On $\ell $-adic Galois periods, relations between coefficients of Galois representations on fundamental groups of a projective line minus a finite number of points

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COEFFICIENTS OF GALOIS REPRESENTATIONS ON

FUNDAMENTAL GROUPS OF A PROJECTIVE LINE

MINUS A FINITE NUMBER OF POINTS

by

Zdzisªaw Wojtkowiak

Contents

1. Introdution... 1

2. Shue relations oftype I(of iterated integralstype)... 4

3.

l

^-adi îteratedîntegrals ôn

P ¹ \ { 0, 1, ∞}

^evaluated ^at

1

^.^.^. ⁷

4.

Z /2

^,

Z /3

^and

Z /5

^-yle^relations^.^.^.^.^.^.^.^.^.^.^.^.^.^.^.^.^.^.^.^.^.^.^.^.^.^. ¹⁴

5. Shue relations of type II and III (multi zeta type and

regularized multi zetatype)... 15

Referenes... 16

Abstrat. TheoeientsoftheDrinfeldassoiator areknowntosatisfy

twokindofshuerelations. Therstrelationsomefromtheformulaforthe

multipliationofiteratedintegrals. Theseondonesomefromthemultipli-

ationofmultizetafuntions. Ouraimistostudytheanalogousrelationsfor

the Ihara-Drinfeldelement desribingthe ation of theGalois groupon the

étalefundamentalgroupof

P _Q ^¯ \ { 0, 1, ∞} .

1. Introdution

ThemixedHodgestrutureofthefundamentalgroupof

P ¹ ( C ) \{ 0, 1, ∞}

^based

at

01 →

^is ^desribed ^by^the^element

a

10 →

01 → (X, Y ) ∈ C{{ X, Y }}

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ommuting variable

X

^and

Y

^given ^expliitly ^by^the ^formula

a ¹⁰ ^→ →

01 (X, Y ) :=

1 + X

w=X ⁱ ¹ · Y ^j ¹ · .. · X ⁱⁿ · Y ^jn

Z 10 ^→ 01 →

− dz z − 1

j n

, − dz z

i n

, . . . , − dz z − 1

j 1

, − dz z

i 1 w

We briey sketh the denition of iterated integrals starting from tangential

points

01 →

^or

10 ^→

^. ^If

j _n > 0

^then^we ^integrate ^from

0

^. ^If

j _n = 0

^and

i _n > 0

^then

weintegratefrom

0

^the^iterated^integral

R _z

0 ( − logz) ⁱⁿ

i n ! ( ^−dz _z−1 ) ^j ⁿ ⁻ ¹ , . . . .

^Similarly^we

deneiteratedintegrals from

10 →

^to

z

^. ^Both^power^series^whose ^oeients^are

iteratedintegralsfrom

01 →

^to

z

^and^from

10 ^→

^to

z

âreât^setionsôf^the^prinipal

C{{ X, Y }} ^∗

^-bre ^bundle ^over

P ¹ ( C ) \ { 0, 1, ∞}

êquipped ^with ân întegrable

onnetiongiven byaoneform

dz

z ⊗ X + _z−1 ^dz ⊗ Y.

^Hene ^omparing^these^two

at setions whih dier bya onstant element we get iterated integrals from

01 →

^to

10 ^→

^and^the ^power^series

a ¹⁰ ^→ _→

01 (X, Y ).

^(See^[22℄^and^[23℄^for^more^details ^as

wellas[4℄and [16℄for dierent approah).

The element

a ¹⁰ ^→ _→

01 (X, Y )

^satises

Z /2, Z /3

^and

Z /5

^-yle ^relations ^(see ^[4℄).

Theoeients of

a ¹⁰ ^→ _→

01 (X, Y )

^satisfy^also ^three ^types ^of ^shue ^relations. ^The

shuerelationsoftypeIoriteratedintegralstypeareonsequeneoftheChen

formula

( Z b

a

ω 1 , . . . , ω p ) · ( Z b

a

ω p+1 , . . . , ω p+q ) = X

π∈Sh(p,q)

Z b a

ω _π(1) , . . . , ω _π(p+q)

foriteratedintegrals(see[2℄)whihremainstrueif

a

^or ^(and)

b

^are^tangential

points (see [22℄and [23℄). The shue relations of type II or multi zeta type

relations aregeneralization ofthefollowing identity

( X ∞ n=1

1 n ² ) · (

X ∞ m=1

1 m ³ ) =

X ∞ n>m=1

1 m ³ · n ² +

X ∞ m>n=1

1 n ² · m ³ +

X ∞ n=1

1 n ⁵ .

Theiterated integrals

Z z 0

− dz

z − 1 , . . . , − dz

z , ( − dz z − 1 ) ^k

aredivergent when

z → 1

^and

k > 0.

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Theiterated integrals

R ^→ 10

→ 01

− dz

z − 1 , . . . , ⁻ _z ^dz , ( ⁻ _z ₋ ^dz ₁ ) ^k

regularized theminsuhaway thatthe Chen formulastill holds (see [22℄, [23℄and [16℄) but the multi zeta

type relations do not hold.

Thesedivergent iteratedintegrals one an also regularized insuh awaythat

the multi zeta type relations hold. The shue relations of type III ompare

thesetwo regularizations.

The absolute Galois group

G _Q

^ats ^on ^the ^étale fundamental group

π 1 ( P ¹ Q ¯ \ { 0, 1, ∞} ; 01) ^→

^,^hene^we ^get^a ^Galois representation

ϕ : G _Q −→ Aut(π ₁ ( P ¹ Q ¯ \ { 0, 1, ∞} ; 01)), ^→

whih was studied by Ihara (see [10℄), Deligne (see [3℄), Grothendiek (see

[8℄) and other persons. The representation

ϕ

^is ^ompletely ^desribed ^by ^the

oyle

G _Q ∋ σ −→ f _p (σ) := p ⁻¹ · σ(p) ∈ π ₁ ( P ¹ Q ¯ \ { 0, 1, ∞} ; 01), ^→

where

p

^is ^the^anonial^path ^from

01 ^→

^to

10, ^→

^the^open ^interval

(0, 1).

We embed the pro-

l

^ompletion ^of

π ₁ ( P ¹ Q ¯ \ { 0, 1, ∞} ; 01) ^→

^into ^the

Q l

^-algebra

of non-ommutative formal power series

Q l {{ X, Y }}

ⁱⁿ ^two non-ommuting variables

X

^and

Y

^sending

x

^onto

e ^X

^and

y

^onto

e ^Y

^(see^[18℄). ^Hene ^we ^get

aGalois representation

ϕ _l : G _Q −→ Aut( Q l {{ X, Y }} ).

