The meta-abelian elliptic KZB associator and periods of Eisenstein series

We compute the image of Enriquez’ elliptic KZB associator in the (maximal) meta-abelian quotient of the fundamental Lie algebra of a once-punctured elliptic curve. Our main result is an explicit formula for this image in terms of Eichler integrals of Eisenstein series, and is analogous to Deligne’s computation of the depth one quotient of the Drinfeld associator. We also show how to retrieve Zagier’s extended period polynomials of Eisenstein series, as well as the values at zero of Beilinson–Levin’s elliptic polylogarithms from the meta-abelian elliptic KZB associator.


Introduction
This paper deals with the computation of some of the coefficients of the elliptic KZB associator defined by Enriquez [14]. In order to put things into context, we first recall the analogous picture in genus zero, due to Deligne, Drinfeld and Ihara.
Let p(U ) := L(x 0 , x 1 ) ∧ be the lower central series completion of the free Lie algebra in variables x 0 , x 1 , and denote by exp p(U ) the associated pro-unipotent algebraic group. The Drinfeld associator (x 0 , x 1 ) is an element of exp p(U ) R := exp(p(U ) ⊗R), which is constructed from the monodromy of the universal Knizhnik-Zamolodchikov (KZ) connection on P 1 C \ {0, 1, ∞} (for this reason, is sometimes B Nils Matthes nilsmath@mpim-bonn.mpg.de 1 Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany called KZ-associator). First introduced in [13], the Drinfeld associator plays a pivotal role in the context of quantum groups and Grothendieck-Teichmüller theory. We are interested in arithmetic properties of (x 0 , x 1 ). The following two aspects, which are in fact closely related to each other, are of particular relevance.
(i) The coefficients of (x 0 , x 1 ) are expressible as Q-linear combinations of multiple zeta values ζ(k 1 , . . . , k n ) = m 1 >···>m n >0 1 m k 1 1 · · · m k n n , k 1 ≥ 2, k 2 , . . . , k n ≥ 1, which are generalizations of the special values of the Riemann zeta function at positive integers. These numbers have (at least conjecturally) a rich algebraic structure [16,20]. (ii) The Lie algebra p(U ) is the de Rham realization of an element of the category MTM of mixed Tate motives over Z ( [12], §5). As a consequence, the unipotent fundamental group U MTM of MTM acts on exp p(U ) (Ihara action), and in particular on (x 0 , x 1 ). 1 The Deligne-Ihara conjecture (proved by Brown in [3]) states that this action is faithful, thus elements of U MTM are completely determined by their action on (x 0 , x 1 ), which can be computed very explicitly [4].
For both (i) and (ii), the archetypal result is due to Deligne ([11], §19), who inspired by unpublished work of Wojtkowiak essentially showed that (1.1) where D 1 p(U ) ⊂ p(U ) denotes the ideal generated by x 1 . On the one hand, this exhibits the Riemann zeta values ζ(k) as coefficients of log( (x 0 , x 1 )). On the other hand, since ζ(k) = 0, one deduces from (1.1) that the generators exp(σ 2n+1 ) of U MTM act non-trivially on exp p(U ) ( [12], §6.8), which was a first step towards establishing the Deligne-Ihara conjecture.
(i) The coefficients of the elliptic KZB associator are the elliptic multiple zeta values, first introduced in [15] and studied in more detail in [2,24,26,27]. They are closely related to both multiple zeta values and to iterated integrals of Eisenstein series [6,25].
(ii) The Lie algebra p(E × τ ), viewed as a local system over the moduli space M 1, − → 1 of elliptic curves with a non-zero tangent vector at the origin, is the de Rham realization of an element of the category MEM− → 1 of universal mixed elliptic motives (over M 1, − → 1 ). This category can be seen as an elliptic enhancement of the category of mixed Tate motives over Z. The corresponding Galois group G MEM− → 1 acts on p(E × τ ) [19], and therefore also on the elliptic KZB associator. In analogy to the Deligne-Ihara conjecture, it is asked in [19], §24.2 whether the action of G MEM− → 1 on p(E × τ ) is faithful. The main goal of this article is to establish an analog of (1.1) for the elliptic KZB associator, i.e. the explicit computation of the images of the formal logarithms A(τ ) := log(A(τ )) and B(τ ) := log(B(τ )) in a certain quotient of p(E × τ ) C . More precisely, ) ∧ be the commutator. Taking its lower central series defines a filtration D • p(E × τ ), the elliptic depth filtration ( [19], §27). In particular, is the double commutator, and our goal is to compute the images A(τ ) met−ab and B(τ ) met−ab of the elliptic KZB associator in the meta-abelian quotient Our main result can then be stated as follows.
Theorem (Theorem 5.6 below) Let U := U 2πi and W := U + τ V. We have Here, ∞ are given by The proof of Theorem 5.6 uses a result of Enriquez [14] to the effect that for certain explicit elements A ∞ , B ∞ ∈ p(E × τ ) C and an automorphism g(τ ) ∈ Aut(exp(p(E × τ ) C )). Then, we separately compute the images of A ∞ and B ∞ in , and from this, we are able to deduce Theorem 5.6.
The series A ∞ and B ∞ are arithmetic: they can be expressed in terms of the Drinfeld associator and therefore come from genus zero. On the other hand, the automorphism g(τ ) is geometric: it describes the action of G . As a byproduct of our proof, we see that already their images in the meta-abelian quotient are interesting objects in their own right. Namely, the automorphism g(τ ) met−ab is essentially the generating series of the special values of elliptic polylogarithms at the zero section of the elliptic curve [1,22]  Finally, we note that Nakamura [28,29] has studied an -adic analog of the metaabelian image of the elliptic KZB associator (called "universal power series for Dedekind sums"), which is a genus one analog of Ihara's universal power series for Jacobi sums [21]. It would be very interesting to compare his results to ours.
The plan of the paper is as follows. In Sects. 2 and 3, we collect some background in order to make the paper self-contained. Then, in Sect. 4, we recall the definition of the elliptic KZB associator [14], but from the point of view of the mixed Hodge structure on the unipotent fundamental group of E × τ [8]. Finally, in Sect. 5, the main results of this paper are proved.

