Depth-graded motivic multiple zeta values

0.36 MB·English

by  Francis Brown

DEPTH-GRADED MOTIVIC MULTIPLE ZETA VALUES
FRANCISBROWN
3
1
0
2 Abstract. Westudythedepthfiltrationonmotivicmultiplezetavalues,andits
n relationtomodularforms. UsingperiodpolynomialsforcuspformsforPSL2(Z),
we construct an explicit Lie algebra of solutions to the linearized double shuffle
a
J equations over the integers, which conjecturally describes all relations between
depth-gradedmotivicmultiplezetavalues(moduloζ(2)). TheBroadhurst-Kreimer
4
conjecture isrecastasastatement aboutthehomologyofthisLiealgebra.
1
To Pierre Cartier, with great admiration, on the occasion of his 80th birthday.

]
T 1. Introduction
N
1.1. Motivicmultiplezeta valuesand the depth filtration. Multiplezetavalues
.
h are defined for integers n ,...,n ≥1 and n ≥2 by
1 r−1 r
t
a 1
m ζ(n1,...,nr)= kn1...knr .
[ 1≤k1X<...<kr 1 r

Their weight is the quantity n1+...+nr, and their depth is the number of indices r.
1
Since the standardtranscendence conjectures for multiple zeta values are inaccessible,
v
3 one can replace the study of multiple zeta values with motivic multiple zeta values
5 (1.1) ζm(n ,...,n )
0 1 r
3 which are elements of a certain algebra H = H over Q, which is graded for the
N N
1. weight. The period homomorphism per : H →LR maps the element ζm(n1,...,nr) to
0 the real number ζ(n ,...,n ). The algebra H carries an action of the motivic Galois
1 r
3 group of the category of mixed Tate motives over Z, and its structure is known [1, 6].
1
: Theorem 1.1. The Hoffman motivic multiple zeta values
v
i (1.2) ζm(n ,...,n ) where n =2 or 3
X 1 r i
r are a basis for H.
a
In particular, the Poincar´e series for motivic multiple zeta values satisfies
1
(1.3) dim (H )sN = ,
Q N 1−s2−s3
X
N≥0
giving an upper bound for the dimension of the space of multiple zeta values of fixed
weight (first proved by Terasoma, and Goncharov). Since the depth filtration is mo-
tivic [8, 7], the algebra H also inherits the depth filtration D, and one can define its
associated graded algebra grDH. The depth-graded motivic multiple zeta value
(1.4) ζm(n ,...,n )∈grDH
D 1 r r
is given by the class of (1.1) modulo elements of lower depth. Note that the depth-
graded Hoffman elements ζm(n ,...,n ), where n = 2 or 3, lie in depths ⌈n⌉ ≤ r ≤
D 1 r i 3
⌊n⌋, and are almost all zero. It follows that theorem 1.1 tells us very little about the
2
structure of depth-graded multiple zeta values.
1
2 FRANCISBROWN
Basedonextensivecomputercalculations(see[5]forthecurrentstateofaffairs)and
some inspired guesswork,Broadhurstand Kreimer made a conjecture on the Poincar´e
seriesforthenumberofmultiplezetavaluesgradedforthedepth[3]. Transposingthis
conjecture to depth-graded motivic multiple zeta values, one obtains:
Conjecture 1. (Broadhurst-Kreimer)
1+E(s)t
(1.5) dim (grDH )sNtd = ,
Q d N 1−O(s)t+S(s)t2−S(s)t4
X
N,d≥0
where, using the notation from ([13], appendix),
s2 s3 s12
(1.6) E(s)= , O(s)= , S(s)= .
1−s2 1−s2 (1−s4)(1−s6)
Note that equation(1.5) specializes to (1.3) onsetting t equalto 1. The series E(s)
and O(s) are the generating series for the dimensions of the spaces of even and odd
singlezetavalues. TheinterpretationofS(s)asthegeneratingseriesforthedimensions
of the space of cusp forms for the full modular group PSL (Z) is due to Zagier. It
2
follows from work of Zagier,and Goncharov,that formula (1.5) has been confirmed in
depths 2 and 3 (i.e., modulo t4).
