Totally odd depth-graded multiple zeta values and period polynomials
Charlotte Dietze, Chokri Manai, Christian Nöbel, Ferdinand Wagner
aa r X i v : . [ m a t h . N T ] A ug TOTALLY ODD DEPTH-GRADED MULTIPLE ZETA VALUES ANDPERIOD POLYNOMIALS
CHARLOTTE DIETZE, CHOKRI MANAI, CHRISTIAN NÖBEL, AND FERDINAND WAGNER
Abstract.
Inspired by the paper of Tasaka [1], we study the relations between totally odd,motivic depth-graded multiple zeta values. Our main objective is to determine the rank ofthe matrix C N,r defined by Brown [2]. We will give new proofs for (conjecturally optimal)upper bounds on rank C N, and rank C N, , which were first obtained by Tasaka [1]. Finally, wepresent a recursive approach to the general problem, which reduces the evaluation of rank C N,r to an isomorphism conjecture. Introduction
In this paper we will be interested in Q -linear relations among totally odd depth-gradedmultiple zeta values (MZVs), for which there conjecturally is a bijection with the kernel of aspecific matrix C N,r connected to restricted even period polynomials (for a definition, see [3]or [4, Section 5]).For integers n , . . . , n r − ≥ n r ≥
2, the MZV of n , . . . , n r is defined as the number ζ ( n , . . . , n r ) := X Conjecture 1.1 (Broadhurst-Kreimer) . The generating function of the dimension of the space Z N,r is given by X N,r ≥ dim Q Z N,r · x N y r ? = 1 − E ( x ) y − O ( x ) y + S ( x ) y − S ( x ) y , Date : September 2016.2010 Mathematics Subject Classification. Primary 11M32, Secondary 11F67. Key words and phrases. Multiple zeta values, period polynomials. where we denote E ( x ) := x − x = x + x + x + · · · , O ( x ) := x − x = x + x + x + · · · , and S ( x ) := x (1 − x )(1 − x ) .Remark. It should be mentioned that S ( x ) = P n> dim S n · x n , where S n denotes the spaceof cusp forms of weight n , for which there is an isomorphism to the space of restricted evenperiod polynomials of degree n − Q -vector space Z odd N,r (respectively H odd N,r ) of totallyodd (motivic) and depth-graded MZVs, that is, ζ D ( n , . . . , n r ) (respectively ζ mD ( n , . . . , n r )) for n i ≥ C N,r , where N = n + · · · + n r denotesthe weight. In particular, he showed that any right annihilator ( a n ,...,n r ) ( n ,...,n r ) ∈ S N,r of C N,r induces a relation X ( n ,...,n r ) ∈ S N,r a n ,...,n r ζ mD ( n , . . . , n r ) = 0 , hence also X ( n ,...,n r ) ∈ S N,r a n ,...,n r ζ D ( n , . . . , n r ) = 0(see Section 2 for the notations) and conjecturally all relations in Z odd N,r arise in this way. Thisled to the following conjecture (the uneven part of the Broadhurst-Kreimer Conjecture). Conjecture 1.2 (Brown [2]) . The generating series of the dimension of Z odd N,r and the rank of C N,r are given by X N,r> rank C N,r · x N y r ? = 1 + X N,r> dim Q Z odd N,r · x N y r ? = 11 − O ( x ) y + S ( x ) y . The contents of this paper are as follows. In Section 2, we explain our notations and definethe matrices C N,r due to Brown [2] as well as E N,r and E ( j ) N,r considered by Tasaka [1]. InSection 3, we briefly state some of Tasaka’s results on the matrix E N,r . Section 4 is devoted tofurther investigate the connection between the left kernel of E N,r and restricted even