A class of non-holomorphic modular forms II : equivariant iterated Eisenstein integrals

We define and study a family of real analytic modular forms on the upper half plane constructed out of real and imaginary parts of iterated integrals of Eisenstein series. They form an algebra of functions, which contains the real analytic Eisenstein series, satisfying many algebraic and differential properties. For example, they admit expansions in $q, \overline{q}$ and $\log |q|$ involving only rational numbers and certain combinations of multiple zeta values. This class is arguably the simplest family of `mixed' versions of modular forms for $\mathrm{SL}_2(\mathbb{Z})$ of level one.


Introduction
We begin by describing a more general framework [1] of real analytic modular forms which contains our class of functions. Let H = {z : Im z > 0} denote the upper half plane, equipped with the standard action of SL 2 (Z): We shall say that a real analytic function is modular of weights (r, s) if for all γ ∈ SL 2 (Z) it satisfies (1.2) f (γz) = (cz + d) r (cz + d) s f (z) .
If the total weight w = r + s is odd then f necessarily vanishes. Let M r,s denote the complex vector space of functions f : H → C which are real analytic and modular of weights (r, s), admitting an expansion at the cusp of the form where q = exp(2πiz) and L := log |q| = −2πIm(z) . Such a function can be written explicitly, for some N ∈ N, in the form (1.4) f (q) = which generate a Lie algebra isomorphic to sl 2 . Out of these one defines a Laplace operator ∆ : M → M which respects the bigradings.
The basic example of a function in M is L ∈ M −1,−1 . Any holomorphic modular form f (z) of weight n and level one defines a pair of elements f (z) ∈ M n,0 and f (z) ∈ M 0,n . The space M also contains the ring of almost holomorphic modular forms and their complex conjugates. These examples all have essentially rational coefficients a (k) m,n . In order to generate non-trivial examples, we must allow the coefficients to be periods [24], which are conjecturally transcendental.
In [1] we suggested constructing new elements in M, starting from holomorphic modular forms, by repeatedly solving the equation for F ∈ M. This equation does not always admit a solution in M (for example, if f is a cusp form 1 ), but does if f is a holomorphic Eisenstein series, in which case F is a real-analytic Eisenstein series. More generally, we show in this paper how to repeat this process and construct a large infinite family of functions in M out of iterated integrals of Eisenstein series. Now although the latter are not themselves modular, one can, under certain conditions, take linear combinations of their real and imaginary parts to force the modularity condition (1.2) to hold. This leads to a space MI E ⊂ M of modular forms whose modular weights (r, s) lie in the first quadrant r, s ≥ 0. The space MI E is equipped with an increasing filtration by subspaces MI E k ⊂ MI E called the length, which has the property that  (4) The elements in MI E satisfy precise differential equations with respect to the operators ∂ and ∂ (theorem 11.3) and an inhomogenous Laplace eigenvalue equation with eigenvalue minus the total modular weight w (corollary 11.8). (5) The subspace lw(MI E ) ⊂ MI E of elements of modular weights (n, 0) forms a subalgebra which is essentially dual to u geom , a Lie algebra of geometric derivations corresponding to the image of the Johnson homomorphism in genus one (theorem 12.1). We proved in [6] that u geom is contained in the space of solutions to linearized double shuffle equations, which describe relations between depth-graded multiple zeta values (theorem 12.3) (6) The images of elements in MI E under the differential operator ∂ are orthogonal to holomorphic cusp forms with respect to the Petersson inner product (this holds in fact for all elements in M by theorem 2.2). (7) Elements of MI E can be assigned 'symbols' (proposition 13.3), which may be useful in applications to physics.
For reasons of space, we did not discuss a number of more sophisticated properties of MI E , including the relationship with the motivic Galois group of the category of mixed Tate motives over Z, which acts on u geom , and the related question of studying motivic versions of functions in MI E .
1.1. Definition of MI E . The space MI E is generated by equivariant solutions of a certain elliptic KZB connection, and provide a modular realisation of Hain and Matsumoto's category of universal mixed elliptic motives [21]. In order to explain this, we briefly recall the theory in genus zero, before turning to genus one.
1.2. Genus zero. Consider the pro-unipotent de Rham fundamental groupoid of P 1 \{0, 1, ∞}. Its graded Lie algebra is isomorphic to Lie (x 0 , x 1 ) , the free graded Lie algebra generated over Q by x 0 , x 1 , which are dual to loops around 0 and 1. The KZ-connection is defined using the formal one-form where x 0 , x 1 act by derivations on Lie (x 0 , x 1 ), by concatentation on the left on its completed universal envelopping algebra. It was shown in [4] that there exists a canonical real-analytic single-valued flat section of this connection. It is a well-defined function taking values in formal non-commutative power series in x 0 , x 1 , which satisfies ∂L ∂z = ω KZ L and L( i.e., its value at the origin, regularised with respect to the unit tangent vector → 1 0 at 0, is equal to 1. Concretely, it can be viewed as a formal power series where the sum ranges over all words w in x 0 , x 1 , and its coefficients (1.8) L w (z) : C × −→ C are single-valued analogues of multiple polylogarithms, and satisfy a system of differential equations with respect to both ∂ ∂z and ∂ ∂z . The regularised values of these functions at the point 1 are and are called single-valued multiple zeta values. They generate a Q-algebra Z sv which is strictly contained in the algebra of all multiple zeta values. Perhaps surprisingly, the numbers ζ sv (w) satisfy all motivic relations between ordinary multiple zeta values.
