:
k(Hm+1,ℓ) = k(MC−1(Hm,ℓ)) = 2(m+ 1) −e1(0,−1,Hm,ℓ)−e1(1,1,Hm,ℓ)−e1(∞,−1,Hm,ℓ)
=










2(m+ 1) −m
2−m
2−0m≡0 mod 4
2(m+ 1) −m+1
2−m−1
2−0m≡1 mod 4
2(m+ 1) −m
2−m
2−0m≡2 mod 4
2(m+ 1) −m+1
2−m−1
2−0m≡3 mod 4











=m+ 2
Fo he alula ion o he loal mono d omy a
0
, we ha e wo ases. I
m
is e en, he lo al
mono d omy o
Hm,ℓ
is o he o m
J1(1)m
2⊕J1(−1)m
2+1
:
J1(1)m
2⊕J1(−1)m
2+1 MC−1
0⊕J2(1)m
2+1 MTi∗L(1,−1)
J2(1)(m+1)+1
2
I
m
is o dd, hen
J2(1)m+1
2MC−1
J1(−1)m+1
2⊕J1(1)m+3
2
MTi∗L(−1,1)
J1(1)m+1
2⊕J1(−1)m+1
2+1.
A
1
we s a wi h he ase
m≡0 mod 4
and he e o e we ge :
J2(1)m
2⊕J1(−1) MC−1
J1(−1)m
2⊕J2(1)⊕J1(1)m
2
MTi∗L(1,−1)
J1(1)(m+1)−1
2⊕J2(−1)⊕J1(−1)(m+1)−1
2
o
m≡1 mod 4
:
J1(1)m−1
2⊕J2(−1) ⊕J1(−1)m−1
2MC−1
0⊕J3(1) ⊕J2(1)m−1
2
MTi∗L(−1,1)
J3(1) ⊕J2(1)m+1
2−1
o
m≡2 mod 4
:
J3(1)⊕J2(1)m
2−1MC−1
J2(−1)⊕J1(−1)m
2−1⊕J1(1)m
2+1 MTi∗L(1,−1)
J2(1)⊕J1(1)(m+1)−3
2⊕J1(−1)(m+1)+1
2
o
m≡3 mod 4
:
J2(1) ⊕J1(1)m−3
2⊕J1(−1)m+1
2MC−1
J1(−1) ⊕0⊕J2(1)m+1
2
MTi∗L(−1,1)
J1(−1) ⊕J2(1)m+1
2.
48

3.3. Cons u ion o
Hm,ℓ
The Jo dan no mal o m o he lo al mono d omy o
i∗Hm,ℓ
a
∞
is o he o m
Jm+1(1)
. Then we
ge by he p e ious ema k ha he lo al mono d omy o
MC−1(Hm,ℓ)
is
Jm+2(1)
. Fo
L(−1,1)
and
L(1,−1)
he lo al mono d omy a
∞
is
−1
. Al oge he we ha e:
Jm+1(1) MC−1
Jm+2(−1) 


MTi∗L(1,−1)
MTi∗L(−1,1)


Jm+2(1).

Co esp onding o he ons u ed lisse
Qℓ
-shea
i∗Hm,ℓ
on
A1
K {0,1}
, we ha e a on inuous
ep esen a ion
ρi∗Hm,ℓ :π´e
1(A1
K {0,1})−→ GL((i∗Hm,ℓ)x)
o
x∈A1
K {0,1}
(see
Co olla y 2.2.12). This map an b e enso ed by he ollowing on inuous one dimensional
ep esen a ion
de (ρi∗Hm,ℓ ) : π´e
1(A1
K {0,1})−→ {±1} ⊂ Qℓ
×, γ 7→ de (ρi∗Hm,ℓ (γ)),
whih is in ou ase
de (ρi∗Hm,ℓ (γ0)) = de (ρi∗Hm,ℓ (γ1)) = (1m6≡ 0 mod 4
−1m≡0 mod 4 .
We ge a ep esen a ion
ρi∗Hm,ℓ ⊗de (ρi∗Hm,ℓ ) : π´e
1(A1
K {0,1})−→ SL((i∗Hm,ℓ)x),
whih a o s h ough
SO((i∗Hm,ℓ)x)
o e en
m
and h ough
Sp((i∗Hm,ℓ)x)
o o dd
m
. Using he
Co olla y again, we ge he
Qℓ
-shea
]
Hm,ℓ := Vρi∗Hm,ℓ ⊗de (ρi∗Hm,ℓ )
on
A1
K {0,1}
. This will lead
us o he wan ed Galois ep esen a ion in Se ion 7.2.
Theo em 3.3.2
Fo
m∈N0
and a p ime numbe
ℓ
, he Za iski losu e o he monod omy g oup o
]
Hm,ℓ
is as
ol lows:
a)
Sp(W)
o
m
odd,
b)
G2(W)
o
m= 6
,
)
SO(W)
o
m
e en and
m6= 6
.
49
3. In odu ion o
MCχ
P oo :
Sine
Hm,ℓ ∈Tℓ(K)
, he
Qℓ
-shea
i∗]
Hm,ℓ
is i eduible on
A1
K {0,1}
. The e o e we ha e
ha
H=ρi∗Hm,ℓ ⊗de (ρi∗Hm,ℓ )(γ0), ρi∗Hm,ℓ ⊗de (ρi∗Hm,ℓ )(γ1)
is i eduible and onne ed by he p esene o he long unip o en elemen . By he onside a ions
ab o e he g oup
H
lea es a symple i o m in a ian i
m
is o dd, i.e.
H≤Sp(W)
, and an
o hogonal o m i
m
is e en, i.e.
H≤SO(W)
.
Mo eo e he p esene o he long unip o en elemen , gi en by he mono d omy a
∞
, implies
ha
H
is enso indeomp osable and by [SS97℄, Theo em B, we ha e he ollowing p ossibili ies o
maximal losed edu i e subg oups o
Sp(W)
esp e i ely
SO(W)
on aining
H
:
(a)
A1<Sp(W)
esp e i ely
SO(W)
( o
p= 0
o
p > h
),
(b)
SO(W).2<Sp(W)
( o
p= 2
),
()
G2<SO7
( esp e i ely
Sp6
i
p= 2
),
(d)
A2.2<Sp8
( o
p= 2
),
(e)
B3<SO8
.
He e
p=
ha
(Qℓ) = 0
, whih on adi s ase (b) and (d). The ase (e) is no p ossible b eause
o
m= 7
a symple i o m is esp e ed.
Fo
m= 6
he laim was p o ed in [DR10℄, Theo em 1.
Fo
m6= 6
he p esene o he unip o en mono d omy elemen a
1
ules ou he ase
H≤A1= PSL2(W)
. This p o es he laim.

50
4 Ho dge S u u es and Middle
Con olu ion
The ollowing in o du ion o Ho dge heo y is aken ou o [PS08℄.
4.1 Ho dge S u u es
Le b e
m∈Z
and
R⊆R
a No e he ian ing, suh ha
R⊗Q
is a eld and
VR
a ni ely gene a ed
R
-mo dule.
Deni ion 4.1.1
A (pu e)
R
-Ho dge s u u e o weigh
m
on
VR
is a di e sum deomposi ion
VC:= VR⊗C=M
p+q=m
Vp,q
wi h
Vp,q =Vq,p
omplex e o spaes o eah
p, q
. The numbe s
hp,q(V) := dimCVp,q
a e al led he
Ho dge numbe s
o he Hodge s u u e. A
mo phism o Ho dge s u u es
:VR−→ WR
is an
R
-linea map suh ha i s omplexia ion
C= ⊗
id
C
p ese es ypes,
i.e.
C(Vp,q)⊆Wp,q
.
Dening a Ho dge s u u e o weigh
m
on a ni e dimensional omplex e o spae
V
is he
same as gi ing a
Hodge l a ion
F•
o
V
. Tha is a de easing l a ion o omplex e o spaes,
suh ha
Fp∩Fq={0}
o
p+q=m+ 1
. A Ho dge l a ion is asso ia ed o a Ho dge s u u e
by
Fp:= M
≥p
V ,s
and ie e sa
Vp,q := Fp∩Fq.
The ee ank one
R
-mo dules whih a y a Ho dge s u u e a e all o e en weigh and up o
isomo phism o he ollowing o m:
Deni ion 4.1.2
A
Ho dge s u u e o Ta e
, deno ed by
Z(n)
, o
n∈Z
is he
Z
-module
(2πi)nZ⊂C
wi h sum
deomposi ion
Z(n)⊗C=V−n,−n
, whih has he e o e weigh
−2n
.
I we ha e an
R
-Hodge s u u e
VR
o weigh
m
, he
Ta e wis
VR(n)
is an
R
-Hodge
s u u e o weigh
m−2n
. I has
VR⊗(2πi)nZ
as unde lying
R
-module, while
VR(n)p,q =Vp−n,q−n
R
.
51
4. Ho dge S u u es and Middle Con olu ion
4.2 Va ia ions o Ho dge S u u e
Ho dge s u u es o u qui e na u ally on he lo al sys ems ons u ed by middle on olu ion.
In his ase, one ge s a whole shea o Ho dge s u u es  ing oge he , whih is alled a a ia ion
o Ho dge s u u e (VHS).
Deni ion 4.2.1
Le
X
be a omplex mani old. A
a ia ion o
R
-Ho dge s u u e
(VR,F•,∇)
o weigh
m
on
X
onsis s o he ol lowing da a:

