Decomposition of elliptic multiple zeta values and iterated Eisenstein integrals
aa r X i v : . [ m a t h . N T ] O c t DECOMPOSITION OF ELLIPTIC MULTIPLE ZETA VALUES ANDITERATED EISENSTEIN INTEGRALS
NILS MATTHES
Abstract.
We describe a decomposition algorithm for elliptic multiple zeta values,which amounts to the construction of an injective map ψ from the algebra of ellipticmultiple zeta values to a space of iterated Eisenstein integrals. We give many examplesof this decomposition, and conclude with a short discussion about the image of ψ . It turnsout that the failure of surjectivity of ψ is in some sense governed by period polynomialsof modular forms. Introduction
The purpose of this paper is to describe a decomposition of elliptic multiple zeta val-ues into linear combinations of iterated Eisenstein integrals, which clarifies the algebraicstructure of elliptic multiple zeta values. This point of view was first taken up in [2], andthe present paper summarizes and extends some of its main results.1.1.
Elliptic multiple zeta values.
The notion of elliptic multiple zeta value first ap-peared explicitly in [13] under the name “analogues elliptiques des nombres multiz´etas”.Elliptic multiple zeta values come in two closely related versions, namely A-elliptic andB-elliptic multiple zeta values, each corresponding to one of the two canonical homologycycles α and β on a once-punctured elliptic curve. Elliptic multiple zeta values are linkedto a variety of other subjects, such as multiple elliptic polylogarithms [7, 18], elliptic braidgroups and elliptic associators [8, 12], as well as mixed elliptic motives [16]. They alsooccur in amplitude computations in string theory [1].Elliptic multiple zeta values are known to satisfy very many Q -linear relations, whichare studied in [1, 2, 22]. Understanding the entirety of all such relations is a delicateproblem, which, despite some advances [22], is not yet fully understood.The main theme of this paper is that the study of relations becomes somewhat simplerwhen one rewrites elliptic multiple zeta values as iterated integrals on the upper half-plane. More precisely, every elliptic multiple zeta value can be decomposed uniquely as alinear combination of iterated Eisenstein integrals [6, 20]. The gain of this representationis that the set of all iterated Eisenstein integrals is linearly independent over C [19], thusfinding relations between elliptic multiple zeta values reduces to solving linear systemsof equations. In fact, this procedure was used in [22] to prove optimal lower bounds forspaces of elliptic double zeta values.1.2. Analogy with decomposition of motivic multiple zeta values.
The decom-position of elliptic multiple zeta values into iterated Eisenstein integrals is in many waysreminiscent of the decomposition algorithm for motivic multiple zeta values into polynomi-als in non-commutative variables f , f , f , . . . , the so-called “ f -alphabet”, which amounts Mathematics Subject Classification.
Key words and phrases.
Modular symbols, elliptic associators, elliptic polylogarithms. to an isomorphism [3, 4] φ : Z m ∼ = −→ T ( F ) ∨ ⊗ Q Q [ f ] . Here Z m is the Q -algebra of motivic multiple zeta values ζ m ( k , . . . , k n ), T ( F ) ∨ denotes thegraded dual of the tensor algebra on the set F = { f k +1 | k ≥ } , and f is an additionalvariable, which commutes with all f k +1 and corresponds to ζ m (2). The main ingredientfor the construction of φ is the motivic coaction for motivic multiple zeta values [3, 15].In addition, one needs to choose a set of free algebra generators for Z m , and therefore theconstruction of φ is not canonical.The analogous situation for elliptic multiple zeta values is similar, but technically sim-pler. Denoting by E Z A the Q -algebra of A-elliptic multiple zeta values, there is aninjection [2, 21] ψ A : E Z A ֒ → T ( E ) ∨ ⊗ Q Z [2 πi ] , where E = { e , e , e . . . } and Z is the Q -algebra of multiple zeta values. Here, the vari-able e k should be thought of as corresponding to the Eisenstein series E k ( τ ) for SL ( Z )(where E ( τ ) := − ψ A is the differential equation forelliptic multiple zeta values, found by Enriquez [12, 13]. In fact, we argue that this differ-ential equation can be seen as an elliptic analogue of the motivic coaction. In contrast tothe φ -map for motivic multiple zeta values, the construction of ψ A is completely canonicaland does not depend on any initial choices. However, unlike φ , the morphism ψ A is not anisomorphism: The failure of surjectivity is related to a certain Lie algebra of derivations,and ultimately to the existence of modular forms for SL ( Z ) [1, 23]. A precise descriptionof the image of ψ A will be given in a joint work with Lochak and Schneps [19].1.3. Overview of the article.
In Section 2, we give a brief introduction to iteratedEisenstein integrals, focusing on their algebraic structure. Section 3 contains the definitionof elliptic multiple zeta values and also a short discussion of their differential equation.The consequences of this differential equation are then studied in the remaining sections:In Section 4, which essentially follows [2], we construct the map ψ A and give many concreteexamples. Then, in Section 5, we turn our attention towards describing the image of ψ A by relating it to the aforementioned Lie algebra of derivations.1.4. Acknowledgments.