Indeed, the relation between the coefficient S(s) of t2 in the denominator of (1.5)
and multiple zeta values of depth 2 is now well-understood by the work of Gangl,
Kaneko and Zagier [10]. They exhibited an infinite family of exotic relations between
multiple zeta values of depth 2, associated to each even period polynomial for cusp
forms on PSL (Z). The smallest such relation occurs in weight twelve:
2
5197
(1.7) 28ζ(3,9)+150ζ(5,7)+168ζ(7,5)= ζ(12) .
691
The aim of these notes is to study the Broadhurst-Kreimer conjecture from the
motivic point of view, and to try to explain the relationship between modular forms
and depth-graded motivic multiple zeta values in all depths.
1.2. The dual Lie algebra. Throughout,letg denote the gradedLie algebraoverQ
freelygeneratedbytwoelementse ,e ofdegree−1(itisthegradedLiealgebraofthe
0 1
de Rham fundamental group of the projective line minus three points). Its universal
enveloping algebra is the tensor algebra:
(1.8) Ug= (Qe ⊕Qe )⊗n ,
0 1
M
n≥0
where the multiplication is given by the concatenation product, and its coproduct is
the unique coproduct for which e and e are primitive.
0 1
Consider the following motivic version of Drinfel’d’s associator
(1.9) Φm = wζm(w) ∈ Hhhe ,e ii
0 1
X
w
where the sum is over all words w in e ,e , and ζm(w) is the (shuffle-regularized)
0 1
motivic multiple zeta value, which is uniquely determined by
ζm(e en1−1...e enr−1)=ζm(n ,...,n ) ,
1 0 1 0 1 r
for n ,...,n asabove,the identities ζm(e )=ζm(e )=0,and the fact thatits linear
1 r 1 0
extension is a homomorphism for the shuffle product: ζm(wxw′) = ζm(w)ζm(w′).1
1Strictlyspeaking,theargumentwofζm(w)shouldbeviewedinthedualalphabet{e0,e1}where
hea,ebi=δa,b. Forthepurposesofthisintroduction,(andalsoin§5)weshallignorethisdistinction.
DEPTH-GRADED MZV’S 3
The element Φm defines a map from the graded dual of H to Ug
(1.10) H∨ −→Ug .
n
M
n≥0
The problem of describing the image of this map is equivalent to describing all rela-
tions between motivic multiple zeta values, and is very far from being solved, even
conjecturally. To simplify this problem, let L denote the largest quotient of H in
>0
which ζm(2), and all non-trivial products are zero. It is graded by the weight.
Definition 1.2. The set of motivic elements gm is the image of L∨ in Ug.
n≥1 n
L
The action of the motivic Galois group on the motivic multiple zeta values makes
L into a Lie coalgebra, and hence gm into a Lie algebra for the Ihara bracket, which
is denoted by {.,.}. The following theorem is a consequence of theorem 1.1, together
with the structure of the category of mixed Tate motives over Z.
Theorem 1.3. The graded Lie algebra gm is non-canonically isomorphic to the free
Lie algebra with one generator σ in each degree −(2i+1) for i≥1.
2i+1
Note that the classes [σ ] in (gm)ab are canonical, the elements σ are not.
2i+1 2i+1
Since H (gm;Q)=(gm)ab, one way to rephrase the previous theorem is to say that
1
(1.11) H (gm;Q) ∼= [σ ]Q
1 2i+1
M
i≥1
H (gm;Q) = 0 for i≥2 .
i
In principle, we could use the known bases of H to define ‘canonical’generators of gm
but very little is known about the coefficients of such generators2. Thus the problem
of describing the image of (1.10) above now becomes the following:
Problem 1. Write down explicit generators for gm as elements of Ug.
A solution to this problem would encode all relations between motivic multiple
zeta values (modulo ζm(2)), and would give an exact algorithm, building on [2], to
write multiple zeta values in terms of any chosen basis modulo ζm(2); bound the
heights of the coefficients, and so on. Note that we do not even have an explicit
conjectural description of the generators of gm. In these notes, we shall propose,
granting conjecture 1.5, a solution to the depth-graded version of problem 1.