periodpolynomials, which was first discovered by Baumard and Schneps [3] and appears again in [1,Theorem 3.6]. In Section 5, we will apply our methods to the cases r = 3 and r = 4. The firstgoal of Section 5 will be to show Theorem 1.3. Assume that the map from Theorem 3.2 is injective. We then have the lowerbound X N> dim Q ker C N, · x N ≥ O ( x ) S ( x ) , where ≥ means that for every N > the coefficient of x N on the right-hand side does notexceed the corresponding one on the left-hand side. This was stated without proof in [1]. Furthermore, we will give a new proof by the polynomialmethods developed in Section 4 for the following result. Theorem 1.4 (Tasaka [1, Theorem 1.3]) . Assume that the map from Theorem 3.2 is injective.We then have the lower bound X N> dim Q ker C N, · x N ≥ O ( x ) S ( x ) − S ( x ) . In the last two subsections of this paper, we will consider the case of depth 5 and give anidea for higher depths. For depth 5, we will prove that upon Conjecture 3.3 due to Tasaka ([1,Section 3]), the lower bound predicted by Conjecture 1.2 holds, i.e. X N> dim Q ker C N, · x N ≥ O ( x ) S ( x ) − O ( x ) S ( x ) . OTALLY ODD DEPTH-GRADED MULTIPLE ZETA VALUES AND PERIOD POLYNOMIALS 3 These bounds are conjecturally sharp (i.e. the ones given by Conjecture 1.2). Finally, we willprove a recursion for value of dim Q ker C N,r under the assumption of a similar isomorphismconjecture stated at the end of Section 4, which was proposed by Claire Glanois. Acknowledgments. This research was conducted as part of the Hospitanzprogramm (intern-ship program) at the Max-Planck-Institut für Mathematik (Bonn). We would like to expressour deepest thanks to our mentor, Claire Glanois, for introducing us into the theory of multi-ple zeta values. We are also grateful to Daniel Harrer, Matthias Paulsen and Jörn Stöhler formany helpful comments. 2. Preliminaries Notations. In this section we introduce our notations and we give some definitions. Asusual, for a matrix A we define ker A to be the set of right annihilators of A . Apart from this,we mostly follow the notations of Tasaka in his paper [1]. Let S N,r := { ( n , . . . , n r ) ∈ Z r | n + · · · + n r = N, n , . . . , n r ≥ } , where N and r are natural numbers. Since the elements of the set S N,r will be used as indicesof matrices and vectors, we usually arrange them in lexicographically decreasing order. Let V N,r := D x m − · · · x m r − r (cid:12)(cid:12)(cid:12) ( m , . . . , m r ) ∈ S N,r E Q denote the vector space of restricted totally even homogeneous polynomials of degree N − r in r variables. There is a natural isomorphism from V N,r to the Q -vector space Vect N,r of n -tuples ( a n ,...,n r ) ( n ,...,n r ) ∈ S N,r indexed by totally odd indices ( n , . . . , n r ) ∈ S N,r , which wedenote π : V N,r ∼ −→ Vect N,r X ( n ,...,n r ) ∈ S N,r a n ,...,n r x n − · · · x n r − r ( a n ,...,n r ) ( n ,...,n r ) ∈ S N,r . (2.1)We assume vectors to be row vectors by default.Finally, let W N,r be the vector subspace of V N,r defined by W N,r := { P ∈ V N,r | P ( x , . . . , x r ) = P ( x − x , x , x , . . . , x r ) − P ( x − x , x , x , . . . , x r ) } . That is, P ( x , x , x , . . . , x r ) is a sum of restricted even