1.3. Genus one. Now consider the unipotent de Rham fundamental group of a punctured elliptic curve E × with respect to a suitable tangent vector at the identity. Its associated graded Lie algebra is the free Lie algebra Lie (H) where H = H 1 dR (E × (C); C) is a two-dimensional complex vector space. When E = E ∂/∂q is the first-order smoothing of the Tate curve, which is defined over Q, the group H has a natural Q-structure. In fact, Hain has shown [20] that when the tangential base point is defined over the integers, the algebra Lie(H) is the de Rham realisation of a mixed Tate motive over Z, its lower central series is canonically split, and one can identify with the free bigraded Lie algebra on two de Rham generators a, b. This Lie algebra is equipped with certain derivations ε 2n ∈ Der Θ Lie(a, b) for all n ≥ 0, where Der Θ Lie(a, b) denotes the Lie algebra of derivations δ : Lie(a, b) → Lie(a, b) which annihilate the element Θ = [a, b]. They were first written down by Tsunogai [34], and studied by Nakamura [31] and Pollack [30]. The derivations ε 2n are uniquely determined by the properties ε 2n Θ = 0, ε 2n (a) ∈ [Lie(a, b), Lie(a, b)] and the formula Define the Lie algebra of geometric derivations to be the graded Lie algebra generated by the ad(ε 0 ) i (ε 2n ), for all i ≥ 0 and all n ≥ 2. The Lie algebra u geom admits an action by a copy of the Lie algebra sl 2 , which is generated by the derivation ε 0 and its opposite ε ∨ 0 = b ∂ ∂a . The Lie algebra u geom is the (bigraded) image of the geometric monodromy, and is a genus one analogue of the Johnson homomorphism. The elliptic KZB connection [8,23,18,7] that we use involves the formal one-form: taking values in sl 2 ⋉ u geom . Let U geom denote the affine group scheme corresponding to u geom . It admits a right action by SL 2 . Our first theorem proves the existence of a genus one analogue of L: There exists a real-analytic function which satisfies the equation ∂ ∂τ J eqv = ωJ eqv and is equivariant for the action of SL 2 (Z), i.e., The generating series J eqv , unlike its genus zero counterpart L, is only well-defined up to right-multiplication by an element a ∈ U geom (C) SL 2 . One can view J eqv as a formal power series whose 'coefficients', which are the analogues of (1.8), are sections where V 2n = Sym n H is the nth symmetric power of the standard representation H of SL 2 . It can be realised explicitly as a space of homogeneous polynomials equipped with a right action of SL 2 , where X, Y correspond to b and a respectively. The coefficients c(J eqv ) are modular equivariant: where c r,s : H → C is real analytic. It follows from the equivariance of J eqv that c r,s transforms like a modular form of weights (r, s), according to definition (1.1). By computing its q-expansion we show that it defines an element of M r,s . Definition 1.2. The space MI E is the free Z sv -module generated by all coefficients c r,s of J eqv , where Z sv is the Q-algebra of single-valued multiple zeta values.
The space MI E is well-defined, and is generated by certain linear combinations of real and imaginary parts of iterated integrals of Eisenstein series. The majority of this paper is devoted to defining and proving properties of this class of functions. Because of potential applications to physics, we decided to emphasize the class of modular components c r,s , rather than the coefficient functions c(J eqv ), which are vector-valued modular forms, but the theory could equivalently be built around the latter objects.
1.4. Motivation. One motivation for this construction was the long standing problem of defining a natural class of functions which contain the modular graph functions arising in genus one closed string perturbation theory [10,14,15,16,36]. This problem was briefly discussed in [1], and has an extensive literature. A relation between modular graph functions and the class MI E [L ± ] should follow by a version of the argument given in the author's thesis. The key point is that the bar de Rham complex defined in [7] has trivial cohomology. The properties of MI E described above should then explain several phenomena which have been observed for modular graph functions.
Another motivation for this paper was to provide an elementary means to access the theory of the relative completion of the fundamental group of the moduli stack of elliptic curves, which is a natural generalisation of the motivic fundamental group of the projective line minus three points. It admits a natural mixed Tate quotient via its action on the unipotent fundamental group of the infinitesimal Tate curve. In this paper we construct a 'modular realisation' of this quotient. In this respect, the algebra of functions MI E correspond to mixed Tate motives over the integers, and suggests a positive answer to the apparently naive question of whether there exist modular forms associated to mixed motives (at least for mixed Tate motives over Z).