1.3. Depth-graded Lie algebra. The depth filtration D is the decreasing filtration
on Ug such that DrUg is spanned by the set of words w in e ,e which have at least
0 1
r occurrences of the letter e . Using the terminology from [8], call the D-degree the
1
degreeinthelettere . Thedepthfiltrationisthe filtrationinducedbyD. Oneverifies
1
that the Ihara bracket is homogeneous with respect to the D-degree.
Define the Lie algebra of depth-graded motivic elements dgm ⊂ Ug to be the asso-
ciated bigraded Lie algebra
dgm =grr gm .
r D
It is the Lie algebra dual to the space of depth-graded motivic multiple zeta values
ζ (w), for w ∈{e ,e }×, modulo non-trivial products, and modulo ζm(2).
D 0 1 D
In depth one, there are canonical elements, for each i≥1:
(1.12) σ =(−1)i(ade )2i(e )∈dgm .
2i+1 0 1 1
2Onecouldtake,forinstance,thepolynomialsystemofgeneratorsforHgivenin[1],theorem8.1,
anddefineσ2i+1 tobethecoefficient ofζm(3,2,...,2),withi−1two’s,inΦm.
4 FRANCISBROWN
Furthermoreit is known[?,19,11,10]thatanexpressionofweight2nanddepthtwo:
(1.13) λ {σ ,σ }∈dgm
i,j 2i+1 2j+1 2
X
i<j
vanishes if and only if the λ are the coefficients of a period polynomial:
ij
P(X,Y)= (λ −λ )X2iY2j ∈S ,
ij ji 2n
X
i,j
where S ⊂ Q[X,Y] is the vector space of antisymmetric homogeneous polynomials
2n
P(X,Y)ofdegree2n−2,whicharedivisiblebyY,satisfyP(±X,±Y)=P(X,Y)and
P(X,Y)+P(X −Y,X)+P(−Y,X−Y)=0 .
The smallest relation (1.13) occurs in weight 12 and was first found by Ihara.
Since the Lie subalgebraofdgm generatedbythe elements σ is notfree,itmust
2i+1
be the case that dgm has exceptional generators in higher depth in order to square
with theorem 1.3. The Broadhurst-Kreimer conjecture suggests that such generators
existin depth 4, and the obviouswayto constructthem wouldbe to consider the first
non-trivialdifferentialinthespectralsequenceongm associatedtothedepthfiltration,
which gives a map d:H (dgm;Q)→H (dgm;Q), and in particular a linear map
2 1
(1.14) d:S −→(dgm)ab .
2n 4
Computing (1.14) is complicated by the fact that the depth 3 partof (some choice of)
generators σ is not known, and have complicated denominators (§8.3).
2i+1
Surprisingly, there is a way to write down candidate generatorsover Z explicitly in
dgm in depth 4 (rather than its abelianization) using modular forms. See definition
8.1. For this, we use the known relations satisfied by multiple zeta values.
1.4. Linearized double shuffle relations. The linearized double shuffle relations
for multiple zeta values were studied in [13], §8. These consistof two sets of relations:
the standardshufflerelationsformultiple zeta values(whicharealreadyhomogeneous
forthe D-grading),andthe termsofhighestD-degreeinthe secondshuffle,or‘stuffle’
relations. For example, the terms of highest D-degree in the relation
ζm(m)ζm(n)=ζm(m,n)+ζm(n,m)+ζm(m+n) for all m,n≥2
taken modulo products, leads to the linearized stuffle relation
(1.15) ζm(m,n)+ζm(n,m)=0 for all m,n≥2
D D
In general, it reduces to the ordinary shuffle relation on the indices n of (1.4). The
i
linearized double shuffle relations are therefore:
(1.16) ζm(wxw′) = 0 for all w,w′ ∈{e ,e }×
D 0 1
ζm(yxy′) = 0 for all y,y′ ∈{1,2,...}×
D
ζm(2n) = 0 for all n≥1
D
The first set of equations are the regularized shuffle relations as before, the second
equations are obtained by shuffling the indices n in ζm(n ,...,n ). Racinet proved
i D 1 r
that the regularizeddouble shuffle relations are motivic [17], and one can deduce that
(1.16) are motivic. Therefore, the solutions to the relations (1.16) define a subspace
ls⊂Ug and there is an inclusion of bigraded Lie algebras for the Ihara bracket:
dgm ⊂ls .
