Pierre Lochak
email: pierre.lochak@imj-prg.fr

AND Nils Matthes
email: nils.matthes@uni-hamburg.de

Leila Schneps
email: leila.schneps@imj-prg.fr

ELLIPTIC MULTIZETAS AND THE ELLIPTIC DOUBLE SHUFFLE RELATIONS

Keywords: 2010 Mathematics Subject Classification. 11M32 Multiple zeta values, Grothendieck-Teichmüller theory, moulds

We define an elliptic generating series whose coefficients, the elliptic multizetas, are related to the elliptic analogs of multiple zeta values introduced by Enriquez as the coefficients of his elliptic associator; both sets of coefficients lie in O(H), the ring of functions on the Poincaré upper halfplane H. The elliptic multizetas generate a Q-algebra E which is an elliptic analog of the algebra of multiple zeta values. Working modulo 2πi, we show that the algebra E decomposes into a geometric and an arithmetic part, and study the precise relationship between the elliptic generating series and the elliptic associator defined by Enriquez. We show that the elliptic multizetas satisfy a double shuffle type family of algebraic relations similar to the double shuffle relations satisfied by multiple zeta values. We prove that these elliptic double shuffle relations give all algebraic relations among elliptic multizetas if (a) the classical double shuffle relations give all algebraic relations among multiple zeta values and (b) the elliptic double shuffle Lie algebra has a certain natural semi-direct product structure analogous to that established by Enriquez for the elliptic Grothendieck-Teichmüller Lie algebra.

1. Introduction 1.1. Elliptic multizetas. An elliptic analog of the multiple zeta values first made an explicit appearance in Enriquez' article [START_REF] Enriquez | Analogues elliptiques des nombres multizétas[END_REF] under the name "analogues elliptiques des nombres multizetas". They arise as coefficients of his elliptic associator constructed in [START_REF] Enriquez | Elliptic associators[END_REF], which is closely related to the elliptic Knizhnik-Zamolodchikov-Bernard (KZB) equation [START_REF] Calaque | Universal KZB equations: the elliptic case[END_REF][START_REF] Levin | Towards multiple elliptic polylogarithms[END_REF] and to multiple elliptic polylogarithms [START_REF] Brown | Multiple elliptic polylogarithms[END_REF][START_REF] Levin | Towards multiple elliptic polylogarithms[END_REF]; more recently, they have even found applications to computations in high energy physics [START_REF] Broedel | Elliptic multiple zeta values and one-loop superstring amplitudes[END_REF]. Taking the regularized limit τ → i∞ of elliptic multizetas, one retrieves the classical multiple zeta values [START_REF] Enriquez | Analogues elliptiques des nombres multizétas[END_REF][START_REF] Matthes | Elliptic multiple zeta values[END_REF], which gives the explicit connection between the genus zero and genus one multizetas. The idea of considering the graded Qalgebra generated by these coefficients, was introduced in [START_REF] Broedel | Relations between elliptic multiple zeta values and a special derivation algebra[END_REF][START_REF] Matthes | Elliptic multiple zeta values[END_REF][START_REF] Matthes | Elliptic double zeta values[END_REF], which provide some explicit dimension results in depth 2.

Recall that the Drinfel'd associator Φ KZ , first introduced in [START_REF] Drinfel | On quasitriangular quasi-Hopf algebras and on a group that is closely connected with Gal(Q/Q)[END_REF], is a power series in two non-commutative variables 1 , which is the generating series for the usual multiple zeta values, by the work of Le and Murakami [?]. In analogy with this, Enriquez's elliptic associator, which is defined as a pair of monodromies (cf. §5.2 of [START_REF] Enriquez | Elliptic associators[END_REF]), takes the form of a pair of group-like power series in two non-commutative variables a and b:

A(τ ), B(τ ) ∈ O(H) a, b ,
where O(H), denotes the ring of holomorphic functions of one variable τ running through the Poincaré upper half-plane. We call the coefficients of A(τ ) and B(τ ) Aelliptic multizetas and B-elliptic multizetas, or A-EMZs and B-EMZs. The acronym EMZ stands for elliptic multizetas; since they are functions of τ and not real numbers like the coefficients of Φ KZ , we drop the word "values" in the elliptic situation 2 .

The main new object introduced in this article is a third power series

C(τ ) ∈ O(H) a, b
and its logarithm E(τ ) = log C(τ ) ( §3.1). The series E(τ ) is called the elliptic generating series, and its coefficients, in O(H), are called E-elliptic multizetas or E-EMZs. We write E, A and B for the vector spaces generated by the coefficients of C(τ ), the A-EMZs and the B-EMZs respectively, which are by definition subspaces of O(H). We show in Lemma 3.2 that like A(τ ) and B(τ ), C(τ ) is a group-like power series, which implies that each of the three subspaces E, A and B of O(H) actually forms a Q-algebra. In Lemma 3.3, we show that that the coefficients of E(τ ), the E-MZVs, form a system of algebra generators for E. This is the generating system we will study in the rest of the article.

One of the main results in [START_REF] Enriquez | Elliptic associators[END_REF] concerning the power series A(τ ), B(τ ) is that they can be written in the form 3 g(τ )•A and g(τ )•B, where g(τ ) is an automorphism of O(H) a, b (introduced in §5.1 of [START_REF] Enriquez | Elliptic associators[END_REF] but recalled in §2 below), and A and B are power series in Z[2πi] a, b (defined in §3.5 of [START_REF] Enriquez | Elliptic associators[END_REF] with the notation A + , A -, but recalled in §3.1 below). This property of the elliptic associator is the motivation for our definition of the power series C(τ ) directly in the form g(τ ) • C, where C is a group-like power series in Z[2πi] a, b closely related to A and B ( §3.1). Since g(τ ) is an automorphism, the power series E(τ ) = log C(τ ) then naturally takes the form g(τ )

• E with E = log C ∈ Z[2πi] a, b .
We let E geom denote the Q-algebra generated by the coefficients of g(τ ) (in a precise sense explained in §2). These coefficients lie in O(H), and are realized as particular linear combinations of iterated integrals of Eisenstein series for SL 2 (Z) (see [START_REF] Brown | Multiple modular values for SL 2 (Z)[END_REF][START_REF] Manin | Iterated integrals of modular forms and noncommutative modular symbols[END_REF]). We note that for any ring R, g(τ ) induces an automorphism of E geom ⊗ R a, b . We use this fact frequently below.

The structure of the Q-algebra E geom is the main topic of §2. It is related to the bigraded Lie algebra u geom of the prounipotent radical of π geom 1 (M EM ), where M EM denotes the Tannakian category of universal mixed elliptic motives [START_REF] Hain | Universal Mixed Elliptic Motives[END_REF]. More precisely, E geom is related to the bigraded Lie algebra u which is the image of u geom under the monodromy representation from u geom to the Lie algebra of derivations of a free Lie algebra on two generators whose existence is shown in [START_REF] Hain | Universal Mixed Elliptic Motives[END_REF], §22. The explicit generators of this Lie algebra are well-known, cf. Definition 2.1. 2 These values, which are Enriquez' "elliptic analogs of MZV's", and the E-MZVs introduced below, are very different from Brown's "multiple modular values" [START_REF] Brown | Multiple modular values for SL 2 (Z)[END_REF], which are complex numbers. 3 Throughout this article we use the dot notation • to indicate the action of automorphisms or derivations on elements.

The first results of this article are summarized in the following theorem, whose proof rests in large part on the C-linear independence of iterated integrals of Eisenstein series proved in Theorem 2.8 and its corollary 2.9 4 .

Theorem. (i) [Thm. 2.6] There is a natural isomorphism

E geom ∼ = U(u) ∨ ,
where U(u) ∨ is the graded dual of the universal enveloping algebra of the Lie algebra u. In particular, E geom is a commutative, graded, Hopf Q-algebra.

(ii) [Cor. 2.10] The subalgebra of O(H) generated by E geom and Z[2πi] is isomorphic to the tensor product

E geom ⊗ Q Z[2πi].
(1.1)

Corollary. The Q-algebras A, B and E are subalgebras of the tensor product (1.1).

Proof. Since A(τ ) has the form g(τ ) • A, the coefficients of A(τ ) are algebraic expressions in elements of E geom and Z[2πi]. The same holds for B(τ ) and C(τ ). The result then follows from (ii) of the Theorem.

1.2. Structure of the elliptic multizeta algebras mod 2πi. For technical reasons linked to our use of Grothendieck-Teichmüller theory and Écalle's mould theory, we restrict our study of the three different types of elliptic multizetas to objects to their reductions modulo 2πi in the following sense. Let Z denote the quotient of Z[2πi] by the ideal generated by 2πi, which is isomorphic to the quotient of the algebra of multiple zeta values Z by the ideal generated by ζ(2) = -(2πi) 2 [START_REF] Levin | Towards multiple elliptic polylogarithms[END_REF] . The quotient of E geom ⊗ Q Z[2πi] by the ideal 1 ⊗ 2πi is isomorphic to

E geom ⊗ Q Z.
Definition 1.1. Let E (resp. B) denote the image of the subalgebra E (resp. B) of E geom ⊗ Z[2πi] in the quotient E geom ⊗ Z. Let the reduced power series E(τ ) (resp. B(τ )) to be obtained from E(τ ) (resp. B(τ )) by reducing the coefficients from E to E (resp. from B to B).

The case of A is slightly different, because it follows from the definition of A(τ ) given in (3.2) and (3.3) that the ring A lies in the ideal Q

• 1 + E geom ⊗ 2πiZ[2πi],
and therefore the image of this ring in the quotient E geom ⊗ Z is just Q • 1. We get around this as follows. We set A = A 1 2πi ; this power series lies in Z a, b by the definition of A (cf. (3.2)). We then set

A (τ ) = g(τ ) • A ∈ E geom ⊗ Z[2πi] a, b .
Definition 1.2. Let A ⊂ E geom ⊗ Z denote the Q-algebra generated by the coefficients of A (τ ). Let A denote the image of A in the reduced ring E geom ⊗ Z. Let A be the power series obtained from A by reducing the coefficients from Z to Z, and A (τ ) to be the power series obtained from A (τ ) by reducing the coefficients from E geom ⊗ Z to E geom ⊗ Z. We have A (τ ) = g(τ ) • A . 4 The second author subsequently generalized this result to the case of arbitrary quasimodular forms for SL 2 (Z).

The coefficients of the three reduced power series E(τ ), B(τ ) and A (τ ), which generate the three subalgebras E, B and A of E geom ⊗ Z, are called E-EMZs, B-EMZs and A-EMZs.

Our first goal is to compare the four algebras E, A, B and E geom ⊗ Q Z. The element 2πiτ ∈ O(H) plays a special role in this comparison. It lies in E geom since it is the coefficient of ε 0 in g(τ ) (see (2.6) below), so 2πiτ ⊗1 lies in the tensor product E geom ⊗ Q Z, but it does not lie in E or A, although it does lie in B. Working mod 2πi allows us to make a much more precise statement than the simple inclusion, which also has the advantage of showing that the three reduced algebras are highly non-trivial.

Theorem (Cor. ??). We have the equalities 5

E[2πiτ ⊗ 1] = A[2πiτ ⊗ 1] = B = E geom ⊗ Q Z.
As noted above, the elliptic generating series E(τ ) has the form g(τ ) • E; thus its reduction E(τ ) has the form g(τ )•E where the reduction E of E has coefficients in Z. Throughout this article, we consider the power series Φ KZ obtained by reducing the coefficients of Φ KZ from Z to Z. This reduced power series, a priori an associator belonging to the torsor of associators M (Z), can be considered as lying in the group GRT 1 (Z) (defined in [START_REF] Drinfel | On quasitriangular quasi-Hopf algebras and on a group that is closely connected with Gal(Q/Q)[END_REF], §5), since the associator relations become equivalent to the group relations for any associator whose degree 2 part is zero (cf. Prop. 5.9 of [START_REF] Drinfel | On quasitriangular quasi-Hopf algebras and on a group that is closely connected with Gal(Q/Q)[END_REF]). The key point (Theorem 3.4) of the proof of the equality

E[2πiτ ⊗ 1] = E ⊗ Z is the fact that E = Γ(Φ KZ ), (1.2)
where Γ is the composition

GRT 1 (Z) Γ → GRT ell (Z) π → Z a, b ,
where Γ is Enriquez' section ( [START_REF] Enriquez | Elliptic associators[END_REF], §4), and the projection π maps an element (λ, f, g + , g -) ∈ GRT ell (Z) to the component g + ∈ Z a, b . In particular, we show that the coefficients of E generate all of Z (Corollary 3.5). The ring E is the Q-algebra generated by the coefficients of E(τ ), which, as for E(τ ), are all algebraic expressions in the coefficients of g(τ ) and those of E. The delicate part of the proof consists in showing that the coefficients of E(τ ) can be "untangled" to separately recover a set of generators of Z and, with the addition of 2πiτ , a set of generators of E geom . The same arguments hold for A and B, except that instead of (1.2), we use [START_REF] Matthes | Elliptic multiple zeta values[END_REF], Theorem 5.4.2, to show that the coefficients of the arithmetic parts A and B generate all of Z and the same untangling argument. In fact, the result can be framed for more general power series, as is done in Theorem 3.6. Thanks to the above theorem, we call the Q-algebra E geom ⊗ Q Z the Q-algebra of elliptic multizetas (modulo 2πi). complete set thereof. The fact that the power series A(τ ) and B(τ ) are group-like provides some relations; another family, the "Fay relations", is partially known for A-EMZs (cf. [START_REF] Matthes | Elliptic multiple zeta values[END_REF]). Enriquez gives a complete set of associator relations satisfied by the elliptic associator, derived from the fact that the power series A(τ ) and B(τ ) induce an automorphism of the prounipotent 2-strand torus braid group. However, these relations mingle the A-EMZs and the B-EMZs along with genus zero multiple zeta values, and do not provide separate relations for each generating set. For the E-EMZs, however, we can say more, both about explicit relations satisfied by E(τ ), and about the question of whether these relations may be a complete set.