Notation and conventions
We start by introducing some general notation, to be used throughout the text.
For any finite set {x 1 , . . . , x n } and a field K , we denote by L(x 1 , . . . , x n ) K the free Lie algebra on X over K (we omit K if K = Q), and by L(x 1 , . . . , x n ) ∧ K the completion for its lower central series. It is a topological Lie algebra over K , whose topology is induced from the lower central series. Its topological universal enveloping algebra is given by K x 1 , . . . , x n , the K -algebra of formal power series in the noncommuting variables x 1 , . . . , x n , and the exponential map exp : L(x 1 , . . . , x n ) ∧ K → K x 1 , . . . , x n defines an isomorphism onto the subspace of K x 1 , . . . , x n of group-like elements, denoted by exp L(x 1 , . . . , x n ) ∧ K . For more background, we refer to [30,31].

Derivations on the fundamental Lie algebra of a once-punctured elliptic curve
Following [19], we will denote by p(E × τ ) the (de Rham) fundamental Lie algebra of the once-punctured elliptic curve E × τ . With notation as above, one has where the generators a, b correspond to the natural homology cycles on E × τ . We will need to consider a special family of derivations on p(E × τ ). Denote by Der 0 (p(E × τ )) the Lie algebra of continuous derivations D, which satisfy D([a, b]) = 0 and such that D(b) has no linear term in a. From these two conditions, it follows easily that every D ∈ Der 0 (p(E × τ )) is uniquely determined by its value on a.
We also let u ⊂ Der 0 (p(E × τ )) be the Lie subalgebra generated by the ε 2k . The derivations ε 2k have first been introduced by Tsunogai ( [32], §3) in the context of Galois actions on fundamental groups of punctured elliptic curves. They also play an important role in the theory of universal mixed elliptic motives, as the relative unipotent completion of SL 2 (Z) acts on p(E × τ ) through them ( [19], §20). Remark 2.2 The value of ε 2k on b is given by In particular, ε 0 (b) = 0.