Let

Λ _p (X, Y )(σ) ∈ Q l {{ X, Y }}

^be ^the ^image ^of

f _p (σ)

ⁱⁿ

Q l {{ X, Y }} .

^The

representation

ϕ _l

îsômpletely ^desribed ^by^the ôyle

G _Q ∋ σ −→ Λ _p (X, Y )(σ) ∈ Q l {{ X, Y }} .

The element

f _p (σ)

^satises ^the

Z /2, Z /3

^and

Z /5

^-yle ^relations ^(see ^[11℄).

Hene after embedding of the pro-

l

^ompletion ^of

π ₁ ( P ¹ Q ¯ \ { 0, 1, ∞} ; 01) ^→

^into

Q l {{ X, Y }}

^we ^get ^some ^kind ^of

Z /2

^and

Z /3

^-yle ^relations ^satised ^by ^the

element

Λ _p (X, Y )(σ)

^(see ^setion³ ^of^this ^note).

The oyles

f _p

^and

Λ _p

^are respetively pro-nite and

l

^-adi ^analogs ^of ^the

Drinfeld assoiator

a ¹⁰ ^→ _→

01 (X, Y ).

^Hene ît îs â ^natural ^question ^to âsk îf ^the

elements

f p (σ)

^and

Λ _p (σ)

^satisfy^shue^relations ôf ^type Î, ÎIândÎII.

In this note we shall show that the oeients of the formal power series

Λ _p (X, Y )(σ)

^satisfy ^trivially ^shue ^relations ôf ^type Î ^beause ôf ^the ^very

denition of the embedding of the pro-

l

^ompletion ^of

π 1 ( P ¹ Q ¯ \ { 0, 1, ∞} ; 01) ^→

intothe

Q l

^-algebra^of non-ommutative formalpowerseries

Q l {{ X, Y }} .

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Theformalpowerseries

Λ p (X, Y )(σ)

^satises

Z /2, Z /3

^and

Z /5

^-yle^relations,

asitdoestheDrinfeldassoiator

a ^→ ¹⁰ _→

01 (X, Y ).

^However^there^are^some^dierenes.

Forexampleinthe

Z /3

^-yle^relation

Z = − X − Y

^is^replaed^by

− (X Y ) :=

− log(e ^X · e ^Y ).

Unfortunatelywedonotknowhowtoformulateshuerelations oftypeIIandIIIin

l

^-adiâse. ^However^whenône^passes^toâssoiated^graded

Lie algebras the

Z /2, Z /3

^and

Z /5

^-yle ^relations ^satised ^by ^the ^element

Λ _p (X, Y )(σ)

^beome^more^familierând^then^weân^formulateânalogsôf^shue

relations of type IIand III for this element. Though we do not knowhow to

provethem.

2. Shue relations of type I (of iterated integrals type)

Let

K

^be^a ^number^eld. ^Let

a ₁ , . . . , a _n

^belong^to

K

^and ^let

V := P ¹ K \ { a ₁ , . . . , a _n , ∞} .

Let

v

^be^a

K

^-point^of

V

ôrâ^tangential ^point^denedôver

K.

^Let

x 1 , . . . , x n

^be

geometri generators of

π ₁ (V K ¯ ; v)

^loops ^around

a ₁ , . . . , a _n

respetively (see [18℄).

Let

X := { X ₁ , . . . , X _n }

^, ^let

Q l {X}

^be ^a

Q l

^-algebra ^of polynomials in non- ommuting variables

X ₁ , . . . , X _n

^and ^let

Q ^l {{X}}

^be ^a

Q ^l

^-algebra ^of ^formal

powerseries innon-ommuting variables

X ₁ , . . . , X _n .

^Let

Lie( X )

^be ^a ^Lie ^al-

gebraof Liepolynomials in

Q l {X}

^and^let

L( X )

^beâ^Lieâlgebraôf ^Lie^formal

powerseries in

Q l {{X}} .

If

A

^and

B

^belong ^to ^a ^Lie ^algebra

L

^then ^we ^set

[A, B ⁽⁰⁾ ] := A

^and

[A, B ⁽ⁿ⁺¹⁾ ] := [[A, B ⁽ⁿ⁾ ], B]

^for

n ≥ 0.

Let

k : π ₁ (V K ¯ ; v) _l −→ Q l {{X}}

be a ontinous multipliative embedding of the pro-

l

^ompletion ^of

π 1 (V K ¯ ; v)

into

Q l {{X}}

^given^by

k(x i ) = e ^X ⁱ

for

i = 1, . . . , n.

Let

z

^be^also^a

K

^-pointôrâ^tangential ^point^denedôver

K.

^Let

γ

^be^an

l

^-adi

pathfrom

v

^to

z

^and^let

σ ∈ G _K .

^W^e^reall^the^denitions^from^[18℄^by^setting

f γ (σ) := γ ⁻ ¹ · σ(γ) ∈ π ₁ (V K ¯ ; v) _l

and

Λ _γ (σ) := k(f _γ (σ)) ∈ Q l {{X}} .

Let

W ( X )

^be â ^set ôf âll ^monomials ⁱⁿ non-ommuting variables

X ₁ , . . . , X _n .

Weinlude

1

ⁱⁿ

W ( X )

^. ^Then

W ( X )

îsâ ^baseôfâ

Q l

^-vetor ^spae

Q l {X} .

^Let

W ( X ) ^∗ := { w ^∗ | w ∈ W ( X ) }

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be the dual base, i.e.,

w ^∗ ∈ Hom( Q l {X} , Q l )

^and

w ^∗ (w ^′ ) = δ ^w _w ′

^for ^any

w ^′ ∈ W ( X ).

^W^e ^extend

w ^∗

^to ^a ^linear ^form ^on

Q l {{X}}

^setting

w ^∗ (Λ)

^to ^be ^equal

aoeient of

Λ

^at ^the^monomial

w.

^Then ^we ^an ^write

Λ = X

w ∈W ( X )

w ^∗ (Λ) · w

forany

Λ ∈ Q l {{X}} .

If

w ∈ W ( X )

^we ^denote^by

| w |

^the ^degree ^of ^the ^monomial

w.

^We ^dene ^the

groupof

(p, q)

^-shuepermutations ofthe set

p + q := { 1, 2, . . . , p + q }

^by

Sh(p, q) := { π ∈ S _p+q | ∀ a, b ∈ p + q, a < b ≤ p or p < a < b implies

π ⁻ ¹ (a) < π ⁻ ¹ (b) } .

If

w = X _i ₁ · X _i ₂ · . . . · X _i _p

^,

w ₁ = X _i _p+1 · X _i _p+2 · . . . · X _i _p+q

^and

π ∈ Sh(p, q)

^then

weset

π(w, w ₁ ) := X _i _π(1) · X _i _π(2) · . . . · X _i _π(p+q) .