The third main result of this article (Theorem 4.4) shows that the E-EMZs satisfy an explicit set of algebraic relations called the elliptic double shuffle relations, given in the form of two elliptic double shuffle equations satisfied by the power series E(τ ).

These elliptic double shuffle relations arise as follows. Since we have E(τ ) = Γ(Φ KZ ), the power series E(τ ) will necessarily satisfy equations that are transports by Γ of the usual double shuffle equations satisfied by Φ KZ (shuffle and stuffle relations, also known as double mélange in Racinet's terminology [START_REF] Racinet | Doubles mélanges des polylogarithmes multiples aux racines de l'unité[END_REF], and symmetrality/symmetrility in Écalle's [START_REF] Ecalle | The flexion structure and dimorphy: flexion units, singulators, generators, and the enumeration of multizeta irreducibles[END_REF]). These transported relations can be determined explicitly thanks to a major theorem of Écalle together with the results of [START_REF] Schneps | Elliptic multiple zeta values, Grothendieck-Teichmüller and mould theory[END_REF]. We call them the elliptic double shuffle equations, and show that they are similar in nature to the well-known (regularized) double shuffle relations for multiple zeta values, but in fact, surprisingly, closer to their depth-graded version.

The proof of the theorem relies on several difficult known results, in particular a crucial theorem of Écalle ( [START_REF] Ecalle | The flexion structure and dimorphy: flexion units, singulators, generators, and the enumeration of multizeta irreducibles[END_REF] but see [START_REF] Schneps | An introduction to Ecalle's theory of moulds[END_REF], Theorem 4.6.1 for a complete proof).

On the question of whether the elliptic double shuffle relations generate all algebraic relations between E-EMZs, we show in §4.2 that the elliptic double shuffle relations are a complete set in depth 2 (Prop. 4.6), thanks to the fact that depth 2 is too small for the real multiple zeta values to occur. In higher depth, however, we naturally encounter problems related to the unknown transcendence properties of the real multiple zeta values, exactly as we do when conjecturing that the usual double shuffle relations generate all algebraic relations between multizeta values. In Prop. 4.5, we show that the elliptic double shuffle relations do form a complete set of algebraic relations between E-EMZs under the following familiar conjectures from multizeta theory:

(a) The double shuffle relations generate all algebraic relations among the multiple zeta values modulo 2πi. (b) The elliptic double shuffle Lie algebra ds ell [START_REF] Schneps | Elliptic multiple zeta values, Grothendieck-Teichmüller and mould theory[END_REF] is isomorphic to a semidirect product ds ell ∼ = u γ(ds), where ds is the usual double shuffle Lie algebra and γ is the extension of Enriquez' section to ds obtained using mould theory (cf.

[?], end of §1).

Conjecture (a) is a standard conjecture in multizeta theory (cf. [START_REF] Ihara | Derivation and double shuffle relations for multiple zeta values[END_REF], Conjecture 1). It would imply strong transcendence results for multiple zeta values, and therefore seems out of reach at the moment. Conjecture (b), however, is purely algebraic, and may therefore be more tractable, although it still seems difficult. It would follow for example from Enriquez' generation conjecture ( [START_REF] Enriquez | Elliptic associators[END_REF], §10) together with the conjecture that grt ell ⊂ ds ell (an elliptic version of Furusho's theorem [START_REF] Furusho | Double shuffle relation for associators[END_REF]).

The last question addressed in the paper, in §4.3, concerns a family of algebraic relations satisfied by the A-EMZs (Theorem 4.8), which are the Fay relations on A-EMZs studied in [START_REF] Broedel | Relations between elliptic multiple zeta values and a special derivation algebra[END_REF][START_REF] Matthes | Elliptic double zeta values[END_REF], but here considered mod 2πi; we compare them to the closely related push-neutrality relations. These families are identical in depth 2 (although not in higher depths). In [START_REF] Broedel | Relations between elliptic multiple zeta values and a special derivation algebra[END_REF][START_REF] Matthes | Elliptic double zeta values[END_REF], the depth 6 2 Fay relations are given explicitly and it is shown that the Fay and shuffle relations give a complete set of Q-linear relations between A-EMZs in depth 2 The possible completeness of the relations in all depths (depending on conjectures such as those cited above), the precise comparison between the algebras E and A, and above all the lifting of the questions considered here to the situation not modulo 2πi are all topics for further research.

1.4. Outline of the article. The contents of this paper are organized as follows.

In §2, we introduce the algebra E geom of geometric elliptic multizetas, describe their relation to iterated integrals of Eisenstein series, and prove the crucial linear independence of iterated Eisenstein integrals, as well as the relation between E geom and the Lie algebra u. In §3 we construct the elliptic generating series E(τ ) and define the E-EMZs to be its coefficients, and E to be the Q-algebra they generate. Passing modulo 2πi, we prove the main structural result E[2πiτ ] E geom ⊗Z and its analogs for A and B. In §4, we study the elliptic double shuffle equations satisfied by the mod 2πi elliptic generating series E(τ ) (or more precisely, the linearized version satisfied by its Lie version), and give evidence for the completeness of the resulting system of algebraic relations between the E-EMZs. Finally, we study a family of relations satisfied by A (τ ). The necessary background concerning moulds is briefly summarized in §4.1.

Acknowledgments. The authors are grateful to the referees for many useful comments. This paper was written while Nils Matthes was a PhD student at Universität Hamburg under the supervision of Ulf Kühn.

Geometric Elliptic Multizetas

In the first two sections, we respectively recall the definition of a certain Lie algebra u of derivations [START_REF] Pollack | Relations between derivations arising from modular forms[END_REF][START_REF] Tsunogai | On some derivations of Lie algebras related to Galois representations[END_REF] and of iterated integrals of Eisenstein series [START_REF] Brown | Multiple modular values for SL 2 (Z)[END_REF][START_REF] Manin | Iterated integrals of modular forms and noncommutative modular symbols[END_REF].

In §2.3, we introduce the algebra of geometric elliptic multizetas, and prove that it is isomorphic to the graded dual of the universal enveloping algebra of u. The crucial step is a linear independence result for iterated integrals of Eisenstein series, which we prove (in slightly greater generality than needed) in §2.4.

2.1.

A family of special derivations. We begin by fixing our notation. For a Qalgebra A, let f 2 (A) = Lie A [[x 1 , y 1 ]] be the completed (with respect to the descending central series) free Lie algebra over A on two generators x 1 , y 1 with Lie bracket [•, •]. Its (topological) universal enveloping algebra will be denoted by U(f 2 ) A , and F 2 (A) := exp(f 2 (A)) ⊂ U(f 2 ) A is the set of exponentials of Lie series. Note that U(f 2 ) A is canonically isomorphic to A x 1 , y 1 , the A-algebra of formal power series in non-commuting variables x 1 , y 1 . Moreover, U(f 2 ) A is a complete Hopf A-algebra, whose (completed) coproduct ∆ is uniquely determined by ∆(w) = w ⊗ 1 + 1 ⊗ w, for w ∈ {x 1 , y 1 }. The group F 2 (A) can also be characterized as the set of group-like elements of U(f 2 ) A . Likewise, the Lie algebra f 2 (A) ⊂ U(f 2 ) A is precisely the subset of Lie-like (or primitive) elements. If A = Q, we will write f 2 instead of f 2 (Q) and likewise U(f 2 ) and F 2 instead of U(f 2 ) A and F 2 (A). Now let Der(f 2 ) denote the Lie algebra of continuous derivations of the completed Lie algebra f 2 , and define Der 0 (f 2 ) as the subalgebra of those D ∈ Der(f 2 ) which (i) annihilate the bracket [x 1 , y 1 ]:

D([x 1 , y 1 ]) = 0
and (ii) are such that D(y 1 ) contains no linear term in x 1 . Since f 2 is the completion of a free Lie algebra, the commutator of y 1 is Q•y 1 , from which it follows easily that every derivation D ∈ Der 0 (f 2 ) is uniquely determined by its value on x 1 . Similarly, the only non-zero derivation D ∈ Der 0 (f 2 ) which annihilates y 1 is the derivation ε 0 defined by x 1 → y 1 , y 1 → 0. We next recall the definition of a family of derivations, which was first considered in [START_REF] Tsunogai | On some derivations of Lie algebras related to Galois representations[END_REF], also played an important role in [START_REF] Calaque | Universal KZB equations: the elliptic case[END_REF], and was studied in detail in [START_REF] Pollack | Relations between derivations arising from modular forms[END_REF].

Definition 2.1. For k ≥ 0, define a derivation ε 2k ∈ Der 0 (f 2 ) by ε 2k (x 1 ) = ad(x 1 ) 2k (y 1 ), and denote

u = Lie(ε 2k ; k ≥ 0) ⊂ Der 0 (f 2 )
the graded completion of the Lie subalgebra generated by the ε 2k . We say that a derivation D of f 2 is of homogeneous degree m ≥ 0 if for every element f ∈ f 2 of homogeneous degree n, D(f ) is of homogeneous degree n + m. Let Der 0 (f 2 ) be the subspace of Der 0 (f 2 ) spanned by derivations of homogeneous degree ≥ 1, and let u = Der 0 (f 2 ) ∩ u. We have isomorphisms Der 0 (f 2 ) Qε 0 ⊕ Der 0 (f 2 ) and u Qε 0 ⊕ u .

(2.1)

Observe that ε 2 = -ad([x 1 , y 1 ]), and thus ε 2 is central in u. Thus we have a generating set for u given by

u = Lie[ad n (ε 0 )(ε 2k ); n ≥ 0, k ≥ 1]. (2.2)
As seen above, every ε 2k is uniquely determined by its value on x 1 , while ε 0 is the only non-zero derivation D ∈ u, which annihilates y 1 . From this, we get Proposition 2.2. The Q-linear evaluation maps

v x1 : Der 0 (f 2 ) → f 2 , D → D(x 1 ), v y1 : Der 0 (f 2 ) → f 2 , D → D(y 1 ), are injective.
For the applications to elliptic multizetas, it will be more natural to scale the derivations ε 2k as follows:

ε 2k := 2 (2k-2)! ε 2k k > 0 -ε 0 k = 0.
In this way, ε 2k is the image of the Eisenstein generator e 2k under the monodromy representation u geom → Der 0 (f 2 ) (cf. [START_REF] Hain | Universal Mixed Elliptic Motives[END_REF], Theorem 22.3).

Iterated Eisenstein Integrals.

In a sense to be made precise below, the derivation ε 2k naturally corresponds to integrals of Hecke-normalized Eisenstein series of weight 2k (for SL 2 (Z)), whereas commutators of ε 2k correspond to iterated integrals of Eisenstein series. These are special cases of iterated Shimura integrals (or iterated Eichler integrals) of modular forms introduced by Manin [START_REF] Manin | Iterated integrals of modular forms and noncommutative modular symbols[END_REF], and later generalized by Brown [START_REF] Brown | Multiple modular values for SL 2 (Z)[END_REF]. 7 For k ≥ 0, let G 2k (q) be the Hecke-normalized Eisenstein series, defined by G 0 (q) := -1 and for k ≥ 1

G 2k (q) = - B 2k 4k + n≥1 σ 2k-1 (n)q n , q = e 2πiτ
Here, σ (n) = d|n d denotes the -th divisor function, and the B 2k are the Bernoulli numbers defined by

z e z -1 = 1 - z 2 + n≥1 B 2n z 2n (2n)! .
Via the exponential map exp : H → D * , τ → q = exp(2πiτ ), from the upper half-plane to the punctured unit disc

D * = {q ∈ C, 0 < |q| < 1},
we may consider G 2k as a function of either variable q or τ , and we shall do so according to context. Next, we define iterated integrals of Eisenstein series. More generally, if f (q) = ∞ n=0 a n q n is such that a 0 = 0, (e.g. f is a cusp form), then the definition of the indefinite integral i∞ τ f (τ 1 )dτ 1 poses no problem, as by definition f vanishes at i∞. This is not the case for the Eisenstein series G 2k , and consequently i∞ τ G 2k (τ 1 )dτ 1 diverges. It can be regularized by setting, for k ≥ 1,

i∞ τ G 2k (τ 1 )dτ 1 := i∞ τ G 2k (τ 1 ) -G ∞ 2k dτ 1 - τ 0 G ∞ 2k dτ 1 ,
where G ∞ 2k = -B 2k 4k is the constant term in the Fourier expansion of G 2k (if k = 0, a similar method works). Note that the integral of G 2k so defined satisfies the differential equation df (τ ) = -G 2k (τ )dτ . The definition of regularized iterated integrals of Eisenstein series in [START_REF] Brown | Multiple modular values for SL 2 (Z)[END_REF], which is a special case of Deligne's tangential base point regularization ( [START_REF] Deligne | Le groupe fondamental de la droite projective moins trois points[END_REF], §15) generalizes this construction, and runs as follows.

