BROWN-ZAGIER RELATION FOR ASSOCIATORS TOMOHIDE TERASOMA 1. Introduction 3 We have big heritage of equalities on hypergeometric funcitons, which can 1 be used for showing many equalities for multiple zeta values. This method 0 2 can be also applicable for showing relations between coefficients of associators n using the theory of Φ-cohomology. A Φ-cohomology is equipped with two a realizations B,dR and a comparison map described by the given associator J Φ. Brown [B] used certain relation between multiple zeta values to show the 0 3 injecctivityofthehomomorphismfromMotivicGaloisgrouptoGrothendieck- Teichmuller group. This relation was proved by Zagier [Z], which we call ] T Brown-Zagier relation. After his work, Li [L] gave another proof of Brown- N Zagier relation using several functional equations of hypergeometric series. . h In this paper, we show that Brown-Zagier relation holds also for the coef- t a ficietns of any associators. In the paper [L], he proved Brown-Zagier relation m using Dixon’s theorem which is equivalent to Selberg integral formula. The [ Selberg integral formula arises from symmetric product construction, which 1 does no exist in the category of moduli space. Even in this case, we can con- v struct isomorphism between Φ-local systems using descent theory. The main 4 7 theorem is steted as follows. 4 7 Theorem 1.1. We use the notation for coefficients ζ (n ,...,n ) of an Φ 1 m . 1 associator Φ. Then we have 0 3 a+b+1 1 ζ (2a,3,2b) = 2 (−1)rcr ζ (2r +1))ζ (2a+b−r+1), φ a,b Φ Φ : v r=1 X i X where ar cr = 2r −(1− 1 ) 2r a,b 2a+2 22r 2b+1 (cid:18) (cid:19) (cid:18) (cid:19) Since the generating series of motivic multiple zeta values satisfies the associator relation, we have the following corollary. Corollary 1.2. The same relation holds in the coordinate ring of mixed Tate motives. The above corollary gives an another proof of a result of Brown. Notation 1.3. The product of Gamma function Γ(a )Γ(a )···Γ(a ) is de- 1 2 n noted by Γ(a ,a ,...,a ) for short. 2 2 n 1 2 TOMOHIDE TERASOMA 2. Differential equations and generating functions In this section, we recall outline of classical theory of hypergoemtric func- tions and Gauss-Manin connections. 2.1. Differentialequation and iterated integral. LetN∗ beaChhe ,e ii 0 1 left module. The action of e and e on N∗ is denoted by P and P , Let 0 1 0 1 O be the ring of analytic functions on an open set U in P1 −{0,1,∞}. We U define a map P : N∗ ⊗O → N∗ ⊗Ω1 an an dx dx N∗ → N∗ ⊗h , i : v 7→ P(x)v x x−1 where P(x) = P dx +P dx . There exists a unique local solution Φ (x) of 0 x 1x−1 u the differential equation dΦ(x) = P(x)Φ(x) for End(N∗,N∗)-valued analytic x functions such that Φ (u) = id . It is denoted by exp( P). For any solu- u V Zu tion of the differential equation dV = PV for End(N∗,N∗)-valued functions, we have x x x V(x) = (I + P + PP +···)V(u) = exp( P)V(u) Zu Zu Zu ′ for t ∈ R,0 < t < ǫ. Fora path γ from u to u′, exp( u P) depends only 0 u on the homotopy class of γ, which is denoted by ρ(γ). Then ρ defines a left R π (M )-module on N∗. 