a loal sys em
VR
o ni ely gene a ed
R
-modules on
X
,

a ni e de easing l a ion
F•
o he holomo phi e o bund le
V:= VR⊗ OX
by
holomo phi subbund les ( he
Ho dge l a ion
).
These da a should sa is y he ol lowing ondi ions:
a)
o eah
x∈ X
he l a ion
F•
x
o
Vx∼
=VR,x ⊗C
denes a Hodge s u u e o weigh
m
on
he ni ely gene a ed
R
-module
VR,x
,
b)
he onne ion
∇:V−→ V⊗Ω1
X
, whose shea o ho izon al se ions is
VC
, sa ises he
G i hs' ans e sali y ondi ion
∇(Fp)⊆ Fp−1⊗Ω1
X.
A
mo phism o a ia ions o Ho dge s u u e
is a mo phism o loal sys ems whih p ese es ypes,
i.e. ag ees wi h he l a ions.
We x a omplex mani old
X
, a base p oin
x∈ X
and a Ho dge s u u e
V
. Fo eah g oup
homomo phism
ρ:π op
1(X, x)−→ Au (V)
, we ge a lo ally ons an a ia ion o Ho dge s u u e.
This is he same me ho d as in Co olla y 2.2.10, gluing
V
as s alk in eah p oin and gluing he
lo al Ho dge l a ion. This p op e y ha a e izes he lo al sys em ob ained by ep esen a ion
p ese ing ypes. The e o e we ge
∇(Fp)⊆ Fp⊗Ω1
X
. By
VX
we deno e he a ia ion o he
i ial ep esen a ion.
Fo a xed Ho dge s u u e
V
, he
Weil ope a o
C
is he
C
-linea au omo phism o
V
, suh ha
o all
∈Vp,q
we ha e
C( ) = ip−q·
.
Deni ion 4.2.2
a)
A
p ola iza ion
o an
R
-Hodge s u u e
VR
o weigh
m
is an
R
- alued bilinea o m
Q:VR⊗VR−→ R
whih is
(−1)m
-symme i and suh ha
1.
The o hogonal omplemen o
Fn
is
Fm−n+1
o al l
n∈Z
,
2.
The he mi ian o m
Q(C(·),·) : VC⊗VC−→ R
on
VC
is posi i e-deni e.
52
4.2. Va ia ions o Ho dge S u u e
b)
A
p ola iza ion
o a a ia ion o
R
-Hodge s u u e
V
o weigh
m
on
X
is a mo phism o
a ia ions
Q:V⊗V−→ R(−m)X
whih indues on eah b e a pola iza ion o he o esponding
R
-Hodge s u u e o
weigh
m
.
Theo em 4.2.3
Le
X
be a ompa Kähle mani old. Le
Hp,q(X)
be he spae o ohomology lasses whose
ha moni ep esen a i e is o ype
(p, q)
. The e is a di e sum deomposi ion
Hm
dR
(X, C) := Hm
dR
(X)⊗C=M
p+q=m
Hp,q(X).
Mo eo e
Hp,q(X) = Hq,p(X)
.
I we deno e he losed Kähle o m o
X
wi h
ω
and he dimension o
X
wi h
n
, hen he
Ho dge-Riemann o m on
Hm
dR
(X, C)
is he bilinea o m
Q(α, β) = (−1)m(m−1)
2ZX
α∧β∧ωn−m,
whih is a p ola iza ion o he p e iously dened pu e
R
-Ho dge s u u e on
Hm
dR
(X, C)
o weigh
m
( . [PS08℄, Theo em 1.33).
The s anda d examples a e geome i a ia ions o Ho dge s u u e (see page 507-508 in [SZ85℄).
Rema k 4.2.4
a)
Gi en a smoo h, p ope holomo phi mapping
:X−→ S
wi h
X
as abo e a Kähle
mani old. Then
Rm ∗Q
is he unde lying sys em o a a ia ion o Hodge s u u e o weigh
m
, dened o e
Q
, in whih
F•(s)
is he usual Hodge l a ion o he ohomology o he
b e
Hm
dR
(Xs,C)
. By adjus ing he up-p odu on ohomology by he use o he Kähle lass
and i s (a ) p imi i e deomposi ion, one ob ains a pola iza ion o e
R
o
Rm ∗R
in he
geome i ase. I
X
is a amily o algeb ai a ie ies, hen he pola iza ion is in a dened
o e
Q
.
b)
By [SZ85℄, Rema k 3.3 , he pola ized s u u e passes on o sub a ia ions, ke nel and images
o un o ial mo phism o ohomology.
Gi en a a ia ion o Hodge s u u e, i is in some ases possible o ex end he s u u e o a
pun u e o he unde lying spae. This s u u e is no pu e any mo e, bu onsis s o he sum o
mul iple Ho dge s u u es o die en weigh s, hene i is a mixed Ho dge s u u e (MHS).
53