This article was written on the occasion of the conference
Var-ious aspects of multiple zeta values , held at the
Research Institute for Mathematical Sci-ences (RIMS) in Kyoto during July 2016. It is my pleasure to thank the organizerHidekazu Furusho for the invitation to talk there. Also, many thanks to the mathemati-cal department of Nagoya University and RIMS for hospitality.2.
Iterated Eisenstein integrals
Iterated Eisenstein integrals are a special case of iterated Shimura integrals [20], whichwere introduced by Manin to study the rational homotopy theory of modular curves.More recently, the theory of iterated Shimura integrals has been thoroughly revisited andextended by Brown [6]. In the context of this paper, iterated Eisenstein integrals will bethe basic building blocks of elliptic multiple zeta values. One also has an injection ψ B : EZ B ֒ → T ( E ) ∨ ⊗ Q Z [2 πi ], where EZ B is the algebra of B-ellipticmultiple zeta values [21]. ECOMPOSITION OF ELLIPTIC MULTIPLE ZETA VALUES 3
Generalities.
We begin by recalling the general notion of an iterated integral, dueto Chen [9]. Let M be a complex manifold. Given a collection of smooth differentialone-forms ω , . . . , ω n ∈ Ω ( M ) and a piecewise smooth path γ : [0 , → M , one definesthe iterated integral Z γ ω . . . ω n := Z ≤ t ≤ ... ≤ t n ≤ γ ∗ ( ω )( t ) . . . γ ∗ ( ω n )( t n ) ∈ C , where γ ∗ ( ω )( t ) ∈ Ω ([0 , ω along γ , and t is the naturalcoordinate on [0 , n = 0, then R γ := 1 (the empty iterated integral).General properties of iterated integrals include the shuffle product formula Z γ ω . . . ω r Z γ ω r +1 . . . ω r + s = X σ ∈ Σ r,s Z γ ω σ (1) . . . ω σ ( r + s ) , (2.1)where Σ r,s ⊂ Σ r + s denotes the set of ( r, s )-shuffle, i.e. the set of all permutations σ of the set { , . . . , r + s } such that σ − is strictly increasing on both { , . . . , r } and on { r + 1 , . . . , r + s } . Also, iterated integrals satisfy the differential equationdd t (cid:12)(cid:12)(cid:12) t = a Z γ t ω . . . ω n = −h ω , γ ′ ( a ) i Z γ a ω . . . ω n , (2.2)where h· , ·i is the natural pairing and for a ∈ [0 , γ a : [0 , → M the path t γ ( t + (1 − t ) a ). For more properties of iterated integrals, we refer to [5, 17].2.2. Iterated integrals on the upper half-plane.
In the definition of iterated inte-grals, we will be mainly interested in the case where M is the upper half-plane H = { z ∈ C | Im( z ) > } . In this case, if the differential one-forms ω , . . . , ω n are holomorphic, thevalue of the iterated integral R γ ω . . . ω n depends only on the start and end point of γ (this holds more generally on every one-dimensional and simply connected complex mani-fold). Hence, given two points a, b ∈ H and ω , . . . , ω n as above, we may write R ba ω . . . ω n without ambiguity.One can also define iterated integrals along a path between a point τ ∈ H and thecusp i ∞ , provided the differential forms ω i have at most simple poles at i ∞ . This usesDeligne’s tangential base points (cf. [10], § H in [6], Section 4. In the sequel, we use the conventions and notationfrom [6], in particular, all our integrals are regularized with respect to the tangent vector −→ ∞ at i ∞ .The iterated integrals on H we are interested in are the iterated Eisenstein integrals E (2 k , . . . , k n ; τ ) := (2 πi ) n Z i ∞ τ E k ( τ )d τ . . . E k n ( τ n )d τ n , (2.3)where k , . . . , k n ≥ τ ∈ H . Here, E ( τ ) := − k ≥ E k ( τ ) denotes theHecke-normalized Eisenstein series E k ( τ ) = (2 k − πi ) k X ( m,n ) ∈ Z \{ (0 , } m + nτ ) k = − B k k + X n ≥ σ k − ( n ) q n , (2.4) The case k = 1 requires Eisenstein summation X ( m,n ) ∈ Z a m,n := lim N →∞ lim M →∞ N X n = − N M X m = − M a m,n . NILS MATTHES where q = e πiτ , B m denotes the m -th Bernoulli number, defined by te t − = P m ≥ B m t m m ! ,and σ m ( n ) := P d | n d m is the m -th divisor function. For notational convenience, we extendthe definition of the Eisenstein series to all non-negative integers by setting E k ( τ ) = 0, if k ≥ E ( k , . . . , k n ; τ ) = 0, if one of the k i is odd.The iterated Eisenstein integrals E ( k , . . . , k n ; τ ) are holomorphic functions of τ andthe analogue of the differential equation (2.2) is (cf. [6], Proposition 4.7)12 πi dd τ (cid:12)(cid:12)(cid:12) τ = ρ E ( k , . . . , k n ; τ ) = − E k ( ρ ) E ( k , . . . , k n ; ρ ) . (2.5)2.3. The algebra of iterated Eisenstein integrals.