Let W = C[[q]] <1 be the C-algebra of formal power series, which converge on

D = {q ∈ C | |q| < 1}. We may decompose W = W 0 ⊕ W ∞ with W 0 = qC[[q]
] and W ∞ = C. For a power series f ∈ W , define f 0 to be its image in W 0 under the natural projection, and define f ∞ ∈ W ∞ likewise. For example, in the case of the Eisenstein series G 2k (q) with k > 0, we have

G ∞ 2k = - B 2k 4k , G 0 2k (q) = n≥1 σ 2k-1 (n)q n .
We denote by T c (W ) the shuffle algebra on the C-vector space W . As a C-vector space, T c (W ) is simply the graded (for the length of tensors) dual of the tensor 7 To be precise, Manin defined iterated Shimura integrals of cusp forms between base points on the upper half-plane (possibly cusps), and the extension to Eisenstein series (which requires a regularization procedure) is due to Brown.

algebra T (W ) = n≥0 W ⊗n . It is customary to write down elements of the dual space (W ⊗n ) ∨ using bar notation

[f 1 |, . . . , |f n ]. Moreover, T c (W ) is naturally a commutative C-algebra, whose product is the shuffle product ¡, defined by [f 1 | . . . |f r ] ¡ [f r+1 | . . . |f r+s ] = σ∈Σr,s f σ -1 (1) . . . f σ -1 (r+s) ,
where Σ r,s denotes the set of permutations σ on {1, . . . , r+s}, such that σ is strictly increasing on both {1, . . . r} and on {r + 1, . . . , r + s}. Now define a map R :

T c (W ) → T c (W ) by the formula R[f 1 | . . . |f n ] = n i=0 (-1) n-i [f 1 | . . . |f i ] ¡ [f ∞ n | . . . |f ∞ i+1 ].
Following [START_REF] Brown | Multiple modular values for SL 2 (Z)[END_REF], eq. ( 4.11), we can now make the Definition 2.3. Given f 1 , . . . , f n ∈ W as above, their regularized iterated integral is defined as

I(f 1 , . . . , f n ; τ ) := (2πi) n n i=0 i∞ τ R[f 1 | . . . |f i ] dτ 0 τ [f ∞ i+1 | . . . |f ∞ n ] dτ , where b a [f 1 | . . . |f n ] dτ := • • • a≤τ1≤...≤τn≤b f 1 (τ 1 ) . . . f n (τ n )dτ 1 . . . dτ n .
Remark 2.4. The reason for the (2πi) n -prefactor is to preserve the rationality of the Fourier coefficients. More precisely, if f 1 , . . . , f n have rational coefficients (i.e.

f i ∈ W Q := Q[[q]] <1 ), then I(f 1 , . . . , f n ; τ ) ∈ W Q [log(q)],
where log(q) := 2πiτ .

As is the case for usual iterated integrals ( [START_REF] Hain | The geometry of the mixed Hodge structure on the fundamental group[END_REF], Sect. 2), regularized iterated integrals satisfy the differential equation

∂ ∂τ τ =τ0 I(f 1 , . . . , f n ; τ ) = -f 1 (τ 0 )I(f 2 , . . . , f n ; τ 0 ), (2.3)
as well as the shuffle product formula

I(f 1 , . . . , f r ; τ )I(f r+1 , . . . , f r+s ; τ ) = σ∈Σr,s I(f σ(1) , . . . , f σ(r+s) ; τ ). (2.4)
The only case of interest for us will be when f 1 , . . . , f n are given by Eisenstein series G 2k1 , . . . , G 2kn . In this case, we set

G k (τ ) := I(G 2k1 , . . . , G 2kn ; τ ), (2.5)
where k = (k 1 , . . . , k n ) and likewise denote by

I Eis := Span Q {G k (τ )} ⊂ O(H)
the Q-span of all iterated Eisenstein integrals G k (τ ) for all multi-indices k (including G ∅ := 1 for the empty index). Note that I Eis is a Q-subalgebra of O(H) by (2.4), and that it contains Q[2πiτ ] as a subalgebra, since by (2.5) we have

G 0 (τ ) = 2πiτ. (2.6) 2.
3. The τ -evolution equation and the algebra of geometric elliptic multizetas. We now put together the special derivations ε 2k and the iterated Eisenstein integrals into a single, formal series

g(τ ) := id + k G k (τ ) ε k , (2.7)
where the sum is over all multi-indices k ∈ Z n ≥0 for n > 0, and for each tuple

k = (k 1 , . . . , k n ), we set ε k := ε 2k1 • . . . • ε 2kn ∈ U(u), the universal enveloping algebra of u. From (2.3), it is clear that g(τ ) satisfies the differential equation 1 2πi ∂ ∂τ g(τ ) = - k≥0 G 2k (τ ) ε 2k g(τ ),
and it follows that g(τ ) is group-like, i.e. it is the exponential g(τ ) = exp(r(τ )) of a Lie series

r(τ ) ∈ u ⊗ Q I Eis . (2.8)
Definition 2.5. Define the Q-algebra E geom of geometric elliptic multizetas to be the Q-algebra generated by the coefficients of r(τ )

• x 1 .
Equivalently, E geom is equal to the Q-vector space linearly spanned by the coefficients of the series g(τ ) • e x1 , because the coefficients of each of the power series r(τ ) • x 1 and g(τ ) • e x1 can be written as algebraic expressions in the coefficients of the other. Also, note that since every derivation in u is uniquely determined by its value on x 1 , the Q-algebra E geom is also the same as the Q-algebra spanned by the coefficients of g(τ ) when written in a basis of the Q-algebra generated by the ε 2k . Thus in particular we have the inclusion of commutative algebras

E geom ⊂ Q[G k (τ ), k ∈ N n , n ≥ 0] ⊂ O(H).
We can now state the main result of §2.

Theorem 2.6. For every Q-subalgebra A ⊂ C, there is an isomorphism

U(u) ∨ ⊗ Q A ∼ = E geom ⊗ Q A of A-algebras.
In particular, E geom is a commutative, graded Hopf algebra in a natural way.

Proof. The main ingredient in the proof is that the iterated Eisenstein integrals G k (τ ) are linearly independent over C, as functions in τ . More precisely, by Corollary 2.9, proved in the next section, there is a natural isomorphism

I Eis ⊗ Q A ∼ = T c (V Eis ) ⊗ Q A,
where T c (V Eis ) is the shuffle algebra on the Q-vector space V Eis spanned by all Eisenstein series G 2k , k ≥ 0.

Assuming Corollary 2.9 for the moment, the proof of Theorem 2.6 proceeds as follows. Since the tensor algebra T (V Eis ) is freely generated by one element in every even degree 2k ≥ 0, we get a canonical surjection T (V Eis ) → U(u) of Q-algebras, which induces by duality an injection

ι : U(u) ∨ → T c (V Eis ) ∼ = I Eis .
On the other hand, choosing a (homogeneous) linear basis B of U(u), the element g(τ ) naturally defines a map

ι : U(u) ∨ → I Eis b ∨ → b ∨ (g(τ )),
where b ∨ ∈ B ∨ are the dual basis elements. Clearly, the image of ι does not depend on the choice of basis, and equals E geom by definition. On the other hand, it is easy to see that the maps ι, ι : U(u) ∨ → I Eis are equal, whence the result for A = Q, and the general case follows simply by extension of scalars. Finally, it is well-known that the universal enveloping algebra of any graded Lie algebra has a natural structure of a (cocommutative) graded Hopf algebra, thus U(u) ∨ is naturally a (commutative) graded Hopf algebra.

2.4. Linear independence. In this subsection, we complete the proof of Theorem 2.6 by showing that the family of iterated Eisenstein integrals is linearly independent over C, and that as a consequence

I Eis ⊗ Q C ∼ = T c (V Eis ) ⊗ Q C as C-algebras.
Although these results can meanwhile also be deduced from [START_REF] Matthes | On the algebraic structure of iterated integrals of quasimodular forms[END_REF], which proves linear independence of iterated integrals of quasimodular forms for SL 2 (Z) over the (fraction field of the) ring of quasimodular forms for SL 2 (Z), we give a slightly different proof here in the special case of Eisenstein series which has the advantage that it works over a larger field of coefficients.

The main idea is to use the following general linear independence result.

Theorem 2.7 ([10, Theorem 2.1]). Let (A, d) be a differential algebra over a field k of characteristic zero, whose ring of constants ker(d) is precisely equal to k. Let C be a differential subfield of A (i.e. a subfield such that dC ⊂ C), X any set with associated free monoid X * . Suppose that S ∈ A X is a solution to the differential equation

dS = M • S,
where M = x∈X u x x ∈ C X is a homogeneous series of degree 1, with initial condition S 1 = 1, where S 1 denotes the coefficient of the empty word in the series S. The following are equivalent:

(i) The family of coefficients (S w ) w∈X * of S is linearly independent over C.

(ii) The family {u x } x∈X is linearly independent over k, and we have

dC ∩ Span k ({u x } x∈X ) = {0}. (2.9)
Using this theorem, we can now prove linear independence of iterated Eisenstein integrals.

Theorem 2.8. The family {G k (τ )} is linearly independent over Frac(Z[[q]]).

Proof. We will apply Theorem 2.7 with the following parameters:

• k = Q, A = Q[log(q)]((q)) with differential d = q ∂
∂q , and C = Frac(Z[[q]]) (the latter is a differential field by the quotient rule for derivatives)

• X = {a 2k } k≥0 , u a 2k = -G 2k (q), hence M (q) = - k≥0 G 2k (q)a 2k .
With these conventions, it follows from (2.3) that the formal series

1 + 0 q [M ] d log q + 0 q [M |M ] d log q + . . . ∈ O(H) X ,
with the iterated integrals regularized as in Section 2.2, is a solution of the differential equation dS = M • S, with S 1 = 1. Consequently, the coefficient of the word w = a 2k1 . . . a 2kn in S is equal to G(2k 1 , . . . , 2k n ; τ ). Moreover, since the Qlinear independence of the Eisenstein series is well-known (cf. e.g. [START_REF] Serre | Cours d'arithmétique, volume 2 of Collection SUP[END_REF], VII.3.2), it remains to verify (2.9) in our situation.

To this end, assume that there exist α 2k ∈ Q, all but finitely many of which are equal to zero, such that k≥0 α 2k G 2k (q) ∈ dC.

(2.10)

Clearing denominators, we may assume that α 2k ∈ Z. Furthermore, from the definition of d = q ∂ ∂q , one sees that the image dC of the differential operator d does not contain any constant except for zero. Therefore, the coefficient of the trivial word 1 in (2.10) vanishes; in other words

k≥0 α 2k G 2k (q) = k≥1 α 2k E 0 2k (q) ∈ qQ[[q]].

Now the differential d is invertible on qQ[[q]

], and inverting d is the same as integrating. Hence (2.10) is equivalent to

k≥1 α 2k G 0 2k (τ ) ∈ C, G 0 2k (τ ) := 0 q E 0 2k (q 1 ) dq 1 q 1 . (2.11)
But this is absurd, unless all the α 2k vanish, as we shall see now. Indeed, if

f ∈ C = Frac(Z[[q]]), then there exists m ∈ Z \ {0} such that f ∈ Z[m -1 ]((q)).
This follows from the well-known inversion formula for power series. On the other hand, the coefficient of q p in G 0 2k (τ ), for p a prime number, is given by

σ 2k-1 (p) p = p 2k-1 + 1 p ≡ 1 p mod Z.
Thus, we must have

1 p k≥1 α 2k ∈ Z[m -1 ],
for every prime number p, in particular k≥1 α 2k is divisible by infinitely many primes (namely, at least all the primes which don't divide m), which implies k≥1 α 2k = 0. Now assume that k 1 is the smallest positive, even integer with the property that α k1 = 0. Consider the coefficient of q p k 1 in G 0 2k (τ ), which is equal to

σ 2k-1 (p k1 ) p k1 = 1 p k1 k1 j=0 p j(2k-1) ≡ 1 p k 1 mod Z if 2k > k 1 1 p k 1 + 1 p mod Z if 2k = k 1 .
By (2.11), we have

α k 1 p + 1 p k 1 k≥1 α 2k ∈ Z[m -1 ],
and by what we have seen before, k≥1 α 2k = 0. Hence

α k 1 p ∈ Z[m -1 ],
for every prime number p, which again implies α k1 = 0, in contradiction to our assumption α k1 = 0. Therefore, in (2.11), we must have α 2k = 0 for all k ≥ 1 and (2.9) is verified.

Corollary 2.9. The iterated Eisenstein integrals G k (τ ) are C-linearly independent, and for every Q-subalgebra A ⊂ C, we have a natural isomorphism of A-algebras Theorem 2.8 shows in particular that the G k are linearly independent over Q. Since the Eisenstein series G 2k have coefficients in Q, it follows from the definition that G k ∈ Q((q))[log(q)], and elements of W Q [log(q)] = Q((q))[log(q)] are linearly independent over Q, if and only they are so over C.

ψ A : T c (V Eis ) ⊗ Q A → I Eis ⊗ Q A [G 2k1 | . . . |G 2kn ] → G k (τ ), where k = (k 1 , . . . , k n ) and V Eis = Span Q {G 2k (τ ) | k ≥ 0} ⊂ O(H). Proof. Since Q ⊂ Frac(Z[[q]]),
For the second statement, it is clear that ψ A is a homomorphism of Q-algebras (since both sides are endowed with the shuffle product) and that it is surjective. The injectivity of ψ A is just the A-linear independence of iterated Eisenstein integrals.

Corollary 2.10. For any Q-subalgebra A ⊂ C (viewed inside O(H) as constant functions), the Q-subalgebra of O(H) generated by I Eis and A is canonically isomorphic to I Eis ⊗ Q A, and the Q-subalgebra generated by E geom and A is canonically isomorphic to E geom ⊗ Q A.

Proof. By the previous corollary, the elements of I Eis are linearly independent over C, so over A. Since both A and I Eis are Q-algebras, any element of the algebra generated by A and I Eis can be expressed as a linear combination of elements of I Eis with coefficients in A which is unique up to scalar multiplication by rationals. This implies the isomorphism with the tensor product. The result carries over to E geom trivially since E geom lies inside I Eis .