Denition 2.1. We say that oeients of a formal power series

Λ ∈ Q l {{X}}

^satisfy^shue^relations ôf ^typeÎ îf

w ^∗ (Λ) · w ^∗ ₁ (Λ) = X

π ∈ Sh( | w | , | w 1 | )

π(w, w ₁ ) ^∗ (Λ)

forany

w, w ₁ ∈ W ( X ).

Let

Q l {{X}} ^⋄

^be ^a

Q l

^-vetor ^subspae ^of

Hom _Q _l ( Q l {{X}} , Q l )

^generated ^by

theset

W ( X ) ^∗ .

^Then ^the^formula

w ^∗ ⊙ w ₁ ^∗ := X

π ∈ Sh( | w | , | w 1 | )

π(w, w ₁ ) ^∗

denes a ommutative produt on

Q l {{X}} ^⋄

^and ^the ^obtained

Q l

^-algebra ^we

denoteby

Q l {{X}} ^⋄ _⊙ .

If

Λ ∈ Q ^l {{X}}

^then^we ^dene ^a^linear^map

ev _Λ : Q l {{X}} ^⋄ → Q l

bysetting

ev Λ (w ^∗ ) := w ^∗ (Λ).

Letus dene aontinous homomorphism of

Q l

^-algebras

∆ : Q l {{X}} → Q l {{X}} ⊗Q ˆ l {{X}}

bysetting

∆(X _i ) := X _i ⊗ 1 + 1 ⊗ X _i

ontopologialgenerators

X ₁ , X ₂ , . . . , X _n , . . .

^of

Q ^l {{X}} .

^W^e ^reall^the^lassi-

alresultof Ree.

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Theorem 2.2. (See [17℄) Let

Λ ∈ Q l {{X}}

^be ^suh ^that

1 ^∗ (Λ) = 1.

^The

followingonditions are equivalent:

i) the oeients of

Λ

^satisfy^shue ^relations ^of ^type ^I;

ii) log

Λ

^is^a ^Lie ^formal^power ^series ⁱⁿ

Q l {{X}}

^;

iii)

∆(logΛ) = logΛ ⊗ 1 + 1 ⊗ logΛ

^;

iv)

∆(Λ) = Λ ⊗ Λ

^;

v) the map

ev _Λ : Q ^l {{X}} ^⋄ ⊙ → Q ^l

^is ^a homomorphism of

Q ^l

^-algebras.

Proposition 2.3. The oeients of the formal power series

Λ _γ (σ) ∈ Q l {{X}}

^satisfy ^shue^relations ôf ^typê Î.

Proof. It follows fromthe Baker-Campbell-Hausdor formulathattheimage

oftheembedding

k

îsôntainedⁱⁿ^the^subgroupêxp

(L( X ))

^of^themultipliative groupof

Q l {{X}} .

^Hene^log

Λ _γ (σ) ∈ L( X )

^,^i.e. ^log

Λ _γ (σ)

^is^a^Lie^formal^power

series. Nowthe propositionfollows fromTheorem 2.2.

Iteratedintegrals satisfythe formula

2.3.1. Z

γ ⁻¹

ω ₁ , . . . , ω _n = ( − 1) ⁿ Z

γ

ω _n , . . . , ω ₁

(see[2℄). Ournext aim is to show theanalogue of this formula for thepower

series

Λ _γ (σ) ∈ Q l {{X}}

^(see ^[19℄,^pages ¹¹⁸ ^and ^119,^where ^this ^problem^was

raised).

Thepath

γ ⁻¹

^is^a ^path ^from

z

^to

v

^,^hene

f _γ − 1 (σ) ∈ π ₁ (V K ¯ ; z).

Lemma 2.4. (see [18℄Lemma 1.0.6.)

We have

(γ ⁻ ¹ · f _γ − 1 (σ) · γ) · f γ (σ) = 1.

The elements

x ^′ _i := γ · x _i · γ ⁻ ¹

^for

i = 1, . . . , n

^are ^geometrial ^generators ^of

π 1 (V K ¯ ; z).

^Let

k ^′ : π ₁ (V K ¯ ; z) _l −→ Q l {{X}}

beaontinousmultipliativeembeddinggivenby

k ^′ (x ^′ _i ) = e ^X ⁱ

^for

i = 1, . . . , n.

Let

Λ _γ − 1 (σ) := k ^′ (f _γ − 1 (σ)).

Lemma 2.5. We have

Λ _γ −1 (σ) · Λ _γ (σ) = 1.

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Proof. Observethat

k(x i ) = k ^′ (x ^′ _i )

^for

i = 1, . . . , n.

^Hene

k(γ ⁻ ¹ · f _γ − 1 (σ) · γ) = k ^′ (f _γ − 1 (σ)).

^Therefore ^it^follows ^from^Lemma ^2.4^that

Λ _γ − 1 (σ) · Λ _γ (σ) = 1.

Let

t : Q ^l {{X}} → Q ^l {{X}}

^be ^a ^ontinous ^linear^mapping ^given ^by

t(X _i ₁ · . . . · X _i _m ) = X _i _m · . . . · X _i ₁

onelements of

W ( X ).

^Then

t

^is^an anti-automorphism of

Q l

^-algebras,^i.e.

t(a · b) = t(b) · t(a).

Proposition 2.6. Let

w ∈ W ( X ).

^Then ^we ^have

w ^∗ (Λ _γ −1 (σ)) = ( − 1) ^| ^w ^| (t(w)) ^∗ (Λ _γ (σ)).

Proof. ItfollowsfromLemma2.5that

Λ _γ −1 (σ) = (Λ _γ (σ)) ⁻ ¹ .

^The^oeients

ofthe powerseries

Λ _γ (σ)

^satisfy^shue^relations ^of^type ^I^byProposition 2.3.

Heneitfollows from[17℄Theorem 2.6that

w ^∗ (Λ _γ − 1 (σ)) = w ^∗ (Λ _γ (σ) ⁻ ¹ ) = ( − 1) ^| ^w ^| (t(w)) ^∗ (Λ _γ (σ)).

Remark 2.7. If

Λ _γ (z, v)

îs â ât ^setion ôf ^the ânonial pro-nilpotent onnetionon

V ( C )

^along^a^path

γ

^from

v

^to

z

^then ^the^formula^2.3.1 ^an ^be

writtenin theform

w ^∗ (Λ _γ − 1 (v, z)) = ( − 1) ^| ^w ^| (t(w)) ^∗ (Λ _γ (z, v)).