Let Q hE i ⊂ O ( H ) be the Q -vectorsubspace spanned by the iterated Eisenstein integrals (where O ( H ) is the C -algebra ofholomorphic functions on H ). By (2.1), we have the shuffle product formula (cf. e.g. [6],Proposition 4.7) E ( k , . . . , k r ; τ ) E ( k r +1 , . . . , k r + s ; τ ) = X σ ∈ Σ r,s E ( k σ (1) , . . . , k σ ( r + s ) ; τ ) . (2.6)In particular, Q hE i is a Q -algebra. In order to describe Q hE i in more detail, let E := { e , e , e , . . . } be a set of variables indexed by the non-negative even integers, and let T ( E ) be the tensor Q -algebra, which is graded by giving the variables e k degree one.In fact, T ( E ) has the natural structure of a graded Hopf algebra: its coproduct ∆ : T ( E ) → T ( E ) ⊗ Q T ( E ) is the unique coproduct such that all the e k are primitive,i.e. ∆( e k ) = e k ⊗ Q ⊗ Q e k for all k ≥
0, and its antipode is the unique anti-homomorphism sending e k
7→ − e k . We denote by T ( E ) ∨ the graded dual of T ( E ),which is the Hopf algebra dual of T ( E ). Its product is the shuffle product (cid:1) : T ( E ) ∨ ⊗ Q T ( E ) ∨ → T ( E ) ∨ e ∨ k . . . e ∨ k r ⊗ Q e ∨ k r +1 . . . e ∨ k r + s X σ ∈ Σ r,s e ∨ k σ (1) . . . e ∨ k σ ( r + s ) , and its coproduct is given by deconcatenation∆ ∨ : T ( E ) ∨ → T ( E ) ∨ ⊗ Q T ( E ) ∨ e ∨ k . . . e ∨ k n n X i =0 e ∨ k . . . e ∨ k i ⊗ Q e ∨ k i +1 . . . e ∨ k n . Given a multi-index 2 k = (2 k , . . . , k n ) ∈ (2 Z ≥ ) n , we will frequently write e k insteadof e ∨ k . . . e ∨ k n .The following theorem is a consequence of C -linear independence of iterated Eisensteinintegrals [19]. It shows in particular that Q hE i is a graded Hopf algebra in a natural way. Theorem 2.1.
For any Q -subalgebra K ⊂ C , the K -linear morphism ψ : Q hE i ⊗ Q K → T ( E ) ∨ ⊗ Q K, E (2 k , . . . , k n ; τ ) e ∨ k . . . e ∨ k n is a well-defined isomorphism of K -algebras. In particular, the only algebraic relations between iterated Eisenstein integrals are givenby (2.6).
ECOMPOSITION OF ELLIPTIC MULTIPLE ZETA VALUES 5 Elliptic multiple zeta values
In this section, we recall the definition of elliptic multiple zeta values [13] (see also[1, 2, 22]). An important role in the definition is played by a certain Jacobi form in twovariables, whose study dates back to Eisenstein and Kronecker [26].3.1.
Differential forms on a once-punctured elliptic curve.
For a point τ ∈ H , wewill denote by E × τ := C / ( Z + Z τ ) \ { } the associated once-punctured complex ellipticcurve, with its canonical coordinate ξ = s + rτ , where r, s ∈ R . In [7], Brown and Levinhave introduced the following differential one-formΩ τ ( ξ, α ) = e πirα θ ′ τ (0) θ τ ( ξ + α ) θ τ ( ξ ) θ τ ( α ) d ξ, which is a variant of the Kronecker-Eisenstein series F τ ( ξ, α ) = θ ′ τ (0) θ τ ( ξ + α ) θ τ ( ξ ) θ τ ( α ) [26, 27]. Here, θ τ ( ξ ) = X n ∈ Z ( − n e πiξ ( n + ) e πiτ ( n + ) is the odd Jacobi theta function. As explained in [7], Section 3.5, Ω τ ( ξ, α ) is invariantunder lattice translations ξ ξ + m + nτ for m, n ∈ Z , and has a formal expansion in α Ω τ ( ξ, α ) = X k ≥ ω ( k ) α k − , where every ω ( k ) is a smooth differential one-form on E × τ .3.2. Definition and first examples of elliptic multiple zeta values.