The generating series of elliptic multizetas

In the first paragraph of this section, we recall an important fact about the elliptic associator defined by Enriquez in [START_REF] Enriquez | Elliptic associators[END_REF], or more precisely, the structure of the group-like power series A(τ ), B(τ ): namely, there exist power series A and B (whose definitions are recalled in (3.2) below) such that

A(τ ) = g(τ ) • A, B(τ ) = g(τ ) • B,
where g(τ ) denotes the automorphism of (2.7). In analogy with this structure, we will define a new power series E(τ ), which will also take the form g(τ )•E for a power series E ∈ F 2 (Z). We call E(τ ) the elliptic generating series, and its coefficients the E-elliptic multizetas or E-EMZs; similarly we call the coefficients of A(τ ) the A-EMZs and those of B(τ ) the B-EMZs. We define E (resp. A, resp. B) to be the Q-algebra generated by the E-EMZs (resp. the A-EMZs, resp. the B-EMZs; these coefficients are Enriquez's "elliptic analogs of multiple zeta values"). The algebras E, A and B are closely related but not equal. More importantly, each family of EMZs satisfies different algebraic relations.

In the remainder of the section, for technical reasons related to the use of mould theory, we work modulo 2πi in the sense explained in §1.2. In particular, we consider the power series Φ KZ and E, which are the power series obtained from Φ KZ and E by reducing the coefficients from Z to Z = Z/ (2πi) 2 . In §3.2, we give an expression for E which relates it explicitly to the Drinfel'd associator Φ KZ , and which will be essential in determining the algebraic relations satisfied by the reduced E-EMZs (denoted E-EMZs) in §4.

In §3.3, we give a result on the structure of the Q-algebra generated by the coefficients of a power series of the form F (τ ) = g(τ ) • F where F ∈ F 2 (Z) and g(τ ) • F ∈ F 2 (E geom ⊗ Z). We define the reduced power series A(τ ) and B(τ ); together with E(τ ), they are all power series of the form g(τ ) • F , so we apply the theorem to obtain our main result of the section, an explicit structural result relating the Q-algebras E, A and B generated by the coefficients of E(τ ), A(τ ) and B(τ ) respectively.

3.1. Definition of the elliptic generating series E(τ ). Throughout this section, we use the following change of variables: a = y 1 and b = x 1 . This change of variables will be applied to all the expressions in x 1 , y 1 encountered in the previous section, such as g(τ ) • y 1 , and we will also express other quantities studied by B. Enriquez in terms of a and b, in particular the elliptic associator. The purpose of this change of variables is for the application of mould theory in §4.

Let Ass µ denote the set of genus zero associators Φ ∈ F 2 (C) such that the coefficient of ab in Φ is equal to µ 2 /24 [START_REF] Drinfel | On quasitriangular quasi-Hopf algebras and on a group that is closely connected with Gal(Q/Q)[END_REF]. We will use the same elements t 01 , t 02 , t 12 as in [START_REF] Enriquez | Elliptic associators[END_REF], §5.3, but rewritten in the variables a, b:

t 01 = Ber b (-a), t 02 = Ber -b (a), t 12 = [a, b], (3.1)
where Ber x (y) = ad(x) e ad(x) -1 (y), so that t 01 + t 02 + t 12 = 0. Recall that Enriquez showed that a section from Ass µ to the set of elliptic associators is given by mapping Φ ∈ Ass µ to the elliptic associator (µ, Φ, A, B) defined by

A = Φ(
A(τ ) = g(τ ) • A, B(τ ) = g(τ ) • B (3.3) (see §5.2 of [15]). The (completed) Lie algebra f 2 = Lie[[a, b]
] is topologically generated by a and b, but since the operator Ber b is invertible, we have

a = -Ber -1 b (t 01 ) = e ad(b) -1 ad(b) (-t 01 ), (3.4)
so that we can just as well take t 01 and b as generators. Similarly, we can take e t01 and e b as topological generators of the prounipotent group We set E = σ(a), C = exp(E) = σ(e a ).

F 2 = F 2 (Q) = exp(f 2 ),
(3.5) The automorphism σ extends to an automorphism of the completed enveloping algebra U(f 2 ), and restricts to an automorphism of f 2 . Thus the power series E = σ(a) is Lie-like. All Lie-like and group-like power series discussed in this section are contained in the free non-commutative power series ring R a, b where R is either E geom ⊗ Z[2πi] or E geom ⊗ Z. When we speak of the ring generated by the coefficients of such a power series, we mean that we take coefficients of the power series written in any linear basis of R a, b (for example the basis of monomials in a and b), all of which lie in R, and consider the subring of R generated by these coefficients. The "degree" is the degree in the variables a, b.

Up to degree 5, the explicit expansion of E is given by

E = a - πi 2 c 3 + πi 12 [c 1 , c 3 ] + ζ(3)c 5 + π 2 36 [c 1 , c 4 ] + π 2 9 [c 3 , c 2 ], (3.6)
where

c i = ad(a) i-1 (b).
In analogy with (3.3), we now set

E(τ ) = g(τ ) • E and C(τ ) = g(τ ) • C = g(τ ) • σ(e a ). (3.7)
These power series lie in Proof. Since g(τ ) can be considered as an automorphism of the universal enveloping algebra of f 2 , it preserves the Lie algebra

E geom ⊗ Z[2πi] a, b .
f 2 ⊗ Q (E geom ⊗ Q Z[2πi])
; thus E(τ ) is Lielike, and C(τ ) is group-like. To check that a rational multiple of 2πi occurs as a coefficient in each of the three power series in the statement, it suffices to give their expansions in the c i in low weights using the explicit formulas (3.2), (3.3), (3.6) and (3.7), together with formula (2.7) defining g(τ ). For the first three, we obtain

E(τ ) = a + G 2 (τ ) - πi 2 c 3 + πi 12 G 0 (τ )c 4 + πi 12 [c 1 , c 3 ] + • • • A(τ ) = 1 -2πic 1 -2π 2 c 2 1 + πic 2 + • • • B(τ ) = 1 + c 1 + 1 2 c 2 1 + πic 2 + π 2 3 c 3 + • • •
which shows that the coefficient 2πi appears as a coefficient in low weight.

For the final statement, it is easy to see that 2πi does not appear in the ring of coefficients of A (τ ) since this power series is given by applying g(τ ) to the product of three terms A = Φ KZ (t 01 , t 12 ) e t01 Φ KZ (t 01 , t 12 ) -1 , none of which have 2πi in their coefficient rings, since these lie in R and 2πi does not. Since Φ KZ has ζ(2) as a coefficient, we may ask whether π 2 lies in the coefficient ring of A (τ ). The expansion of the power series A (τ ) is quite complicated and necessitates the help of a computer. It is necessary to go up to weight 5 in order to find enough coefficients to isolate π 2 . In weight 5, however, we find that the sum of the coefficient of c 3 1 c 2 and c 2 c 3 1 in the expansion of A (τ ) is equal to 4π 2 -1 24 , which shows that π 2 does lie in the coefficient ring A .

Lemma 3.3. The underlying vector space of E is spanned by the coefficients of C(τ ).

Proof. Let E denote the Q-vector space spanned by the coefficients of C(τ ). We first note that E is in fact a Q-algebra, because C(τ ) is a group-like power series, which means that the product of two of its coefficients can be written as a linear combination of such by using the (multiplicative) shuffle relations. Next, we note that since C(τ ) = exp E(τ ), the coefficients of C(τ ) can be expressed as algebraic combinations of the coefficients of E(τ ), so they lie inside the subring E ⊂ E geom ⊗ Z[2πi]. Thus E ⊂ E. But conversely, since E(τ ) = log C(τ ), the coefficients of E(τ ) are all algebraic and thus linear combinations of those of C(τ ), so they lie in E , so E ⊂ E . This completes the proof.

3.

2. An expression for E modulo 2πi. From now until the end of this article, we work modulo 2πi, in the sense that if a series has coefficients in Z[2πi], we reduce these modulo the ideal generated by 2πi. The quotient ring Z is equal to the quotient of Z by (2πi) 2 , or equivalently, by ζ(2). We use overlining to denote the reduced objects. The goal of the section is to obtain an expression for E that relates it directly to the reduced Drinfeld associator Φ KZ .

In order to approach this result, we will move from the Lie algebra of derivations over to power series in a and b by using the map given by evaluation at a. This is important because it allows us to compare derivations with power series in a and b such as Φ KZ .

Let v a denote the linear map given by evaluation at a. In Prop. 2.2 we considered this map restricted to D ∈ Der 0 (f 2 ); we have v a (D) = D(a) for D ∈ Der 0 (f 2 ). Let the push-operator be the linear automorphism of Q a, b defined by cyclically permuting the powers of a between the letters b in a monomial:

push(a k0 b • • • ba kr ) = a kr ba k0 b • • • a kr-1 , (3.8)
extended to polynomials and power series by linearity. A power series is said to be push-invariant if push(p) = p. It is shown in [START_REF] Schneps | Elliptic multiple zeta values, Grothendieck-Teichmüller and mould theory[END_REF], Lemma 2.1.1, that the restriction of v a to the Lie subalgebra Der 0 (f 2 ), which is injective by Prop. 2.2, has image equal to the space of push-invariant Lie series f push (3.9)

We also use v a to transport the exponential map exp : Der 0 (f 2 ) → Aut(f 2 ) to an exponential map exp a which makes the following diagram commute:

Der 0 (f 2 ) exp / / va Aut 0 (f 2 ) va f push 2 exp a / / f 2 , (3.10)
where Aut 0 (f 2 ) = exp Der 0 (f 2 ) . We observe that the right-hand vertical map v a is injective on Aut 0 (f 2 ). Indeed, if exp(D) This shows that v a is injective on Aut 0 (f 2 ), which by the diagram (3.10) then shows that exp a is also injective. Let G a denote the image of f push 2 under exp a , or equivalently, the image of Aut 0 (f 2 ) under v a . Then G a is a set of elements in f 2 , which forms a group when equipped with the group law transported from Aut 0 (f 2 ) by v a . This group law, which we denote by a , is compatible with the Campbell-Hausdorff law on the Lie algebra f push 2 , since for two derivations D, D ∈ Der 0 (f 2 ), we have

exp a D(a) a exp a D (a) = exp(D) • a a exp(D ) • a = exp(D) • exp(D ) • a = exp ch Der 0 (f2) (D, D ) • a = exp a ch f push 2 D(a), D (a) . (3.12)
We also have exp a (0) = v a exp(0) = v a (id) = a, so in fact the element a is the unit element of the group G a equipped with the multiplication a . Explicitly, for D ∈ Der 0 (f 2 ), we have

exp a D(a) = v a exp(D) = exp(D) • a = a + D(a) + 1 2 D 2 (a) + • • • (3.13)
Let grt ell be the elliptic Grothendieck-Teichmüller Lie algebra defined by B. Enriquez in §5.6 of [START_REF] Enriquez | Elliptic associators[END_REF]. Not surprisingly, this Lie algebra will be an essential tool in proving our results. Let us recall some of the basic facts concerning it. Firstly, Enriquez showed that there is a natural Lie morphism grt ell → Der 0 (f 2 ). It was further shown in [START_REF] Schneps | Elliptic multiple zeta values, Grothendieck-Teichmüller and mould theory[END_REF], equation (1.2.4), that this map is injective. 8 We will identify grt ell with its image in Der 0 (f 2 ).

Enriquez also proved the following results. There is a canonical surjection grt ell → grt. Let r ell denote the kernel; then it is easy to see that u ⊂ r ell . Finally, Enriquez gave a section γ : grt → grt ell of the canonical surjection, and showed that grt ell has the form of a semi-direct product grt ell ∼ = r ell γ(grt).

We write γ a for the composition map v a • γ, so that Ihara [START_REF] Ihara | The Galois representation arising from P 1 -{0, 1, ∞} and Tate twists of even degree[END_REF][START_REF] Ihara | On the stable derivation algebra associated with some braid groups[END_REF] studied these derivations in detail, and in particular, he showed that if Ψ = exp (ψ) and A Ψ denotes the automorphism exp(D ψ ) of U(Lie[[x, y]]), then

γ a : grt → f push 2 . ( 3
A Ψ (x) = x, A Ψ (y) = Ψ y Ψ -1 .
(3.17)

We can now state the main result of this subsection.

Theorem 3.4. Let E be obtained from E by reducing the coefficients from Z to Z/ (2πi) 2 . Then E = Γ(Φ KZ ) • a.

Proof. Let ψ ∈ grt, and let Ψ = exp (ψ) ∈ GRT . Then γ(ψ) ∈ grt ell ⊂ Der 0 (f 2 ) and Γ(Ψ) = exp γ(ψ) Recall that Φ KZ ∈ GRT (Z). (This is the reason for which we work mod 2πi, since the term -ζ(2)[x, y] in Φ KZ means that it does not lie in GRT , preventing us from taking advantage of the results on grt ell .) Let σ be the automorphism of F 2 (Z) obtained from σ by reducing modulo 2πi, i.e.

∈ GRT ell ⊂ Aut 0 (f 2 ),
σ(e t01 ) = Φ KZ (t 01 , t 12 )e t01 Φ KZ (t 01 , t 12

) -1 = A (3.19) σ(e b ) = Φ KZ (t 02 , t 12 )e b Φ KZ (t 01 , t 12 ) -1 = B,
where we set A = A 1/2πi and A and B denote the reductions of A and B mod 2πi.

Comparing with the values of Γ(Φ KZ ) from (3.18) on the generators t 01 , b of f 2 , we find that σ = Γ(Φ KZ ). Since E = σ(a) by (3.5), we find that

E = σ(a) = Γ(Φ KZ ) • a,
which concludes the proof.

Corollary 3.5. The Q-algebra generated by the coefficients of E is all of Z.

