Weighted Completion of Galois Groups

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Weighted Completion of Galois Groups

the Deligne-Ihara and Conjecture

Makoto Matsumoto, Kyushu University

(*)joint work with Richard Hain

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1. Deligne-Ihara Conjecture

Ihara stated a conjecture in

"Galois group over

^Q

" Publ. MSRI 16 (1989),

attributed to Deligne.

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Conjecture (Deligne-Ihara)

The Lie algebra

Gr^Ih G^Q ^Z_` ^Q _` has the following properties.

(G)

^as ^Q `-Lie algebra, it has a gener- ating set consisting of one element in each odd grade 3.

(F)

these are free generators.

We shall prove (G) by weighted completion of Galois groups. The proof

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Notation:

P101¹: the projective line over ^Q mi- nus three points.

₁ =< x;y >: its topological funda- mental group (free with two generators),

₁^: its pronite completion,

₁^`: its pro-` completion, (`: a prime)

G^Q := Gal(^Q =^Q ).

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We have the pro-l nonabelian Galois representation

^` : G^Q ^! Out ₁^^! Out ₁^`:

Let L^m₁^` be the lower central series L¹₁^` := ₁^`

L²₁^` := [₁^`;₁^`]

L³₁^` := [₁^`; [₁^` ;₁^` ]]

::: :

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Then, induce a descending ltration on G^Q by

Ih^m(G^Q ) := ker[G^Q ^! Out(₁^`=L^m⁺¹₁^`)]: By group theory, Ih^m(m 1) is a central ltration, and

Gr^Ih_m G^Q := Ih^mG^Q =Ih^m⁺¹G^Q

is abelian and a free ^Z_`-module of nite rank.

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The direct sum of ^Z_`-modules Gr^Ih G^Q := ^M

m1Gr^Ih_m G^Q

has a graded Lie algebra structure:

sm ² Ih^m(G^Q );sn ² Ihⁿ(G^Q )

7! smsns^,_m¹s^,_n¹ ² Ihⁿ⁺^m(G^Q ) gives Lie product

GrÎh_m G^Q ^Z_` GrÎh_n G^Q ^! GrÎh_m₊_n G^Q :

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Known:

G^Q ^!⁹ Aut ₁^` ^! Aut(₁^`=L^m⁺¹₁^`) which denes the same Ih.

Ih¹G^Q = Ih²G^Q = Ih³G^Q

is the kernel of : G^Q ^! ^Z_`.

Soule's elements: ⁹m ⁶= 0 ² Gr^Ih_m G^Q (m 3, odd). (Ihara, Soule).

Some nonvanishing of their prod- ucts.

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Gr^Ih G^Q ,^! ^gt,

Lie version of Grothendieck-Teichmuller group. (Deligne, Drinfeld, Ihara)

dimGr^Ih_m G^Q = dim ^gt_m (m 12) ts to Conjecture (Tsunogai)

(F) implies (G). (Ihara)

The Lie algebra appears in Gr ₁^arith of the moduli stack ^Mg;n

(H. Nakamura, T. Oda).

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Outline of Proof of (G)

It factors through G_` := ₁(^Z[¹_`]).

Construct the weighted completion

A` of G_` with Zar.dense G_` ^! ^A_`(^Q _`) which is \relative-unipotent group closure =^Q _`" (cf. Deligne, same book).

A` has a natural weight ltration.

The kernel ^K_` of ^A_` ^! ^G m is free unipotent group generated by Soule's elements. This follows from Soule's computation of Hⁱ(G_`; ^Q _`(m)).

So generated is ^k_` := Lie(^K_`) = Gr^ ^K_`.

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Let ^P be the \unipotent group closure" of ₁^l . Then, we have

^unip : G_` ^! ^A_` ^! Aut ^P:

The image Im^A_` Aut ^P has two ltrations:

1. image of the weight ltration ^A_` 2. one from L^m⁺¹^P (induces Ih^mG_`).

They coincide.

Gr^Ih G_` ^Q _` = Gr(Im^K_`) is generated by Soule's elements.

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2. Weighted Completion of

G`

:=

₁

(Spec

^Z

[

¹_l

])

Known: ^` factors through

^` : G^Q ^f G_` := Gal(^Q ^ur;l=^Q ) ^! Out ₁^`; where ^Q ^ur;l is a maximal algebraic ex- tension of ^Q unramied outside `.