Elliptic mul-tiple zeta values will be defined as regularized iterated integrals of the forms ω ( k ) alongpaths on E × τ . There are two natural such choices, namely the images α and β of the (open)straight line paths from 0 to 1 resp. from 0 to τ under the projection C \ ( Z + Z τ ) → E × τ .Corresponding to the two natural paths α and β on the once-punctured elliptic curve E × τ , there are two types of elliptic multiple zeta values, namely A-elliptic and B-ellipticmultiple zeta values, which are related to one another by a certain modular transforma-tion formula (cf. [13], Section 2.5). For simplicity, we will consider in this paper only theA-elliptic multiple zeta values. Definition 3.1.
For integers k , . . . , k n ≥
0, define the
A-elliptic multiple zeta value I A ( k , . . . , k r ; τ ) to be the regularized iterated integral I A ( k , . . . , k n ; τ ) = (2 πi ) − ( k + ... + k n − n ) Z α ω ( k ) . . . ω ( k n ) . The length of I A ( k , . . . , k n ; τ ) is defined to be n . Remark 3.2.
The original reference for elliptic multiple zeta values is [13], with additionalreferences being [1, 2, 22]. Note that the pre-factor (2 πi ) − ( k + ... + k n − n ) is not included inthe original definition of A-elliptic multiple zeta values. In the context of this paper,introducing this factor has the benefit of removing many cumbersome powers of 2 πi fromthe formulas, which will make the algebraic structure of A-elliptic multiple zeta valuesmore transparent. See [22], Definition 2.1 for the details, which employs Deligne’s regularization prescription usingtangential base points ([10], § NILS MATTHES
As functions of τ , A-elliptic multiple zeta values are holomorphic on the upper half-plane. In fact, more is true. Proposition 3.3 ([13], Proposition 5.3) . Every A-elliptic multiple zeta value has a con-vergent Fourier expansion X m ≥ a m q m , q = e πiτ , such that a m ∈ Z [2 πi ] , where Z denotes the Q -algebra of multiple zeta values. Example 3.4.
In length one, we have I A ( k ; τ ) = ( πiB k k ! if k is even,0 if k is odd, (3.1)where B k denotes the k -th Bernoulli number. This is straightforward to verify using thedefinition I A ( k ; τ ) = R α ω ( k ) and the Fourier expansion of the Kronecker-Eisenstein series F τ ( ξ, α ) (cf. [27], Theorem 3).3.3. Differential equation and constant term.
In [13], Th´eor`eme 3.3, Enriquez hasfound a differential equation for elliptic multiple zeta values, viewed as holomorphic func-tions in the coordinate τ on H . This differential equation is recursive for the length ofthe elliptic multiple zeta values, and can be expressed using Eisenstein series. Theorem 3.5 (Enriquez) . We have πi dd τ I A ( k , . . . , k n ; τ ) = α k +1 E k +1 ( τ ) I A ( k , . . . , k n ; τ ) − α k n +1 E k n +1 ( τ ) I A ( k , . . . , k n − ; τ )+ n X i =2 ( ( − k i α k i − + k i +1 E k i − + k i +1 ( τ ) I A ( k , . . . , k i − , , k i +1 , . . . , k n ; τ ) (3.2) − k i − +1 X m =0 (cid:18) k i + m − m (cid:19) α k i − − m +1 E k i − − m +1 ( τ ) I A ( k , . . . , k i − , m + k i , k i +1 , . . . , k n ; τ )+ k i +1 X m =0 (cid:18) k i − + m − m (cid:19) α k i − m +1 E k i − m +1 ( τ ) I A ( k , . . . , k i − , m + k i − , k i +1 , . . . , k n ; τ ) ) , where α n := − if n = 0 , if n = 1 , n − if n ≥ . Given I A ( k , . . . , k n ; τ ) with Fourier expansion P m ≥ a m q m , we have12 πi dd τ I A ( k , . . . , k n ; τ ) = X m ≥ ma m q m . Thus, (3.2) gives a recursive formula for the Fourier coefficients a m for m ≥
1. On the otherhand, the constant term a in the Fourier expansion is given by lim τ → i ∞ I A ( k , . . . , k n ; τ ). To be precise, the result in [13] is expressed in terms of the (not normalized) Eisenstein series G k ( τ ) = πi ) k (2 k − E k ( τ ). ECOMPOSITION OF ELLIPTIC MULTIPLE ZETA VALUES 7