Proof. Set φ KZ = log (Φ KZ ), so that φ KZ ∈ grt ⊗ Q Z. We first show that the coefficients of φ KZ (written in a basis of grt) multiplicatively generate the same ring as the coefficients of Φ KZ , namely all of Z. To do this, we use an argument analogous to the one in the proof of Lemma 3.3. Let Z denote the Q-algebra generated multiplicatively by the coefficients of φ KZ . We of course know that the Q-vector space Z spanned by the (reduced) multizeta values which are the coefficients of Φ KZ is actually a Q-algebra, since Φ KZ is group-like. The definition of the twisted Magnus exponential (cf. [START_REF] Racinet | Séries génératrices non commutatives de polyzêtas et associateurs de Drinfeld[END_REF], (2.14), or [START_REF] Schneps | An introduction to Ecalle's theory of moulds[END_REF], (3.5.2)) shows that the coefficients of Φ KZ are all algebraic expressions in the coefficients of φ KZ ; thus Z ⊂ Z . But similarly, since φ KZ = log Φ KZ , the coefficients of φ KZ are also all algebraic expressions in elements of Z; thus Z = Z; in other words, the coefficients of φ KZ multiplicatively generate Z.

Since γ is an injective map from grt to grt ell , the coefficients of γ(φ KZ ) in a basis of grt ell also generate all of Z. Recall that Enriquez showed that grt ell is isomorphic to a semi-direct product of two of its subspaces, γ(grt) r ell , and that ε 2k ∈ r ell for k ≥ 0. Therefore we see that ε 0 / ∈ γ(grt) ⊂ grt ell ⊂ Der 0 (f 2 ). Since the natural bigrading on Der 0 (f 2 ) restricts to a bigrading on γ(grt) (cf. [START_REF] Enriquez | Elliptic associators[END_REF]), we find that in fact γ(grt) ⊂ Der 0 (f 2 ). Therefore, by Prop. 2.2, the evaluation map v a is injective on γ(grt), so the coefficients of γ(φ KZ ) • a in a basis of v a (grt ell ) also generate Z. By the same argument as above, thanks to the definition of exp a in (3.13), the coefficients of exp a γ(φ KZ ) • a then span Z. But by (3.13) and the diagram (3.15), we have

exp a γ(φ KZ • a = exp γ(φ KZ ) • a = Γ(Φ KZ ) • a = E, (3.20)
which completes the proof. , which as we saw is isomorphic to the tensor product of these two rings:

E, A, B ⊂ E geom ⊗ Q Z[2πi].
We have

E geom ⊗ Q Z[2πi] → E geom ⊗ Q Z[2πi] / 1 ⊗ 2πi E geom ⊗ Q Z.
We saw in Lemma 3.3 that 2πi ∈ E. The Q-algebra E generated by the coefficients of E(τ ) is equal to the quotient of E ⊂ E geom ⊗ Z[2πi] by the intersection of E with the ideal 1 ⊗ 2πi , so we have an inclusion

E ⊂ E geom ⊗ Q Z.
Recall from Definitions 1.1 and 1.2 that we set A (τ ) = g(τ ) • A and B(τ ) = g(τ ) • B, where A and B are as in (3.19) above. We write A for the Q-algebra generated by the coefficients of A (τ ) and B for that generated by the coefficients of B(τ ). Then like E, we have inclusions Definition 1.2). The goal of this paragraph is to compare the subrings E, A and B of E geom ⊗ Q Z. Theorem 3.6. We have the following equalities:

A, B ⊂ E geom ⊗ Q Z (see
E[2πiτ ] = A[2πiτ ] = B = E geom ⊗ Q Z.
(f 2 ). We set δ(τ ) = ch [ , ] r(τ ), γ(φ KZ ) ∈ Der 0 (f 2 ). Then we have      log A (τ ) = g(τ ) • log A = exp r(τ )) • exp γ(φ KZ ) • t 01 = exp δ(τ ) • t 01 log B(τ ) = g(τ ) • log B = exp r(τ )) • exp γ(φ KZ ) • b = exp δ(τ ) • b E(τ ) = g(τ ) • E = exp r(τ )) • exp γ(φ KZ ) • a = exp ) • a. (3.21)
We also set

a(τ ) = δ(τ ) • t 01 , b(τ ) = δ(τ ) • b, e(τ ) = δ(τ ) • a.
Step 1: The case of B. This case turns out to be the easiest one of the three, for the reason that by Prop. 2.2, the map v b evaluating derivations on b ∈ f 2 is injective on all of u, which is not the case for v a or v t01 . Let V b ⊂ f 2 denote the vector space v b Der 0 (f 2 ) . Recall the whole situation with the exponential map exp a and the group G a that we introduced in (3.9), (3.10), (3.12), (3.13). Thanks to the injectivity of v b , we can set up the analogous situation for b instead of a, but now using all Der 0 (f 2 ). We first transport the Lie bracket from Der 0 (f 2 ) onto V b via v b , setting

D(b), D (b) b = [D, D ](b),
which makes V b into a Lie algebra. We then define exp b to be the map that makes the diagram

Der 0 (f 2 ) exp / / v b Aut 0 (f 2 ) v b V b exp b / / f 2 (3.22) commute.
Exactly as in the case of a, we can show that the right-hand vertical map v b induced on Aut 0 (f 2 ) is still injective. Thus exp b is also injective on V b . We write

G b = exp b (V b ) = v b Aut 0 (f 2 ) ⊂ f 2 ,
and equip this set with a group law b in analogy with (3.12), transported from Aut 0 (f 2 ) by v b : We will use all this information in the following calculation. By (3.21) and the diagram (3.22), we have log B(τ

exp b D(f ) b exp b D (b) = v b exp(D) b v b exp(D ) = v b exp(D) • exp(D ) = exp(D) • exp(D ) • b. ( 3
) = exp δ(τ ) • b = exp b δ(τ ) • b ∈ G b , (3.25) so b(τ ) := δ(τ ) • b = log b log B(τ ) ∈ V b . (3.26)
Thus by (3.24) with

D = δ(τ ) = ch [ , ] r(τ ), γ(φ KZ ) , we have b(τ ) = δ(τ ) • b = ch [ , ] r(τ ), γ(φ KZ ) • b = r(τ ) • b + γ(φ KZ ) • b + 1 2 [r(τ ), γ(φ KZ )] • b + • • • = r(τ ) • b + γ(φ KZ ) • b + 1 2 r(τ ) • b, γ(φ KZ ) • b b + • • • , which we rewrite as b(τ ) = r(τ ) • b + γ(φ KZ ) • b + s(τ ) • b, (3.27)
where s(τ ) is the sum of all the bracketed terms in ch [ , ] r(τ ), γ(φ KZ ) .

By the argument of Lemma 3.3, the Q-algebra generated by the coefficients of b(τ ) = log b log B(τ ) is equal to the one generated by the coefficients of log B(τ ), which in turn is equal to the one generated by the coefficients of B(τ ), namely B ⊂ E geom ⊗ Z. In order to show that these two algebras are equal, we will consider B as the Q-algebra generated by the coefficients of b(τ ), and use properties of the right-hand side of (3.27) to show separately that it contains Z and E geom .

Let us write Z >0 for the Q-vector space spanned by the images in Z of the multizeta values in Z under the surjection Z → → Z. Then Z is generated by Q = Z 0 and Z >0 . Let us write nz for the vector space quotient Z >0 /Z 2 >0 , where Z 2 denotes the vector subspace of Z generated by products of elements of Z >0 , which can be viewed as linear combinations thanks to the shuffle multiplication of multizetas. The vector space nz is called the space of new multiple zeta values.

Let MZ denote the Q-algebra of motivic multiple zeta values defined by Goncharov (see [START_REF] Goncharov | Galois symmetries of fundamental groupoids and noncommutative geometry[END_REF]); let MZ >0 denote the Q-vector subspace generated by the motivic multizeta values, and let nmz = MZ >0 /MZ 2 >0 be the space of new motivic multizeta values. Goncharov showed that MZ is a Hopf algebra, so that nmz is a Lie co-algebra. He further showed that the motivic ζ m (2) = 0 in MZ, that there is a surjection MZ → → Z, and that the motivic multizeta values satisfy the associator relations of Φ KZ . It follows that there are injective maps in the dual situation nz ∨ ⊂ nmz ∨ ⊂ grt.

(3.28)

All three spaces are vector subspaces of f 2 , so that these injections can be considered as inclusions. The Lie series φ KZ lies in the vector space nz ∨ ⊗ Q Z, so by (3.28), it can also be considered as lying in the Lie algebras

nmz ∨ ⊗ Q Z ⊂ grt ⊗ Q Z. Thus γ(φ KZ ) ∈ γ(nmz ∨ ) ⊗ Q Z and γ(φ KZ ) • b ∈ v b γ(nmz ∨ ) ⊗ Q Z.
An important theorem by Brown ([5]) identified the Lie algebra nmz ∨ with the fundamental Lie algebra of the category of mixed Tate motives over Z, which is free on one generator in each odd weight ≥ 3 (the weight used here corresponds to the degree of the Lie polynomials, and is the negative of the usual motivic weight). In [START_REF] Hain | Universal Mixed Elliptic Motives[END_REF], Hain and Matsumoto defined a category of universal mixed elliptic motives, and they showed that the fundamental Lie algebra of that category has a monodromy representation in Der 0 (f 2 ) whose image Π is isomorphic to a semidirect product Π ∼ = u γ(nmz ∨ ). Thus D 1 ∈ u and D 2 ∈ γ(nmz ∨ ), then any bracketed word in these two derivations lies in u . R and

D 2 ∈ γ(nmz ∨ ) ⊗ Q R,
In fact, we can actually say more, and show that even if D 1 ∈ u and not just u , all bracketed words of D 1 and D 2 still lie in u ; in other words, the Lie algebra generated inside Der 0 (f 2 ) by u and γ(nmz ∨ ) has the structure of a semi-direct product u γ(nmz ∨ ). Indeed, we have ε 0 ∈ sl 2 , and Hain and Matsumoto show that there is an action of sl 2 on the semi-direct product u γ(nmz ∨ ). But sl 2 is included in the Lie algebra r ell defined by Enriquez, who shows that r ell is a normal Lie subalgebra of grt ell , and in fact that grt ell has the structure of a semidirect product grt ell = r ell γ(grt). Enriquez's result shows that a bracketed word combining ε 0 with any derivations in grt ell lies in r ell . In particular, a bracketed word combining ε 0 with any derivations in the subspace γ(nmz ∨ ) ⊂ γ(grt) lies in r ell . But Hain-Matsumoto's result that u γ(nmz ∨ ) is an sl 2 -module shows that such a bracket lies inside u γ(nmz ∨ ). Thus it lies inside the intersection of u γ(nmz ∨ ) with r ell , which is just u . The above statements all hold when the Lie algebra u, u and γ(nmz ∨ ) are tensored with any Q-algebra R.

In our situation, we set

D 1 = r(τ ) ∈ u ⊗ Q E geom and D 2 = γ(φ KZ ) ∈ γ(nmz ∨ ) ⊗ Q Z.
Then from the above, any bracketed word combining these two derivations lies in the space u ⊗ Q E geom ⊗ Q Z . Thus in particular, we have s(τ (3.27). Altogether, we thus have

) ∈ u ⊗ Q E geom ⊗ Q Z and s(τ ) • b ∈ v b (u ) ⊗ Q E geom ⊗ Q Z , where s(τ ) is the sum of bracketed terms in ch [ , ] (D 1 , D 2 ) as in
     γ(φ KZ ) • b ∈ v b γ(nmz ∨ ) ⊗ Q Z r(τ ) • b ∈ v b (u) ⊗ Q E geom s(τ ) ∈ v b (u ) ⊗ Q E geom ⊗ Q Z (3.29)
for the three terms in the right-hand side of (3.27).

We are now ready to show that B ⊃ Z. Since the evaluation map v b is injective on Der 0 (f 2 ), it is in particular injective on the subspace u γ(nmz ∨ ). Let V denote the underlying vector space of v b (u), and W that of v b γ(nmz ∨ ) . Then the underlying vector space of the semi-direct product

v b u γ(nmz ∨ ) is the direct sum V ⊕ W . Writing R = E geom ⊗ Q Z, we deduce from (3.29) that γ(φ KZ ) • b ∈ W ⊗ Q R and r(τ ) • b, s(τ ) • b ∈ V ⊗ Q R. (3.30)
Let us take a linear basis of V ⊕ W adapted to the direct product, i.e. in which every element belongs either to V or to W . Write b(τ ) in this basis, and consider the coefficient of a basis element w ∈ W . By (3.30), b(τ ) decomposes as a sum of two terms, γ(φ

KZ ) • b ∈ W ⊗ Q R and r(τ ) • b + s(τ ) • b ∈ V ⊗ Q R.
Therefore, the coefficient in b(τ ) of any basis element w ∈ W is equal to the coefficient of w in γ(φ KZ ) • b written in the same basis of W . But since we know that the coefficients of φ KZ written in a basis of nmz ∨ multiplicatively generate all of Z, and since γ is injective and defined over Q, the same holds for the coefficients of γ(φ KZ ) written in a basis of γ(nmz ∨ ), and then since v b is injective on this space and defined over Q, the same again holds for the coefficients of v b γ(φ KZ ) written in a basis of W . Thus the coefficients in b(τ ) of the elements of the basis of W span all of Z, so B ⊃ Z. Now we will show that B ⊃ E geom . Here we need to deal with the s(τ ) term. For this, we will proceed by induction on the weight. We take a basis for V which is the image under v b of a weight-graded basis of u. The Lie series b(τ ) starts in weight 1 with the term 2πiτ a, which comes from the 2πiτ ε 0 term of r(τ ) acting on b. Since r (τ ) := r(τ ) -2πiτ ε 0 and s(τ ) lie in u , these derivations are strictly weight-increasing, so there are no other weight 1 terms in b(τ ). Thus 2πiτ ∈ B. We use this result as the base case, fix n > 1, and make the induction hypothesis that for all m < n, the coefficients in b(τ ) of the weight m basis elements of V span the weight m graded part of E geom , so that B contains the weight graded parts of

E geom for all weight m < n. Consider the coefficient in b(τ ) of a weight n basis element v ∈ V . Each coefficient is of the form r v + s v where r v is the coefficient of v in r(τ ) • b and s v is the coefficient of v in s(τ ) • b.
But the part of the derivation s(τ ) that takes b to a weight n polynomial is made up of brackets of parts of r(τ ) and of γ(φ KZ ) of strictly smaller weight, and whose coefficients are thus algebraic combinations of coefficients of r(τ ) of lower weight, which already appear in B by the induction hypothesis, and of coefficients of γ(φ KZ ), i.e. elements of Z, which already lie in B by the result above that B ⊃ Z. Thus not just the coefficient r v +s v , but also the term s v lies in B, which proves that r v ∈ B. Thus all the coefficients of the weight n part of r(τ ) • b lie in B, so by induction, all the coefficients of r(τ ) • b lie in B; since v b is injective, these coefficients generate the same Q-algebra as the coefficients of r(τ ), namely E geom . This proves that B ⊃ E geom , and completes the proof of the desired result B E geom ⊗ Q Z.