period polynomials in x , x multipliedby monomials in x , . . . , x r . More precisely, one can decompose W N,r = M 2. Since W n, isisomorphic to the space S n of cusp forms of weight n by Eichler-Shimura correspondence (see[6]), (2.2) leads to the following dimension formula. Lemma 2.1 (Tasaka [1, equation 3.10]) . For all r ≥ , X N> dim Q W N,r · x N = O ( x ) r − S ( x ) . OTALLY ODD DEPTH-GRADED MULTIPLE ZETA VALUES AND PERIOD POLYNOMIALS 4 Ihara action and the matrices E N,r and C N,r . We use Tasaka’s notation (from [1])for the polynomial representation of the Ihara action defined by Brown [2, Section 6]. Let ◦ : Q [ x ] ⊗ Q [ x , . . . , x r ] −→ Q [ x , . . . , x r ] f ⊗ g f ◦ g , where f ◦ g denotes the polynomial( f ◦ g )( x , . . . , x r ) := f ( x ) g ( x , . . . , x r ) + r − X i =1 (cid:16) f ( x i +1 − x i ) g ( x , . . . , ˆ x i +1 , . . . , x r ) − ( − deg f f ( x i − x i +1 ) g ( x , . . . , ˆ x i , . . . , x r ) (cid:17) . (the hats are to indicate, that x i +1 and x i resp. are omitted in the above expression).For integers m , . . . , m r , n , · · · , n r ≥ 1, let furthermore the integer e (cid:0) m ,...,m r n ,...,n r (cid:1) denote thecoefficient of x n − · · · x n r − r in x m − ◦ (cid:16) x m − · · · x m r − r − (cid:17) , i. e. x m − ◦ (cid:16) x m − · · · x m r − r − (cid:17) = X n + ··· + n r = m + ··· + m r n , ··· ,n r ≥ e (cid:0) m ,...,m r n ,...,n r (cid:1) x n − · · · x n r − r . (2.3)Note that e (cid:0) m ,...,m r n ,...,n r (cid:1) = 0 if m + · · · + m r = n + · · · + n r . Remark. One can explicitly compute the integers e (cid:0) m ,...,m r n , ··· ,n r (cid:1) by the following formula: ([1,Lemma 3.1]) e (cid:0) m ,...,m r n ,...,n r (cid:1) = δ (cid:0) m ,...,m r n ...,n r (cid:1) + r − X i =1 δ (cid:0) ˆ m ,m ,...,m i , ˆ m i +1 ,m i +2 ,...,m r n ,...,n i − , ˆ n i , ˆ n i +1 ,n i +2 ,...,n r (cid:1) · ( − n i m − n i − ! + ( − m − n i +1 m − n i +1 − !! (again, the hats are to indicate that m , m i +1 , n i , n i +1 are omitted), where δ (cid:0) m ,...,m s n ,...,n s (cid:1) := ( m i = n i for all i ∈ { , . . . , s } Definition 2.2. Let N, r be positive integers.(i) We define the | S N,r | × | S N,r | matrix E N,r := (cid:16) e (cid:0) m ,...,m r n ,...,n r (cid:1)(cid:17) ( m ,...,m r ) , ( n ,...,n r ) ∈ S N,r . (ii) For integers r ≥ j ≥ | S N,r | × | S N,r | matrix E ( j ) N,r := (cid:16) δ (cid:0) m ,...,m r − j n ,...,n r − j (cid:1) e (cid:0) m r − j +1 ,...,m r n r − j +1 ,...,n r (cid:1)(cid:17) ( m ,...,m r ) , ( n ,...,n r ) ∈ S N,r . Definition 2.3 ([1, Definition 2.3 and Proposition 3.3]) . The | S N,r | × | S N,r | matrix C N,r isdefined as C N,r := E (2) N,r · E (3) N,r · · · E ( r − N,r · E N,r . OTALLY ODD DEPTH-GRADED MULTIPLE ZETA VALUES AND PERIOD POLYNOMIALS 5 Known Results Recall the map π : V N,r → Vect N,r (equation (2.1)). Theorem 3.1 due to Baumard andSchneps [3] establishes a connection between the left kernel of the matrix E N, and the space W N, of restricted even period polynomials. This connection was further investigated byTasaka [1], relating W N,r and the left kernel of E N,r for arbitrary r ≥ Theorem 3.1 (Baumard-Schneps [3, Proposition 3.2]) . For each integer N > we have π ( W N, ) = ker