1 4 2.2. Differential equation of Gauss hypergeometric functions. In this section, we recall the differential equations satisfied by hypergoemtric func- tions. 2.2.1. We define hypergeometric function by ∞ (a) (b) F(a,b;c;x) = n nxn, n!(c) n n=0 X where (a) = a(a+1)···(a+ n−1). The hypergoemetric function has the n following integral expression. 1 B(a,c−a)F(a,b;c;x)= ta−1(1−t)c−a−1(1−xt)−bdt. Z0 2.2.2. The differential is denoted by D = ∂ . We define a matrix V = ∂x 0 (v ) , where ij 1≤i,j≤2 Γ(a,c−a+1) (0) v = F(a,b;c+1;x), 11 Γ(c+1) Γ(b−c,1−b) v(0) =x−c F(b−c,a−c;1−c;x), 12 Γ(1−c) 1 1 (0) (0) v = xD(v ), v = xD(v ) 22 a 12 21 a 11 BROWN-ZAGIER RELATION FOR ASSOCIATORS 3 Let P be a matrix defined by dx dx (2.1) P = P + P 0 1 x x−1 where 0 a 0 0 P = ,P = . 0 0 −c 1 −b c−a−b (cid:18) (cid:19) (cid:18) (cid:19) Then the matrix V satisfies the differential equation 0 (2.2) dV = PV. Let V be matrix defined by 1 Γ(a,b−c) (1) v = F(a,b;a+b−c;1−x), 11 Γ(a+b−c) Γ(c+1−a,1−b) v(1) =(1−x)c−a−b+1 F(c+1−a,c+1−b;2+c−a−b;1−x), 12 Γ(c+2−a−b) 1 1 (1) (1) v = xD(v ) v = xD(v ), 22 a 12 21 a 11 Then V also satisfies the differential equation (2.2). 1 2.2.3. Connections and differential equations for coefficients. Let N∗ be the vector space generated by ω∗,ω∗, N be its dual and ω ,ω be the dual basis 1 2 1 2 of ω ,ω . We define a linear maps ∇ : N → N ⊗hdx, dx i and ∇∗ : N∗ → 1 2 x x−1 N∗ ⊗hdx, dx i by x x−1 ω ω (2.3) ∇ 1 = P 1 ω ω 2 2 (cid:18) (cid:19) (cid:18) (cid:19) ∇∗ ω∗ ω∗ = − ω∗ ω∗ P 1 2 1 2 The map ∇∗ can be extende(cid:0)d to a co(cid:1)nnect(cid:0)ion on th(cid:1)e N∗⊗C[x, 1, 1 ], which x x−1 is also denoted by ∇∗. We use the following identification f (x) f (x)ω∗ +f (x)ω∗ = 1 1 1 2 2 f (x) 2 (cid:18) (cid:19) Then we have f (x)dx f (x) f (x) ∇ 1 = d 1 −P 1 f (x)dx f (x) f (x) 2 2 2 (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) on N∗ ⊗ C[x, 1, 1 ]. Let γ be an N∗-valued analytic function. Using the x x−1 pairing h , i, γ is written as hγ,ω i γ = hγ,ω iω∗ +hγ,ω iω∗ = 1 , 1 1 2 2 hγ,ω i 2 (cid:18) (cid:19) Therefore γ is a horizontal section for ∇ if and only if hγ,ω i hγ,ω i d 1 = P 1 hγ,ω i hγ,ω i 2 2 (cid:18) (cid:19) (cid:18) (cid:19) 4 TOMOHIDE TERASOMA 2.2.4. Hypergeometric function and its integral expression. By the integral expression of hypergoemetric functions, the matrix elements of V is written 0 as (0) (0) (0) (0) (2.4) v = ω ,v = ω ,v = ω ,v = ω , 11 1 12 1 22 2 22 2 Zγ1 Zγ2 Zγ1 Zγ2 (1) (1) (1) (1) v = ω ,v = ω ,v = ω ,v = ω , 11 1 12 1 22 2 22 2 Zγ1# Zγ2# Zγ1# Zγ2# where ω ,ω are the relative twisted de Rham cohomology classes defined by 1 2 dt bxdt (2.5) ω = , ω = , 1 2 t a(1−xt) (cid:2) (cid:3) (cid:2) (cid:3) and γ ,γ be twisted cycle defined by 1 2 (2.6) γ = ta(1−t)c−a(1−xt)−b , 1 [0,1] γ = (cid:2)ta(t−1)c−a(xt−1)−b(cid:3) 2 [1,∞] x γ# =(cid:2) (−t)a(1−t)c−a(1−x(cid:3)t)−b 1 [−∞,0] γ# = (cid:2)ta(t−1)c−a(1−xt)−b (cid:3) 2 [1,1] x We have the following eq(cid:2)uality of cycles. (cid:3) (2.7) s(c)γ# =s(c−a)γ +s(b)γ , 1 1 2 s(c−a−b)γ =s(c−b)γ# +s(b)γ#. 