A conjecture in [13], §8 amounts to the statement dgm ∼= ls. The main result of this
paper is the construction of the missing exceptional generators in the algebra ls.
DEPTH-GRADED MZV’S 5
Theorem 1.4. There is an explicit injective linear map
(1.17) e:S −→ls .
2n 4
The formula for the map e is given in §8, and associates to every even period poly-
nomialP anintegralsolutionofthelinearizeddoubleshufflerelations. Thequestionof
whether the elements e are motivic, i.e., whether they lie in dgm, is open. Assuming
P
this, we propose the following reformulation of (1.5):
Conjecture 2. (Broadhurst-Kreimerv2). The image of e lies in dgm, and
(1.18) H (dgm;Q) ∼= σ Q⊕ e(S )
1 2i+1 2n
M M
i≥1 n
H (dgm;Q) ∼= S
2 2n
M
n
H (dgm;Q) = 0 for i≥3 .
i
Infact,bytheorem1.3,conjecture2holdsifthee lieindgm,andiftheLiesubalge-
P
bra of dgm generated by σ ,e has the homology (1.18). Since the elements σ ,
2i+1 P 2i+1
e and the Lie bracket are totally explicit, this part of the conjecture is elementary.
P
For the first part, which claims that the elements e are motivic, it is enoughto show
P
that their class in lsab lies in the image of (1.14). This is closely related to a question
4
posed by Ihara (see the examples 8.4). In small weights, we find that the two maps
eab,d:S→ls differ by an automorphismof S whichinvolvesnumeratorsofBernoulli
4
coefficients, and should be related to arithmetic questions in Galois cohomology [16].
We show in §9.2 that conjecture 2 implies (1.5). If true, then (1.18) describes all
relations between depth-graded motivic multiple zeta values (modulo ζm(2)): a linear
relationofweightN anddepthdis true ifandonlyifitholds amongstthe coefficients
of all Lie brackets of the elements σ and e of the same weight and depth (see
2i+1 P
example 8.3).
1.5. Contents of the paper. In sections §§2-4 we recall some background on the
motivic fundamental group of P1\{0,1,∞}, the Ihara action and the depth filtration.
In §5, we discuss the linearized double shuffle relations from the Hopf algebrapoint of
view. In §6, and throughout the rest of the paper, we use polynomial representations
to replace words of fixed D-degree r in e ,e with polynomials in r variables:
0 1
ρ:e en1...e enr 7→ xn1...xnr .
1 0 1 0 1 r
This replaces identities between non-commutative formal power series with functional
equationsincommutativepolynomials,reminiscentoftheworkofEcalle. Inparticular,
the depth r part of the motivic Drinfel’d associator yields generating series
ρ(grr Φm)= ζm(n ,...,n )xn1−1...xnr−1
D D 1 r 1 r
X
n1,...,nr
similartothoseconsideredin[13]. Weshowthatinthepolynomialrepresentation,the
Ihara bracket has an extremely simple form (§6.5). The generators (1.12) are simply
ρ(σ )=x2n .
2n+1 1
In §7 we review the relation between period polynomials and depth 2 multiple zeta
values, and in §8, we define for each period polynomial P, an element
ρ(e )∈Q[x ,x ,x ,x ] ,
P 1 2 3 4
whichdefines the exceptionalelements in ls . These elements satisfy some remarkable
4
properties,ofwhichweonlyscratchthesurfacehere.In§9wediscussconjecture2and
6 FRANCISBROWN
its consequences, and in §10 we discuss some applications for the enumeration of the
totally odd multiple zeta values ζm(2n +1,...,2n +1).