In order to retrieve the constant terms of A-elliptic multiple zeta values in a systematicway, we consider the generating series of A-elliptic multiple zeta values A ( τ ) := X n ≥ ( − n X k ,...,k n ≥ I A ( k , . . . , k n ; τ ) ad k n ( a )( b ) . . . ad k ( a )( b ) ∈ C hh a, b ii . Here, C hh a, b ii is the C -algebra of formal power series in the non-commuting variables a and b , and ad k ( a ) denotes the k -fold iterate of the adjoint action ad( a )( p ) = ap − pa on C hh a, b ii . The series A ( τ ) is related to the series A ( τ ) ∈ C hh a, b ii occurring as a datum ofEnriquez’s elliptic KZB associator [12] by A ( τ ) = e − πi [ a,b ] A ( τ ). Theorem 3.6 ([12], Proposition 6.3) . The limit lim τ → i ∞ A ( τ ) exists and we have lim τ → i ∞ A ( τ ) = e πit Φ(˜ y, t ) e πi ˜ y Φ(˜ y, t ) − , (3.3) where t = − [ a, b ] and ˜ y = − ad( a ) e ad( a ) − ( b ) and Φ is the Drinfeld associator [11, 14] . The coefficients of the Drinfeld associator Φ are given by Q -linear combinations ofmultiple zeta values; see [14], Proposition 3.2.3 for an explicit formula. Comparing co-efficients on both sides of (3.3), one therefore obtains a formula for the constant termlim τ → i ∞ I A ( k , . . . , k n ; τ ) of I A ( k , . . . , k n ; τ ) in terms of Q [2 πi ]-linear combinations ofmultiple zeta values, which is however rather cumbersome to write down in practice (cf.[2], Section 2.3.1 for some examples).It turns out that the differential equation for A-elliptic multiple zeta values (i.e. The-orem 3.5) can also be expressed using the generating series A ( τ ). We will come back tothis in Section 5.4. Decomposition of elliptic multiple zeta values
We will show how the results of the last section can be used to rewrite A-elliptic multiplezeta values as iterated Eisenstein integrals. This has the crucial advantage that, by The-orem 2.1, the algebraic relations satisfied by iterated Eisenstein integrals are completelyunder control.4.1.
The decomposition map.
The starting point is the interpretation of the differen-tial equation (3.2) as a statement about the algebraic structure of A-elliptic multiple zetavalues. Let
E Z A be the Q -vector space spanned by the A-elliptic multiple zeta values E Z A := Span Q { I A ( k , . . . , k n ; τ ) | n ≥ , k i ≥ } . By the shuffle product formula for iterated integrals (2.1),
E Z A is a Q -algebra. Proposition 4.1.
There is a natural embedding of Q -algebras ψ A : E Z A ֒ → T ( E ) ∨ ⊗ Q Z [2 πi ] . Proof:
We first claim that every A-elliptic multiple zeta value can be written as a Z [2 πi ]-linear combination of iterated Eisenstein integrals (2.3). By Example 3.4, wehave I A (2 k ; τ ) = πiB k (2 k )! and I A (2 k + 1; τ ) = 0, which are Z [2 πi ]-linear combinations ofthe empty iterated Eisenstein integral E ( ∅ ; τ ) = 1, hence the claim is true for A-ellipticmultiple zeta values of length one. Now assume the claim for all A-elliptic multiple zetavalues up to and including length n −
1. By the differential equation (3.2), we know that To be precise, Enriquez writes the elliptic KZB associator in variables x , y , which are related to thevariables a , b introduced here by a = 2 πix , b = (2 πi ) − y . This also slightly changes the appearance,though not the essential content, of several results concerning A ( τ ) such as Theorems 3.6 and 5.2. NILS MATTHES πi dd τ I A ( k , . . . , k n ; τ ) is a Q -linear combination of products E l ( τ ) I A ( m , . . . , m n − ; τ ),for l ≥ m , . . . , m n − ≥
0. Using the differential equation for iterated Eisensteinintegrals (2.5) and the induction hypothesis, one sees that I A ( k , . . . , k n ; τ ) is a Z [2 πi ]-linear combination of iterated Eisenstein integrals, plus a constant of integration, whichis given by lim τ → i ∞ I A ( k , . . . , k n ; τ ) ∈ Z [2 πi ], by Theorem 3.6. In conclusion, we have aunique representation I A ( k , . . . , k n ; τ ) = X k ′ α k ′ E ( k ′ ; τ ) , for a finite number of multi-indices k ′ = ( k ′ , . . . , k ′ n ) ∈ ( Z ≥ ) n , and α k ′ ∈ Z [2 πi ]. Applyingthe isomorphism of Theorem 2.1 in the case K = Z [2 πi ], we get the result. (cid:3) Remark 4.2.
Recall from Section 2.3 that T ( E ) ∨ is a Hopf algebra, whose coproduct∆ ∨ is given by deconcatenation. By base-extension, ∆ ∨ naturally defines a coproduct on T ( E ) ∨ ⊗ Q Z [2 πi ], which restricts to a coaction E Z A → ( T ( E ) ∨ ⊗ Q Z [2 πi ]) ⊗ Q E Z A . This coaction on
E Z A can be seen as an elliptic analogue of the motivic coaction formotivic multiple zeta values Z m [3, 15]. In fact, it is known that under a suitable isomor-phism φ : Z m ∼ = → T ( F ) ∨ ⊗ Q Q [ f ] (cf. Section 1.2), the motivic coproduct on the Hopfalgebra Z m /ζ m (2) corresponds precisely to the deconcatenation coproduct on T ( F ) ∨ (cf.[4], Section 3).4.2. Examples.