1.5.
Acknowledgements. This work was partly supported by ERC grant 724638 and written during a stay at the IHES. Many thanks to Richard Hain for ongoing discussions about relative completion and mixed elliptic motives and to Nils Matthes, for corrections. Many thanks also to Andrey Levin, who suggested several years ago that double elliptic polylogarithms could be orthogonal to cusp forms.

Notations and background on the class of functions M
We recall some notations and background from [1].
2.1. General definitions. We write z = x + iy, q = exp(2πiz) and set Denote the bigraded algebra of functions defined in the introduction by It is stable under complex conjugation. Define where w is called the total modular weight. Since M r,s = 0 if r + s is odd, we shall always take w, h to be even. Every f ∈ M admits a unique expansion of the form (1.3). The constant part of f (also called the Laurent polynomial or zeroth Fourier mode in the physics literature) is defined to be The subspace of cusp forms S ⊂ M is defined to be ker(f → f 0 : M → C[L ± ]). There is a decreasing filtration by the order of poles in L: In this paper we focus on the space of functions whose modular weights lie in the first quadrant, with poles in L of order at most the total modular weight.

Differential operators.
For all r, s ∈ Z, there exist operators which are a variant of the Maass raising and lowering operators, defined by If we extend the action of ∂ r , ∂ s to all of M to be the zero map on all components M i,j for i = r or j = s respectively, we obtain bigraded differential operators of modular weights (1, −1) and (−1, 1) respectively. They satisfy ∂(L) = ∂(L) = 0, and generate a copy of sl 2 acting upon M: for all r, s by the equivalent formulae are the interior of the standard fundamental domain for the action of SL 2 (Z) on H, and the SL 2 (Z)-invariant volume form on H in its standard normalisation. Restricting to holomorphic cusp forms S n ⊂ M n , we deduce pairings Since the Petersson inner product is non-degenerate on holomorphic cusp forms, these two pairings are equivalent to a linear map whose components are called the holomorphic and anti-holomorphic projections [33].
Theorem 2.2. Let f ∈ M r,s . If f = ∂F for some F ∈ M then (2.9) f, g = 0 for all g ∈ S r−s holomorphic .
In particular, f is in the kernel of the holomorphic projection (2.8).
This can be written p h ∂ = 0. By taking the complex conjugate, p a ∂ = 0.