3.

l

^-adi îterated întegrals ôn

P ¹ \ { 0, 1, ∞}

^evaluated ^at

1

The oeient of the Drinfeld assoiator

a ¹⁰ ^→ _→

01 (X, Y )

^at

X ⁿ ⁻ ¹ Y

^is ^equal

R 1

0 − dz

z − 1 , ⁻ _z ^dz , . . . , ⁻ _z ^dz = ( − 1) ⁿ⁻¹ R 1

0 − log(1 − z) ^dz _z , . . . , ^dz _z = ( − 1) ⁿ⁻¹ Li _n (1) = ( − 1) ⁿ⁻¹ ζ(n).

^It ^was ^shown ^by ^Euler ^that

ζ(2k) = ( − 1) ^{k−1 (2} ₂ _· _(2k)! ^π) ^2k b _2k =

− ^(2πi) ₂ _· _(2k)! ^2k b _2k

^,^where

b ₁ = − ¹ ₂

^,

b ₂ = ¹ ₆

^,

. . .

^are^Bernoulli^numbers. ^Using

Z /2

^and

Z /3

^symmetries ^of

P ¹ \ { 0, 1, ∞}

^and ^the Baker-Campbell-Hausdor formula onean reprove theEulerresult (see[3℄).

We shallprove herethe analogous resultinthe

l

^-adi ^setting.

Let

V := P ¹ _Q \{ 0, 1, ∞} .

^Let

g : V → V

^and

h : V → V

^be^given^by

g(z) = 1 − z

and

h(z) = _z ₋ ^z ₁ .

^Let

p

^be^the^anonial^path^from

01 ^→

^to

10 ^→

^the^interval

(0, 1).

Let

x 0

^,

y 1

^and

z _∞

^be ^loops ^around

0

^,

1

^and

∞

^based ^at

01 ^→

^,

10 ^→

^and

∞ ^→ 0

respetively (see Piture 1).

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b

x ₀ y ₁

z _∞

0 1 0 1

∞ 0

b

Picture 1

Letus set

x := x ₀ y := p ⁻¹ · y ₁ · p and z := s ⁻¹ · h(p) ⁻¹ · z _∞ · h(p) · s,

where

s

^is^a ^path ^from

01 ^→

^to

0 ^→ ∞

^as^on ^Piture ^2.

∞ 0 b 1

s

Picture 2

Observe that

x · y · z = 1

ⁱⁿ

π ₁ (V Q ¯ ; 01) ^→

^,^hene

3.1. s ⁻¹ · h(p) ⁻¹ · z _∞ · h(p) · s = (p ⁻¹ · y ₁ · p) ⁻¹ · x ⁻ ₀ ¹

(seePiture 3).

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b b

∞ 0

0 1

Picture 3

We reallthatfor any

σ ∈ G _Q

^,

f _p (σ) := p ⁻¹ · σ(p).

Theelement

f _p

^has^been^studied^byÎhara^(see^[11℄)ândâlso^by^Nakamuraând

Shneps (see [14℄). Observe that

g(p) = p ⁻¹ .

^Hene ^we ^get

g _∗ (f _p ) = f _g(p) = f _p − 1 = p · f ⁻ _p ¹ · p ⁻ ¹

^,^i.e.

p ⁻ ¹ · (g _∗ (f _p )) · p = f ⁻ _p ¹ .

^The ^last^equality^implies

3.2. f _p (y, x) = f _p (x, y) ⁻ ¹ .

(This is of ourse the famous

Z /2

^-yle ^relation ^of ^Drinfeld ^(see ^for ^example

[11℄). Observe that

s ⁻¹ · h(x) · s = x and s ⁻¹ · h(y) · s = z.

Thisimplies

3.3. s ⁻ ¹ · f _h(p) · s = s ⁻ ¹ · h _∗ (f _p ) · s = f p (x, z).

Let

χ : G _Q −→ Z ^∗ l

betheylotomiharater. Itfollowsfrom[18℄Lemma1.0.6andtheequality

3.3that

3.4. f _(p ₋₁ _· _y −1

1 · p · x ⁻¹ ₀ ) (σ) = x · y · (f _p (x, y)(σ)) ⁻¹ · y ^−χ(σ) · (f _p (x, y)(σ)) · x ^−χ(σ)

and

3.5. f _(s − 1 · h(p) ⁻ ¹ · z _∞ · h(p) · s) (σ) = z ⁻ ¹ · x ^(χ(σ) ² ⁻ ¹⁾ · (f _p (x, z)(σ)) ⁻ ¹ · z ^χ(σ) · f p (x, z)(σ) · x ⁻ ^(χ(σ) ² ⁻ ¹⁾ .

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Let

k : π ₁ (V Q ¯ ; 01) ^→ _l → Q l {{ X, Y }}

^be ^a ^ontinous multipliative embedding given by

k(x) = e ^X

^and

k(y) = e ^Y .

^Then

Λ _p (X, Y )(σ) := k(f _p (x, y)(σ)).

Itfollows from3.2that

3.6. Λ _p (Y, X ) = Λ _p (X, Y ) ⁻¹ .

Let

I ₂

^(resp.

J ₂

⁾ ^be â ^losed ^Lie îdeal ôf

L(X, Y )

^generated ^by ^Lie ^brakets

with2or more

Y

^'s^(resp.

X

^'s). ^We^reall ^that

logΛ _p (X, Y ) ≡

X ∞ n=2

l _n (1)[Y, X ⁽ⁿ⁻¹⁾ ] mod I ₂

by the very denition of

l

^-adi polylogarithms (see [19℄). It follows from 3.6 that

3.7. logΛ _p (X, Y ) ≡ l ₂ (1)[Y, X ]+

X ∞ n=2

l _n (1)[Y, X ⁽ⁿ⁻¹⁾ ]+

X ∞ n=3

− l _n (1)[X, Y ⁽ⁿ⁻¹⁾ ] mod I ₂ ∩ J ₂ .

Letus set

X Y := log (e ^X · e ^Y ).