t E N, . Theorem 3.2 (Tasaka [1, Theorem 3.6]) . Let r ≥ be a positive integer and F N,r = E N,r − id Vect N,r . Then, the following Q -linear map is well-defined: W N,r −→ ker t E N,r P ( x , . . . , x r ) π ( P ) F N,r . (3.1) Conjecture 3.3 (Tasaka [1, Section 3.3]) . For all r ≥ , the map described in Theorem 3.2is an isomorphism.Remark. For now, only the case r = 2 is known, which is an immediate consequence of The-orem 3.1. In [1], Tasaka suggests a proof of injectivity, but it seems to contain a gap, which,as far as the authors are aware, couldn’t be fixed yet. However, assuming the injectivity ofmorphisms (3.1) one has the following relation. Corollary 3.4 (Tasaka [1, Corollary 3.7]) . For all r ≥ , X N> dim Q ker t E N,r · x N ≥ O ( x ) r − S ( x ) . Main Tools Decompositions of E ( j ) N,r . We use the following decomposition lemma: Lemma 4.1. Let ≤ j ≤ r − and arrange the indices ( m , . . . , m r ) , ( n , . . . , n r ) ∈ S N,r of E ( j ) N,r in lexicographically decreasing order. Then, the matrix E ( j ) N,r has block diagonal structure E ( j ) N,r = diag (cid:16) E ( j )3 r − ,r − , E ( j )3 r − ,r − , . . . , E ( j ) N − ,r − (cid:17) . Proof. This follows directly from Definition 2.2. (cid:3) Corollary 4.2. We have E (2) N,r E (3) N,r · · · E ( r − N,r = diag ( C r − ,r − , C r − ,r − , . . . , C N − ,r − ) . Proof. Multiplying the block diagonal representations of E (2) N,r , E (3) N,r , . . . , E ( r − N,r block by blocktogether with Definition 2.3 yields the desired result. (cid:3) Corollary 4.3. For all r ≥ , X N> dim Q ker (cid:16) E (2) N,r · · · E ( r − N,r (cid:17) · x N = O ( x ) X N> dim Q ker C N,r − · x N . OTALLY ODD DEPTH-GRADED MULTIPLE ZETA VALUES AND PERIOD POLYNOMIALS 6 Proof. According to Corollary 4.2, the matrix E (2) N,r · · · E ( r − N,r has block diagonal structure, theblocks being C r − ,r − , C r − ,r − , . . . , C N − ,r − . Hence, X N> dim Q ker (cid:16) E (2) N,r · · · E ( r − N,r (cid:17) · x N = X N> N − X k =3 r − dim Q ker C k,r − · x N = X N> dim Q ker C N,r − (cid:16) x N +3 + x N +5 + x N +7 + · · · (cid:17) = O ( x ) X N> dim Q ker C N,r − · x N , thus proving the assertion. (cid:3) Connection to polynomials. Motivated by Theorem 3.2, we interpret the right actionof the matrices E (2) N,r , . . . , E ( r − N,r , E ( r ) N,r = E N,r on Vect N,r as endomorphisms of the polynomialspace V N,r . Having established this, we will prove Theorems 1.3 and 1.4 from a polynomialpoint of view. Definition 4.4. The restricted totally even part of a polynomial Q ( x , . . . , x r ) ∈ V N,r is thesum of all of its monomials, in which each exponent of x , . . . , x r is even and at least 2. Let r ≥ j . We define the Q -linear map ϕ ( r ) j : V N,r −→ V N,r , which maps each polynomial Q ( x , . . . , x r ) ∈ V N,r to the restricted totally even part of Q ( x , . . . , x r ) + r − X i = r − j +1 (cid:16) Q ( x , . . . , x r − j , x i +1 − x i , x r − j +1 , . . . , ˆ x i +1 , . . . , x r ) − Q ( x , . . . , x r − j , x i +1 − x i , x r − j +1 , . . . , ˆ x i , . . . , x r ) (cid:17) . Remark. Note that ϕ ( r )1 ≡ id V N,r .The following lemma shows that the map ϕ ( r ) j corresponds to the right action of the matrix E ( j ) N,r