4. Ho dge S u u es and Middle Con olu ion
Deni ion 4.2.5
An
R
-mixed Ho dge s u u e
on
VR
onsis s o wo l a ions:

an in easing l a ion by a ional e o spaes on
VR⊗Q
, he
weigh l a ion
W•
and

a de easing l a ion
F•
by omplex e o spaes on
VC=VR⊗C
, he
Ho dge l a ion
.
The Hodge l a ion indues a pu e
K:= (R⊗Q)
-Hodge s u u e o weigh
m
on eah g aded
piee
G W
m(VR⊗Q) = Wm/Wm−1
by
Fp(G W
m(VR⊗Q)⊗C) = (Fp∩Wm⊗C+Wm−1⊗C)/(Wm−1⊗C).
The
Ho dge numb e s
a e he dimensions o he g aded piees o his indued g ading:
hp,q(V) := dimCG p
F(G W
p+q(VR⊗Q)⊗C)
= dimCFp(G W
p+q(VR⊗Q)⊗C)/Fp+1(G W
p+q(VR⊗Q)⊗C).
Fo wo ni ely gene a ed
R
-modules
VR, VR
′
wi h
R
-mixed Hodge s u u es a
mo phism
:VR−→ VR
′
(o weigh
0
) is an
R
-linea map, whih indues o
m∈Z
mo phisms o Hodge
s u u es by
G W
m( ) : G W
m(VR⊗Q)−→ G W
m(VR
′⊗Q).
A
g aded p ola iza ion
on an
R
-mixed Hodge s u u e is a pola iza ion o eah
G W
m(VR⊗Q)
,
whih has a pu e
(R⊗Q)
-Hodge s u u e.
4.3 Ex ensions o Va ia ions o Ho dge S u u e
In o de o ge some in o ma ion on he Ho dge s u u e ons u ed by middle on olu ion, whih
will b e in odued in he nex se ion, i is help ul o ha e a lo ok a he limi s u u e a he
singula i ies. This is p ossible as he s u u e an b e ex ended in some ases by he wo k o
Shmid. We will summa ize he esul s o [Sh73℄. E e y omplex lo al sys em an b e seen as a
holomo phi e o bundle wi h a a (and he e o e in eg able) onne ion. In a , i he g ound
spae is omplex analy i b o h a ego ies a e equi alen .
Le
(V,∇)
b e a holomo phi e o bundle on he pun u ed disk
D∗
equipp ed wi h an in eg able
onne ion. An ex ension
(e
V,e
∇)
o he bundle o
D
is said o b e
loga i hmi a
0
i
∇
ex ends o
a mo phism
e
∇:e
V −→ e
V ⊗Ω1
D(log z)
whih sa ises Leibniz' ule, i.e.
e
∇( s) = e
∇(s) + s⊗
d
o e e y lo al se ion
o
OD
and
s
o
e
V
. The
Poina é esidue map
an b e dened as
R: Ω1
D(log z)−→ Oz∼
=C, ω =η∧
d
z
z+η′7→ η(0),
suh ha
z= 0
is an equa ion o
D
and
η, η′
no on aining d
z
. This indues a
C
-linea
endomo phism es
0(e
∇)
o
e
V0
, he esidue a
0
.
54
4.3. Ex ensions o Va ia ions o Ho dge S u u e
I we x he anonial on inuous se ion
τ:C/Z−→ [0,1) + iR⊂C
, we ge he ollowing
p op osi ion due o Manin (see [Del70℄).
P op osi ion 4.3.1
Le
(V,∇)
be a holomo phi e o bund le on he pun u ed disk
D∗
equipped wi h an in eg able
onne ion. The e exis s a unique ex ension
e
V
o
V
, al led he
anonial ex ension
, o a e o
bund le on
D
suh ha
∇
ex ends o a loga i hmi onne ion
e
∇
on
e
V
whose esidue a
0
has i s
eigen alues in he image o
τ
, i.e. hei eal pa is g ea e o equal o
0
and less han
1
.
Le
V
b e a p ola ized a ia ion o
C
-Ho dge s u u es o weigh
m
on
D∗
. Supp ose ha he lo al
mono d omy op e a o is
T∈GLn(C)
, whe e
T
is unip o en and we ha e a de easing l a ion o
holomo phi e o bundles
F•
. Now we wan o ex end his Ho dge l a ion
F•
o
D
so ha we
ge some hing lose o a VHS. This ex ended l a ion gi es a mixed Ho dge s u u e in
0
, whe e
he weigh l a ion an b e des ib ed e y explii ly in he ollowing way. As
T
is unip o en he e
is a nilpo en ma ix
N
, suh ha
T= exp N
.
On a ni e dimensional e o spae e e y nilpo en endomo phism has a Jo dan deomp osi ion
and he e o e an b e w i en as sum o Jo dan blo ks o he eigen alue
0
. The e is an app op ia e
basis
( 1,..., j)
o leng h
j
o eah Jo dan blo k
Jj(0)
in he Jo dan no mal o m o
N
. We an
dene an in easing l a ion on he e o spae by pu ing
Wo:= 




{0}o≤ −j,
h 1,..., ⌊o+j+1
2⌋i −j < o < j −1,
h 1,..., jij−1≤o.
By adding hese o he die en blo ks and shi ing i by an in ege
m
, we ge he ollowing
p op e ies o he weigh l a ion, whih des ib e i uniquely.
Deni ion and ema k 4.3.2
Gi en a nilpo en endomo phism
N
o a ni e dimensional e o spae
V
, he e exis s a unique
in easing l a ion
W•=W•(N, m)
o
V
, al led he
weigh l a ion o
N
en ed a
m
, wi h he
p ope ies
a)
N(Wo+2)⊆Wo, o ∈N0
b)
he map
No: G W
m+oV−→ G W
m−oV
is an isomo phism o al l
o∈N0
.
Mo eo e , he e is a Le she z- yp e deomp osi ion
G WV=
m
M
o=0
o
M
=0
N PVm+o
wi h
PVm+o:=
Ke
(No+1 : G W
m+oV−→ G W
m−o−2V)
and he endomo phism
N
has
dimCPVm+o
Jo dan blo ks o size
o+ 1, o = 0,...,m
.
55
4. Ho dge S u u es and Middle Con olu ion
As [Sh73℄, Theo em 6.16 we nd he ollowing esul :
Theo em 4.3.3
Le
V
be a pola ized a ia ion o
C
-Hodge s u u es o weigh
m
on
D∗
wi h loal monod omy
ope a o
T∈GLn(C)
, suh ha
T
is unipo en . Choose
N∈GLn(C)
nilpo en suh ha
exp N=T
.
The Hodge bund les
F•
o
V
ex end o holomo phi subbund les
e
F•
o he anonial ex ension
e
V
,
and he iple
V
Hdg
0:= (e
V0, W•(N, m),e
F•
0)
is a mixed Hodge s u u e, al led he
anonial b e
.
This is an impo an ool o he de e mina ion o he o iginal a ia ion o
C
-Ho dge s u u es.
Fo a unip o en
T
a nilp o en ma ix
N
wi h
exp N=T
has he same Jo dan blo k s u u e
bu eigen alue
0
ins ead o
1
. The e o e he a ional dimension o
G W
m±o(e
V0⊗Q)
o
o∈N0
is
he numb e o Jo dan blo ks o o dd leng h whose leng h is g ea e han
o
, i
o
is e en, and he
numb e o Jo dan blo ks o e en leng h whose leng h is g ea e han
o
, i
o
is o dd.
Le us assume om now on ha
T
is a long unip o en elemen , i.e. has Jo dan no mal o m
Jn(1)
,
whe e
n
is he ank o
V
. By he las heo em, we an ex end a pola ized a ia ion
V
o
C
-Ho dge
s u u e o weigh
m
on
P1
C S
(
S
ni e) o a xed
s∈S
by a mixed Ho dge s u u e
V
Hdg
s= (e
Vs, W•(1
2πi log T, m),e
F•
s).
The weigh l a ion
W•:= W•(1
2πi log T, m)
on
Vs
indues he e o e he ollowing dimensions o
he g ading:
dimQG W
j(Vs) = (1−n < j −m < n
and
2|j−m−(n−1),
0
else
.
In gene al his do es no dene
e
F•
s
uniquely. Bu exa ly in his s i ly one and ze o dimensional
ase he dimensions o he Ho dge l a ion a e de e mined by he dimensions o he weigh
l a ion. This is done by alula ing he Ho dge numb e s o he mixed Ho dge s u u es
V
Hdg
s
.
As eah g adua ed piee is o dimension ei he
0
o
1
, he e is exa ly one Ho dge s u u e o e
Q
o a gi en e en weigh
m
, namely
Q(−m
2)
wi h Ho dge ype
(m
2,m
2)
. Sine
N
lowe s he weigh
s i ly by
2
, he Ho dge l a ion has maximal leng h and he mixed Ho dge s u u e is uniquely
dened by he weigh l a ion. Summa izing we ob ain he ollowing esul :
Co olla y 4.3.4
Le
V
be a pola ized a ia ion o
C
-Hodge s u u e o weigh
m
and ank
n
on
P1
C S
(
S
ni e)
and
s∈S
suh ha he monod omy ope a o a
s
is a long unipo en elemen . Then he Hodge
l a ion o
V
has maximal leng h.
56
5 Mo i i Des ip ion o
Hm,ℓ
Fo a gene al in o du ion o mo i es, ha e a look a [Jan94℄ o [And04℄. The ollowing pa is a
lose adap ion o hap e 8 o [Ka 96℄.
5.1 Se ing
Le
K
b e an algeb aially losed eld and ho ose
o≥2
dis in p oin s
s1,...,so∈A1
K
. Fo a
p ime numb e
ℓ
and an in ege
N∈N
, suh ha ha
(K)∤Nℓ
, we x a p imi i e
N
- h o o o
uni y in
K
and a p imi i e
N
- h o o o uni y
ζN
in
Q
. We dene he ings
RN,ℓ := Z[ζN,(Nℓ)−1]⊂Q
and
SN,o,ℓ := RN,ℓ[T1,...,To][∆−1]⊂Q(T1,...,To),
whe e
∆ := Q
i<j
(Ti−Tj)
. By he xa ion ab o e, we ge a unique ing homomo phism
ϕ:SN,o,ℓ −→ K
wi h he p op e y, ha
ϕ(Ti) = si
o
i= 1,...,o
and ha
ζN
is mapp ed
o he hosen
N
- h o o o uni y in
K
. Fo
m∈N0
we onside he ollowing ane spaes
A(o, m + 1)RN,ℓ := spec RN,ℓ[T1,...,To, X1,...,Xm+1][∆−1
m+1]
whe e
∆m+1 := Y
i<j
(Ti−Tj)·
m+1
Y
i=1
o
Y
j=1
(Xi−Tj)·
m
Y
i=1
(Xi+1 −Xi)
wi h na u al p o je ions p
i:A(o, m + 1)RN,ℓ ////A1
SN,o,ℓ {T1,...,To}
indued by he
emb edding
SN,o,ℓ[X]//RN,ℓ[T1,...,To, X1,...,Xm+1][∆−1
m+1]
wi h
X7→ Xi
.
On
Gm,RN,ℓ
wi h o o dina e
Z
, one has he Kumme o e ing o deg ee
N
o he equa ion
YN=Z
.
Le
µN(RN,ℓ)
deno e he
N
- h o o s o uni y, hen ho osing an
N
- h p imi i e o o o uni y in
Qℓ
is he same as ho osing an emb edding
χ:µN(RN,ℓ)//Qℓ
×
. Then he o e ing denes a
onne ed
µN(RN,ℓ)
- o so , whih gi es a ep esen a ion
π´e
1(Gm,RN,ℓ )ε////µN(RN,ℓ)χ
//Qℓ
×
and he e o e he o esp onding Kumme shea
Lχ
. Fo any sheme
G
and any mo phism
:G −→ Gm,RN,ℓ
, we dene
Lχ( ):= ∗Lχ
. Le
:A(o, 2)RN,ℓ −→ Gm,RN,ℓ
b e indued by he
ing homomo phism
X7→ X2−X1
.
57
5. Mo i i Des ip ion o
Hm,ℓ
The un o iali y o (
−1
)-omp onen o he highe di e image in he sense o Se ion 5.2 ( he
no ion ex ends in an ob ious way o
X
and o
D
) and again he exa ness o he sequene yield
he ollowing hain o isomo phisms
K=m= (Rm(φA)!Qℓ)−1
=m∼
=im Rm(φA)!Qℓ→Rm(φXA)∗Qℓ−1
∼
=ke Rm(φXA)∗Qℓ→Rm(φDA)∗Qℓ−1.
By Co olla y 5.3.1 he shea
K=m
is lisse and he isomo phisms imply ha
ke Rm(φX)∗Qℓ→Rm(φD)∗Qℓ−1
is lisse o o. I ollows om p op e base hange ha
ke Rm(φXA)∗Qℓ→Rm(φDA)∗Qℓ−1
is lisse, whe e
DA:= `
i∈I
Di,A
and
φDA=`
i∈I
φXA|Di,A
.
We laim ha he na u al map
ψ: ke Rm(φXA)∗Qℓ→Rm(φDA)∗Qℓ−1−→ ke Rm(φXA)∗Qℓ→Rm(φDA)∗Qℓ−1,
whe e
φDA:= `
i∈I
φDi,A :DA−→ A1
A {0,1}
, is an isomo phism. In o de o p o e ha by he
Sp eializa ion Theo em (see [Ka 90℄, 8.18.2), i sues o show his o any losed geome i p oin
x
o
HypA
. As
(Rm(φA)!Qℓ)−1
=m∼
=ke Rm(φXA)∗Qℓ→Rm(φDA)∗Qℓ−1
, we ha e o show ha
(Hm
c(HypA,x,Qℓ))−1
=m−→ ke Hm
´e (XA,x,Qℓ)→Hm
´e (DA,x,Qℓ)−1
is an isomo phism o
DA,x := `
i∈I
Di,A,x
. We dene he ollowing sequene o s alks
X0
A,x := XA,x
, and o na u al numb e s
i
, le
Xi
A,x
deno e he disjoin union o
he i eduible omponen s o he lo us, whe e
i
pai wise die en omp onen s o
DA,x
mee . I ollows om he Weil onje u es [Del74℄ ha he sp e al sequene
E1=Hj
´e (Xi
A,x,Qℓ)x⇒Hi+j
c(UA,x,Qℓ)
degene a es a
E2
. Consequen ly, we ha e
(Hm
c(HypA,x,Qℓ))=m∼
=ke Hm
´e (XA,x,Qℓ)→Hm
´e (DA,x,Qℓ),
whih p o es ha he map
ψ
is an isomo phism as laimed. So,
(Rm(φHypA)∗Qℓ)−1
=m∼
=ke Rm(φXA)∗Qℓ→Rm(φDA)∗Qℓ−1
=1
2(1 −σ)ke Rm(φXA)∗Qℓ→Rm(φDA)∗Qℓ,
whe e he las equali y is by using ep esen a ion heo y o ni e (yli) g oups. I ollows ha
φ∗K=m=φ∗(Rm(φHyp)!Qℓ)−1
=m∼
=1
2(1 −σ)ke Rm(φX)∗Qℓ→Rm(φD)∗Qℓ,
as laimed.