Step 2. The case of E. The argument is similar to the one for B, but there is an added subtlety coming from the fact that r(τ ) lies in Der 0 (f 2 ), but v a is not injective on Der 0 (f 2 ) since ε 0 (a) = 0. To get around this, we will use the fact that u u Qε 0 (see (2.1)). The universal enveloping algebra of a semi-direct product of Lie algebras is isomorphic to the tensor product of the two Lie algebras, and its graded dual is isomorphic to the tensor product of the two duals. Therefore we have

(Uu) ∨ (Uu ) ∨ ⊗ Q (UQε 0 ) ∨ .
Under the identification (Uu) ∨ E geom of Theorem 2.6, this translates to

E geom E geom 0 ⊗ Q Q[2πiτ ],
where E geom 0 is multiplicatively generated by the coefficients of r (τ ) = r(τ )-2πiτ ε 0 . In particular, the subspace inclusion u ⊂ u corresponds in the dual to the surjection

E geom → → E geom / 2πiτ E geom 0 . The derivation δ(τ ) = ch [ , ] r(τ ), γ(φ KZ ) lies in u ⊗ Q E geom ⊗ Z
, but if we consider the derivation δ(τ ) obtained by reducing its coefficients mod 2πiτ , we see that

δ(τ ) = ch [ , ] r (τ ), γ(φ KZ ) ∈ u ⊗ Q E geom 0 ⊗ Q Z,
where r (τ ) = r(τ ) -2πiτ ε 0 .

Let E(τ ) and ê(τ ) be the power series obtained from E(τ ) and e(τ ) ∈ f 2 ⊗ Q E geom ⊗ Q Z by reducing the coefficients mod 2πiτ . Then we have

E(τ ), ê(τ ) ∈ f 2 ⊗ E geom 0 ⊗ Q Z and E(τ ) = exp δ(τ ) • a, ê(τ ) = δ(τ ) • a, so ê(τ ) = log a E(τ ).
Thus the Q-algebras generated by the coefficients of e(τ ) and by E(τ

) are equal. Denote this Q-algebra by E ⊂ E geom 0 ⊗ Q Z. Since ê(τ ) = ch [ , ] r (τ ), γ(φ KZ ) • a = r (τ ) • a + γ(φ KZ ) • a + ŝ(τ ) • a
where ŝ(τ ) denotes the bracketed terms in the Campbell-Hausdorff product, we can use the identical arguments to the case of b(τ ) above to prove that E ⊃ Z. We also again use induction on the weight to prove that E contains all the coefficients of r (τ ). The only difference with the case of b(τ ) is the base case, which is no longer in weight 1. The lowest weight term of ê(τ ) is of weight 3, and it comes from the term G 2 (τ )ε 2 of r (τ ) acting on a (note that ŝ(τ ) • a has no terms of weight lower than 7). Thus the same induction as above works to prove that every coefficient of r (τ ) lies in E, so we find that E E geom 0 ⊗ Q Z. Since E is the Q-algebra generated by the reduction of E(τ ) mod 2πiτ and it is equal to E geom 0 ⊗ Q Z, we see that E[2πiτ ] E geom ⊗ Q Z. This means that the composition map

E → E geom ⊗ Q Z → → E geom ⊗ Q Z / 2πiτ ⊗ 1 E geom 0 ⊗ Q Z
is surjective, which gives us the desired equality E[2πiτ ] E geom ⊗ Q Z.

Step 3. The case of A. The argument here is identical to the one for E. We again have the problem that, as discovered independently by Enriquez and by Hain and Matsumoto, there exists a unique derivation η ∈ u that annihilates t 01 ; η is defined over Q and is a linear combination of the ε 2k , k ≥ 0, with rational coefficients

η = ε 0 - 1 12 ε 2 + 1 240 ε 4 - 1 6048 ε 6 + • • •
The exact nature of η is not important, only the fact that it has a term in ε 0 and is defined over Q; this shows that v t01 is injective on u . We let W = v t01 (nmz ∨ ) and V = v t01 (u ), and choose a basis of V which is the image under v t01 of a weightgraded basis of u as above. We then proceed exactly as in the case of E to show that the Q-algebra generated by the coefficients of exp δ(τ ) •t 01 , (which is the reduction of log A(τ ) mod 2πiτ ) contains Z. For the induction argument, even if V is not itself graded by the weight, we can simply transport the weight-grading of u to V and use induction on that (or equivalently, do the induction on the lowest weight parts of the basis elements). Since t 01 starts with -a, the argument is identical to the one for a above, and shows that the Q-algebra generated by the coefficients of δ(τ ) • t 01 , and thus also by those of exp δ(τ ) • t 01 , is isomorphic to E geom 0 ⊗ Q Z, so that again we have A[2πiτ ] E geom ⊗ Q Z as desired. This concludes the proof.

The elliptic double shuffle and push-neutrality relations

In this section we use mould theory to explore and compare algebraic relations between the E-EMZs with algebraic relations between the A-EMZs. The first paragraph, §4.1, gives a brief exposition of the necessary definitions and results from mould theory.

Our main result on E-elliptic multizetas, in §4.2, arises as a corollary of the preceding theorem and the main result of [START_REF] Schneps | Elliptic multiple zeta values, Grothendieck-Teichmüller and mould theory[END_REF]. We show that E(τ ) satisfies a certain double family of algebraic relations called the elliptic double shuffle relations, related to the familiar double shuffle properties of Φ KZ , but more similar to the graded double shuffle relations studied for example in [START_REF] Brown | Depth-graded motivic multiple zeta values[END_REF]. Further, we show that if one assumes certain reasonable conjectures from multizeta and Grothendieck-Teichmüller theory, the elliptic double shuffle relations can be expected to form a complete set of algebraic relations for the E-EMZs. We compute these relations and the associated dimensions in detail in depth 2.

Finally, in §4.3 we consider a double family of relations satisfied by A (τ ) (or more precisely by the log of this series). The first family is just the usual shuffle, but the second is very different from the second shuffle relation satisfied by E(τ ). We call it the family of push-neutrality relations, and show that it is related to the Fay relations studied in [START_REF] Matthes | Elliptic double zeta values[END_REF]. We compute the relations and the associated dimensions in depth 2 and show that they are different from those of E(τ ), which means that while we know by Theorem 3.6 that E[2πiτ ] = A[2πiτ ], the algebras E and A themselves are not equal nor even isomorphic as filtered algebras (i.e. the dimensions of the associated gradeds are not equal).

4.1.

A very brief introduction to moulds. We recall some notions from Ecalle's theory of moulds [START_REF] Ecalle | ARI/GARI, la dimorphie et l'arithmétique des multizêtas: un premier bilan[END_REF][START_REF] Ecalle | The flexion structure and dimorphy: flexion units, singulators, generators, and the enumeration of multizeta irreducibles[END_REF] that we will need in order to study algebraic relations between elliptic multizetas. Besides the original references, a more detailed introduction to moulds can be found in [START_REF] Schneps | An introduction to Ecalle's theory of moulds[END_REF].

4.1.1. Moulds and bialternality. In this article, we use the term 'mould' to refer only to rational-function valued moulds with coefficients in Q. Thus, a mould is a family of functions {P (u 1 , . . . , u r ) | r ≥ 0} with P (u 1 , . . . , u r ) ∈ Q(u 1 , . . . , u r ). In particular P (∅) is a constant. The depth r part of a mould is the function P (u 1 , . . . , u r ) in r variables. By defining addition and scalar multiplication of moulds in the obvious way, i.e. depth by depth, moulds form a Q-vector space that we call M oulds. We write M oulds pol for the subspace of polynomial-valued moulds. The vector space ARI is the subspace of M oulds consisting of moulds P with constant term P (∅) = 0, and ARI pol is again the subspace of polynomial-valued moulds in ARI.

The standard mould multiplication mu is given by

mu(P, Q)(u 1 , . . . , u r ) = r i=0 P (u 1 , . . . , u i )Q(u i+1 , . . . , u r ). (4.1)
For simplicity, we write P Q = mu(P, Q). This multiplication defines a Lie algebra structure on ARI with Lie bracket lu defined by lu(P, Q) = mu(P, Q) -mu(Q, P ).

We now introduce four operators on moulds. The ∆-operator on moulds is defined as follows: if P ∈ ARI, then

∆(P )(u 1 , . . . , u r ) = u 1 • • • u r (u 1 + • • • + u r )P (u 1 , . . . , u r ). (4.2)
The dar-operator is defined by

dar(P )(u 1 , . . . , u r ) = u 1 • • • u r P (u 1 , . . . , u r ). (4.3)
The push-operator is defined by

push(B)(u 1 , . . . , u r ) = B(u 2 , . . . , u r , -u 1 -• • • -u r ). (4.4)
Finally, the swap operator is defined by

swap(A)(v 1 , . . . , v r ) = A(v r , v r-1 -v r , . . . , v 1 -v 2 ). (4.5)
Here the use of the alphabet v 1 , v 2 , . . . instead of u 1 , . . . , u r is purely a convenient way to distinguish a mould from its swap.

The main property on moulds that we will need to consider is alternality. A mould P is said to be alternal if for all r > 1 and for 1 ≤ i ≤ [r/2], we have u∈sh((u1,...,ui),(ui+1,...,ur))

P (u) = 0, (4.6)
where the set of r-tuples sh (u 1 , . . . , u i ), (u i+1 , . . . , u r ) is the set

(u σ -1 (1) , . . . , u σ -1 (r) ) σ ∈ S r such that σ(1) < • • • < σ(i), σ(i+1) < • • • < σ(r) .
The mould swap(A) is alternal if it satisfies the same property (4.6) in the variables v i .

We write ARI al for the space of alternal moulds in ARI, and ARI al/al for the space of moulds which are alternal and whose swap is also alternal. We also consider moulds which are alternal and whose swap is alternal up to addition of a constantvalued mould. The space of these moulds is denoted ARI al * al and we call them bialternal.

We say that a mould P is ∆-bialternal if ∆ -1 (P ) is bialternal, and we write ARI ∆-al * al for the space of such moulds.

Qa ⊕ L[[C]] ∼ = f 2 = Lie[[a, b]].
Following Écalle, let ma denote the standard vector space isomorphism from Q C to the space (M oulds) pol defined by

ma : Q C ∼ → (M oulds) pol c k1 • • • c kr → (-1) k1+•••+kr-r u k1-1 1 • • • u kr-1 r (4.7)
on monomials, extended by linearity to all power series. . Finally, we recall that for any mould P ∈ ARI, Écalle defines a derivation arit(P ) of the Lie algebra ARI lu . We do not need to recall the definition of arit here (but it is given in §4.4 below where we prove a technical lemma). For now it is enough to know that when restricted to polynomial-valued moulds, it is related to the Ihara derivations (3.16) via the morphism ma:

ma D f (g) = -arit ma(f ) • ma(f ).
For each P ∈ ARI, we also define the derivation arat(P ) = -arit(P ) + ad(P ), (

where ad(P ) • Q = lu(P, Q).

i.e. ds ell consists of the Lie power series f ∈ f 2 such that ma(f ) is ∆-bialternal.

The following results are essentially contained in [?] and [START_REF] Schneps | Elliptic multiple zeta values, Grothendieck-Teichmüller and mould theory[END_REF]. We give some details of the proofs for the convenience of the reader. Proposition 4.2. The space ds ell satisfies the following properties.

(i) ds ell ⊂ f push 2
, where f push 2 has been defined in Section 3.2; (ii) ds ell is a Lie algebra under the bracket , on f push 2 defined in (3.9). (iii) There is a Lie algebra inclusion grt ell ⊂ ds ell , where grt ell is the Lie subalgebra of grt ell generated by γ(grt) and u.

Proof. For (i), by definition, elements of ma(ds ell ) are alternal moulds whose swap is also alternal up to the addition of a constant-valued mould. It is shown in Lemma 2.5.5 of [START_REF] Schneps | An introduction to Ecalle's theory of moulds[END_REF] that such moulds are push-invariant and ∆ trivially respects push-invariance which shows that ds ell ⊂ f push 2 . For (ii), first note that ARI ∆-al * al is a Lie algebra under the ari-bracket ([?], Theorem 3.3; note that ∆ is denoted "sing" there). Therefore by definition, ds ell = ∆(ARI ∆-al * al ) is a Lie algebra under the Dari-bracket, which is the transport of the ari-bracket via the map ∆, by Proposition 3.2.1 of [START_REF] Schneps | Elliptic multiple zeta values, Grothendieck-Teichmüller and mould theory[END_REF]. But when restricted to polynomials, the Dari-bracket is nothing but the Lie-bracket of derivations, i.e. Darit(Dari(P, Q)) = [Darit(P ), Darit(Q)], so it is the same as the bracket of derivations in Der 0 (f 2 ).