We describe some explicit examples of the decomposition map in lowlengths. The case of length one is clear from Example 3.4: we have ψ A ( I A (2 k ; τ )) = 2 πiB k (2 k )! , ψ A ( I A (2 k + 1; τ )) = 0 . In what follows, we will set γ k ,...,k n = lim τ → i ∞ I A ( k , . . . , k n ; τ ). Length two:
It follows from the differential equation (3.2) together with (3.1) that I A ( k , k ; τ ) = γ k ,k − β k +1 ,k E ( k + 1; τ )+ β k +1 ,k E ( k + 1; τ ) − ( − k β k + k +1 , E ( k + k + 1; τ ) (4.1)+ k +1 X m =0 (cid:18) k + m − m (cid:19) β k − m +1 ,m + k E ( k − m + 1; τ ) − k +1 X m =0 (cid:18) k + m − m (cid:19) β k − m +1 ,m + k E ( k − m + 1; τ ) , where β i,j = α i πiB j j ! if j is even and β i,j = 0 if j is odd (recall that α i was defined inTheorem 3.5). In addition, comparing coefficients on both sides of (3.3), we get γ k ,k = ( − k (2 πi ) B k B k k ! k ! if k = 1 or k = 1,0 if k = k = 1.One now obtains ψ A ( I A ( k , k ; τ )) by replacing in (4.1) every E (2 m, n ; τ ) by e ∨ m e ∨ n (recall that E ( m, n ; τ ) = 0, if m or n is odd). ECOMPOSITION OF ELLIPTIC MULTIPLE ZETA VALUES 9
Note that, since there are no Eisenstein series of odd weight, and also since β i,j = 0, if j is odd, we see that I A ( k , k ; τ ) = γ k ,k ∈ Q · (2 πi ) , if k + k is even. In particular, I A ( k , k ; τ ) is, up to a power of 2 πi , a rational multiple of an A-elliptic multiple zeta valueof length one. This is a special case of the “length-parity theorem” for elliptic multiplezeta values (cf. [2], Appendix A.1). Length three:
Instead of giving a closed formula, which would be cumbersome to writedown, we give a few typical examples.Consider the A-elliptic multiple zeta value I A (2 , , τ ). From (3.2), we get12 πi dd τ I A (2 , , τ ) = 2 I A (3 , τ ) . On the other hand, by (3.3), the constant term γ , , = (2 πi ) B = (2 πi ) . Since I A (3 , τ ) = − πi E (4; τ ) + E (0; τ ) ! , by (4.1), it follows that I A (2 , , τ ) = 2 πi (2 πi ) − E (0 , τ ) − E (0 , τ ) ! . Thus, we have ψ A ( I A (2 , , τ )) = 2 πi (2 πi ) − e ∨ e ∨ − e ∨ e ∨ ! . Similarly, one shows that I A (0 , , τ ) = 2 πi (2 πi )
72 + 4 E (0 , τ ) + 160 E (0 , τ ) ! , (4.2) I A (0 , , τ ) = 2 πi (2 πi ) − E (0 , τ ) − E (0 , τ ) ! , (4.3)so that ψ A ( I A (0 , , τ )) = 2 πi (2 πi )
72 + 4 e ∨ e ∨ + 160 e ∨ e ∨ ! ,ψ A ( I A (0 , , τ )) = 2 πi (2 πi ) − e ∨ e ∨ − e ∨ e ∨ ! . Note that I A (2 , , τ ) = I A (0 , , τ ), which is an example of the reflection relationsbetween elliptic multiple zeta values [2, 22]. Length four:
We end this section with the decomposition of I A (0 , , , τ ). This is thesmallest example in which a non-trivial multiple zeta value occurs as a coefficient. Usingthe same procedure as before, we have by (3.2)12 πi dd τ I A (0 , , , τ ) = I A (0 , , τ ) − I A (0 , , τ ) , and γ , , , = − πiζ (3) by (3.3). Using (4.2) and (4.3), we then get I A (0 , , , τ ) = 2 πi − ζ (3) + 6 E (0 , , τ ) + 140 E (0 , , τ ) ! , which yields ψ A ( I A (0 , , , τ )) = 2 πi − ζ (3) + 6 e ∨ e ∨ e ∨ + 140 e ∨ e ∨ e ∨ ! . The image of the decomposition map
In the last section, we have constructed an embedding ψ A : E Z A ֒ → T ( E ) ∨ ⊗ Q Z [2 πi ]by rewriting A-elliptic multiple zeta values as iterated Eisenstein integrals. In this section,we will see that ψ A is not surjective, and that its image lies in a subspace associated to acertain Lie algebra of derivations u [23, 25] (see below). The key to establish this resultis the differential equation for the generating series A ( τ ) of A-elliptic multiple zeta values[12].5.1. A Lie algebra of derivations.