Example: real analytic Eisenstein series.
The simplest examples of modular forms that we shall consider are given by real analytic Eisenstein series. They are defined for any integers r, s ≥ 0 such that w = r + s > 0 is even, by They satisfy E r,s ∈ P −w M r,s together with the system of differential equations They are eigenfunctions of the Laplacian: ∆E r,s = −w E r,s , with constant part which involves an odd value of the Riemann zeta function ζ(w + 1) (which lies in the space of single-valued multiple zeta values Z sv ). This example is fairly atypical -in general the differential equations with respect to the holomorphic and anti-holomorphic differentials ∂ and ∂ will not be symmetric. The theory is set up in such a manner as to make the different structure with respect to ∂ as simple as possible, at the cost of losing explicit control over the action of ∂. This is entirely analogous to the situation in genus 0, since L satisfies a complicated differential equation with respect to ∂ ∂z whose coefficients involve multiple zeta values.
Let us write It is equivariant: E(γτ ) = E| γ for all γ ∈ SL 2 (Z). Consider the equivariant 1-form Note that the normalisation of the power of 2πi in this expression can be different in different contexts [2]. The systems of equations (2.11) and (2.12) are equivalent to the differential equation: The functions E r,s are thus obtained from the real part of an indefinite integral of a holomorphic Eisenstein series. This is the prototype for the general theory.

Reminders on SL 2 -representations
Throughout this paper, all tensors are over Q unless otherwise indicated.

3.1.
Definitions. For all n ≥ 0 define equipped with the right action of SL 2 , and hence SL 2 (Z), given by for γ of the form (1.1). There is an isomorphism of SL 2 -representations Consider the SL 2 -equivariant projector is the multiplication map.
Recall the standard notation for generators of SL 2 (Z):

de Rham version.
It is convenient to define a de Rham version V dR 2n of the (Betti) vector space V 2n generated by elements denoted by X and Y.
There is a comparison isomorphism The reason for this is that X and Y will span copies of Q(0) and Q(1) respectively: for example, X is a Betti basis for Q(0) and X a de Rham basis.
The vector space V dR 2n admits a (de Rham) action of SL 2 (Z) on the right via (X, Y) γ = (aX + bY, cX + dY) for γ of the form (1.1). When the context is not clear, we shall denote this copy of SL 2 by SL dR 2 . The comparison isomorphism is an isomorphism of group schemes comp B,dR : which on the level of points is given by conjugation by ( Consequently, the Betti action of SL 2 (Z) on V dR 2n ⊗ C is twisted by powers of 2πi. The images of the Betti elements S and T under comp B,dR are If bar denotes complex conjugation, these satisfy S ′ = −S ′ and T ′ = (T ′ ) −1 .
Remark 3.1. There is a de Rham version of the projection δ dR : V dR 2m ⊗ V dR 2n → V 2n+2m−k defined in an identical manner to (3.1), except that we replace X, Y with X, Y. Under the comparison isomorphism, it differs from δ by a factor of 2πi.

Iterated integrals of Eisenstein series
4.1. Preamble on filtrations. In order to keep the paper as accessible as possible, we work mostly with Lie algebras and iterated integrals and gloss over the geometric foundations. From time to time, a short paragraph marked with a star explains the geometric background with references for the interested reader. Only in section 7, which is considerably more technical, do we require any substantial Hodge theory and Tannakian theory of fundamental groups and their completions. As a result, the objects described in this and later sections are equipped with a limiting mixed Hodge structure and in particular, possess three filtrations: W, M and F . Since all mixed Hodge structures considered here are of mixed Tate type, the monodromyweight filtration M is split in the de Rham realisation by the Hodge filtration F , and is therefore associated to a grading which we call the M -degree. It also determines F , which will not be discussed again. The geometric weight filtration W plays a relatively minor role in this paper. Indeed, it is canonically split in the de Rham realisation, and the W -degrees can be deduced from the M -degrees and sl 2 -weights. It is related to the depth filtration [6], which is briefly discussed in §5.4. For this reason, we mainly emphasize the M -degree, and mention the W -weights only in passing.