Finally, for (iii), we first show that both γ(grt) and u lie in ds ell . The inclusion γ(grt) ⊂ ds ell is Theorem 1.3.1 of [START_REF] Schneps | Elliptic multiple zeta values, Grothendieck-Teichmüller and mould theory[END_REF]. For the other inclusion u ⊂ ds ell , it is shown in Corollary 3.6 of [?] that ∆ -1 •ma gives an injective map from u (which is called E in [?]) into ARI ∆-al/al which is clearly equivalent to ma giving an injection from u to ds ell = ∆(ARI ∆-al/al ). Finally, the semi-direct action is nothing other than the Dari-bracket, and we saw in (ii) that this is the same as the bracket of derivations in Der 0 .

Remark 4.3. In [START_REF] Brown | Zeta elements in depth 3 and the fundamental Lie algebra of the infinitesimal Tate curve[END_REF], a Lie algebra called pls (for "polar linearized shuffle") is introduced, which is essentially equivalent to ds ell . It is also shown that u embeds into pls ([6], Proposition 4.6) and, moreover, it is asked whether the equality u = pls holds. Proposition 4.2.(iii) implies that ds ell is, in fact, much larger than u. More precisely, Enriquez ([14], §7) has shown that u lies in the kernel of the surjection grt ell → grt from which it follows that the image γ(grt) ⊂ grt ell of grt under the splitting γ is disjoint from u. In particular, the Lie algebra u cannot equal ds ell . 4.2. The elliptic double shuffle relations. We can now give the elliptic double shuffle property satisfied by the reduced elliptic generating series E(τ ). It is in fact phrased more directly as a property of the log power series

e(τ ) = log a E(τ ) -a + 1 ,
where log a is the inverse of the exponential exp a defined in (3.9), or rather, on the mould version of this power series e m (τ ) = ma e(τ ) . Proof. Consider equation (3.27) from the proof of Theorem 3.6 in the case where f = a, F = E, F (τ ) = g(τ ) • E. The left-hand side of (3.27) is equal to e(τ ), so we find that e(τ ) = r(τ ) • a + γ(φ KZ ) • a + s(τ ). Let e = γ(φ KZ ) • a, so that e ∈ v a γ(grt) ⊗ Q E. By the proof of Theorem 3.6, we have r(τ

) • a + s(τ ) ∈ V ⊗ Q E where V = v a (u ).
Therefore, e(τ ) ∈ grt ell by the definition of grt ell , and since grt ell ⊂ ds ell by Proposition 4.2 (iii), we also have e(τ ) ∈ ds ell ⊗ Q E, which proves the theorem thanks to (4.9).

We conjecture that the elliptic double shuffle relations form a complete set of algebraic relations between the E-elliptic multizetas. This statement really breaks down into two statements, one concerning the arithmetic part Z of E and the other the geometric part E geom = U(u) ∨ . We show in Proposition 4.5 that indeed, the completeness follows from two conjectures: the first one a standard conjecture from multizeta theory, and the second a similar conjecture from elliptic multizeta theory. Due to the fact that it is much easier to work in the geometric situation than the arithmetic situation (as there are no problems of transcendence), we are actually able to prove that the elliptic double shuffle relations are complete in depth 2, without any recourse to conjectures (see Proposition 4.6).

The first conjecture amounts to the inclusions in (3.28) being all isomorphisms as well as the standard conjecture that the inclusion grt ⊂ ds (proved by Furusho in [START_REF] Furusho | Double shuffle relation for associators[END_REF]) is actually also an isomorphism. All these are implied by the simplified statement:

Conjecture 1: nz ∨ ∼ = ds.
This is the standard conjecture that the double shuffle relations suffice to generate all the algebraic relations satisfied by multiple zeta values [START_REF] Ihara | Derivation and double shuffle relations for multiple zeta values[END_REF].

The second conjecture amounts to the existence of a canonical semi-direct product structure on the elliptic double shuffle Lie algebra ds ell . This is inspired by Enriquez result that the elliptic Grothendieck-Teichmüller Lie algebra grt ell is isomorphic to a semi-direct product r ell γ(grt) where r ell is a certain Lie ideal of grt ell containing u. Analogously, we have This conjecture is closely related to Enriquez' "generation conjecture" for grt ell [START_REF] Enriquez | Elliptic associators[END_REF], §10, namely that u ∼ = r ell . If Enriquez' conjecture were true, then the left hand side of our Conjecture 2 would be isomorphic to grt ell , and Conjecture 2 would reduce to showing that grt ell ∼ = ds ell which is the elliptic analog of the well-known conjecture grt ∼ = ds.

One can also merge Conjectures 1 and 2 into a single conjecture, thereby extending (3.28) to the elliptic setting. The elliptic analog of nmz ∨ is the elliptic motivic fundamental Lie algebra, which is conjecturally isomorphic to its image Π = V nmz ∨ in the derivation algebra Der 0 (f 2 ) (cf. the proof of Theorem 3.6). Then we get inclusions

V nz ∨ ⊂ V nmz ∨ ∼ = Π ⊂ grt ell , (4.10)
which conjecturally are all equalities. Note that the first equality would also follow from Conjecture 1 above.

Proposition 4.5. If Conjectures 1 and 2 are true, then the elliptic double shuffle relations generate all algebraic relations between elliptic multizetas.

Proof. By Conjecture 1, we would have Z ∼ = U(ds) ∨ , so since E geom ∼ = U(u) ∨ ∼ = U(V ) ∨ by Theorem 2.6, we would have

E[2πiτ ] ∼ = U(V ) ∨ ⊗ Q U(ds) ∨ .
It is known that the underlying vector space of the universal enveloping algebra U(R L) of a semi-direct product of Lie algebras R L is the space U(R) ⊗ Q U(L); in fact U(R L) is a Hopf algebra equipped with the smash product ( [START_REF] Molnar | Semi-direct products of Hopf algebras[END_REF]) and with the standard coproduct for which elements of R L are primitive. The dual U(R L) ∨ has underlying Q-algebra U(R) ∨ ⊗ Q U(L) ∨ (and is equipped with the smash coproduct). Thus by Conjecture 2, we would have the isomorphism of Q-algebras

E[2πiτ ] ∼ = U(u) ∨ ⊗ Q U(ds) ∨ ∼ = U(ds ell ) ∨ .
Now, for any Lie algebra g defined over Q and any Q-algebra R, if f is an element of g ⊗ Q R, then the subring of R generated by the coefficients of f (in a linear basis of g) generate a subring of R which is necessarily isomorphic to a quotient of U(g) ∨ ; in other words, the coefficients of f satisfy relations that are imposed by the fact that f lies in the Lie algebra g, and possibly others. If this quotient is actually isomorphic to U(g) ∨ , this signifies that the coefficients do not satisfy any further algebraic relations than those imposed on them by the fact that f lies in g.

In our case, we have e(τ ) ∈ ds ell ⊗ Q E, and the coefficients of e(τ ), together with 2πiτ , generate E[2πiτ ], which by the conjectures is isomorphic to U(ds ell ) ∨ , implying that the coefficients of e(τ ) do not satisfy any other algebraic relations than those imposed by the fact that e(τ ) lies in ds ell , i.e. is ∆-bialternal.

Explicit elliptic double shuffle relations. Let us take a closer look at what the ∆bialternality properties are. The first property is that e m (τ ) is ∆-alternal, i.e. that ∆ -1 (e m (τ )) is alternal. But ∆ trivially preserves alternality, so this is equivalent to saying that e m (τ ) is alternal, i.e. that for each r > 1, (EDS.1) u∈sh (u1,...,u k ),(u k+1 ,...,ur) e m (τ )(u) = 0 for 1 ≤ k ≤ [r/2]. This condition is equivalent to the statement that the power series e(τ ) is a Lie series.

The new relations on e m (τ ) are the second set, which say that up to adding on a constant-valued mould, the swap of the mould ∆ -1 e m (τ ) is also alternal, where the swap-operator is defined in (4.5). This alternality is given by the equalities for r > 1 Thus the alternality conditions in (EDS.2) are all sums of rational functions with denominators that are products of terms of the form v i and (v i -v j ), which sum to zero. Therefore, by multiplying through by the common denominator

v 1 • • • v r i>j (v i -v j ),
the second elliptic shuffle equation can be expressed as a family of polynomial conditions on the mould swap(e m (τ )). directly on e m (τ ). Applying the depth 2 swap operator from ARI to ARI (given by v 1 → u 1 + u 2 , v 2 → u 1 ), we transform this relation into e m (τ )(u 1 , u 2 ) = e m (τ )(u 1 + u 2 , -u 2 ).

Finally, e m (τ ) is of odd degree, so by the depth 2 version of (EDS.1), we have e m (τ )(-u 2 , -u 1 ) = e m (τ )(u 1 , u 2 ), which gives (EDS.2-depth 2) e m (τ )(u 1 , u 2 ) = e m (τ )(u 2 , -u 1 -u 2 ).

Note that this is nothing other than e m (τ )(u 1 , u 2 ) = push e m (τ ) (u 1 , u 2 ) where the push-operator is defined in (4.4). Thus in depth 2, the ∆-bialternality conditions correspond to alternality and push-invariance of e m (τ ) (which in turn correspond to the fact that e(τ ) is a Lie series that is push-invariant in depth 2 in the sense of power series, as in (3.8)). This simple reformulation is special to depth 2; the ∆bialternal property does not lend itself so easily to a direct expression as a property of e(τ ) in higher depths.

We end this subsection by showing that the conjecture that the ∆-bialternal relations are sufficient holds in depth 2. Proof. We can prove this result without recourse to any conjectures, essentially because depth 2 is too small to contain any of the arithmetic part of e m (τ ) (we qualify this statement below), and the geometric part V = v a (u) is well-understood in depth two. We know that e(τ ) ∈ ds ell ⊂ f push •P (u i+1-k , . . . , u i+t-k ) = 0.

We will show that the coefficients of each term P (u m+1 , . . . , u m+t ) sums to zero due to the push-neutrality of A. In fact it is enough to show that the coefficient of P (u 1 , . . . , u t ) sums to zero, as all the other terms are obtained from this one by applying powers of the push-operator.

The terms containing P (u 1 , . . . , u t ) are those for which the index k = i, so that k ∈ {0, . . . , r -t = s}, and we must show that the sum s k=0 A(u r-k+2 , . . . , u r , u 0 , u 1 + • • • + u t+1 , u t+2 , . . . , u r-k ) vanishes, where u 0 = -u 1 -• • • -u r and we have shifted some of the indices modulo (r + 1) in order to make them positive. Note now that

u 1 + • • • + u t+1 = -u 0 -u t+2 -• • • + u r .
As a result the last sum runs over the (s + 1) cyclic permutations of u t+2 , . . . , u r , u 0 and -u t+2 -• • • -u r -u 0 , so it is equal to the sum over the push s -orbit of just one term, say the one with k = s, i.e. to s k=0 A(u t+2 , . . . , u r , u 0 ), which indeed vanishes since A is push-neutral. This concludes the proof of Lemma 4.9.

1. 3 .5

 3 The elliptic double shuffle relations for E-EMZs. The theorem in the previous section shows that the E-EMZs together with 2πiτ form a set of generators of E geom ⊗ Z, as do the A-EMZs and the B-EMZs taken together. These three sets of generators are quite different from each other. A natural question, given a set of generators of a ring, is to try to establish the relations they satisfy, and if possible a For two Q-algebras A ⊂ B and an element b ∈ B, the notation A[b] here denotes the subring of B generated by A and b.

Definition 3 . 1 .Lemma 3 . 2 .

 3132 The Lie-like power series E(τ ) is called the elliptic generating series, and its coefficients are called the E-EMZs or E-elliptic multizetas.For k = (k 1 , . . . , k r ) we write E(k) for the coefficient in E(τ ) of the monomial c k1 • • • c kr , which lies in the tensor product E geom ⊗ Z[2πi]. The Q-algebra generated inside E geom ⊗ Z[2πi]by the E-elliptic multizetas E(k) is denoted by E. The power series E(τ ) is Lie-like and C(τ ) is group-like. The element 2πi appears as a coefficient in each of the three power series A(τ ), B(τ ) and E(τ ) expanded in the variables c i . The element 2πi does not lie in the Q-algebra A generated by the coefficients of A (τ ), but π 2 does.

2 ⊂ f 2 .

 22 The map v a transports the Lie bracket on Der 0 (f 2 ) to a Lie bracket •, • on f push 2 as follows: D(a), D (a) = [D, D ](a).

  • a = exp(D ) • a for two derivations D = D ∈ Der 0 (f 2 ), then letting D = ch Der 0 (f2) (-D , D), where ch Der 0 (f2) denotes the Campbell-Hausdorff law on Der 0 (f 2 ), we would have exp( D) • a = a = a + D(athat D = D , we have D(a) = 0. Let Let d denotes the lowest degree term occurring in D(a). Then d > 1, and the degree d part of exp( D) • a can only come from the term D(a) in (3.11), contradicting the desired identity exp( D) • a = a.

. 14 )

 14 Let exp denote the ("twisted Magnus") exponential map exp : grt → GRT [[31], (2.14), where it is denoted * ]. Recall that Der * Lie[[x, y]] is the space of derivations that annihilate x and take y to a bracket [y, f ] and z = -x -y to a bracket [z, g] for some f, g ∈ Lie[x, y]. Writing Aut * Lie[[x, y]] for the group of automorphisms exp(D) with D ∈ Der * Lie[[x, y]] , we have the commutative diagram Der * (Lie[[x, y]]) the group homomorphism that makes the middle square commute, and the upper map grt → Der * (Lie[[x, y]]) in the left-hand square is the map that takes a Lie element ψ ∈ f 2 to the associated Ihara derivation D ψ defined by D ψ (x) = 0, D ψ (y) = [ψ(x, y), y].(3.16)

3. 3 .