1 1 2 1 sin(πz) where s(z) = = . Let N be the vector space generated Γ(z)Γ(1−z) π byω ,ω . ThentheGauss-Maninconnectionisgivenbythemap(2.3). Under 1 2 the comparison map, γ ,γ defines a horizontal N∗-valued analytic map on 1 2 (0,1). By the expression 2.4, we have 1 lim γ (ǫ) =B(a,c−a+1) , 1 0 ǫ→+0 (cid:18) (cid:19) 1 lim (ǫcγ )(ǫ) =B(b−c,1−b) . 2 −c ǫ→+0 (cid:18) a(cid:19) 2.3. Gauss-Manin connection and horizontal section on the daul. 2.3.1. Dual differential equations. We construct solutions of the dual differ- ential equation around 1. We define a matrix W = (w ) by 1 ij 1≤i,j≤2 Γ(−a,c−b+1) w = F(−a,−b;c−a−b+1;1−x), 11 Γ(c−a−b+1) Γ(a−c,b+1)) w = (1−x)−c+a+bF(a−c,b−c;1−c+a+b;1−x) 21 Γ(a+b−c+1) 1 1 w = − (1−x)D(w ), w = − (1−x)D(w ). 12 11 22 21 b b BROWN-ZAGIER RELATION FOR ASSOCIATORS 5 The matrix elements of W is written as 1 v = ω∗,v = ω∗,v = ω∗,v = ω∗, 11 1 12 1 22 2 22 2 Zγ1∗ Zγ2∗ Zγ1∗ Zγ2∗ where ω∗,ω∗ are 1 2 dt (x−1)dt (2.8) ω∗ = , ω∗ = , 1 t 2 (1−xt) and γ∗,γ∗ are (cid:2) (cid:3) (cid:2) (cid:3) 1 2 γ∗ = (−t)−a(1−t)−c+a(1−xt)b 1 [−∞,0] γ∗ = (cid:2)t−a(t−1)−c+a(1−xt)b (cid:3) 2 [1,1] x Then the matrix W sat(cid:2)isfies the differential equ(cid:3)ation dW = −W P. There- 1 1 1 fore we have t1 W = W (t )exp( P). 1 1 1 Zx 2.3.2. Duality and exponential map around 1. Let V and W be matrix de- 1 1 fined in §2.2, §2.3.1. Since ∂ ∂W ∂V 1 1 (W V ) = V +W = −W PV +W PV = 0 1 1 1 1 1 1 1 1 ∂x ∂x ∂x the matrix W V does not depends on x. By considering the limit for x → 0, 1 1 we have s(a+b−c) 0 (2.9) W V = as(−a)s(b−c) = D 1 1 s(c−a−b) 1 0 ! as(a−c)s(b) and as a consequence, we have y V (y)D−1W (x) = exp P. 1 1 1 Zx 2.4. Generating function of multiple zeta values ζ(2,...,3,...,2). We specialize to the case c = 0,a = −b. Then the matrix P , P of (2.1) becomes 0 1 0 a 0 0 (2.10) P = ,P = . 0 0 0 1 a 0 (cid:18) (cid:19) (cid:18) (cid:19) Using the limit computation of the last subsection, we have x 1 1 s(a) 0 V (x) = 1 0 exp( P) = F(a,−a;1;x) 0 0 0 (cid:18) (cid:19) Z0 (cid:18) (cid:19) (cid:0) (cid:1) (cid:0) (cid:1) Proposition 2.1. Let P and P be matrices defined in (2.10). 0 1 (1) For I = (i ,...,i ) ∈ {0,1}n, we define E = P ···P . Then we 1 n I i1 in have 1 a2n if I = (10)n (1,0)P = I 0 (0 otherwise. (cid:18) (cid:19) 6 TOMOHIDE TERASOMA (2) Let ϕ(e ,e ) be an element Chhe ,e ii given by 0 1 0 1 (2.11) ϕ(e ,e ) = c e ...e 0 1 i1,...,in i1 in nX≥0i1∈{0,1}X,...,in∈{0,1} where c ∈ C. Then we have i1,...,in 1 (2.12) (1,0)ϕ(P ,P ) = 1+ c a2n. 0 1 0 (01)n (cid:18) (cid:19) n>0 X Proof. This is an easy consequence of the equalities P2 = P2 = 0, and 0 1 a2 0 0 0 P P = , P P = . 