Easy to see:

Gr^Ih G^Q = Gr^Ih G_`;

where Ih^mG_` := ker G_` ^! Out(₁^`=L^m⁺¹₁^`), since Ih^mG^Q = f^,¹(Ih^mG_`).

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Representation Theory

V : nite dimensional ^Q _`-vector space.

G m ^! Aut V : a rational action of ^G m. decomposition of ^G m-representations

+

V = ^L_m^2Z Vm (unique),

where ^G m acts on Vm by m-th power multiplication, i.e.

c ² ^G m acts by c^m (^,) : Vm ^! Vm.

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By Levi-decomposition Theorem, any () 1 ^! U ^! G ^! ^G m ^! 1

(U unipotent, G algebraic =^Q _`) has a section G ^G m unique upto the inner automorphisms of U. Thus

G m ^! G conjugation

! Aut U = Aut ^u (^u: the Lie algebra of U) gives

u = ^M

m^2Z ^um:

The sequence (*) is negatively weighted if, for some (hence any) section,

um = 0 if m 0:

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We equip the weight ltration on ^u by W^,_N^u := ^M

m^N²

um:

Facts:

Is independent of choice of section.

u = W⁰û = W^,1û W^,2û = W^,3û

is a central ltration of Lie ideals.

) Gr^W^u: a graded ^Q _`-Lie algebra.

9 A bijection exp : ^u ^! U,

gives a central ltration on group

U = W = W U W U = W U

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If we have two negative sequences:

1 ^! U ^! G ^! ^G m ^! 1

f ^# ^# ^#

1 ^! U⁰ ^! G⁰ ^! ^G m ^! 1; Then f : U ^! U⁰ is strictly weight preserving, i.e.:

f(W_N(U)) = f(U)^\W_NÛ⁰ (N ^,1) (^* Decompose û; û⁰ by ^G m-action.

Then f restricts to f : ^um ^! ^u⁰m.) In particular, f: surjective ⁾

Gr(f) : Gr ^U ^! Gr ^U⁰ : surjective.

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For a pronite group , with continuous homomorphism : , ^! ^G m(^Q _`), the weighted completion of , relative to is the projective limit of the following objects.

1 ^! U ^! G ^! ^G m ^! 1

is negative, with a : , ^! G(^Q _`) is specied so that:

, = , ^# ^#

1 U G m 1:

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Apply this to , = G_` and the cyclo- tomic character : G_` ^! ^Z_` ^G m(^Q _`).

We get the weighted completion of G_`: G_` = G_`

^# ^#

1 ^! ^K_` ^! Â_` ^! ^G m ^! 1 with : G_` ^! Â_`(^Q _`) Zariski dense in the proalgebraic group Â_`.

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It has the following universality:

For any G ^! ^G m with unipotent kernel U and for any : , ^! G(^Q _`) with

G_` = G_`

^# ^#

G(^Q _`) ^! ^G m(^Q _`);

we have unique ' : ^A_` ^! G such that G_` = G_`

^# ^#

1 ^! ^K_` ^! ^A_` ^! ^G m ^! 1

# ' ^# ^k

1 ^! U ^! G ^! ^G m ^! 1:

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k`: Lie algebra of ^K_`.

Theorem 1

^k` is a pronilpotent free ^Q _`- Lie algebra. We can choose free generators m ² W^,₂_m^k_` (m = 1; 3; 5;::: ), such that their emages freely generate Gr ^k_`.

* (Wt. compl.) ^k_`ab is the product of H_cts¹ (G_`; ^Q _`(m))^Q _`(m) (m 1).

Soule: = ^Q _`; 0 (m:odd, even).

k`: pronilpotent ⁾ generated by these.

Free ⁽ H²(^k_`) = 0 since

H_cts² (G_`; ^Q _`(m)) = 0 (m 2).

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* Fix ^G m-section to ^A_`. Then ^k_` = ^Q_m>⁰ ^k_`m. We can show for m 1,

(^k_`ab)m = H_cts¹ (G_`;^Q _`(m)) = ⁽ ^Q ^` (m : odd) 0 (m : even) (the last step is Soule's result + some).