4.2.
A tensor algebra. The generators X, Y of the vector space V dR 2n lie in W -degree zero, and satisfy Let e 2n+2 be a symbol which corresponds to the Eisenstein series of weight 2n + 2. It is assigned a W -degree of −2n − 2 and an M -degree of −2 (it spans a copy of Q(1)).
and for all n ≥ 1. Equivalently, it is the free graded Lie algebra generated by e 2n+2 V dR 2n , for every n ≥ 1. It is naturally equipped with a right action of SL dR 2 (Z).
The completed Lie algebra u E is the Lie algebra of a pro-unipotent affine group scheme U E over Q. Its affine ring is the tensor coalgebra where E 2n+2 are symbols dual to e 2n+2 , and where for any (graded) vector space W , the tensor coalgebra is defined to be the (bigraded) vector space equipped with the shuffle product x , and the deconcatenation coproduct The ring O(U E ) and the group scheme U E inherit a left (respectively right) action of SL dR 2 . Let us denote also by U E the completed universal envelopping algebra of u E . It is the ring of non-commutative formal power series in e 2n+2 X i Y 2n−i , for 0 ≤ i ≤ 2n.
Proof. The definition of the tangential base point → 1 ∞ is given in [2] §2, as well as a concrete explanation of the regularised iterated integrals. Parts (i), (ii) and (iii) are proved in [2], propositions 1 and 2. Only the last part remains to be proven. For  Furthermore, such a primitive of q i log j (q) dq is again a polynomial in q and log q of degree at most j + 1 in log(q). Since the coefficients of E 2n+2 (τ ) lie in C[[q]][log(q)]dq and have degree at most 2n in log q, the statement follows by induction on r.
Properties (i) and (ii) in the previous proposition determine I E (τ ) uniquely. Property (iii) is equivalent to a shuffle product relation between iterated integrals of Eisenstein series, which is spelt out in [2] (3.8).
It satisfies the cocycle equation Proof. Identical to [2] lemma 5.1, bearing in mind that the right action of SL 2 (Z) on U E (C) is given by (3.4).
Since SL 2 (Z) is finitely presented, and generated by the two elements T, S, any cocycle C is uniquely determined ([2] lemma 5.5) by its values C S and C T , which satisfy where U = T S and C U = C T S C S . The series C E T was computed explicitly in [2] §6, and its coefficients involve only powers of 2πi. The coefficients of C E S are numbers which we called multiple modular values in [2]. 4.3.1. Geometric background*. The affine group scheme U E is not a good object in the sense that it does not admit a natural mixed Hodge structure. A closely related object is the de Rham relative completion of the fundamental group of M 1,1 with respect to the unit tangent vector at the cusp [17,2]. The group scheme U E is a quotient of it (but not in the category of mixed Hodge structures). The reason for working with U E is for simplicity of exposition, since under the monodromy homomorphism, discussed below, all generators of relative completion which correspond to cusp forms act trivially, leaving only the Eisenstein series. These are artificially captured by U E .

4.4.
Totally holomorphic iterated integrals. In this section we work over the ring of complex numbers, and all tensors are over C. Consider the tensor coalgebra where M 2n+2 (C) denotes the complex vector space of holomorphic modular forms of weight 2n+2. Elements in this coalgebra define 'totally holomorphic' iterated integrals on H. They are linear combinations of the following iterated integrals considered by Manin [27,28]: where for any modular form f ∈ M 2n+2 (C) of weight 2n + 2, we write A different normalisation of the power of 2πi was used in the first chapter of [2]. From the definition of iterated integrals one has: The corresponding integrals regularised with respect to a tangent vector at the cusp were studied in [2] §4.
is injective. It follows that iterated integrals are linearly independent over the ring M [τ ] generated by holomorphic modular forms and the function τ .
We give two different proofs of linear independence. The first is conceptual and uses properties of relative completion. It is a generalisation to vector-valued iterated integrals of a well-known theorem on the linear independence of iterated integrals due to Chen [9]. The second proof is elementary but more computational. 2 We first deduce the second part of the proposition from the linear independence.
Proof. Since holomorphic modular forms are an algebra, we can multiply any linear relation amongst iterated integrals with coefficients in M [τ ] by a non-zero modular form of sufficiently large weight to obtain a relation of the form where λ I ∈ C and every ω i is of the form τ k f (τ ) where k + 2 is bounded above by the weight of f . This in turn implies, by integrating, a linear relation of the form where λ ∈ C. Since the right-hand side is a multiple of the empty iterated integral 1, this is relation between iterated integrals. Using the fact that they are linearly independent, we deduce that all coefficients λ vanish.
The following corollary follows immediately from the proposition by applying it term by term in q, τ .
is injective.