Eichler integrals of Eisenstein series
Consider the Hecke-normalized Eisenstein series for SL 2 (Z) of weight 2k: where B 2k denotes the 2k-th Bernoulli number and q = e 2πiτ . Extending earlier work of Manin [25], Brown [6] introduced (regularized) iterated integrals of (2.1) (or iterated Eisenstein integrals for short) where − → 1 ∞ denotes the tangential base point 1 at i∞. We refer to [6], §4, for the general definition, and only note the special case where {0} n denotes an n-tuple of zeros, and a 0 (G 2k ) = − B 2k 4k is the constant term in the Fourier expansion (2.1) of G 2k . From the shuffle product formula for (regularized) iterated integrals ( [6], Proposition 4.7), we further deduce with the classical Eichler integral of G 2k being the special case n = 2k − 2 and k ≥ 2 (cf. e.g. [34], §1).

5)
and for k, n ≥ 1: Proof The first equality is immediate from the definition (2.2). The second equality (2.5) is trivial for n = 0, and the general case is easy to prove from (2.4) by induction on n. Finally, (2.6) follows directly from (2.4), (2.5) and the definition (2.3).

The elliptic KZB connection and the associated transport map
We recall the definition of the elliptic KZB (Knizhnik-Zamolodchikov-Bernard) connection ∇ KZB on E × τ , whose monodromy will give rise to the elliptic KZB associator. Originally, ∇ KZB was defined as a meromorphic connection on C (cf. [9,17,23]). Here, we will instead follow [8], which consider a certain C ∞ -trivialization of ∇ KZB , which is defined on the quotient be the classical Jacobi theta function.

Proposition 2.5
The connection ∇ KZB satisfies the following properties.
The connection ∇ KZB has a simple pole at ξ = 0 with residue which in turn follows from a direct computation: (ii) The residue of the connection ∇ KZB is just the residue of the one-form ω KZB . But the computation of the latter is easy from the definition, using the fact that the residue of 2πi F τ (2πiξ, η) at ξ = 0 is equal to one (cf. [17], eqn. (8)).
Now for any two base points ρ 1 , ρ 2 , let π 1 (E × τ ; ρ 2 , ρ 1 ) be the fundamental torsor of paths from ρ 1 to ρ 2 . The integrability of ∇ KZB implies that the transport function is well-defined, where γ ω k KZB denotes the iterated integral in the sense of Chen [10] γ ω k KZB := In other words, γ ω k KZB depends only on the homotopy class of γ . Rather than choosing points ρ 1 , ρ 2 ∈ E × τ , which is not canonical, we work with tangential base points, in the sense of [11], §15, at the puncture 0. Since ∇ KZB has only a simple pole at ξ = 0, one can extend the definition of the transport function to the case of tangential base points as in [11], Proposition 15.45. More precisely, for any two non-zero tangent vectors − → v 0 = λ ∂ ∂ξ and − → w 0 = μ ∂ ∂ξ at 0, there is a well-defined function given by 2 ) and the branches of the logarithms are determined by the path γ . For arithmetic applications, it will be important that the tangent vectors are integral on the Tate curve C × /q Z and moreover non-zero modulo every prime number p, which fixes them uniquely (up to a sign):

The elliptic depth filtration
Consider the canonical embedding of the once-punctured elliptic curve E × τ into the (complete) elliptic curve E τ . On fundamental Lie algebras, it induces the abelianization map Also, let gr • D p(E × τ ) be the associated graded Lie algebra.
It is clear from the definition that the elliptic depth filtration is the lower central series on the commutator of p(E × τ ). Therefore, the quotient Lie algebra is the (maximal) meta-abelian quotient of p(E × τ ). The following proposition is well-known.