 3 Structure of the Q-algebras E, A and B. Since the rings generated by the coefficients of E, A and B all lie inside Z[2πi] and the ring generated by the coefficients of g(τ ) is E geom , and since E(τ ) = g(τ ) • E, A(τ ) = g(τ ) • A and B(τ ) = g(τ ) • B, the rings E, A and B generated by the coefficients of E(τ ), A(τ ) and B(τ ) respectively are all contained inside the ring generated inside O(H) by the subrings E geom and Z[2πi]

Proof.

  Let r(τ ) = log g(τ ) ∈ Der 0 (f 2 ). Recall from (3.19) that A = σ(e t01 ), B = σ(e b ), and E = σ(a), where σ = Γ(Φ KZ ) = exp γ(φ KZ ) . Let us write ch [,] = ch Der0(f2) for the Campbell-Hausdorff law in the derivation algebra Der 0

. 23 )

 23 We write log b : G b → V b for the inverse of exp b , so that log b exp(D) • b = D(b).(3.24)

4. 1 . 2 .

 12 From power series to moulds. Let c i = ad(a) i-1 (b) for i ≥ 1 as in §3.1. Let the depth of a monomial c i1 • • • c ir be the number r of c i in the monomial; the weight (degree in a and b) and the depth form a topological bigrading on the formal power series ring Q C = Q c 1 , c 2 , . . . on the free variables c i . Here, by "topological bigrading" we mean that Q C is the direct product (not the direct sum) n,d≥0 V n,d of its components of weight n and depth d. Similarly, we writeL[[C]] = Lie[[c 1 , c 2 , . . .]]for the corresponding Lie algebra. By Lazard elimination, we have an isomorphism

4. 1 . 3 .Definition 4 . 1 .

 1341 Reminders on the elliptic double shuffle Lie algebra ds ell . We end this subsection by recalling the definition and a few facts about the elliptic double shuffle Lie algebra ds ell from[START_REF] Schneps | Elliptic multiple zeta values, Grothendieck-Teichmüller and mould theory[END_REF]. The elliptic double shuffle Lie algebra ds ell is the subspace of f 2 such that ma ds ell = ARI ∆-al * al pol ,

Theorem 4 . 4 .

 44 The mould e m (τ ) is ∆-bialternal, i.e. ∆ -1 e m (τ ) is a bialternal mould.

Conjecture 2 :

 2 u γ(ds) ∼ = ds ell .

(EDS. 2 )

 2 v∈sh (v1,...,v k ),(v k+1 ,...,vr)swap ∆ -1 e m (τ ) (v) = 0 for 1 ≤ k ≤ [r/2].The swapped mould is given explicitly byswap ∆ -1 e m (τ ) = 1 v 1 (v 1 -v 2 ) • • • (v r-1 -v r )v r e m (τ )(v r , v r-1 -v r , . . . , v 1 -v 2 ).

Elliptic double shuffle relations in depth 2 .

 2 Let us work this out explicitly in depth 2. The usual alternality condition reduces to(EDS.1-depth 2) e m (τ )(u 1 , u 2 ) + e m (τ )(u 2 , u 1 ) = 0.The swap alternality condition reads1 v 1 (v 1 -v 2 )v 2 swap(e m (τ ))(v 1 , v 2 ) + 1 v 1 (v 2 -v 1 )v 2swap(e m (τ ))(v 2 , v 1 ) = 0, which, clearing denominators, reduces simply toswap(e m (τ ))(v 1 , v 2 ) -swap(e m (τ ))(v 2 , v 1 ) = 0. Since swap(e m (τ ))(v 1 , v 2 ) = e m (v 2 , v 1 -v 2 ), this is given by the relatione m (τ )(v 2 , v 1 -v 2 ) = e m (τ )(v 1 , v 2 -v 1 )

Proposition 4 . 6 .

 46 The relations (EDS.1) and (EDS.2) in odd degrees are the only relations satisfied by e m (τ ) in depth 2.

  which is a priori topologically generated by e a and e b . Let us define an automorphism σ of F 2 (Z[2πi]) by

σ(e t01 ) = Φ KZ (t 01 , t 12 )e t01 Φ KZ (t 01 , t 12 ) -1 σ(e b ) = e πit12 Φ KZ (t 02 , t 12 )e b Φ KZ (t 01 , t 12 ) -1 .

  It is well-known that p ∈ Q C satisfies the shuffle relations if and only if p is a Lie series, i.e. p ∈ Lie[[C]]. The alternality property on moulds is analogous to these shuffle relations, that is a series p ∈ Q C satisfies the shuffle relations if and only if ma(p) is alternal (see e.g. [33], §2.3 and Lemma 3.4.1]). Writing ARI al for the subspace of alternal moulds and ARI al pol for the subspace of alternal polynomial-valued moulds, this shows that the map ma restricts to a Lie algebra isomorphism ma : Lie[[C]]

	ma -→ ARI al lu,pol

Note that what is denoted Der 0 (f 2 ) in[START_REF] Schneps | Elliptic multiple zeta values, Grothendieck-Teichmüller and mould theory[END_REF] is denoted here by Der 0 (f 2 ).

2

. The graded dimensions of f 2 in depth 2 are given by dim(f push + 1 if n is odd ≥ 5 0 otherwise. (4.12)

Indeed, the last equality follows from the fact that in depth 2, V is spanned by the [ε 2j , ε 2k ](a) with j < k, j, k = 1, which are all of odd weight, and the fact that, as shown in [START_REF] Pollack | Relations between derivations arising from modular forms[END_REF], the only relations between these n-3 4 brackets come from period polynomials, whose number is given by n-7

4

n-5

6

. Thus V 2 = ds 2 ell = (f push 2 ) 2 , so the Lie relation (EDS.1) and the push-invariance relation (EDS.2) suffice to characterize elements of ds ell in depth 2.

Depth 2 elements of ds ell in low weights:

• in weight 7,

• in weight 9,

• in weight 11,

The coefficients of a(τ ) generate the Q-algebra A of A-EMZs. In this paragraph we will consider certain relations satisfied by the coefficients of a(τ ), different from the linearized elliptic double shuffle relations satisfied by e(τ ). The first family of relations on the coefficients of a(τ ) is the usual family of alternality relations, but the second is the family of push-neutrality relations. These relations are related (mod 2πi) to the Fay-shuffle relations introduced in [START_REF] Matthes | Elliptic double zeta values[END_REF], and studied explicitly in depth 2. We show that in depth 2, the push-neutrality relations are identical to the Fay-shuffle relations. We also show that even in depth 2 and mod 2πi, the alternality and push-neutrality relations are strictly weaker than the linearized elliptic double shuffle relations.

We will give our relations in terms of mould theory (but see Corollary 4.11 for a translation into power series terms at the end). For this we recall the push and dar-operators defined in (4.4) and (4.3). We will say that a mould B is push-neutral if B(u 1 , . . . , u r ) + push(B)(u 1 , . . . , u r ) + • • • + push r (B)(u 1 , . . . , u r ) = 0 (4.13) for all r ≥ 1, where push denotes the push-operator on moulds defined in (4.4).

Theorem 4.8. Let a m (τ ) = ma a(τ ) . Then a m (τ ) is alternal and dar -1 a m (τ ) is push-neutral in depth r > 1.

Proof. Recall the derivation arat defined in (4.8). For any P ∈ ARI, set

It is shown in [START_REF] Schneps | Elliptic multiple zeta values, Grothendieck-Teichmüller and mould theory[END_REF], Lemma 3.1.2, that the map

is an injective Lie morphism, so that we have

Let a m = ma(a), a m (τ ) = ma a(τ ) , and r m (τ ) = ma r a (τ ) . Under the map (4.15), we have r(τ

we have

Let σ denote the automorphism of f 2 defined in §3.2. We have

which means that a m (τ ) is alternal. This settles the first property of a m (τ ) stated in the theorem.

Let us consider the second property. Since γ(φ KZ ) ∈ Der 0 (f 2 ), it annihilates t 12 . Therefore, setting t 01 = t 01 + 1 2 t 12 , we have

Set T 01 = ma(t 01 ), and set

Then by (4.16), the equality (4.19) translates into moulds as

To complete the proof of the second property, we will use the following lemma, whose proof is deferred to the final subsection of this paper. Lemma 4.9. Let A ∈ ARI. If A is push-neutral, then arat(P ) • A is push-neutral for all P ∈ ARI. If dar -1 A is push-neutral, then dar -1 •Darit(P )•A is push-neutral for all P ∈ ARI.

It is easy to see that if A is a push-invariant mould, then dar -1 A is push-neutral, since

where

By Proposition 4.10 below, dar -1 T 01 is push-neutral and by Lemma 4.9, so is

To show that dar -1 a m (τ ) is push-neutral we use the same lemma again. Since dar -1 a m is push-neutral, so is dar -1 • Darit r m (τ ) • a m , and then successively, so is dar -1 • Darit r m (τ ) n • a m for all n ≥ 1. Thus dar -1 a m (τ ) is push-neutral by (4.18). This proves the theorem.

The following proposition was used in the proof of Theorem 4.8.

Proposition 4.10. The mould

is push-neutral.

Proof. It is enough to show the push-neutrality of f n := ma([ad n (b)(a), a]) for all n ≥ 2 separately. Using the definition of ma (cf. Section 4.1), we see that

Now in depth n, the operator ad(a) on Q C corresponds to multiplication by

On the other hand, by the definition of the push-operator (4.4), we have push(

, where the indices are to be taken mod n (so that u k+n = u k ). Using the elementary fact that

i.e. f n is push-neutral for all n ≥ 2, as was to be shown.

We end this subsection by studying these relations more explicitly in depth 2 and comparing them with the elliptic double shuffle relations on e m (τ ). The alternality relation is of course the same:

The push-neutrality relation in depth 2 is given by

Multiplying by the common denominator u 0 u 1 u 2 yields the polynomial relation

It was shown in [START_REF] Matthes | Elliptic double zeta values[END_REF] Theorem 3.11, that the dimension of the space of polynomials in u 1 , u 2 of odd degree d satisfying (FS.1) and (FS.2) is given by d 3 + 1. In terms of the weight n = d + 2 of the corresponding polynomials in f 2 , this is n -2 3 + 1.

In weight 5, for example, there are two independent such polynomials:

In weight 7, there are again two independent polynomials, given by

In weight 9, the space is three-dimensional, given by

Finally, we work out the case of weight 11, where the dimension is four:

Observe that these dimensions are significantly bigger than those given by the elliptic double shuffle equations (EDS.1) and (EDS.2) in depth 2. This is explained by the fact that the vector space generated by the coefficients of a m (τ ) in a given weight and depth is not equal to the one generated by the analogous coefficients of e m (τ ), or in terms of the algebras, that while A[2πiτ ] = E[2πiτ ] by virtue of Corollary ??, the Q-algebras E and A are quite different and do not even have the same graded dimensions.

Under the conjecture Z ∼ = U(grt) ∨ , the Q-algebra E is isomorphic to U(grt ell ) ∨ , and thus inherits a natural bigrading dual to that of grt ell . Together with products of elements of E of smaller depth and weight (including G 0 ), the coefficients of e m (τ ) in a given weight n and depth d span the bigraded part E d n , whereas those of a m (τ ) do not.

For example, in weight 5 and depth 2, the coefficients of e m (τ ) generate the 1-dimensional space 2G 0,4 + G 0 G 4 . The bigraded subspace E

We end this subsection with a power series statement of the alternality and push-neutrality relations on a m (τ ).

Corollary 4.11. The power series A = [a, a(τ )] is push-neutral in the sense that, if A r denotes the depth r part of A for r > 1, then

where push denotes the push-operator on power series defined in (3.8).

Proof. By Theorem 4.8, the mould dar -1 a m (τ ) is push-neutral. Consider the operator

Since the factor u 1 . . . u r (-u 1 -. . . -u r ) is push-invariant, the mould -∆(A) is push-neutral if A is. Therefore in particular -∆ dar -1 a m (τ ) is push-neutral. But this mould is given by

where the last equality is a standard identity (see Appendix A of [START_REF] Racinet | Séries génératrices non commutatives de polyzêtas et associateurs de Drinfeld[END_REF] or (3.3.1) of [START_REF] Schneps | An introduction to Ecalle's theory of moulds[END_REF]). Therefore the mould ma([a, a(τ )]) is a push-neutral mould, i.e. [a, a(τ )] is push-neutral as a power series. 4.4. Proof of Lemma 4.9. In order to prove this lemma, we need to have recourse to the complete formula for the action of arat. We first recall Écalle's formula for arit (cf. [START_REF] Ecalle | The flexion structure and dimorphy: flexion units, singulators, generators, and the enumeration of multizeta irreducibles[END_REF] or [START_REF] Schneps | An introduction to Ecalle's theory of moulds[END_REF]), which is given as Now let A be push-neutral, and let P ∈ ARI. We need to show that (4.24) is push-neutral. In fact we will show that the two terms of (4.24) are separately push-neutral.

Because the push-neutrality relations take place in fixed depth, we may assume that A is concentrated in depth s and P in depth t, with s + t = r. We will prove the push-neurality of the first term in (4.25); the proof for the second term is completely analogous.

Therefore the decompositions w = abc we need to consider are those of the form w = abc = (u 1 , . . . , u i )(u i+1 , . . . , u i+t )(u i+t+1 , . . . , u r ), and we can rewrite the first term of (4.25) as r-t i=0 A(u 1 , . . . , u i , u i+1 + • • • + u i+t+1 , u i+t+2 , . . . , u r )P (u i+1 , . . . , u i+t ).

The k-th power of the push-operator acts by u i → u i-k , with indices considered modulo (r + 1). The push-neutrality condition thus reads