Let L be the free C -Lie algebra on the set { x, y } (cf. e.g. [24], Chapter IV). For every k ≥
0, define a derivation ε k : L → L by theformula ε k ( x ) = ad k ( x )( y ) , ε k ( y ) = X ≤ j 0, there existsa canonical surjection of Hopf Q -algebras T ( E ) → U ( u ) e k ε k , and the fact that u is not freely generated by the ε k means that this morphism is notinjective. Equivalently, the dual morphism ι : U ( u ) ∨ ֒ → T ( E ) ∨ (5.1)is not surjective, where U ( u ) ∨ denotes the graded dual of U ( u ) (all ε k have degree one). Example 5.1. In u , we have for example the relation (cf. [23], eq. (3))[ ε k , ε ] = 0 , ∀ k ≥ , (5.2) ECOMPOSITION OF ELLIPTIC MULTIPLE ZETA VALUES 11 which follows from ε = − ad([ x, y ]), and from the fact that every ε k annihilates [ x, y ].Since [ ε k , ε ] = ε k ◦ ε − ε ◦ ε k , the relation (5.2) implies that a linear combination P λ k e ∨ k is contained in ι ( U ( u ) ∨ ), only if λ , k = λ k, for every k ≥ ε , ε ] − ε , ε ] = ε ◦ ε − ε ◦ ε − ε ◦ ε − ε ◦ ε ) = 0 , (5.3)which essentially goes back to Ihara and Takao. Let W ⊂ T ( E ) ∨ be the four-dimensionalsubspace spanned by e ∨ e ∨ , e ∨ e ∨ , e ∨ e ∨ and e ∨ e ∨ . Then the intersection ι ( U ( u ) ∨ ) ∩ W is contained in the (three-dimensional) annihilator of (5.3), viewed as the column vector(1 , − , − , t ∈ Q . Explicitly ι ( U ( u ) ∨ ) ∩ W ⊆ Span Q { e ∨ (cid:1) e ∨ , e ∨ (cid:1) e ∨ , e ∨ e ∨ + e ∨ e ∨ } , and one can show that equality holds.5.2. The differential equation revisited. As was already mentioned at the end ofSection 3, the differential equation for A-elliptic multiple zeta values can be reformulatedas a differential equation for its generating series A ( τ ). The precise result is the following Theorem 5.2 ([12], Proposition 6.2, and [13], eq. (7)) . The series A ( τ ) satisfies thedifferential equation πi dd τ A ( τ ) = − X k ≥ E k ( τ ) e ε k !(cid:16) A ( τ ) (cid:17) , (5.4) where E k ( τ ) denotes the Eisenstein series (2.4) , and e ε k = ( k − ε k if k > , − ε if k = 0 . As shown in [13], Section 4, Theorem 5.2 is equivalent to Theorem 3.5. Solving (5.4)iteratively, using in addition the initial condition A ∞ := lim τ → i ∞ A ( τ ), which is knownexplicitly by Theorem 3.6, we get that A ( τ ) = g ( τ )( A ∞ ) = g ( τ )( e πit Φ(˜ y, t ) e πi ˜ y Φ(˜ y, t ) − ) , with g ( τ ) = P E (2 k ; τ ) e ε k , where the sum is over all multi-indices (2 k , . . . , k n ) ∈ (2 Z ≥ ) n (for all n ≥ e ε k = e ε k ◦ . . . ◦ e ε k n .Theorem 5.2 is the key to relate A-elliptic multiple zeta values to the Lie algebra u . Theorem 5.3. The decomposition map ψ A : E Z A ֒ → T ( E ) ∨ ⊗ Q Z [2 πi ] factors throughthe subspace ι ( U ( u ) ∨ ) ⊗ Q Z [2 πi ] , where ι : U ( u ) ∨ ֒ → T ( E ) ∨ is the natural dual injection (5.1) .Proof: Let B be a homogeneous vector space basis for U ( u ), and write g ( τ ) = X b ∈B " X λ k,b E (2 k ; τ ) · b, where λ k,b ∈ Q and the innermost sum is finite for every b . Under the isomorphism ψ : Q hE i → T ( E ) ∨ of Theorem 2.1, the element g ( τ ) corresponds to ψ ( g ( τ )) = X b ∈B " X λ k,b e ∨ k · b, See also the footnote on page 7. and ψ ( g ( τ )) can be seen as a morphism U ( u ) ∨ → T ( E ) ∨ b ∨ b ∨ ( ψ ( g ( τ ))) = X λ k,b E (2 k ; τ ) . (5.5)This morphism is clearly dual to the natural surjection T ( E ) → U ( u ) given by e k e ε k ,thus, comparing with (5.1), we see that the image of (5.5) is equal to ι ( U ( u ) ∨ ).Now since A ( τ ) is the generating series of the I A ( k , . . . , k n ; τ ), the Q -span of thecoefficients of A ( τ ) = g ( τ )( A ∞ ) equals E Z A . On the other