The right hand sides of 3.4 and 3.5 are equal by 3.1. Hene applying

k

^and

thentakinglogarithm we gettheequality

3.8. ( − logΛ _p (X, Y )) ( − χ · Y ) (logΛ _p (X, Y )) ( − χ · X) =

( χ − 1

2 X) ( − logΛ _p (X, − (X Y )) ( − χ(X Y )) (logΛ _p (X, − (X Y )) ( − χ − 1 2 X).

Itfollows fromtheformulas

( − X) Y X = Y + X ∞ n=1

1 n! [Y, X ⁽ⁿ⁾ ]

(see[13℄) and

X Y ≡ X+Y + 1

2 [X, Y ]+

X ∞ n=1

b _2n

(2n)! [Y, X ⁽²ⁿ⁾ ]+

X ∞ n=1

b _2n

(2n)! [X, Y ⁽²ⁿ⁾ ] mod I ₂ ∩ J ₂

(see[1℄or [6℄)and fromthe ongruenes 3.7and

logΛ _p (X, − (X Y )) ≡ X ∞ n=2

( − 1) ⁿ l _n (1)[X, Y ⁽ⁿ ⁻ ¹⁾ ] mod J ₂

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3.9. − χ · X − χ · Y − X ∞ n=2

χ · l n (1)[X, Y ⁽ⁿ⁾ ] − 1

2 χ ² [X, Y ] − X ∞ k=1

b _2k

(2k)! χ ^2k+1 [X, Y ^(2k) ] mod J 2

andthe right handside ofthe equality3.8is ongruent to

3.10. − χ · X − χ · Y − 1

2 χ ² [X, Y ] − χ X ∞ k=1

b _2k

(2k)! [X, Y ^(2k) ]+

X ∞ n=2

( − 1) ⁿ χ · l _n (1)[X, Y ⁽ⁿ⁾ ] mod J ₂

Comparing 3.9and3.10 we getthe following result.

Proposition 3.1.

l _2k (1) = − b _2k

2 · (2k)! (χ ^2k − 1).

(Thisresultisstated withoutproofin[11℄andinreferenesgiven byIhara in

[11℄only

l _n (1)

^for

n

ôddîs âlulated.)

The

l

^-adi polylogarithm

l _2k (1)

^is ^a ^funtion ^from

G _Q

^to

Q l (2k)

^, ^hene ^its

ohomology lass in

H ¹ (G _Q ; Q l (2k))

^is ^zero. ^However ^the ^related ^funtion

studied in [15℄ and in [21℄, whih takes values in

Z l (2k)

^need ^not ^be ^zero ⁱⁿ

H ¹ (G _Q ; Z ^l (2k))

^. ^We ^shall ^show ^below ^that ^it ^determines ^a ^torsion ^lass ⁱⁿ

H ¹ (G _Q ; Z l (2k))

ând ^we ^shallâlulate îts ôrder^(see âlso^[3℄).

We start by realling the arithmeti formula for

l

^-adi polylogarithms from [15℄. Let

z

^be ^a

Q

^-point ^of

V

ôr â ^tangential ^point ^dened ôver

Q .

^Let

q

^be

apath on

V Q ¯

^from

01 →

^to

z.

^Let

ϕ _n : π ₁ (V Q ¯ ; 01) ^→ _l → Z /l ⁿ

^be ^a homomorphism

given by

ϕ _n (x) = 1

^and

ϕ _n (y) = 0.

^Let ^us^set

H _n := ker(ϕ _n : π ₁ (V Q ¯ ; 01) ^→ _l → Z /l ⁿ ).

Thenwe have

3.11. x ⁻ ^l(z) ^q ^(σ) · f _q (σ) ≡

l ⁿ Y −1 i=0

(x ⁱ · y · x ⁻ ⁱ ) ^κ(1 ⁻ ^ξ ⁱ ^ln ^z

ln 1 )(σ) mod (H _n , H _n )

where

(H n , H n )

îs^theômmutator ^subgroup ôf

H n

^and^where ^the^oeients

κ(1 − ξ ⁱ _l n z ^ln ¹ )(σ) ∈ Z ^l

^are^dened ^by ^the ^formula

ξ ^κ(1−ξ ^ln ⁱ ^z

1 ln )(σ)

l ^k = σ((1 − ξ _l ^iχ(σ) n ⁻ ¹ · z ^ln ¹ ) ^lk ¹ ) (1 − ξ ^i+l(z) _l n ^q ^(σ) · z ^ln ¹ ) ^lk ¹

.

(12)

After the embedding

k

^of

π ₁ (V Q ¯ ; 01) ^→

^into

Q l {{ X, Y }}

^and ^then ^taking ^loga-

rithmwe get

3.12. ( − l(z) _q (σ)X) logΛ _q (X, Y )(σ) ≡ X ∞

m=0

( − 1) ^m m! (

l X ⁿ −1 i=0

i ^m κ(1 − ξ _l ⁱ n z ^ln ¹ )(σ))[Y, X ^(m) ] mod log(k((H _n , H _n ))).

Letus denoteby

c _m+1 (z)(σ)

^theôeient ôf^the ^left^hand^side ôf^3.12ât ^the

term

[Y, X ^(m) ].

^Then^we^have

3.13. c _m+1 (z)(σ) ≡ ( − 1) ^m m! (

l X ⁿ − 1 i=0

i ^m κ(1 − ξ _l ⁱ n z ^ln ¹ )(σ))[Y, X ^(m) ] mod l ⁿ ⁻ ^v ^l ^((m ⁻ ^1)!) .

(The formula on the right hand side is related to the Gabberonstrution of

theHeisenbergoverof

P ¹ \ { 0, 1, ∞}

^(see^[5℄).

It isshown in[15℄that

κ(z)(σ) := { κ(1 − ξ ⁱ _l n z ^ln ¹ )(σ) } i ∈Z /l ⁿ , n ∈N

^is^a ^measure

on

Z l .

^W^e ^set

ℓ _m+1 (z)(σ) :=

Z

Z l

x ^m dκ(z)(σ).

Therefore

ℓ m+1 (z)

^is^a ^funtion^from

G Q ¯

^to

Z l .

^It^follows ^from^3.13^that

3.14. c _m+1 (z)(σ) = ( − 1) ^m

m! ℓ _m+1 (z)(σ).

Let

q

^be^the ^path

p

^from

01 ^→

^to

10 ^→

^. ^Then

l(1)(σ) = 0.

^Hene ^we^get

3.15. c _m+1 (1)(σ) = l _m+1 (1)(σ).

Theorem 3.2. i) We have

ℓ _2k (1) = ₂ _· ^b _(2k) ^2k · (χ ^2k − 1)

ⁱⁿ

Z ¹ (G _Q ; Z l (2k))

^;

ii) Let us write

b _2k

2 · (2k) = ^a _b

^, ^where

a, b ∈ Z

^are ^relatively ^prime. ^The ^lass ^of

the oyle

ℓ _2k (1)

îs â ^torsionêlement ôf

H ¹ (G _Q ; Z ^l (2k))

^of ^order

l ^v ^l ^(b) .

iii) The lass of the oyle

ℓ _2k (1)

îs â ^torsion êlement ôf ^maximal ôrder ⁱⁿ

H ¹ (G _Q ; Z l (2k)).

Proof. We have already observed that

ℓ _2k (1)

^is^a ^funtion ^from

G Q ¯

^to

Z l .

^It

followsimmediatelyfromProposition3.1and theequalities3.14and3.15that

ℓ _2k (1) = _2·2k ^b ^2k (χ ^2k − 1).