D 1 r
We do not discuss conjecture 2 per se in this paper, and we make no mention of
Goncharov’s somewhat different conjectural description [11] of the depth filtration in
terms of the cohomology of GL (Z), since we have no understanding of its connec-
d
tion to (1.18). Ultimately, we believe that a full, geometric, understanding of con-
jecture 2 should come from studying the motivic fundamental groups of M (via
1,n
Grothendieck’s two-tower principle).
2. Reminders on πm(P1\{0,1,∞})
1
→ →
2.1. The motivic π of P1\{0,1,∞}. Let X =P1\{0,1,∞},and let 1 ,−1 denote
1 0 1
the tangential base points on X given by the tangent vector 1 at 0, and the tangent
vector −1 at 1. Denote the de Rham realization of the motivic fundamental torsor of
paths on X with respect to these basepoints by:
→ →
Π =πdR(X, 1 ,−1 ) .
0 1 1 0 1
ItistheaffineschemeoverQwhichtoanycommutativeunitaryQ-algebraRassociates
the set of group-like formal power series in two non-commuting variables e and e
0 1
{S ∈Rhhe ,e ii× :∆S =S⊗S} ,
0 1
where ∆ is the completed coproduct for which the elbements e0 and e1 are primitive:
∆e =1⊗e +e ⊗1fori=0,1. Theringofregularfunctions on Π is the Q-algebra
i i i 0 1
O( Π )∼=Qhe0,e1i
0 1
whose underlying vector space is spanned by the set of words w in the letters e0,e1,
together with the empty word, and whose multiplication is given by the shuffle prod-
uct x : Qhe0,e1i⊗ Qhe0,e1i → Qhe0,e1i. The deconcatenation of words defines a
Q
coproduct,makingO( Π )intoaHopfalgebra. Thisgivesrisetoagroupstructureon
0 1
Π (Q)(correspondingtothe factthat inthe de Rhamrealisationthere isa canonical
0 1
path between any two points on X). Any word w in e0,e1 defines a function
Π (R)−→R
0 1
whichextractsthe coefficientS ofthe wordw (viewedin e ,e ) inagroup-likeseries
w 0 1
S ∈ Rhhe ,e ii×. The Lie algebra of Π (Q) is the completed Lie algebra g∧ of the
0 1 0 1
graded Lie algebra g which is freely generated by the two elements e ,e in degree
0 1
minus one. The universal enveloping algebra of g is the tensor algebra on e ,e :
0 1
(2.1) Ug= (Qe ⊕Qe )⊗n .
0 1
M
n≥0
It is the graded cocommutative Hopf algebra which is the graded dual of O( Π ). Its
0 1
multiplication is given by the concatenation product, and its coproduct is the unique
coproduct for which e and e are primitive.
0 1
2.2. Action of the motivic Galois group. Now let MT(Z) denote the Tannakian
category of mixed Tate motives over Z, with canonical fiber functor given by the de
Rham realization. Let G denote the group of automorphisms of the fiber functor,
MT
whichisanaffinegroupschemeoverQ. Ithasadecompositionasasemi-directproduct
G ∼=U ⋊G ,
MT MT m
where U is pro-unipotent. Furthermore, one knows from the relationship between
MT
the Ext groups in MT(Z) and Borel’s results on the rationalalgebraic K-theory of Q
that the graded Lie algebra of U is non-canonically isomorphic to the Lie algebra
MT
DEPTH-GRADED MZV’S 7
freely generated by one generator σ in degree −(2i+ 1) for every i ≥ 1. It is
2i+1
important to note that only the classes of the elements σ in the abelianization
2i+1
Uab are canonical, the elements σ themselves are not.
MT 2i+1
Since O( Π ) is the de Rham realization of an Ind-object in the category MT(Z),
0 1
there is an action of the motivic Galois group
(2.2) U × Π −→ Π .