0 1 0 0 1 0 0 a2 (cid:18) (cid:19) (cid:18) (cid:19) (cid:3) Since V(x) is expressed by using iterated integral, we have ∞ x du du 1 F(a,−a;1;x) = 1 0 (P +P )n 0 u 1u−1 0 n=0Z0 (cid:18) (cid:19) (cid:0) (cid:1)X ∞ x du du = ( )na2n u u−1 n=0Z0 X by Proposition 2.1. By setting x = 1, we have ∞ 1 du du sin(πa) ( )na2n = F(a,−a;1;1)= u u−1 πa n=0Z0 X by the equality (??). Similarly, we have 1 1 1 B(−a,a+1)−1 1 0 W (x) = 1 0 exp( P) = F(−a,a,1,1−x), 1 0 0 (cid:18) (cid:19) Zx (cid:18) (cid:19) (cid:0) (cid:1) (cid:0) (cid:1) and it is equal to ∞ 1 du du ( )ma2m u u−1 m=0Zx X We have the following proposition. Proposition 2.2. We set ∞ ∞ 1 du du du du du (2.13) φ(a,b) = ( )m ( )na2nb2m u u−1 u u u−1 m=0n=1Z0 X X Then we have 1 dw φ(a,b) = F(−b,b,1,1−w)(F(a,−a,1,w)−1) w Z0 BROWN-ZAGIER RELATION FOR ASSOCIATORS 7 Proof. By the definition of iterated integral, we have 1 du du du du du ( )m ( )n u u−1 u u u−1 Z0 1 1 du du dx x dv dv = ( )m ( )n dx. u u−1 x v v −1 Z0 (cid:20)Zx (cid:21) (cid:20)Z1 (cid:21) By taking the generating function on m and n, we get the proposition. (cid:3) Remark 2.3. Zagier showed that φ(a,b) is also equal to sin(πb) d (2.14) | F (a,−a,z;1+b,1−b;1). z=0 3 2 πb dz In §6, we show that associator versions of the formal power series (2.13) and (2.14) coincides. 3. Associator and Hopf algebroid 3.1. Fundamental algebroid of moduli spaces. Werecallthestructureof Hopf algebroids A ,A of the moduli space M = M of n-punctured n,dR n,B n 0,n genus zero curves in this subsection. Definition 3.1. We define the set of tangential points T of n points in genus n zero curve as the set of planer trivalent tree with n terminals. For example T = {01,10,0∞,∞0,1∞,∞1} 4 Thus #T = 3×2, #T = 15×4, etc. 4 5 Then we can define the pro-nilpotent algebroid A ,A over the set n,dR n,B T as follows. n Definition 3.2. For two points a,b ∈ T , the bifiber of the algebroid A = n n,dR {A } is defined as the following generators and reltaions. n,dR,ab ab (1) (Genrators) t with 1 ≤ i < j ≤ n. We use the notation t = t for ij ji ij i < j. (2) (Relations) (a) [t ,t ] = 0 ij kl (b) [t ,t +t ] = 0 ij ik kj (c) t = 0 j6=i ij Then A is the completed de Rhan fundamental group algebra of M and n,dR P n has a standard coproduct ∆(t ) = t ⊗1+1⊗t . ij ij ij Definition 3.3. (1) Two tangential base points a,b ∈ T are adjacent if n it can be transformed by elementary change H ↔ I. (2) Two tangentail base points a,b ∈ T are neighbours if it can be trans- n formed by twisting with respect to a edge. (3) A = {A } is a pro-nilpontent algebroid generated by two n,B n,B,ab ab type of generators: (a) path p connecting two adjacent tangential base points. ab (b) small circle c connecting two neibours. ab 8 TOMOHIDE TERASOMA (c) Relations on A are generated by 2-cycle relations, 3-cycle re- n,B lations, 5-cycle