^Hene

ℓ _2k (1)

^is ⁱⁿ

Z ¹ (G _Q ; Z l (2k))

^. ^The ^point ⁱⁱ⁾ ^is ^a

onsequeneimmediate ofthe point i).

Let us suppose that

l

îs ân ôdd ^prime. ^The ^ylotomi ^harater

χ : G _Q →

Z ^∗ l = (1 + l Z l ) × µ _l ₋ ₁

îs ^surjetive. ^Therefore ^there âre ^torsion êlements ⁱⁿ

(13)

H ¹ (G _Q ; Z l (2k))

îfând ônly îf

2k ≡ 0

^mod

(l − 1)

^. ^Let ^us ^suppose ^that

2k = (l − 1) · l ^m ⁻ ¹ · q

^with

(q, l) = 1.

^Then ^it ^follows immediately that

χ ^2k − 1

^is

divisibleby

l ^m

^but^not ^by

l ^m+1 .

^On^the^other ^side^Von^Staudt^Congruee^(see

[12℄, hapter 2, setion 2, Corollary 2) implies that

v _l ( ₂ ^b _· ^2k _2k ) = − m.

^Hene

ℓ _2k (1)

îsâ^torsion êlement ⁱⁿ

H ¹ (G _Q ; Z l (2k))

^of ^maximal^order.

Theproof inase

l = 2

îs ^similarând ^we ômit ît.

Asa orollaryweget awell known resultabout the Bernoullinumbers.

Corollary 3.3. Let

2k 6≡ 0

^mo^d

(l − 1).

^Then

v _l ( ₂ ^b _· ^2k _2k ) ≥ 0.

Proof. If

2k 6≡ 0

^mod

l − 1

^then

χ ^2k − 1

^is^not ^divisible^by

l

^. ^Hene ^the^point

i)ofthe theorem impliesthat

v _l ( ₂ ^b _· ^2k _2k ) ≥ 0.

Let

s : Q l {{ X, Y }} → Q l {{ X, Y }}

^be^an isomorphism of

Q l

^-algebras ^given^by

s(X) = Y

^and

s(Y ) = X.

^We ^reall ^from ^setion ¹ ^that

t : Q l {{ X, Y }} → Q l {{ X, Y }}

^is ^ananti-isomorphism. Letus set

τ := s ◦ t.

Nowweshallshowthe analogofthedualitytheoremformulti zetavalues(see

[9℄page 53).

w ∈ W (X, Y ).

^We ^have

w ^∗ (Λ _p (X, Y )(σ)) = ( − 1) ^|w| (τ (w)) ^∗ (Λ _p (X, Y )(σ)).

Proof. It followsfrom Proposition2.6 andLemma 2.5that

w ^∗ (Λ p (X, Y )(σ)) = ( − 1) ^| ^w ^| (t(w)) ^∗ ((Λ p (X, Y )(σ)) ⁻ ¹ ).

Itfollows from

Z /2

^-yle ^relation ^that

( − 1) ^| ^w ^| (t(w)) ^∗ ((Λ _p (X, Y )(σ)) ⁻ ¹ ) = ( − 1) ^| ^w ^| (t(w)) ^∗ (Λ _p (Y, X)(σ)).

Observethatforany

u ∈ W (X, Y )

^we^have

u ^∗ (Λ _p (Y, X)(σ)) = s(u) ^∗ (Λ _p (X, Y )(σ).

Henewe get

w ^∗ (Λ _p (X, Y )(σ)) = ( − 1) ^| ^w ^| (τ (w)) ^∗ (Λ _p (X, Y )(σ)).

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4.

Z /2

^,

Z /3

^and

Z /5

^-yle ^relations

Letus hose generatorsof

π ₁ (V Q ¯ ; 01) ^→

^as^on^the^piture

b

b b

x

z

∞ 0 1 y

Picture 4

Thenwe have

x · y · z = 1.

b b

b

∞ 0 p 1

Picture 5

Calulatingtheelement

f

^along^the^path ^on^the ^Piture⁵^we^get^the

Z /3

^-yle

relationof Drinfeldin

π ₁ (V Q ¯ ; 01) ^→

x ^χ ⁻ ² ¹ · f _p (z, x) · z ^χ ⁻ ² ¹ · f _p (y, z) · y ^χ ⁻ ² ¹ · f _p (x, y) = 1.

(seefor example [11℄). After theembedding

k

^of

π ₁ (V Q ¯ ; 01) ^→

^into

Q l {{ X, Y }}

the

Z /2

^and

Z /3

^-yle ^relations ^have^the^following ^form

∗ 2 Λ _p (X, Y ) · Λ _p (Y, X ) = 1,

and

∗ 3

e ^χ−1 ² ^X · Λ _p ( − (X Y ), X) · e ^χ−1 ² ⁽ ⁻ ^(X ^Y ⁾⁾ · Λ _p (Y, − (X Y )) · e ^χ−1 ² ^Y · Λ _p (X, Y ) = 1.

Inthismoment weannot expetthatshuerelations oftype IIand IIIhave

thesame formasinDeRhamBettisetting(see [16℄). However whenwepass

to assoiated graded Lie algebras the situation beomes more familiar. We

reallthatwiththeationofaGaloisgroupon

π 1

ôrônâ^torsorôf^paths^there

isassoiateda ltration oftheGalois group (see [18℄setion3).

(15)

Let

I := ker(ε : Q l {{ X, Y }} −→ Q l )

^be^an augmentation ideal.

Inthe aseof theation of

G Q ¯

^on

π ₁ (V Q ¯ ; 01) ^→

ôr ôn ^the^torsorôf ^paths ôn

V Q ¯

from

01 →

^to

10 ^→

^this ^ltration ^is^dened ^as^follows

G 0 ( Q ) := G Q ¯ , G 1 ( Q ) := G Q ¯ (µ l∞ ) ,

G _n ( Q ) := { σ ∈ G Q ¯ (µ l∞ ) | Λ _p (X, Y )(σ) ≡ 1 mod I ⁿ }

for

n > 1.

Ifwe passto a gradedLiealgebra

LieG Q ¯ :=

M ∞ i=1

G _i ( Q )/G _i+1 ( Q )

⊗ Q

assoiated withthe ation of

G Q ¯

^on

π ₁ (V Q ¯ ; 01) ^→

^the ^equations

∗ 2

^and

∗ 3

^have

morefamilierform aswe an seeinthenext proposition.

σ ∈ G _n ( Q ).

^Let^us ^set

Z := − X − Y.

^Then^we ^have

∗∗ 2 Λ _p (X, Y )(σ) + Λ _p (Y, X)(σ) = 0 mod I ⁿ⁺¹ ,

∗∗ 3 Λ p (Z, Y )(σ) + Λ p (Y, Z)(σ) + Λ p (X, Y ) = 0 mod I ⁿ⁺¹ .