MT 0 1 0 1
The action of U on the unit element 1∈ Π defines a map
MT 0 1
(2.3) g 7→g(1):U −→ Π ,
MT 0 1
and the action (2.2) factors through a map
(2.4) ◦: Π × Π −→ Π
0 1 0 1 0 1
first computed by Y. Ihara. It is obtained from [8], §5.9, §5.13 by reversing all words
in order to be consistent with our conventions for iterated integrals. Given a ∈ Π ,
0 1
its action on an element g ∈ Π is
0 1
(2.5) a◦g =(hai (g))a ,
0
where hai acts on the generators exp(e ), for i=0,1, by
0 i
(2.6) hai (exp(e )) = exp(e )
0 0 0
hai (exp(e )) = aexp(e )a−1 .
0 1 1
Definition 2.1. Define a Q-bilinear map
◦ :Ug⊗ Ug→Ug
Q
inductively as follows. For any words a,w in e ,e , and for any integer n≥0, let
0 1
(2.7) a◦(ene w)=enae w+ene a∗w+ene (a◦w)
0 1 0 1 0 1 0 1
where a◦en =ena, and for any a ∈{e ,e }×, (a ...a )∗ =(−1)na ...a .
0 0 i 0 1 1 n n 1
The map ◦ is a linearisationofthe full action◦:Ug⊗ Ug→Ug induced by (2.2).
Q
Proposition 2.2. Let ◦ also denote the action
(2.8) g⊗ Ug→Ug
Q
induced by (2.4). If i:g→Ug is the natural map, then a◦b=i(a)◦b.
Proof. For any S ∈ Π , the coefficient of w in S−1 is equal to the coefficient of w∗ in
0 1
S, since the map ∗ is the antipode in O( Π ). Identifying g with its image in Ug, it
0 1
followsthatthe infinitesimal, weight-graded,versionofthe map(2.6) is the derivation
hfi :g→g which for any f ∈g is given by
0
e 7→0 , e 7→fe +e f∗ .
0 1 1 1
Thus hfi :Ug→Ug is the map
0
hfi (em0e ...e emr)=em0fe ...e emr +...+em0e ...e f∗emr .
0 0 1 1 0 0 1 1 0 0 1 1 0
Concatentating on the right by f gives precisely the map defined by (2.7). (cid:3)
8 FRANCISBROWN
2.3. The motivic Lie algebra. By equation (2.3), we obtain a map of Lie algebras
(2.9) Liegr(U )−→g .
MT
Itfollowsfromtheorem1.1thatthismapisinjective[1]. Inthispaper,weshallidentify
Liegr(U ) with its image, and abusively call it the motivic Lie algebra.
MT
Definition 2.3. The motivic Lie algebra gm ⊆g is the image of the map (2.9).
The Lie algebra gm is non-canonically isomorphic to the free Lie algebra with one
generator σ in each degree −(2i+1) for i≥1.
2i+1
3. Motivic multiple zeta values
Let AMT denote the graded Hopf algebra of functions on U . Dualizing (2.2)
MT
gives the motivic coaction:
∆M :O( Π )−→AMT ⊗ O( Π ) .
0 1 Q 0 1
Furthermore,the deRhamimageofthe straightpathdchfrom0 to1inX definesthe
Drinfel’d associator element Φ∈ Π (R) which begins
0 1
Φ=1+ζ(2)[e ,e ]+ζ(3)([e ,[e ,e ]]+[e ,[e ,e ]])+...
1 0 1 1 0 0 0 1
ThemapwhichtakesthecoefficientofawordwinΦdefinestheperiodhomomorphism
per:O( Π )−→R .
0 1
Here, we use the convention from [1]: the coefficient of the word e ...e in Φ, for
a1 an
a ∈{0,1},is the iterated integral
i
1
ω ...ω
Z a1 an
0
regularizedwithrespecttothetangentvector1at0,and−1at1,wheretheintegration
begins on the left, and ω = dt and ω = dt .