H²(^k_`) ,^! ^M

m²H_cts² (G_`;^Q _`(m)) = 0 (the right 0 is by Soule + some).

k`, acted by ^G m, is automatically graded:

k` = ^Y

m>⁰(^k_`)m = (Gr ^k_`)^

gives a (completed) graded Lie algebra structure on ^k_`. General theory on nilpotent Lie algebra says that if we take _m ² ^k_`m ^k_`ab_m whose image generates _`ab_m for odd m, then

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Malcev completion of

₁

⁽

^P₁₀₁¹

⁾

Recall ₁ =< x;y >.

Consider A := lim m ^Q `[₁]=J^m;

J = (x^,1;y^,1) is the augmentation ideal.

A is a complete Hopf-algebra.

P 1 + J : the set of group-like elements.

Is a subgroup of A.

p J : the set of Lie-like elements.

Is a ^Q _`-Lie algebra, & ⁹ bijection:

log : ^P 1 + J ^! ^p J:

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Proposition

1. ₁ ^! A continuously extends to ₁^l ^! A, and gives

Aut ₁^l ^! Aut_Hopf A = Aut _Lie ^p: 2. ^[¹^l ^! ^A^[

L^m₁^l ^! 1 + J^m: 3. W^,⁰ ₂_m Aut ^p

:= ker[Aut ^p ^! Aut(^p=^p ^\ J^m⁺¹)]: ThenG_` Aut ₁^l Aut

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Proposition

1. There is a proalgebraic group ^Aut ^p over ^Q _` such that Aut ^p = ( ^Aut ^p)(^Q _`).

2. Aut ^p^ab = GL(2; ^Q _`).

By scaler embedding ^G m ^! GL(2), we pull back

1 ^! K ^! Âut ^p ^! Âut^pâb ^! 1

" " "

1 ^! K⁰ ^! ^Aut ^p ^! ^G m ^! 1:

Then, the lower exact sequence is a negative sequence.

3. The weight ltration on K⁰ coin- cides with W⁰ (m 1):

W^,₂_mK⁰ = ker[Aut ^p ^! Aut ^p=(^p\J^m)]:

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Lemma 2

G_` ^! Aut ₁^l ^! Aut ^p ^! Aut ^p^ab is G_` ^! ^Z_` ^G m(^Q _`) ^diag^! GL(2; ^Q _`).

* G_` ^! Aut ₁^ab is -multiplication.

Now, by the universality of G_` ^! ^A_`(^Q _`), we have

G_` = G_`

^# ^#

1 ^! ^K_` ^! ^A_` ^! ^G m ^! 1

# ' ^# ^k

1 K ut m 1:

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Strictness of ltration in ^K_` Im(^K_`) says Gr ^K_` Gr Im(^K_`);

so

Gr Im(^K_`) is generated by the images ₁;₃;₅;::: as Lie algebra.

Only two steps left for proving (G):

Step 1.

^Gr ^Im^K` = Gr ^Ih G_` ^Z_` ^Q _`.

Step 2.

(Already commented:)

Gr^Ih₁ G_` = 0, hence ₁ vanishes there.

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Lemma 3

U: a prounipotent group over ^Q _`,

W^,m^U: ltration such that every Gr^,m ^U

is nite dimensional abel.

,: a pronite group,

: , ^! ^U(^Q _`): a continuous map.

I^,m, := ^,¹(W^,m^U(^Q _`)).

If the image of is Zariski dense, then Gr^I, ^Z_` ^Q _` = (Gr ^W ^U)(^Q _`):

( induction).

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For , = Ih¹G_` and

U := Im^K_` W^,₁Im^K_` , we saw that I^,₂_m, = Ih^m,. Thus, for Step 1, we need only:

Ih¹G_` ^! Im^K_` is Zariski dense.

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Lemma 4

^Let ^S be the weighted completion of ^Z_` ^! ^G m(^Q _`). Then (*) in the below is Zariski dense.

1 ^! Ih¹G_` ^! G_` ^! ^Z_` ^! 1

() ^# ^# ^#

1 ^! ^K ^! ^A_` ^! ^S ^! 1

# k #

1 ^! ^K_` ^! ^A_` ^! ^G m ^! 1 Moreover, by

H_cts¹ (^Z_`; ^Q _`(m)) = 0 (m ⁶= 0);

S = ^G m and consequently Ih¹G_` = _` is dense.

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Concluding Remarks

Our proof of (G) says nothing on (F).

(F) says the faithfullness of the \mo- tivic Galois group" action

A`= < exp(₁) >^! ^Aut ^P

(^P : completion of ₁^l . c.f. Bely's injectivity of G^Q ^! Aut ₁^).