The

Z 5

^-yle ^relation ^an ^also ^be ^written ⁱⁿ ^this ^form ^but ^we ^do ^not ^state

it here in order not to make this paper too heavy. In the next setion we

formulate shuerelations of typeIIandIIIfor oeients of

Λ _p (X, Y ).

5. Shue relations of type II and III (multi zeta type and

regularized multi zeta type)

We start thissetion byrealling some notations from[16℄.

Let

Y := { y 1 , y 2 , . . . , y n , . . . }

^be ^a^set^of non-ommuting variables. We setdeg

y _i := i.

^Then^the^degree^of^a^monomial

y _i ₁ · y _i ₂ · . . . · y _i _k

^is

P _k

α=1 i _α .

^We^denote^by

Q << Y >>

^the

Q

^-algebra^of^formal^power^seriesⁱⁿnon-ommutingvariables

y ₁ , y ₂ , . . . , y _n , . . .

^Wê ^dene âôprodutôf

Q

^-algebras

∆ _∗ : Q << Y >> −→ Q << Y >> ⊗Q ˆ << Y >>

bysetting

∆ _∗ (y _n ) :=

X n i=0

y _n ₋ _i ⊗ y _i

ongenerators (withtheonvention that

y ₀ = 1

^).

(16)

Let

W (X, Y ) _X _ˆ

^be^a^subset ^of

W (X, Y )

^ontaining

1

^and^all^monomials ^whose

last term is

Y

^. ^Let

Q{{ X, Y }} X ˆ

^be ^a ^subalgebra ^of

Q{{ X, Y }}

^generated

topologially bythe set

W (X, Y ) _X _ˆ

^.

We identify the

Q

^-algebra

Q << Y >>

^with ^the ^subalgebra

Q{{ X, Y }} X ˆ

^of

Q{{ X, Y }}

^sending

y _i → X ⁱ⁻¹ Y

for

i = 1, 2, . . .

^. ^Hene^we ^get^a ^oprodut

∆ _∗ : Q{{ X, Y }} X ˆ → Q{{ X, Y }} X ˆ ⊗Q{{ ˆ X, Y }} X ˆ ,

whih indues a ommutative produt

⋆ : ( Q{{ X, Y }} X ˆ ) ^⋄ ⊗ ( Q{{ X, Y }} X ˆ ) ^⋄ −→ ( Q{{ X, Y }} X ˆ ) ^⋄ .

The

Q

^-vetor ^spae

( Q{{ X, Y }} X ˆ ) ^⋄

^equipped ^with ^the ^produt

⋆

^we ^de-

note by

( Q{{ X, Y }} X ˆ ) ^⋄ _⋆

^. ^If ^we ^tensored ^by

Q l

^or ^by

C

^we ^obtain ^algebras

( Q ^l {{ X, Y }} X ˆ ) ^⋄

^and

( C{{ X, Y }} X ˆ ) ^⋄ _⋆ .

Conjeture 5.1. (assoiated graded Liealgebraversionofshuerelations

oftype IIand III.)Let

σ ∈ G _n ( Q ).

^Then^for ^any

w, w ₁ ∈ W (X, Y ) _X _ˆ

^suh ^that

| w | + | w ₁ | = n

^we ^have

X

u∈W(X,Y ) X ˆ

( − 1) ^deg ^Y ^(u) (w ^∗ ⋆ w ^∗ ₁ )(u) · u ^∗ (Λ p (X, Y )(σ)) = 0 mod I ⁿ⁺¹ .

We point out that the shue relations dedued from produts of multi zeta

funtionsarealso studied ingreater generality in[7℄.

Referenes

[1℄ N.Bourbaki,Elémentsdemathématiques,GroupesetalgèbresdeLie,Diusion

C.C.L.S.Paris1972.

[2℄ Kuo-Tsai Chen, Algebra of iterated path integrals and fundamental groups,

Transationsof theAmerian Mathematial Soiety, Vol. 156,May 1971, pp.359-

379.

[3℄ P.Deligne,Legroupefondamentaldeladroiteprojetivemoinstroispoints,in

Galoisgroupsover

Q

^(ed.^Y.^Ihara,^K.^Ribet^and^J.-P.^Serre),MathematialSienes Researh InstitutePubliations 16(1989),pp.79-297.

[4℄ V.G.Drinfeld,Onquasi-triangulatedquasi-Hopfalgebrasandonagroupthat

isloselyonnetedwith

Gal( ¯ Q / Q )

^,^Leningrad^Math. ^J.²^(4),^(1991),^pp.^829-860.

[5℄ O. Gabber,notesofDeligne,notpublished.

(17)

[6℄ K. Goldberg, The formalpowerseries for log

e ^x e ^y

^, ^Duke^Math. ^J., ²³ ^(1956),

pp.13-21.

[7℄ A. B. Gonharov,Thedihedral LiealgebraandGaloissymmetriesof

π ^(l) ₁ P ¹ \ ( { 0, ∞} ∪ µ N )

^,^Duke^Math. ^J.^Vol.^110,^No.^3,^2001,^pp.^397-487.

[8℄ A.Grothendiek,Esquissed'unprogramme,inGeometriGaloisAtions,(ed.

L.Shneps andP. Lohak),L.M.S.Leture NoteSeries243,CambridgeUniversity

Press,pp.5-48.

[9℄ M. E.Hoffman, AlgebraiAspets ofMultipleZetaValues,inZetaFuntions,

TopologyandQuantum Physis,Springer,2005,pp.51-73.

[10℄ Y. Ihara,Pronitebraidgroups,Galoisrepresentationsandomplexmultipli-

ations,Annals of Math.123(1986),pp.43-106.

[11℄ Y. Ihara,Braids,GaloisGroupsandSomeArithmeti Funtions,Pro.ofthe

Int.CongressofMath. Kyoto1990,Springer-Verlagpp.99-120.

[12℄ S.Lang, CylotomiFieldsIandII,1990,Springer-VerlagNewYorkIn.

[13℄ W. Magnus, A. Karrass, D. Solitar, Combinatorial Group Theory, Pure

andApplied MathematisXIII,IntersienePublishers(1966).

[14℄ H. Nakamura, L. Shneps, On a subgroup of the Grothendiek-teihmuller

groupating on thetowerof pronite Teihmuller modular groups, Invent. Math.

141,(2000),pp.503-560.

[15℄ H. Nakamura,Z. Wojtkowiak,On theexpliitformula for

ℓ

^-adi^polyloga-

rithms,inArithmetiFundamentalGroupsandNonommutativeAlgebra,Pro. of

Symposia inPureMath.70,AMS 2002,pp.285-294.