0 t 1 1−t
Definition 3.1. Thealgebraofmotivicmultiplezetavaluesisdefinedasfollows. The
ideal of motivic relations between MZV’s is defined to be JMT ≤O( Π ), the largest
0 1
graded ideal contained in the kernel of per which is stable under ∆M. We set
H=O( Π )/JMT ,
0 1
and let ζm(n1,...,nr) denote the image of the word e1(e0)n1−1...e1(e0)nr−1 in H.
Likewise,foranya ,...,a ∈{0,1},weletIm(0;a ,...,a ;1)denotetheimageofthe
1 n 1 n
word ea1...ean in H.
Then H is naturally graded by the weight, has a graded coaction
(3.1) ∆M :H−→AMT ⊗ H
Q
and a period map per : H → R. The period of ζm(n ,...,n ) is ζ(n ,...,n ). One
1 r 1 r
obtains partial information about the motivic coaction (3.1) using the fact that it
factors through the coaction which is dual to the Ihara action (2.4).
DEPTH-GRADED MZV’S 9
3.1. The Ihara coaction. For any graded Hopf algebra H, let IH = H /H2
>0 >0
denote the Lie coalgebra of indecomposable elements of H, and let π : H → IH
>0
denote the natural map. Dualizing (2.8) gives an infinitesimal coaction
(3.2) O( Π )−→IO( Π )⊗ O( Π ) .
0 1 0 1 Q 0 1
LetD :O( Π )→IO( Π ) ⊗ O( Π )denoteits componentofdegree(r,·), andlet
r 0 1 0 1 r Q 0 1
us denote the element ea1...ean in O(0Π1) by I(0;a1,...,an;1), where ai ∈{0,1}.
Proposition 3.2. For all r≥1, and a ,...,a ∈{0,1}, it is given by:
1 n
(3.3) D I(0;a ,...,a ;1)=
r 1 n
n−r
π(I(a ;a ,..,a ;a ))⊗I(0;a ,..,a ,a ,..,a ;1) .
p p+1 p+r p+r+1 1 p p+r+1 n
X
p=0
where I(a ;a ,..,a ;a ) ∈ O( Π ) is defined to be zero if a = a , and
p p+1 p+r p+r+1 0 1 p p+r+1
equal to (−1)rI(a ;a ,..,a ;a ) if a =1 and a =0.
p+r+1 p+r p+1 p p p+r+1
Proof. It is almost immediate from (2.7) that this formula is dual to ◦. (cid:3)
Since the motivic coaction on H factors through the Ihara coaction, it follows that
the degree (r,·) component factors through operators
D :H−→IA⊗ H
r Q
givenby the same formulaas (3.3) in which eachterm I is replacedby its image Im in
H (resp. A). Since AMT is cogenerated in odd degrees only, the motivic coaction on
H is completely determined by the set of operators D for all r≥1 (see [1]).
2r+1
4. The depth filtration
TheinclusionP1\{0,1,∞}֒→P1\{0,∞}inducesamaponthemotivicfundamental
groupoids ([8], §6.1)
→ → → →
(4.1) πmot(X, 1 ,−1 )→πmot(G , 1 ,−1 ) ,
1 0 1 1 m 0 1
and hence on the de Rham realisations
→ →
O(πdR(G , 1 ,−1 ))−→O( Π )
1 m 0 1 0 1
which is the inclusion of Qhe0i into Qhe0,e1i. Define the image to be D O( Π ). It
0 0 1
generates,viathedeconcatenationcoproduct,anincreasingfiltrationcalledthedepth:
(4.2) D O( Π )=hwords w such that deg w ≤ii ,
i 0 1 e1 Q
with respect to which O( Π ) is a filtered Hopf algebra. Furthermore, since the map
0 1
(4.1) is motivic, the depth filtration is preserved by the coaction (3.1) (which also
follows by direct computation), and descends to the algebra H. By definition 3.1 the
depth filtration D H is the increasing filtration defined by
d
D H=hζm(n ,...,n ):i≤di .
d 1 i Q
Following [7], it is convenient to define the D-degree on O( Π ) to be the degree
0 1
in e1. It defines a grading on Π which is not motivic. By (4.2), the depth filtration
0 1
(which is motivic) is simply the increasing filtration associated to the D-degree.