64

5.4. Analy ia ion o
Hm,ℓ
5.4 Analy ia ion o
Hm,ℓ
Le
K
b e a numbe eld and
S⊆K
a ni e se . We x an emb edding
K//C
. This
yields a on inuous mo phism
ι:π op
1(C S)//
π op
1(C S) = π´e
1(A1
C S)−→ π´e
1(A1
K S)
.
A lisse
Qℓ
-shea
V
on
A1
K S
o esp onds by Co olla y 2.2.12 o a on inuous ep esen a ion
ρV:π´e
1(A1
K S)−→ GLn(Qℓ)
.
Deni ion 5.4.1
The
analy ia ion
Van
o
V
is he loal sys em o
Qℓ
-modules
VρV◦ι
on
C S
o esponding o he
ep esen a ion
ρV◦ι:π op
1(C S)−→ GLn(Qℓ)
by Co ol la y 2.2.10 .
The ompa ison isomo phism b e ween é ale and singula ohomology implies (a e xing an
isomo phism
C∼
=Qℓ
o elds), ha
(φ∗K=m)an ∼
=1
2(1 −σ)an(ke Rm(φX)an
∗Qℓ→Rm(φD)an
∗Qℓ),
as
1
2(1 −σ)
is an algeb ai p o je o and hene deRham. Fu he we ha e a mo phism b e ween
smo o h p o je i e a ie ies on he igh hand side, whih is gi en by es i ion o inlusions o
he smo o h di iso s
Di
. Applying [Del87℄, P op osi ion 1.13. o he i eduible shea
(φ∗K=m)an
,
i omes om a a ia ion o Ho dge s u u e in he ollowing sense:
Rema k 5.4.2
The loal sys em o
Q
- e o spaes
Gm:= 1
2(1 −σ)an
|{z }
=1
2(1−σan)
ke (Rm(φX)an
∗Q→Rm(φD)an
∗Q)
on
A1
C {0,1}
is a pola ized a ia ion o Hodge s u u e, whih is pu e o weigh
m
, sine i is a
sub a ia ion o
Rm(φX)∗Z
by Rema k 4.2.4 b). Mo eo e by Co ol la y 5.3.1 and Co ol la y 5.3.2 ,
we ha e
Gm⊗Qℓ∼
=(φ∗K=m)an = (Hm,ℓ|A1
C {0,1})an.
Conluding we ha e:
Theo em 5.4.3
Le
Hm,ℓ
be as in Theo em 3.3.1 . Then he e exis s a loal sys em o
Z
-modules
Gm
on
A1
C {0,1}
unde lying a pola ized a ia ion o
Z
-Hodge s u u e
(Gm,F•,∇)
on
C {0,1}
pu e o weigh
m
suh ha
(i∗Hm,ℓ)an ∼
=Gm⊗Qℓ.
The indued isomo phism on he s alks
(i∗Hm,ℓ)an
x∼
=Gm⊗Qℓx
o
x∈C {0,1}
is gi en by he
65
5. Mo i i Des ip ion o
Hm,ℓ
ompa ison isomo phism be ween é ale ohomology and singula ohomology
1
2(1 −σ)ke Hm
´e (Xx,Qℓ)→Hm
´e (Dx,Qℓ)∼
=1
2(1 −σ)ke (Hm
B(X(C)x,Z)→Hm
B(D(C)x,Z)) ⊗Qℓ.
Mo eo e , he Hodge l a ion o
Gm
has maximal leng h.
P oo :
All laims bu he las ollow om Rema k 5.4.2. The las laim ollows om he long
unip o en lo al mono d omy o
Hm,ℓ
using Theo em 3.3.1 and Co olla y 4.3.4.