on Vect N,r via the isomorphism π . Lemma 4.5. Let r ≥ j . Then, for each polynomial Q ∈ V N,r , π (cid:16) ϕ ( r ) j ( Q ) (cid:17) = π ( Q ) E ( j ) N,r . or equivalently, the following diagram commutes: V N,r V N,r Vect N,r Vect N,r v v · E ( j ) N,r ϕ ( r ) j ∼ π ∼ π Proof. We proceed by induction on r . Let r = j and Q ( x , . . . , x j ) = X ( n ,...,n j ) ∈ S N,j q n ,...,n j x n − · · · x n j − r . OTALLY ODD DEPTH-GRADED MULTIPLE ZETA VALUES AND PERIOD POLYNOMIALS 7 Then, E ( j ) N,j = E N,j and thus π ( Q ) E N,j = X ( m ,...,m j ) ∈ S N,j q n ,...,n j e (cid:0) m ,...,m j n ,...,n j (cid:1) ( n ,...,n j ) ∈ S N,j . By (2.3) and linearity of the Ihara action ◦ , the row vector on the right-hand side correspondsto π applied to the restricted totally even part of the polynomial X ( n ,...,n j ) ∈ S N,j q n ,...,n j x n − ◦ (cid:16) x n − · · · x n j − j − (cid:17) . (4.1)On the other hand, plugging r = j into Definition 4.4 yields that ϕ ( j ) j ( Q ( x , . . . , x j )) corre-sponds to the restricted totally even part of some polynomial, which by definition of the Iharaaction ◦ coincides with the polynomial defined in (4.1). Thus, the claim holds for r = j .Now suppose that r ≥ j + 1 and the claim is proven for all smaller r . Let us decompose Q ( x , . . . , x r ) = N − (3 r − X n =3 n odd x n − · Q N − n ( x , . . . , x r ) , where the Q k are restricted totally even homogeneous polynomials in r − Q k ∈ V k,r − for all k . Arrange the indices of π ( Q ) in lexicographically de-creasing order. Then, by grouping consecutive entries, π ( Q ) is the list-like concatenationof π ( Q r − ) , . . . , π ( Q N − ), which we denote by π ( Q ) = (cid:0) π ( Q r − ) , π ( Q r − ) , . . . , π ( Q N − ) (cid:1) . Since we have lexicographically decreasing order of indices, the block diagonal structure of E ( j ) N,r stated in Lemma 4.1 yields π ( Q ) E ( j ) N,r = (cid:16) π ( Q r − ) E ( j )3 r − ,r − , π ( Q r − ) E ( j )3 r − ,r − , . . . , π ( Q N − ) E ( j ) N − ,r − (cid:17) = (cid:16) π (cid:16) ϕ ( r − j ( Q r − ) (cid:17) , π (cid:16) ϕ ( r − j ( Q r − ) (cid:17) , . . . , π (cid:16) ϕ ( r − j ( Q N − ) (cid:17)(cid:17) = π (cid:16) ϕ ( r ) j ( Q ) (cid:17) by linearity of ϕ ( r ) j and the induction hypothesis. This shows the assertion. (cid:3) Corollary 4.6. For all r ≥ , Im t (cid:16) E (2) N,r · · · E ( r − N,r (cid:17) ∩ ker t E N,r ∼ = ker ϕ ( r ) r ∩ Im (cid:16) ϕ ( r ) r − ◦ · · · ◦ ϕ ( r )2 (cid:17) . Proof. By the previous Lemma 4.5, the following diagram commutes: V N,r V N,r · · · V N,r V N,r Vect N,r Vect N,r · · · Vect N,r Vect N,r ∼ π ∼ π ∼ π ∼ πϕ ( r )2 ϕ ( r )3 ϕ ( r ) r − ϕ ( r ) r · E (2) N,r · E (3) N,r · E ( r − N,r · E ( r ) N,r From this, we have Im t (cid:16) E (2) N,r · · · E ( r − N,r (cid:17) ∼ = Im (cid:16) ϕ ( r ) r − ◦ · · · ◦ ϕ ( r )2 (cid:17) and ker t E N,r ∼ = ker ϕ ( r ) r .Thereby, the claim is established. (cid:3) OTALLY ODD DEPTH-GRADED MULTIPLE ZETA VALUES AND PERIOD POLYNOMIALS 8 Lemma 4.7. Let j ≤ r − . Then, ker ϕ ( r ) j ∼ = M n Let Q ∈ V N,r . We may decompose Q ( x , . . . , x r ) = X n Let ≤ j ≤ r − . The restricted map ϕ ( r ) j (cid:12)(cid:12) W N,r : W N,r −→ W N,r is well-defined and satisfies ker ϕ ( r ) j (cid:12)(cid:12) W N,r ∼ = M n Since j ≤ r − 2, for each Q ∈ V N,r the map Q ( x , . . . , x r ) ϕ ( r ) j ( Q ) does not interferewith x or x and thus not with the defining property of W N,r . Hence, ϕ ( r ) j (cid:12)(cid:12) W N,r is well-defined.The second assertion is done just like in the previous Lemma 4.7. (cid:3) Lemma 4.9. Let r ≥ . For all P ∈ W N,r , π ( − P ) E ( r − N,r = π ( P ) F N,r . Proof. Recall that by Lemma 4.5, π ( − P ) E ( r − N,r = π (cid:16) ϕ ( r ) r − (cid:0) − P ( x , . . . , x r ) (cid:1)(cid:17) = π (cid:18) − P ( x , . . . , x r ) + r − X i =2 (cid:16) − P ( x , x i +1 − x i , x , . . . , ˆ x i +1 , . . . , x r )+ P ( x , x i +1 − x i , x , . . . , ˆ x i , . . . , x r ) (cid:17)(cid:19) = π (cid:18) − P ( x , . . . , x r ) + r − X i =2 (cid:16) P ( x i +1 − x i , x , . . . , ˆ x i +1 , . . . , x r ) − P ( x i +1 − x i , x , . . . , ˆ x i , . . . , x r ) (cid:17)(cid:19) , since − P is antisymmetric with respect to x ↔ x . In the same way we compute π ( P ) F N,r = π ( P ) (cid:16) E N,r − id Vect N,r (cid:17) = π (cid:0) ϕ ( r ) r ( P ( x , . . . , x r )) (cid:1) − π ( P )= π (cid:18) P ( x , . . . , x r ) + r − X i =1 (cid:16) P ( x i +1 − x i , x , . . . , ˆ x i +1 , . . . , x r ) − P ( x i +1 − x i , x , . . . , ˆ x i , . . . , x r ) (cid:17)(cid:19) − π ( P ) . Now the desired result follows from P ( x , x , . . . , x r ) + P ( x − x , x , x , . . . , x r ) − P ( x − x , x , . . . , x r ) = 0 , OTALLY ODD DEPTH-GRADED MULTIPLE ZETA VALUES AND PERIOD POLYNOMIALS 9 since P is in W N,r . (cid:3) Corollary 4.10. Assume that the map from Theorem 3.2 is injective. Then, for all r ≥ , dim Q (cid:16) Im t E ( r − N,r ∩ ker t E N,r (cid:17) ≥ dim Q W N,r . Proof. This is immediate by the previous Lemma 4.9. (cid:3) Lemma 4.11. For all r ≥ , Im (cid:16) ϕ ( r ) r − ◦ ϕ ( r ) r − (cid:12)(cid:12) W N,r ◦ · · · ◦ ϕ ( r )2 (cid:12)(cid:12) W N,r (cid:17) ⊆ ker ϕ ( r ) r ∩ Im (cid:16) ϕ ( r ) r − ◦ · · · ◦ ϕ ( r )2 (cid:17) . Proof. We may replace the right-hand side by just ker ϕ ( r ) r . Note that by Corollary 4.8 thecomposition of restricted ϕ ( r ) j (cid:12)(cid:12) W N,r on the left-hand side is well-defined. Moreover, each Q ∈ Im (cid:16) ϕ ( r ) r − ◦ ϕ ( r ) r − (cid:12)(cid:12) W N,r ◦ · · · ◦ ϕ ( r )2 (cid:12)(cid:12) W N,r (cid:17) can be represented as Q = ϕ ( r ) r − ( P ) for some P ∈ W N,r and thus Q ∈ ker ϕ ( r ) r according to Lemma 4.9 and Theorem 3.2. (cid:3) Similar to Conjecture 3.3 we expect a stronger result to be true, which is stated in thefollowing conjecture due to Claire Glanois: Conjecture 4.12. For all r ≥ , Im (cid:16) ϕ ( r ) r − ◦ ϕ ( r ) r − (cid:12)(cid:12) W N,r ◦ · · · ◦ ϕ ( r )2 (cid:12)(cid:12) W N,r (cid:17) = ker ϕ ( r ) r ∩ Im (cid:16) ϕ ( r ) r − ◦ · · · ◦ ϕ ( r )2 (cid:17) . Remark. Note that intersecting ker ϕ ( r ) r ∩ Im (cid:16) ϕ ( r ) r − ◦ · · · ◦ ϕ ( r )2 (cid:17) , Conjecture 4.12 does not needthe injectivity from Conjecture 3.3. However, we haven’t been able to derive Conjecture 4.12from Conjecture 3.3, so it is not necessarily weaker.5. Main Results Throughout this section we will assume that the map from Theorem 3.2 is injective, i.e. theinjectivity part of Conjecture 3.3 is true. This