[16℄ G. Rainet, Doubles mélange des polylogarithmes multiples aux raines de

l'unité,Publ.Math. Inst.Hautes EtudesSi.No.95(2002),pp.185-231.

[17℄ R.Ree,Lieelementsandanalgebraassoiatedwithshues,Ann.ofMath.(2)

68(1958),pp.210-220.

[18℄ Z. Wojtkowiak,On

ℓ

^-adi îteratedîntegrals, Î Ânalog ôf ^Zagier ^Conjeture,

NagoyaMath.Journal,Vol.176(2004),pp.113-158.

[19℄ Z.Wojtkowiak,On

ℓ

^-adiîteratedîntegrals,ÎI^Fûntionalêquationsând

ℓ

^-adi

polylogarithms,NagoyaMath. Journal,Vol.177(2005),pp.117-153.

[20℄ Z.Wojtkowiak,On

ℓ

^-adiîteratedîntegrals,ÎII^Galoisâtionsônfundamental groups,NagoyaMath. Journal,Vol.178(2005),pp.1-36.

[21℄ Z. Wojtkowiak, A Note on Funtional Equations of

ℓ

^-adi Polylogarithms, JournaloftheInst.ofMath. Jussieu(2004)(3),pp.461-471.

[22℄ Z.Wojtkowiak,MonodromyofIteratedIntegralsandNonabelianUnipotent

Periods, in GeometriGaloisAtions LondonMath. So.Leture Note Series243,

CambridgeUniversityPress1997,219-289.

[23℄ Z. Wojtkowiak,MixedHodgeStruturesand IteratedIntegralsI inMotives,

Polylogarithms and HodgeTheory. Part I: Motives and Polylogarithms. F. Bogo-

molovandL.Katzarkov,eds.InternationalPressLetureSeriesVol.3,partI,2002,

pp.121-208.

(18)

Zdzisªaw Wojtkowiak, Université deNie-Sophia Antipolis, Département de Mathé-

matiques,LaboratoireJeanAlexandreDieudonné,U.R.A. auC.N.R.S.,N

o

168,Par

ValroseB.P.N

o

71,06108NieCedex2,Frane

•

^E-mail^:wojtkowmath.unie.fr

On $\ell $-adic Galois periods, relations between coefficients of Galois representations on fundamental groups of a projective line minus a finite number of points

l

P 1 \ { 0, 1, ∞}

1

Z /2

Z /3

Z /5

P Q ¯ \ { 0, 1, ∞} .

P 1 ( C ) \{ 0, 1, ∞}

01 →

a

10 →

01 → (X, Y ) ∈ C{{ X, Y }}

X

Y

a 10 → →

01 (X, Y ) :=

1 + X

w=X i 1 · Y j 1 · .. · X in · Y jn

Z 10 → 01 →

− dz z − 1

j n

, − dz z

i n

, . . . , − dz z − 1

j 1

, − dz z

i 1 w

01 →

10 →

j n > 0

0

j n = 0

i n > 0

0

R z

0

( − logz) in

i n ! ( −dz z−1 ) j n − 1 , . . . .

10 →

z

01 →

z

10 →

z

C{{ X, Y }} ∗

P 1 ( C ) \ { 0, 1, ∞}

dz

z ⊗ X + z−1 dz ⊗ Y.

01 →

10 →

a 10 → →

01 (X, Y ).

a 10 → →

01 (X, Y )

Z /2, Z /3

Z /5

a 10 → →

01 (X, Y )

( Z b

a

ω 1 , . . . , ω p ) · ( Z b

a

ω p+1 , . . . , ω p+q ) = X

π∈Sh(p,q)

Z b a

ω π(1) , . . . , ω π(p+q)

a

b

( X ∞ n=1

1 n 2 ) · (

X ∞ m=1

1 m 3 ) =

X ∞ n>m=1

1 m 3 · n 2 +

X ∞ m>n=1

1 n 2 · m 3 +

X ∞ n=1

1 n 5 .

Z z 0

P ¹ \ { 0, 1, ∞}

P _Q ^¯ \ { 0, 1, ∞} .

P ¹ ( C ) \{ 0, 1, ∞}

a ¹⁰ ^→ →

w=X ⁱ ¹ · Y ^j ¹ · .. · X ⁱⁿ · Y ^jn

Z 10 ^→ 01 →

10 ^→

j _n > 0

j _n = 0

i _n > 0

R _z

( − logz) ⁱⁿ

i n ! ( ^−dz _z−1 ) ^j ⁿ ⁻ ¹ , . . . .

10 ^→

C{{ X, Y }} ^∗

P ¹ ( C ) \ { 0, 1, ∞}

z ⊗ X + _z−1 ^dz ⊗ Y.

10 ^→

a ¹⁰ ^→ _→

a ¹⁰ ^→ _→

a ¹⁰ ^→ _→

ω _π(1) , . . . , ω _π(p+q)

1 n ² ) · (

1 m ³ ) =

1 m ³ · n ² +

1 n ² · m ³ +

1 n ⁵ .

z , ( − dz z − 1 ) ^k

R ^→ 10

z − 1 , . . . , ⁻ _z ^dz , ( ⁻ _z ₋ ^dz ₁ ) ^k

G _Q

π 1 ( P ¹ Q ¯ \ { 0, 1, ∞} ; 01) ^→

ϕ : G _Q −→ Aut(π ₁ ( P ¹ Q ¯ \ { 0, 1, ∞} ; 01)), ^→

G _Q ∋ σ −→ f _p (σ) := p ⁻¹ · σ(p) ∈ π ₁ ( P ¹ Q ¯ \ { 0, 1, ∞} ; 01), ^→

01 ^→

10, ^→

π ₁ ( P ¹ Q ¯ \ { 0, 1, ∞} ; 01) ^→

e ^X

e ^Y

ϕ _l : G _Q −→ Aut( Q l {{ X, Y }} ).

Λ _p (X, Y )(σ) ∈ Q l {{ X, Y }}

f _p (σ)

ϕ _l

G _Q ∋ σ −→ Λ _p (X, Y )(σ) ∈ Q l {{ X, Y }} .

f _p (σ)

π ₁ ( P ¹ Q ¯ \ { 0, 1, ∞} ; 01) ^→

Λ _p (X, Y )(σ)

f _p

Λ _p

a ¹⁰ ^→ _→

Λ _p (σ)

Λ _p (X, Y )(σ)

π 1 ( P ¹ Q ¯ \ { 0, 1, ∞} ; 01) ^→

a ^→ ¹⁰ _→

− log(e ^X · e ^Y ).

Λ _p (X, Y )(σ)

a ₁ , . . . , a _n