relations. Then the A is the completed groupoind algebra of M . n,B n Definition 3.4. (Category C) We define the abelian category C as follows. An object V of C is a triple (V ,V ,c ) consisting of dR B V (1) Q-vector space V , dR (2) Q-vector space V , and B (3) an isomorphism V ⊗C ≃ V ⊗C B dR Sometimes one consider profinite version. In this case, ⊗C means the com- pleted tensor product. Morphism form f : V → W is a pair of morphisms f : V → W and f : V → W compatible with the comparison maps. dR dR dR B B B The category C becomes a tensor category by tensoring each dR and B com- ponents Definition 3.5. We define the category Minf be the category whose objects are M and morphisms are generated by infinitesimal inclusions. n Definition 3.6. We can define two functors A ,A : Minf → Hopf from dR B Q Minf to the category of Hopf algebroids by attaching de Rham fundamental groups and Betti fundamental groups. 3.2. Choiceof coordinate. LetC beagenuszerocurveandP = (C,p ,...,p ) 1 n (p ∈ C) an element in M . We choose a coordinate t of C such that i n t(p ) = 0,t(p ) = 0,t(p ) = 0. Using the coordinate t, M is iden- n−2 n−1 n n tified with an open set of An−3 defined by {(x ,...,x ) | x 6= x for i 6= j,x 6= 0,1 for all i} 1 n−3 i j i by setting x = t(p ). This coordinate is called the distinguished coordi- k k nate. By taking the distinguished coordinate of M , the underlying curve is 4 identified with P1 −{0,1,∞}. Definition 3.7 (admissible function, admissible differential form). (1) Let S = (i,j,k,l) be a ordered subset of distinct elements in [1,n]. For an element P = (C,p ,...,p ) be an element of M . There is a unique 1 n n coordinate t of C such that t(p ) = 0,t(p ) = 1,t(p ) = ∞. The value i j k t(p ) at p gives rise to an algebraic function on M , which is denoted l l n by ϕ . The set of admissible functions is denoted by Ad(M ). S n (2) Let x ,...,x −3 be the distinguished coordinate. An element in the 1 n linear span of dxi, dxi , d(xi−xj) is called an admissible differential xi xi−1 xi−xj form. Remark 3.8. (1) ϕ ∈ Ad(M ) defines a morphism M → M . and a n n 4 morphism of algebroids A → A . n 4 (2) If S∩{n−2,n−1,n} = ∅, using the distinguised coordinates of M , n we have (x −x )(x −x ) l i j k ϕ (P) = . S (x −x )(x −x ) l k j i BROWN-ZAGIER RELATION FOR ASSOCIATORS 9 Therefore ϕ is invariant under substitutions i ↔ l,j ↔ k and i ↔ S j,k ↔ l. (3) The following functions are admissible functions. x (x −0)(x −∞) x (x −x )(∞−0) i i j i j i = , 1− = . x (x −0)(x −∞) x (x −0)(∞−x ) j j i j j j (1−x )(∞−0) i 1−x = i (1−0)(∞−x ) i Proposition 3.9. The set of functorial isomophisms from A ⊗C to A ⊗C B dR sending small half circle log(c ) to πit is identified with the set of assoica- ij ij tors. The one to one correspondence is given by A ∋ [0,1] 7→ Φ ∈ A = Chhe ,e ii 4,B,01,10 dR,4 0 1 dx dx Here e and e are the dual basis of ω = and ω = , respectively. 