was also the precondition for Tasaka’s originalproof of Theorem 1.4.5.1. Proof of Theorem 1.3. By Corollary 4.3, Remark 3 and the fact that E N, = C N, weobtain X N> dim Q ker E (2) N, · x N = O ( x ) X N> dim Q ker C N, · x N = O ( x ) S ( x ) . (5.1)We use Corollary 4.10 and Lemma 2.1 to obtain X N> dim Q (cid:16) Im t E (2) N, ∩ ker t E N, (cid:17) · x N ≥ X N> dim Q W N, · x N = O ( x ) S ( x ) . (5.2)Now observe that since C N, = E (2) N, E N, , we havedim Q ker C N, = dim Q ker t E (2) N, + dim Q (cid:16) Im t E (2) N, ∩ ker t E N, (cid:17) . By (5.1) and (5.2), the assertion is proven. (cid:3) OTALLY ODD DEPTH-GRADED MULTIPLE ZETA VALUES AND PERIOD POLYNOMIALS 10 Proof of Theorem 1.4. Since C N, = E (2) N, E (3) N, E N, , we may split dim Q ker C N, intodim Q ker C N, = dim Q ker t (cid:16) E (2) N, E (3) N, (cid:17) + dim Q (cid:18) Im t (cid:16) E (2) N, E (3) N, (cid:17) ∩ ker t E N, (cid:19) . The two summands on the right-hand side are treated separately. For the first one, by Corol-lary 4.3 and Theorem 1.3 one has X N> dim Q ker t (cid:16) E (2) N, E (3) N, (cid:17) · x N ≥ O ( x ) S ( x ) . (5.3)For the second one, we use Corollary 4.6 and Lemma 4.11 to obtaindim Q (cid:18) Im t (cid:16) E (2) N, E (3) N, (cid:17) ∩ ker t E N, (cid:19) ≥ dim Q Im (cid:16) ϕ (4)3 ◦ ϕ (4)2 (cid:12)(cid:12) W N, (cid:17) = dim Q Im ϕ (4)2 (cid:12)(cid:12) W N, , since we assume ϕ (4)3 to be injective on W N,r according to Conjecture 3.3. According toCorollary 4.8 and Theorem 3.1,ker ϕ (4)2 (cid:12)(cid:12) W N, ∼ = M n Definition 5.1. For r ≥ 2, let us define the formal series B r ( x ) := X N> dim Q Im (cid:16) ϕ ( r ) r − ◦ ϕ ( r ) r − (cid:12)(cid:12) W N,r ◦ · · · ◦ ϕ ( r )2 (cid:12)(cid:12) W N,r (cid:17) · x N (i) T r ( x ) := X N> dim Q ker C N,r · x N . (ii)We set T ( x ) , T ( x ) := 0.The main observation is the following lemma: Lemma 5.2. Assume that Conjecture 4.12 is true and that the map from Theorem 3.2 isinjective. Then, for r ≥ the following recursion holds: B r ( x ) = O ( x ) r − S ( x ) − r − X j =2 O ( x ) r − j − S ( x ) B j ( x ) . Proof. We havedim Q Im (cid:16) ϕ ( r ) r − ◦ ϕ ( r ) r − (cid:12)(cid:12) W N,r ◦ · · · ◦ ϕ ( r )2 (cid:12)(cid:12) W N,r (cid:17) = dim Q W N,r − r − X j =2 dim Q ker ϕ ( r ) j (cid:12)(cid:12) W N,r ∩ Im (cid:16) ϕ ( r ) j − (cid:12)(cid:12) W N,r ◦ · · · ◦ ϕ ( r )2 (cid:12)(cid:12) W N,r (cid:17) − dim Q ker ϕ ( r ) r − ∩ Im (cid:16) ϕ ( r ) r − (cid:12)(cid:12) W N,r ◦ · · · ◦ ϕ ( r )2 (cid:12)(cid:12) W N,r (cid:17) . Since we assume ϕ ( r ) r − to be injective on W N,r , the last summand on the right-hand sidevanishes. Let 2 ≤ j ≤ r − 2. As the restriction to W N,r only affects x and x , whereas ϕ ( r ) j OTALLY ODD DEPTH-GRADED MULTIPLE ZETA VALUES AND PERIOD POLYNOMIALS 12 acts on x r − j +1 , . . . , x r , we obtainker ϕ ( r ) j (cid:12)(cid:12) W N,r ∩ Im (cid:16) ϕ ( r ) j − (cid:12)(cid:12) W N,r ◦ · · · ◦ ϕ ( r )2 (cid:12)(cid:12) W N,r (cid:17) = M n Upon Conjecture 4.12 and the injectivity of (3.1) , for all r ≥ the followingrecursion is satisfied: T r ( x ) = O ( x ) T r − ( x ) − S ( x ) T r − ( x ) + O ( x ) r − S ( x ) . Proof. As we assume Conjecture 4.12, we get from Definition 2.3 and Corollary 