In [Fal88℄, Se ion 4(a) Fal ings gi es he ons u ion o na u al isomo phisms o
ℓ
-adi é ale
ohomology and deRham ohomology.
Rema k 5.4.4
Le
∈ΣK {0}
and
ℓ=
ha
(k )
and
X
a p ope a
OK
-sheme, hen we ha e an isomo phism
Hm
´e (X⊗K ,Qℓ)⊗C
∼
−→ M
p+q=m
Hq(X, Ωp
X/OK )⊗C (−q),
whih p ese e up p odu s,
GK
-a ion, ha a e is i lasses o yles and Che n lasses o
e o bund les.
This ema k o Fal ings has imp o an onsequenes o he onne ion b e ween
τ
-Ho dge-Ta e
numb e s o an
ℓ
-adi ep esen a ion
ρℓ
and Ho dge numbe s o
XC
, ela ed by he a ion o
GK
on
X
. Illusie a ies ou he essen ial pa o he exa onne ion a e [Ill94℄, Theo em 3.1.2:
hj,m−j(XC) = dimQℓ(C ⊗Hm
´e (XQℓ,Qℓ)(j))GK .
The e o e we ha e o
Vℓ:= Hm
´e (XQℓ,Qℓ)⊗Qℓ
ha
h ,j(ρℓ) = dimQℓ(C (j)⊗Vℓ)GK =h−j,m+j(XC).
Co olla y 5.4.5
Fo
X:= Gm,x
as in Theo em 5.4.3 endowed wi h a
GK
-a ion, o esponding o he
ep esen a ion
ρℓ:GK −→ GL(Gm,x)
, we ge he ol lowing esul :
h ,j(ρℓ) = (1−m≤j≤0,
0
else
.
Mo eo e
ρ= (ρℓ)ℓ
p ime
ull ls egula i y in he sense o Deni ion 7.2.1 .
By Theo em 5.4.3 hese
τ
-Ho dge-Ta e numb e s oinide wi h he
τ
-Ho dge-Ta e numb e s o he
sys em o
ℓ
-adi ep esen a ions
ρm= (ρm,ℓ)ℓ
p ime
dened in Se ion 7.2.
66
6 I eduibili y o
ρm
6.1 Li ing I eduibili y
Le
G
b e a g oup,
n∈N
and
ℓ
a p ime numb e .
Lemma 6.1.1
Le
ρFℓ:G−→ GLn(Fℓ)
be an i eduible ep esen a ion wi h a long unipo en elemen . Then he
ex ension
ρFℓ:G−→ GLn(Fℓ), g 7→ ρFℓ(g)
is i eduible (i.e.
ρFℓ
is absolu ely i eduible).
P oo :
We ake a minimal
ρFℓ
-in a ian subspae
{0} 6=W⊆Fℓ
n
. We ha e he omp onen -wise
Galois a ion o
GFℓ
on
Fℓ
n
, whih maps
W
o an o bi o subspaes. As he ep esen a ion is
dened o e
Fℓ
b o h a ions ommu e and he e o e eah subspae in he o bi is
ρFℓ
-in a ian .
By he minimali y o
W
hese a e all linea ly disjoin and on he o he hand xed by he long
unip o en elemen . Eah spae xed by his elemen inludes he eigen e o and he e o e he
o bi has jus one elemen . As
W
is in a ian unde b o h a ions, he e is a subspae
U⊆Fn
ℓ
whih is
ρFℓ
-in a ian and o whih
W=Fℓ⊗U
. As
ρFℓ
is i eduible and
{0} 6=W
, we ha e
U=Fn
ℓ
and he e o e
W=Fℓ
n
.

The lo al ing
Zℓ=x∈Qℓ| (x)≥0
has he maximal ideal
x∈Zℓ| (x)>0
wi h esidue
eld
Fℓ
. In he ase o he lemma ab o e, we ha e hene ha
ρZℓ:G−→ GLn(Zℓ), g 7→ ρZℓ(g)
is i eduible as well. He e we all a
Zℓ[G]
-mo dule i eduible i and only i i has no non- i ial
Zℓ[G]
-submo dules. A his p oin I wan o hank S e an Rei e and And eas Mau isha o elling
me ab ou he nex well-known s a emen .
Lemma 6.1.2
I
ρZℓ:G−→ GLn(Zℓ)
is an i eduible ep esen a ion, hen he ex ension
ρQℓ:G−→ GLn(Qℓ), g 7→ ρZℓ(g)
is i eduible.
P oo :
Assume ha
ρQℓ
is eduible and has he in a ian subspae
06=V6=Qℓ
n
. The e o e he
Zℓ
-mo dule
W:= V∩Zn
ℓ
is in a ian and
W6=Zn
ℓ
, b eause
V6=Qℓ
n
. Addi ionally i is non- i ial
as o all
∈Qℓ
n
he e exis s a
λ∈Qℓ
×
, suh ha
λ ∈Zn
ℓ
, and we ha e
06=W
. This is a
on adi ion sine
ρZℓ
is i eduible.

This lemma an easily b e adap ed o ings and hei quo ien elds, esp eially o alua ion ings.
67
6. I eduibili y o
ρm
6.2 Se e's Resul s on Cha a e s o
GQ
In he nex pa , we es a e some esul s o Se e. The main one is ha a sys em o homomo phisms
(θℓ:GQ−→ F×
ℓ)ℓ∈L
ullling some ompa ibili y ela ions is he edu ion o he p o du o a
ni e ha a e and a p owe o he ylo omi ha a e . This will b e ano he ing edien o he
p o o o Theo em 6.4.1.
Le
K
b e a numbe eld and
IK
he idele g oup o
K
. Fo a ni e se
S
o plaes o
K
, we dene
a
modulus
m
wi h suppo
S
as a amily
m:= (m ) ∈S∈NS
. Then we ge an op en subg oup
Um
o eah mo dulus
m
, by
Um:= Q
plae o
K
Um, ⊆IK
, whe e
Um,
is as ollows:
onne ed omp onen o
1
in
K×
o an inni e plae
6∈ S
,
K×
o a ni e plae
6∈ S
,
{x∈K×
suh ha
(1 −x)≥m }
o
∈S
.
On he o he hand, we ha e wo asso ia ed algeb ai g oups
Tm
and
Sm
o e
Q
wi h an algeb ai
mo phism
Tm//Sm
( o mo e de ails see [Se 68℄, Se ion 2.2.).
Le
E
b e a numbe eld. I is shown in [Se 68℄ how o a ah a s i ly ompa ible sys em
(ψℓ:GQ−→ E×
λ)λ∈ΣE {0}
o one dimensional
λ
-adi
Q
- a ional Galois ep esen a ions o any
ha a e
ψ:Sm−→ E×.
P op osi ion 6.2.1 ( [Se 72℄, P op.20 )
Le
L⊆ΣQ∪ {∞} {0}
be an inni e se and le
θℓ:GQ−→ F×
ℓ, ℓ ∈L,
be a ol le ion o
homomo phisms. Assume ha he e exis s a modulus
m
and
j∈Z
suh ha o al l
ℓ∈L
and o
al l
a∈Um
one has
θℓ(c −1[a]) ≡a−j
ℓmod ℓ,
whe e
c : Gab
Q−→ IQ/Q×
is he lass eld isomo phism and
aℓ
is he omponen o
a
a
ℓ.
Then
he e exis s a numbe eld
E
and a Heke ha a e
ψ:Sm−→ E×
suh ha
ψλ=θℓ
o inni ely
many
ℓ∈L
and
λ
a ni e plae o
E
abo e
ℓ
.
The ollowing esul is a onsequene o [Sh88℄, P op osi ion 1.4, and Se e's heo y o ab elian
ep esen a ions [Se 68℄:
P op osi ion 6.2.2
Le
(ψλ:GQ−→ E×
λ)λ∈ΣE {0}
be a s i ly ompa ible sys em o one dimensional
λ
-adi
E
- a ional
Galois ep esen a ions whih a e assoia ed o a Heke ha a e
ψ:Sm−→ E×.
Then he e exis s
a ni e ha a e
ǫ:GQ−→ E×
and an in ege
k∈Z
suh ha
ψλ=ǫ·χk
ℓ,
whe e
λ|ℓ.
Combining b o h esul s, we ge he ollowing:
68
6.3. G oups o Lie Type
Co olla y 6.2.3
Le
L⊆ΣQ∪ {∞} {0}
be an inni e se and le
θℓ:GQ−→ F×
ℓ, ℓ ∈L,
be a ol le ion o
homomo phisms. Assume ha he e exis s a modulus
m
and
j∈Z
suh ha o al l
ℓ∈L
and o
al l
a∈Um
one has
θℓ(c −1[a]) ≡a−j
ℓmod ℓ,
whe e
c : Gab
Q−→ IQ/Q×
is he lass eld isomo phism and
aℓ
is he omponen o
a
a
ℓ.
Then
he e exis s a numbe eld
E
, a ni e ha a e
ǫ:GQ−→ E×
and an in ege
k∈Z
suh ha
ǫ·χk
ℓ=θℓ
o inni ely many
ℓ∈L.
6.3 G oups o Lie Typ e
In he yea 1972 Go ens ein announed a p og am o he omple e lassia ion o ni e simple
g oups. Many ma hema iians wo ked on i and he las gap was lled 2004 by Ashbahe and
Smi h ( . [As04℄). Beside he well-known abelian ni e simple g oups, i.e. yli g oups o p ime
o de , he e a e he ollowing p ossibili ies:

he al e na ing g oups
An
(
n≥5
),

he ni e lassial g oups - ha is, he linea , symple i, uni a y and o hogonal g oups o
ni e e o spaes,

he exep ional g oups o Lie ype,

he
26
sp o adi g oups
(see [Go 85℄). This shows ha g oups o Lie yp e play an impo an ole in he unde s anding o
g oups and hei subg oup s u u e.
Le
K
b e a eld o ha a e is i
p
,
q
a
p
-p owe and
n∈N
, hen we ha e he homomo phism
F obq: GLn(K)−→ GLn(K),(aij)7→ (aq
ij)
. I
K
is algeb aially losed and
G
is a linea
algeb ai g oup o e
K
a
s anda d F obenius
F:G−→ G
is a map suh ha he e exis s an
n∈N
, an inlusion
ι:G//GLn(K)
and a p owe
q
o
p
suh ha o all
g∈G
we ha e
ι(F(g)) = F obq(ι(g)).
A homomo phism
F:G−→ G
is alled a
F obenius mo phism
, i some
p owe o
F
is a s anda d F ob enius.
Deni ion 6.3.1
Le
G
be a onne ed edu i e algeb ai g oup o e
Fℓ
and le
F:G−→ G
be a F obenius
mo phism. The ni e g oup o xed poin s
GF
is al led
g oup o Lie yp e
and some imes also
i s ommu a o subg oup
(GF)′:= [GF, GF]
and i s en al quo ien
G/Z(G)
, whe e
Z(G)
is he
en e .
69

6. I eduibili y o
ρm
In his se ing, a maximal losed onne ed sol able algeb ai subg oup is alled
Bo el subg oup
and an algeb ai subg oup whih on ains a Bo el subg oup is alled
pa aboli
.
Rema k 6.3.2
a)
The s anda d F obenius mo phism
F obq:Fq−→ Fq, α 7→ αq
, gi es ise o he so al led
Che alley g oups
(un wis ed g oups o Lie ype), whe eas he p odu o some
F obq
wi h
o he au omo phisms leads o wis ed g oups o Lie ype (e.g.
SUn(Fq2) = SLn(Fq)δ◦F obq
,
whe e
δ
is he in e se anspose map).
b)
Eah g oup o Lie ype is he quo ien o
Fq
-poin s
G(Fq)
o an algeb ai g oup sheme o e
Fq
. (Fo he de i ed g oup o an o hogonal g oup, we ake he spin g oup,  . [Wil09 ℄).
Deni ion and ema k 6.3.3
To any onne ed Dynkin diag am he e is an assoia ed simple algeb ai g oup
G
o e
Fq
(wi h
he exep ions
A1(F2), A1(F3), A2(F3), B2(F2)
and
G2(F2)
).
Ca an yp e Che alley g oup Dynkin diag am
Al(Fq) SLl+1,PGLl+1
Bl(Fq) SO2l+1
Cl(Fq) Sp2l
Dl(Fq) SO2l
E6(Fq)
E7(Fq)
E8(Fq)
F4(Fq)
G2(Fq)
Fo he lassia ion i is neessa y o ake a Dynkin diag am oge he wi h an au omo phism o
he g aph, whih hen is in o espondene o simple g oup o Lie ype. This leads o he ol lowing
ypes
2Al(Fq2),2B2(F22n+1 ),2Dl(Fq2),3D4(Fq3),2E6(Fq2),2F4(F22n+1 ),2G2(F32n+1 )
.
70
6.3. G oups o Lie Type
Nex we will need he s u u e o he maximal subg oups o he ommu a o subg oup
Ωn(Fq) := [SOn(Fq),SOn(Fq)]
o
n∈N
o dd and
q
an o dd p ime p owe . In [KL90℄, we ha e
he ollowing deni ion o se s
S,C
o maximal subg oups o
Ωn(Fq)
:
Deni ion o
S
A maximal subg oup
H
o
Ωn(Fq)
lies in
S:= S(Ωn(Fq))
i and only i he ollowing holds.
a) The so le
S
o
H
, ha is he subg oup gene a ed by he minimal non- i ial no mal
subg oups o
H
, is a non-ab elian simple g oup - i.e.
H
has a unique minimal non- i ial
no mal subg oup, whih is non-ab elian and simple.
b) I
L
is he ull o e ing g oup o
S
, and i
ρ:L−→ GL(V)
is a ep esen a ion o
L
suh ha
ρ(L) = S
, hen
ρ
is absolu ely i eduible.
)
ρ(L)
anno b e ealized o e a p op e subeld o
Fq
.
( ough) Deni ion o
C
Fo he subg oup
Ωn(Fq)
o
SOn(Fq)
we dene
Ci(Ωn(Fq)) := {C∩Ωn(Fq)|C∈ Ci(SOn(Fq))}
o
i= 1,...,8
, and le
C:= C(Ωn(Fq)) :=
8
[
i=1 Ci(Ωn(Fq)).
C1:
s abilize s o o ally singula o non-singula subspaes
C2:
s abilize s o deomp osi ions
V=
L
j=1
Vj
wi h
dimFqVj= dimFqV=n
C3:
s abilize s o ex ension elds o
Fq
o p ime index
C4:
s abilize s o enso p o du deomp osi ions
V=V1⊗V2
C5:
s abilize s o subelds o
Fq
o p ime index
C6:
no malize s o symple i- yp e
-g oups (
p ime) in absolu ely i eduible ep esen a ions
C7:
s abilize s o deomp osi ions
V=
N
j=1
Vj
wi h
(dimFqVj) = dimFqV=n
C8:
lassial subg oups
71
6. I eduibili y o
ρm
The [KL90℄, Main Theo em (C) and [KL90℄, Table 3.5.D e eals he s u u e o
Ωn(Fq)
o
n, q
o dd:
Theo em 6.3.4 ( [KL90℄, Main Theo em (C) )
Assume ha
n > 12
is odd an
q
an odd p ime powe . Fo a membe
H∈ C
, he p eise ondi ions
unde whih
H
is maximal in
Ωn(Fq)
a e de e mined by he ol lowing able. Mo eo e , his able
also de e mines he se o o e g oups o
H
lying in
C ∪S
.
Ci
yp e ondi ions
C1Pm1≤m≤n−1
2
Om(Fq)⊥Oǫ
n−m(Fq) 1 ≤m < n
,
m
o dd,
ǫ=±
C2Om(Fq)≀S n=m , m, ≥2
O1(Fq)≀Snq
p ime
C3On
(Fq ) |n
,
p ime,
6=n
C4Om(Fq)⊗On
m(Fq)m|n
,
m < √n
C5On(Fq0)q=q
0
,
p ime
C6
do es no o u
C7Om(Fq)≀S n=m
,
(q, m)6= (3,3)
C8
do es no o u
Table 6.1: The maximal subg oups o
Ωn(Fq)
o
n, q
o dd ([KL90℄, Table 3.5.D)
He e
Pm
is a s abilize o an
m
-dimensional o al ly singula spae, i.e. a pa aboli subg oup,
Om
and
Oǫ
m
a e o hogonal g oups espe ing some symme i bilinea o m and
Sm
is a symme i
g oup.
Lemma 6.3.5
Le
n∈N
be a xed odd in ege . Fo almos al l p ime numbe s
ℓ
le
G(Fq)⊆SOn(Fℓ)
be a g oup o
Lie ype o
q
a powe o
ℓ
on aining a long unipo en elemen , i.e. an elemen o Jo dan no mal
o m
Jn(1)
, and one non- i ial unipo en elemen wi h die en Jo dan no mal o m. Then o
almos al l
ℓ
as abo e, we ha e he inlusion
Ωn(Fℓ)⊆G(Fq)
i
n6= 7
and
G2(Fℓ)⊆G(Fq)
i
n= 7
.
P oo :
The e exis s a simply onne ed g oup o Lie yp e
G(Fq)
and an epimo phism
G(Fq)////G(Fq)
. By [S e63℄ we ha e a mo phism
L(λ) : G(Fq)−→ Ωn(Fq)⊆GLn(Fq)
o
Fq
-p oin s o algeb ai g oup shemes on
Fq
dened o e
Fq
(whe e
λ
deno es he highes weigh
o he ep esen a ion), i
ℓ
is la ge ompa ed o
n
. The long unip o en elemen
u
is asso ia ed o
a e ain o o
α
in he o o sys em o
G
. The ansp osed elemen
u
is hen asso ia ed o he
nega i e o o
−α
o
G
. Then he g oup
H=hu, u i
is an i eduible subg oup o
G
o ype
A1
,
again i
ℓ
is la ge enough.
72
6.4. I eduibili y o he mo d-
ℓ
Rep esen a ion
Now he exis ene o he non- i ial unip o en elemen wi h Jo dan no mal o m die en om
Jn(1)
implies ha
H
is p op e ly on ained in
G(Fq)
and ha
G
is o die en ype hen
A1
(i
ℓ
is la ge enough).
By S einbe g's Tenso P o du Theo em ( . [MT11℄, Theo em 29.6), i
ℓ
is la ge ompa ed o
n
any i eduible ep esen a ion o
G
is gi en by a highes weigh ep esen a ion and hene was
indued by a mo phism o onne ed g oup shemes o e
Fq
. Base hange o
L(λ)
o
Fq
denes a
mo phism o algeb ai g oups and hene a ep esen a ion
L(λ)⊗Fq:G(Fq)−→ GLn(Fq)
, whih
a o s o e
SOn(Fq)
. By [SS97℄, Theo em B, any algeb ai g oup on aining a long unip o en
elemen inside an unde lying gene al linea g oup
GL(W)
die en om
GL(W)
is ei he o yp e
A1,SO(W),Sp(W), G2
o
B3
. As
G(Fq)
denes
G
uniquely i he ha a e is i
ℓ
and he e o e
q
is la ge enough and sine he e a e no non- i ial wis s o o dd dimensional o hogonal g oups,
his implies ha
G
lies b e ween
Ωn(Fℓ)
and
SOn(Fℓ)
i
n6= 7
. In he hase
n= 7
, hene we ha e
G2(Fℓ)⊆G(Fq)
.