Proposition 3.2 We have isomorphisms of (abelian) Lie algebras
and ] by the adjoint action.
Proof The first isomorphism is clear, since the right hand side of (3.1) is just the abelianization of p(E × τ ). It follows from the Jacobi identity that every element of gr 1 D p(E × τ ) is a series in the elements ad k (a) ad l (b) ([a, b]), and then the isomorphism (3.2) is a consequence of the universal property of free Lie algebras. Finally, the last statement of the proposition follows from the fact that the adjoint action splits the short exact sequence of Lie algebras

Proof
The action of ε 0 on gr 0 D p(E × τ ) is clear from the definition (cf. Definition 2.1). For the action on gr 1 D p(E × τ ), by the Jacobi identity, the linear operators ad(a), ad(b) ∈ End(gr 1 D p(E × τ )) commute with each other. Consequently, we have Therefore, under the isomorphism gr 1 of Proposition 3.2, the derivation ε 0 corresponds to −V ∂ ∂U . As for (ii), the triviality of ε 2k , for k > 0, on gr 0 D p(E × τ ) is clear from Definition 2.1, and triviality on gr i D p(E × τ ) follows by induction on i. Finally, (iii) follows easily from (i) and (ii).

The elliptic KZB associator
In this section, we define Enriquez's elliptic KZB associator [14], which is an elliptic analogue of the Drinfeld associator [13]. Our approach differs slightly from [14] in that we define the elliptic KZB associator using the "elliptic transport isomorphism" of Brown-Levin. This definition is analogous to the definition of the Drinfeld associator using parallel transport along the KZ-connection [12]. We also recall an important result of Enriquez (cf. [14], §6) which describes the variation of the elliptic KZB associator in the modulus of the once-punctured elliptic curve.

Definition via the transport function
In Sect. 2.4, we have defined a transport function T KZB ρ 2 ,ρ 1 on a once-punctured elliptic curve for any choice of base points ρ 1 , ρ 2 (possibly tangential), using the elliptic KZB connection. We now specialize these base points to be ± In particular, − → v 0 is defined over Z on the Tate curve.

Remark 4.2
The definition of the elliptic KZB associator given here is not exactly the same as the one given in [14], but equivalent. Using the elliptic transport map, Enriquez definition is Explicitly, the relation between the two versions is given by

Variation in the modulus
An important property of the elliptic KZB associator is that it satisfies a linear differential equation, which relates it to iterated Eisenstein integrals and the special derivations ε 2k reviewed in Sect. 2. The boundary condition of this differential equation establishes a relation between the series A(τ ), B(τ ) and the Drinfeld associator . More precisely, we have the following theorem, due to Enriquez.

Theorem 4.3 ([15], §5.2) We have
the sum being over all multi-indices (k 1 , . . . , k n ) ∈ Z n ≥0 , for n ≥ 0, and The element g(τ ) defines an automorphism of exp p(E × τ ). Letting since g(τ ) commutes with exponential and logarithm functions. The next corollary follows immediately from Proposition 3.4. where M is the scheme of classical associators in the sense of [13], and Ell is its elliptic counterpart [14]. A geometric way of interpreting this morphism is via the degeneration of the once-punctured Tate curve to P 1 \ {0, 1, ∞} (cf. Remark 3.3).

Elliptic KZB associator in depth zero
The following proposition shows that A(τ ) 0 and B(τ ) 0 precisely retrieve the periods of H 1 (E × τ ).

Proposition 4.6 We have
Proof We only prove the result for A(τ ) (0) , the formula for B(τ ) (0) is proved analogously. By Theorem 4.3, we know that A(τ ) = g(τ )(A ∞ ), and since g(τ ) is an automorphism, we also have On the other hand, it follows directly from the explicit formula for A ∞ given in Theorem 4.3 that . But as every derivation ε 2k annihilates b, we finally get A(τ ) (0) = g(τ )(2πib) = 2πib.