hand, by definition the imageof ψ A is equal to the Q -span of the coefficients of ψ ( g ( τ ))( A ∞ ), which, by the precedingdiscussion and the fact (cf. Theorem 3.6) that the coefficients of A ∞ lie in Z [2 πi ], arecontained in ι ( U ( u ) ∨ ) ⊗ Q Z [2 πi ]. Thus, the image of ψ A is indeed contained in ι ( U ( u ) ∨ ) ⊗ Q Z [2 πi ]. (cid:3) Remark 5.4. Essentially the same result holds for the algebra E Z B of B-elliptic multiplezeta values. More precisely, one has a canonical embedding [21] ψ B : E Z B ֒ → T ( E ) ∨ ⊗ Q Z [2 πi ] . The Fourier subspace. In the last subsection, we have seen that the relation be-tween the differential equation for A ( τ ) and the derivations e ε k implies that the image of ψ A lies in ι ( U ( u ) ∨ ) ⊗ Q Z [2 πi ]. In this subsection, we will see that the Fourier expansion ofA-elliptic multiple zeta values (cf. Proposition 3.3) further constrains the image of ψ A . Definition 5.5. The Fourier subspace Q hE i Fou ⊂ Q hE i is the Q -linear subspace definedby Q hE i Fou := Span Q {E (2 k , . . . , k n ; τ ) | n ≥ , k i ≥ } , where E (2 k , . . . , k n − , τ ) := 0 and for k n = 0, we set E (2 k , . . . , k n ; τ ) := E (2 k , . . . , k n ; τ ) − B k n k n E (2 k , . . . , k n − , τ ) . We will denote by T ( E ) ∨ Fou the subspace of T ( E ) ∨ , which is the image of Q hE i Fou underthe isomorphism ψ : Q hE i ∼ = → T ( E ) ∨ of Theorem 2.1. Note that T ( E ) ∨ Fou is a Q -subalgebraof T ( E ) ∨ and a left comodule under T ( E ) ∨ , i.e. the coproduct ∆ ∨ on T ( E ) ∨ restricts toa morphism T ( E ) ∨ Fou → T ( E ) ∨ ⊗ Q T ( E ) ∨ Fou . Remark 5.6. The name “Fourier subspace” is motivated by the fact that a Q -linearcombination of iterated Eisenstein integrals E (2 k , . . . , k n ; τ ) has a Fourier expansion in q = e πiτ , if and only if it is contained in Q hE i Fou . This follows easily from E k ( τ ) = − B k k + O ( q ), valid for k > 0, which together with E ( τ ) := − E (2 k , . . . , k n ; τ ) ∈ O ( q ) (since the ideal q · C [[ q ]] ⊂ C [[ q ]] is closed under integration with respect to themeasure 2 πi d τ = d(log q )). Theorem 5.7. The morphism ψ A maps E Z A into the Fourier subspace; more precisely ψ A : E Z A ֒ → ι ( U ( u ) ∨ ) Fou ⊗ Q Z [2 πi ] , where ι ( U ( u ) ∨ ) Fou := ι ( U ( u ) ∨ ) ∩ T ( E ) ∨ Fou , and ι : U ( u ) ∨ ֒ → T ( E ) ∨ is the natural injection (5.1) . We should note that this additional constraint is a particular feature of A-elliptic multiple zeta values.More precisely, the analogue of Theorem 5.7 for B-elliptic multiple zeta values is false, since B-ellipticmultiple zeta values in general do not have a Fourier expansion. ECOMPOSITION OF ELLIPTIC MULTIPLE ZETA VALUES 13 Proof: We can rewrite g ( τ ) using the E (2 k , . . . , k n ; τ ) as follows: g ( τ ) = X E (2 k ; τ ) e ε k = X k n =0 (cid:16) E (2 k ; τ ) + B k n k n E (2 k , . . . , k n − , τ ) (cid:17)e ε k + X k n =0 E (2 k , . . . , k n − , τ ) e ε k ◦ . . . ◦ e ε k n − ◦ e ε = X E (2 k ; τ ) e ε k + X E (2 k , . . . , k n − , τ ) e ε k ◦ . . . ◦ e ε k n − ◦ (cid:16) e ε + X k n ≥ B k n k n e ε k n | {z } =: D (cid:17) , where all sums are over the multi-indices 2 k = (2 k , . . . , k n ) ∈ (2 Z ≥ ) n , for all n ≥ 0. It isshown in the proof of [12], Proposition 6.3, that D is a derivation of C hh a, b ii that annihilatesboth ˜ y = − ad( a ) e ad( a ) − ( b ) and t = − [ a, b ], thus it annihilates every word in ˜ y and t . Since A ∞ = e πit Φ(˜ y, t ) e πi ˜ y Φ(˜ y, t ) − is a power series in ˜ y and t , it follows that D ( A ∞ ) = 0. 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