4.6dim Q ker C N,r = dim Q ker t (cid:16) E (2) N,r · · · E ( r − N,r (cid:17) + dim Q (cid:18) Im t (cid:16) E (2) N,r · · · E ( r − N,r (cid:17) ∩ ker t E N,r (cid:19) = dim Q ker t (cid:16) E (2) N,r · · · E ( r − N,r (cid:17) + dim Q Im (cid:16) ϕ ( r ) r − ◦ ϕ ( r ) r − (cid:12)(cid:12) W N,r ◦ · · · ◦ ϕ ( r )2 (cid:12)(cid:12) W N,r (cid:17) and thus, by Corollary 4.3, T r ( x ) = O ( x ) T r − ( x ) + B r ( x ) . (5.10)Using Lemma 5.2, we obtain T r ( x ) = O ( x ) T r − ( x ) + O ( x ) r − S ( x ) − r − X j =2 O ( x ) r − j − S ( x ) (cid:0) T j ( x ) − O ( x ) T j − ( x ) (cid:1) = O ( x ) T r − ( x ) + O ( x ) r − S ( x ) − S ( x ) T r − ( x ) + O ( x ) r − S ( x ) T ( x )= O ( x ) T r − ( x ) − S ( x ) T r − ( x ) + O ( x ) r − S ( x ) , where by definition T ( x ) = 0. The conclusion follows. (cid:3) Note that by our choice of T ( x ) and T ( x ), Theorem 5.3 remains true for r = 2 since weknow from [3] that T ( x ) = S ( x ). Under the assumption of Conjecture 4.12 and injectivityin (3.1), we are now ready to prove that the generating series of rank C N,r equals the explicitseries − O ( x ) y + S ( x ) y as was claimed in Conjecture 1.2. This (under the same assumptionsthough) proves the motivic version of Conjecture 1.2 (i.e. with Z odd N,r replaced by H odd N,r ).Let R r ( x ) = O ( x ) r − T r ( x ) and note that by Theorem 5.3 R r ( x ) = O ( x ) R r − ( x ) − S ( x ) R r − ( x ) OTALLY ODD DEPTH-GRADED MULTIPLE ZETA VALUES AND PERIOD POLYNOMIALS 13 for all r ≥ 2. Hence, (cid:16) − O ( x ) y + S ( x ) y (cid:17) X r ≥ R r ( x ) y r = X r ≥ (cid:0) R r ( x ) − O ( x ) R r − ( x ) + S ( x ) R r − ( x ) (cid:1) y r + R ( x ) + R ( x ) y − O ( x ) R ( x ) y = R ( x ) + O ( x ) y − O ( x ) y = 1and thus 1 + X N,r> rank C N,r · x N y r = X r ≥ R r ( x ) y r = 11 − O ( x ) y + S ( x ) y , which is the desired result. EFERENCES 14 References [1] Koji Tasaka. On linear relations among totally odd multiple zeta values related to periodpolynomials . arXiv: .[2] Francis Brown. Depth-graded motivic multiple zeta values . arXiv: .[3] Samuel Baumard; Leila Schneps. Period polynomial relations between double zeta values .arXiv: .[4] Herbert Gangl; Masanobu Kaneko; Don Zagier. “Double zeta values and modular forms”.In: Automorphic forms and zeta functions: proceedings of the conference in memory ofTsuneo Arakawa . 2006, pp. 71–106.[5] Francis Brown. “Mixed Tate motives over Z ”. In: Annals of Mathematics Modular forms (1984). Charlotte Dietze, Max-Planck-Institut für Mathematik,Vivatsgasse 7,53111 Bonn, Germany Current address : Ludwig-Maximilians-Universität München, Mathematisches Institut, Theresienstr. 39,80333 München E-mail address : [email protected] Chorkri Manai, Max-Planck-Institut für Mathematik,Vivatsgasse 7,53111 Bonn, GermanyChristian Nöbel, Max-Planck-Institut für Mathematik,Vivatsgasse 7,53111 Bonn, Germany Current address : Rheinische Friedrich-Wilhelms-Universität Bonn, Mathematisches Institut, EndenicherAllee 60, 53115 Bonn Ferdinand Wagner, Max-Planck-Institut für Mathematik,Vivatsgasse 7,53111 Bonn, Germany Current address : Rheinische Friedrich-Wilhelms-Universität Bonn, Mathematisches Institut, EndenicherAllee 60, 53115 Bonn E-mail address ::