The meta-abelian elliptic KZB associator
In this section, we compute the image of A(τ ) and B(τ ) in the meta-abelian quotient The strategy is to use Theorem 4.3 which yields that and then to compute the images of A ∞ and B ∞ in the meta-abelian quotient separately. This is done in Sect. 5.1. In Sect. 5.2, we then compute the action of g(τ ) on the metaabelian quotient. The two computations are then combined in Sect. 5.3 to yield our formula for A(τ ) met−ab and B(τ ) met−ab .

The arithmetic piece: periods of Eisenstein series
Let A met−ab ∞ (resp. B met−ab ∞ ) be the image of A ∞ (resp. the image of B ∞ ) in the meta-abelian quotient p(E × τ ) met−ab The computation of the depth zero component was already carried out in Proposition 4.6 so that it remains to compute the depth one contribution. For this, we need a short lemma about the Drinfeld associator. Remark 3.3). In particular, we have ϕ(ι(x 0 ), ι(x 1 )) ∈ D 1 p(E × τ ) C . Proof It is well-known (cf. [12], §6.7) that Applying ι to both sides, we get the result.

Theorem 5.2 We have
Using a "truncated" version of the Baker-Campbell-Hausdorff formula (cf. [30], Corollary 3.24) and Lemma 5.1, we get Combining (5.3) and (5.4) and again applying [30], Corollary 3.24, we get (5.5) where in the last line, we have used that B 1 = − 1 2 and that B 2n+1 = 0 for all n ≥ 1. Using Lemma 5.1 together with Euler's formula − ζ(k) (−2πi) k = B k 2k! for k ≥ 2 even, it follows that (5.5) equals ∞ is very similar, so we will omit some details. First, by definition We obtain where the last equality follows from the fact that T ≡ a mod D 1 p(E × τ ) C . A short calculation shows that The term in brackets is equal to The first term a belongs to B ∞ are closely related to the extended period polynomials of Eisenstein series r G 2k (X, Y) [33]. Precisely, for k ≥ 2, one has where These . Comparing now (5.9) with Theorem 5.2, we get Corollary 5. 3 We have where U = U 2πi .

The geometric piece: special values of elliptic polylogarithms
Recall from Sect. 4.2 the definition of the automorphism g(τ ) : It naturally extends to the topological enveloping algebra Q a, b of p(E × τ ) C . In this section, we compute the images of g(τ )(a), g(τ )(b) in the meta-abelian quotient p(E × τ ) met−ab C of p(E × τ ) C , and relate the result to special values of Beilinson-Levin's elliptic polylogarithms [1,22]. and Proof By Corollary 4.4, we have Now we apply the differential operator V ∂ ∂U and split the first and the last sum to obtain From the definition of I n (G 2k ; τ ), it is easy to see that the third sum equals On the other hand, the first, second and fourth sum give Combining the two equations and setting W = U 2πi + τ V, the first equality (5.10) follows. Since g(τ ) is uniquely determined by its value on e a , the second statement (5.11) follows from the first, but can also be proved directly along similar lines.

Putting the pieces together
We can now complete the computation of A(τ ) met−ab and B(τ ) met−ab by combining the results of the previous sections.
Theorem 5. 6 We have where U = U 2πi , W = U + τ V and A (1) ∞ and B (1) ∞ are as given in Theorem 5.2 Proof We only prove the first equality, the second one is shown analogously. By The only derivation which acts non-trivially on gr 1 D p(E × τ ) C is ε 0 which itself acts as − ∂ ∂U V = 1 2πi ∂ ∂U V. Combining this with Theorem 5.4, we get the result: Remark 5.7 The value for A(τ ) met−ab given in Theorem 5.6 can be further simplified. To this end, recall from Theorem 5.2 that Note that − B 2k 4k = a 0 (G 2k ), the zeroth Fourier coefficient of G 2k . Consequently, we obtain