Definition 4.1. Let D A be the induced filtration on the quotient A = H/ζm(2)H.
d
Wedefinethedepth-gradedmotivicmultiplezetavalueζm(n ,...,n )tobetheimage
D 1 r
of ζm(n ,...,n ) in grDH.
1 r r
10 FRANCISBROWN
Now let us choose a homomorphism π :H →Q[ζm(2)] which respects the weight-
2
grading and such that π (ζm(2)) = ζm(2). Such a homomorphism exists by [8], §5.20
2
(see [1], §2.3). Composing with the coaction (3.1), we obtain a map
H−∆→M A⊗ Hid−⊗→π2 A⊗ Q[ζm(2)] .
Q Q
Since D H = 0, and because ζm(2)n lies in depth 1 (by Euler’s theorem, ζm(2)n is a
0
rational multiple of ζm(2n)), the map π preserves the depth filtration, as does ∆M.
2
Thereforethe previous map H→A⊗ Q[ζm(2)] respects the depth filtration too. We
Q
also know that it is an isomorphism (by (2.13) in [1] or 5.8 in [7]). In conclusion,
(4.3) grDH∼=grDA⊗QQ[ζDm(2)] ,
where ζm(2)n ∈ grDH for all n ≥ 1. We will show in proposition 6.4 that grDIA
D 1
vanishes in bidegree (N,r) if N and r have different parity. In particular,
(4.4) ζm(n ,...,n )≡0 (mod products)
D 1 r
if n +...+n 6≡r (mod 2).
1 r
Thedepthfiltrationdefinesanincreasingfiltrationongm andUgm,whichisinduced
by the grading on T(Qe ⊕Qe ) for which e has degree 0, and e has degree −1. We
0 1 0 1
denote the associated graded Lie algebra by dgm.
Remark 4.2. From now on, we shall replace the depth on the Lie algebra dgm with its
opposite, giving a decreasing filtration Dr on gm consisting of words with at least r
occurrences of e . This is more or less standard in the literature. Therefore:
1
Drgm =hw ∈gm :deg w ≥ri ,
e1
and all depths will be non-negative. By (4.4), it follows that the components of dgm
for which the weight and depth have different parity are zero.
5. Linearized double shuffle relations
5.1. Reminders on the standard relations. We briefly review the double shuffle
relationsandtheirdepth-linearizedversions. Oneshouldemphasizethattheparticular
formoftheseequationsisnotofanyimportancehere: inthispapertheyareonlyused
as a means of finding candidate motivic elements amongst their solutions.
5.1.1. Shuffleproduct. ConsiderthealgebraQhe ,e iofwordsinthetwoletterse ,e ,
0 1 0 1
equipped with the shuffle product x. It is defined recursively by
(5.1) e wxe w′ =e (wxe w′)+e (e wxw′)
i j i j j i
for all w,w′ ∈ {e ,e }× and i,j ∈ {0,1}, and the property that the empty word 1
0 1
satisfies 1xw = wx1 = w. It is a Hopf algebra for the deconcatenation coproduct.
A linear map Φ : Qhe ,e i → Q is a homomorphism for the shuffle multiplication, or
0 1
ΦwΦw′ =Φwxw′ for all w,w′ ∈{e0,e1}× and Φ1 =1, if and only if the series
Φ= Φ w ∈Qhhe ,e ii
w 0 1
X
w
is invertible and group-like for the (completed) coproduct ∆x with respect to which
e and e are primitive (compare §2.1). In other words, there is an equivalence:
0 1
(5.2) Φ homomorphism for x ⇐⇒ Φ∈Qhhe ,e ii× and ∆xΦ=Φ⊗Φ
0 1
OnesaysthatΦsatisfiestheshufflerelationsifeitheroftheequivalentconditiobns(5.2)
holds. Passing to the corresponding Lie algebra, we have an equivalence
(5.3) Φwxw′ =0 for all w,w′ ⇐⇒ Φ∈Qhhe0,e1ii and ∆xΦ=1⊗Φ+Φ⊗1
b b