0 1 0 1 x x−1 By the above proposition, we have an isomorphism of Hopf algebra ≃ c : A ⊗C −→ A ⊗C. Φ,n n,B n,dR associated to a given assoicator Φ. This isomorphism gives an object AΦ = n (A ,A ,c ). The isomorphism c is called the Φ-comparison map. n,dR n,B Φ,n Φ,n Proposition 3.10. (1) Let 3 ≤ m < n be integers and f morphsim de- fined by f : M → M : (x ,...,x ) → (x ,...,x ) n m 1 n−3 1 m−3 Then for ⋆ = dR,B, the induced maps of algebroids A → A n,⋆ m,⋆ are compatible with the Φ-comparison maps. (2) Let 3 ≤ m,n be integers. Then a morphsim f : M → M ×M n+m−3 n m (x ,...,x ,y ,...,y ) 7→ (x ,...,x )×(y ,...,y ) 1 n−3 1 m−3 1 n−3 1 m−3 induces a morphism of algebroids f : A → A ⊗A n+m−3 n m in C. (3) Let 3 ≤ m < n ,n be integers. Then the natural morphsim 1 2 f : M × M → M ×M n1 Mm n2 n1 n2 induces a morphism of algebroids in C. Φ Thecoefficientc ofe e ...e inc ([0,1])iswrittenas ω ...ω . Φ,I i1 i2 ik Φ,4 [0,1] i1 ik We define Φ-multiple zeta value similarly. A Φ-multiple zeta value is written R as Φ ζ (m ,...,m ) = ωmk−1ω ...ωm1−1ω Φ 1 k 0 1 0 1 Z[0,1] It is a coefficint of the associator Φ. 10 TOMOHIDE TERASOMA 3.3. A-module. Let T be a set and A a Hopf algbroid object in C over T. We define the notion of A-module. Definition 3.11. Let M = (M ) = (M ,M ,c ) be an object a a∈T dR,a B,a M,a a∈T in C indexed by a ∈ T. M is called an A-module if it is equipped with an action of A in C µ : A⊗M → M M which is associative and unitary. Here action of algebroid is given by a mor- phism A ⊗M → M . ab a b in C. Remark 3.12. Let M,N be A module. Then using coproduct structure of A, M ⊗N is equipped with A module. Example 3.13. (1) Let 4 ≤ m < n and f : M → M be the map n m defined by (x ,...,x ) 7→ (x ,··· ,x ). Then we have an alge- 1 n−3 1 m−3 broid homomorphism f : AΦ → AΦ. Therefore for a fixed p ∈ T , n m m by setting M = AΦ we have an AΦ-module. It is called a pull a m,p,f(a) n back of the map f. (2) By taking an abelianization AΦ,ab of AΦ, we have a homomorphism n n of Hopf algebroids AΦ → AΦ,ab. n n By choosing a base point p ∈ T , we have have an AΦ-module AΦ,ab. n n n,p∗ In particular, by using the distinguished coordinate x, A module 4 xαQ[[a]] is defined by taking the base point as 01, (3) Let ϕ be an admissible function on M and α formal parameter. The n morphism M → M induced by ϕ is also denoted by ϕ and x be the n 4 distinguished coordinate of A . We define A [[α]]-module 4 n ϕαQ[[α]] by the pull back ϕ∗(xαQ[[a]]) of xαQ[[a]]. We define m ϕαi Q[[α ,...,α ]] = ϕα1Q[[α ]]⊗···⊗ϕαmQ[[α ]] 1 m 1 1 m m (cid:18)i=1 (cid:19) Y Proposition 3.14. Let ϕ , (i = 1,...,m), ψ , b(j =b1,...,l) be admissible i j functions on M and a ∈ Z. We assume that ψ = m ϕaij We set n ij j i=1 m Q L = a α j ij i i X for j = 1,...,l. Then m l ϕαi Q[[α ]] = ψLj Q[[α ]]. i i j i (cid:18)i=1 (cid:19) (cid:18)j=1 (cid:19) Y Y as AΦ module. n