6.4 I eduibili y o he mo d-
ℓ
Rep esen a ion
In his se ion we ha e
K=Q
,
m∈N0
e en and
ℓ
a p ime numb e . Then we ge he
lisse
Qℓ
-shea
i∗Hm,ℓ
o ank
m+ 1
o
i:A1
Q {0,1}//A1
Q
and
Hm,ℓ
like in Se ion 3.3.
By xing an
s∈A1
Q {0,1}
his o esp onds o a on inuous ep esen a ion o ank
m+ 1
ρi∗Hm,ℓ :π1
´e (A1
K {0,1})−→ GL((Hm,ℓ)s)
, whih a o s h ough
Zℓ
and esp e s a symme i
bilinea o m. This ep esen a ion an b e enso ed wi h he de e minan as in Theo em 3.3.2, o
ob ain a on inuous ep esen a ion
ρi∗Hm,ℓ ⊗de (ρi∗Hm,ℓ ) : π1
´e (A1
Q {0,1})−→ GL((Hm,ℓ)s),
a o ing o e
SL
and espe ing a symme i bilinea o m. This ep esen a ion is o weigh
m
by Delignes' wo k ( . Theo em 2.3.4), i.e. maps
F obq
o
q−m
2
. Then we ge a weigh
0
ep esen a ion by enso ing wi h he
m
2
h p owe o he ylo omi ha a e
χℓ
. Now we will show
ha he edu ion mo d
ℓ
dρm,ℓ := ρi∗Hm,ℓ ⊗de (ρi∗Hm,ℓ )⊗χm
2
ℓ:π1
´e (A1
Q {0,1})−→ SOm+1(Fℓ)
is i eduible o almos all
ℓ
. This will lead us in Se ion 7.2 o he a ha
ρm,ℓ
is i eduible
as a weakly ompa ible sys em o Galois ep esen a ions o
Q
. Fo any
x∈A1
Q {0,1}
, we ge he
sp eializa ion map
ιx:π´e
1({x})∼
=GQ//π´e
1(A1
Q {0,1})
as dened in Se ion 2.2.
73
7. Po en ial Au omo phy o Sp eializa ions
whe e
p
is he ha a e is i o he esidue eld
k
as b e o e. Fo
∈S
he deni ion o
L (π, s)
is no so s aigh o wa d, bu an b e ob ained by he lo al Langlands o espondene [HT01℄ and
[Hen00℄. I is known ha hese analy i
L
- un ions sa is y a o able p op e ies like me omo phi
(mos ly e en holomo phi) on inua ion o he whole omplex plane and ulll un ional equa ions
e ..
On he o he hand, o any i eduible, weakly ompa ible sys em
ρ= (ρℓ)ℓ
p ime
o Galois
ep esen a ions
ρℓ:GK−→ GL(Vℓ)
(see Deni ion 2.1.4), we an also dene he
es i ed
L
- un ion
o
ρ
by
L(ρ, s) := Y
∈S
L (ρ, s)·Y
∈ΣK S
de (1−p−s
F ob ,ρℓ)−1:= Y
∈S
p−sn
˜
,ρℓ(ps
)·Y
∈ΣK S
psn
,ρℓ(ps
)−1,
whe e
n
is he ank o he ep esen a ion and
,ρℓ(x)∈Q[x]⊂Qℓ[x]
he ha a e is i p olynomial
o he F ob enius (see page 22) o
|ℓ
. S i ly sp eaking
F ob ,ρℓ
is no well-dened and should
b e seen as symbol o he e y las p o du on he igh .
In he ase
∈S
, we ha e a p eimage o he F ob enius in he deomp osi ion g oup o eah elemen
o he ine ia g oup
I := lim
←− Iw
( o he sys em
w|
). The a ion o hese p eimages ia he
Galois ep esen a ion
ρℓ
is unique up o onjuga ion on he spae
Vρℓ(I )
ℓ
xed by
ρℓ(I )
. The e o e
he ha a e is i p olynomial
˜
,ρℓ(x) := de (1 ·x−ρℓ(F ob )|Vρℓ(I )
ℓ
)∈Qℓ[x]
is well-dened and
an b e in e p e ed as a omplex p olynomial by ho osing an emb edding o elds
ιℓ:Qℓ//C
.
We dene
L (ρ, s) := de (1 −p−s
ρℓ(F ob )|Vρℓ(I )
ℓ
) := p−sn
˜
,ρℓ(ps
),
whe e
n
is he dimension o
Vρℓ(I )
ℓ
.
Deni ion 7.1.1
Le
ρ= (ρℓ)ℓ
p ime
be an i eduible, weakly ompa ible sys em o
ℓ
-adi ep esen a ions
o a numbe eld
K
, hen
ρ
is al led
au omo phi
, i he e exis s an i eduible omplex
sub ep esen a ion
π
o
GLn(AK)
, suh ha
L(ρ, s) = L(π, s).
In his ase, we say
L(ρ, s)
is
au omo phi
as wel l.
I he e is a ni e Galois ex ension
L|K
suh ha he es i ion
(ρℓ|GL)ℓ
p ime
is au omo phi,
ρ
is al led
p o en ially au omo phi
.
An imp o an onje u e whih is a pa o he amous Langlands p og am [Lan79℄, s a es ha
any i eduible uspidal sys em o Galois ep esen a ions is au omo phi.
7.2 Mo dula Li ing
Due o Ba ne -Lamb, Gee, Ge agh y and Taylo ( . [BLGGT10℄) one has a  i e ion o
au omo phy o whih we will ha e o in o due some mo e no a ion.
80

7.2. Mo dula Li ing
Fo a numb e eld
K
, we ha e he maximal o ally eal subeld
K+
. The inni e plaes o
K+
o esp ond o he emb eddings
K+//R
. I we x suh an inni e plae
, we ge an embedding
o
GR={1, c}//GK+
. The image o he omplex onjuga ion
c
will b e deno ed by
c
.
Deni ion 7.2.1
Le
K
be a numbe eld and
(ρℓ)ℓ
p ime
a weakly ompa ible sys em o Galois ep esen a ions o
K
o ank
n
(see Deni ion 2.1.4 ). The sys em
(ρℓ)ℓ
p ime
is al led
a) o ally o dd, essen ially onjuga e sel -dual
in he ase
K
is o al ly eal o CM, i a weakly
ompa ible sys em
(εℓ)ℓ
p ime
o Galois ep esen a ions o
K+
o ank one wi h he ol lowing
p ope y exis s. Fo al l p ime numbe s
ℓ
he e is a non-degene a e, symme i pai ing
h·,·iℓ
on
Qℓ
n
, suh ha o al l
σ∈GK
and
x, y ∈Qℓ
n
we ha e
hρℓ(σ)x, ρℓ(cℓσcℓ)yiℓ=εℓ(σ)hx, yiℓ.
b) egula
, i o eah
τ:K//Q
we ha e
n
dis in
τ
-Hodge-Ta e numbe s (wi h
mul iplii y one).
A p owe ul o ol o p o ing he p o en ial au omo phy o sys ems o
ℓ
-adi Galois ep esen a ions
is he ollowing heo em.
Theo em 7.2.2 ( [BLGGT10℄, Thm.5.3.1 )
Suppose ha
K
is a CM (o o al ly eal) eld and ha
(ρℓ)ℓ
p ime
is an i eduible, o al ly odd,
essen ial ly onjuga e sel -dual, egula , weakly ompa ible sys em o
ℓ
-adi ep esen a ions o
K
.
Then he e is a ni e, CM (o o al ly eal), Galois ex ension
L|K
suh ha he es i ion o
(ρℓ)ℓ
p ime
o
GL
is au omo phi.
Fo
m∈N0
and a p ime numb e
ℓ
he
Qℓ
-shea
Hm,ℓ
on
A1
K
ons u ed in 3.3 (page 47), we ge a
lisse
Qℓ
-shea
i∗Hm,ℓ
on
A1
K {0,1}
by pulling bak along he inlusion
i:A1
K {0,1}//A1
K
.
I is by deni ion he enso p o du o a lisse
Zℓ
-shea and
Qℓ
o e
Zℓ
. This o esp onds by
Co olla y 2.2.12 o a on inuous ep esen a ion
ρi∗Hm,ℓ :π´e
1(A1
K {0,1})−→ GL(W)
on he s alk
W
o
i∗Hm,ℓ
a a hosen p oin in
A1
K {0,1}
, whih a o s h ough
Zℓ
. The enso
p o du o his ep esen a ion wi h he one dimensional ep esen a ion
de (ρi∗Hm,ℓ ) : π´e
1(A1
K {0,1})−→ {±1} ⊂ Qℓ
×,
whih is ob ained by aking he omp osi ion o he ep esen a ion and he de e minan a o s as
well h ough
SL(W)
. By a hoie o
x∈A1
K {0,1}
ao ding o he es i ions o Theo em 6.4.1
81
7. Po en ial Au omo phy o Sp eializa ions
and sp eializing
ρi∗Hm,ℓ ⊗de (ρi∗Hm,ℓ )
o
x
(see page 29)
ρi∗Hm,ℓ ⊗de (ρi∗Hm,ℓ )◦ιx:GK
ιx
//π´e
1(A1
K {0,1})−→ GL(W),
we ge an
ℓ
-adi Galois ep esen a ion o
K
o eah p ime numb e
ℓ
. Ou o hese maps we wan
o ons u a weakly ompa ible sys em by semi-simplia ion (see page 39) o his sys em.
Lemma 7.2.3
Fo
m∈N0
,
K=Q
and
x∈A1
Q {0,1}
, he sys em
ρm= (ρm,ℓ)ℓ
p ime
:= ((ρι∗Hm,ℓ ⊗de (ρι∗Hm,ℓ )) ◦ιx)ss ℓ
p ime
is a weakly ompa ible sys em o
ℓ
-adi Galois ep esen a ions, whih espe s an o hogonal
espe i ely symple i o m i
m
is e en espe i ely odd.
Le
m= 6
o
m
e en and
m≥12
o
x∈A1
Q {0,1}
, suh ha he e exis odd p ime numbe s
p, q
sa is ying
νp(x)<0
bu
ℓ∤νp(x)
and
νq(x−1) >0
bu
ℓ∤νq(x−1)
(see Theo em 6.4.1 ), his
sys em is i eduible.
P oo :
By Se ion 3.3 we ge ep esen a ions
ρm,ℓ :GK−→ SOm+1(Qℓ)
o
m
e en and
ρm,ℓ :GK−→ Spm+1(Qℓ)
o
m
o dd. The a ionali y and ompa ibili y o he sys em
ρm
is
a di e onsequene o [Ka 96℄, Theo em 5.5.4.
F om Co olla y 5.3.1 and Co olla y 5.3.2 we ha e ha
i∗Hm,ℓ ∼
=1
2(1 −σ) ke Rm(φX)∗Qℓ→Rm(φD)∗Qℓ.
We ha e go o d edu ion o almos all
ℓ
. The de i ed un o s a e  ys alline as ohomology and
he e o e he ke nel is  ys alline as his ommu es wi h mo phisms. As he p o je o
1
2(1 −σ)
is
algeb ai his is  ys alline.
The laim ollows om he  ys alline ompa ison isomo phism sine o
ℓ
la ge enough, he b e
W
is smo o h o e
Zℓ
i
ℓ
is la ge enough ( . [Fal89℄). Hene o
∈ΣK {0}
and
ℓ
equal o he
ha a e is i
k
, he ep esen a ion
ρℓ
is deRham in
and o almos all
e en  ys alline.
By Co olla y 5.4.5 we see ha he
τ
-Ho dge-Ta e numb e s a e independen o
ℓ
o any emb edding
τ:K//Q
.
Combining he esul s o his hap e , we see ha o e e y e en
m∈N0
he se o p ime numb e s
o whih
ρm,ℓ
is eduible is ni e. The e o e he Di ihle densi y o he o he p ime numb e s is
1
and he sys em
ρm= (ρm,ℓ)ℓ
p ime
o
ℓ
-adi Galois ep esen a ions, dened in 7.2, is i eduible
in he sense o Deni ion 2.1.4.

82
Theo em 7.2.4
Fo
m= 6
o
m∈N0
e en,
m≥12
and
K=Q
he i eduible, weakly ompa ible sys em
ρm= (ρm,ℓ)ℓ
p ime
o Galois ep esen a ions is po en ial ly au omo phi.
P oo :
Q
is a o ally eal eld and has exa ly one inni e plae
∞
, he usual absolu e
alue, wi h
c∞∈GQ
he usual omplex onjuga ion. We dene
(εℓ)ℓ
p ime
as he i ial
sys em, whih is weakly ompa ible. As b e o e, we ha e
im (ρm,ℓ)⊆SOm+1(Qℓ)
. The pai ing
hx, yi∞:= x ρm,ℓ(c∞)y
o
x, y ∈Qℓ
m+1
is non-degene a e, symme i and sa ises
ρm,ℓ(c∞) =ρm,ℓ(c∞)−1=ρm,ℓ(c−1
∞) = ρm,ℓ(c∞).
Fo
σ∈GQ
, we ha e
hρm,ℓ(σ)x, ρm,ℓ(c∞σc∞)yi∞=x ρm,ℓ(σ) ρm,ℓ(c∞)ρm,ℓ(c∞)
|{z }
=1
ρm,ℓ(σ)ρm,ℓ(c∞)y
=x ρm,ℓ(c∞)y= 1 ·hx, yi∞
and
ρm
is o ally o dd, essen ially onjuga e sel -dual. The egula i y is a di e onsequene o
Co olla y 5.4.5.
This shows ha o
m
e en we a e in he si ua ion o Theo em 7.2.2 and p o es ha
ρm
is
p o en ially au omo phi.

83
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Con a :
E-Mail: mihael.maie iw .uni-heidelb e g.de
89