General proof of linear independence using relative completion. Suppose that
for all τ ∈ H. In particular, for any γ ∈ SL 2 (Z), the integral from τ = γ −1 τ 0 to τ 0 also vanishes. It follows that ω vanishes on where π rel 1 denotes the de Rham version of relative completion [17,19,2]. Since the topological fundamental group π top 1 (M 1,1 (C), x 0 ) is Zariski-dense in its relative completion, we have ω = 0. Therefore iterated integrals are linearly independent. Note that the proof works more generally for the whole of the affine ring of de Rham relative completion, and not just for totally holomorphic iterated integrals.

An elementary proof of linear independence.
For the benefit of the reader not familiar with relative completion, we spell out the above proof of linear independence of iterated integrals in elementary terms. It only uses the Eichler-Shimura theorem.
Fix τ 0 ∈ H. Given a modular form f of weight w and 0 ≤ i ≤ w − 2, the differential one form τ i f (τ )dτ on H defines a function where γ denotes the geodesic path from γ −1 τ 0 to τ 0 . Lemma 4.6. Let f 1 , . . . , f n be linearly independent holomorphic modular forms of weights w 1 , . . . , w n . Then the functions τ i f j (τ )dτ : SL 2 (Z) → C, where 1 ≤ j ≤ n and 0 ≤ i ≤ w j − 2, are linearly independent over C.
Proof. An element g ∈ SL 2 (Z) defines an automorphism of H. By functoriality of integration with respect to automorphisms (change of variables formula), a linear combination of one-forms of the above type ω vanishes on all γ ∈ SL 2 (Z) if and only if the same is true of g * ω. Now observe by modularity of f that Equivalently, the complex vector space spanned by the τ i f (τ )dτ for 0 ≤ i ≤ w − 2 is an irreducible SL 2 (Z)-representation, and any non-zero vector in it generates the entire space under the action of S and T . The following arguments are easily translated into representation-theoretic language, but we give a long-winded account for explicitness. Suppose that there exists an ω = λ ij τ i f j (τ )dτ which vanishes on SL 2 (Z) for some λ ij ∈ C not all zero. By applying (T * − id) N for a sufficiently large N , we can assume that all powers of τ are zero, i.e., λ i,j = 0 for i > 0. By applying S, and once again (T * − id) N ′ for some N ′ , we can assume in addition that all f j have the same weight w. Thus we can write ω = λ j f j (τ )dτ where λ j ∈ C are not all zero. Via the action of g * for g ∈ SL 2 (Z), we deduce that τ i ω vanishes as a function on SL 2 (Z) for all 0 ≤ i ≤ w − 2. The same is therefore true of the formal linear combination associated to f . But the Eichler-Shimura theorem [11,32] implies that the map is injective, so a fortiori the cocycles C fj are independent. Hence λ j = 0 for all j, a contradiction.
The following lemma is an exercise, but included for completeness.
where the functions φ i are extended linearly to linear combinations in S.
Proof. Consider the matrix-valued function P = (φ j (s i )) i,j on S m . Its determinant is not identically zero on S m . For otherwise, a row expansion would yield a relation m i=1 λ i φ i (s m ) = 0, for all s m , where the λ i are minors of P . By linear independence, the λ i vanish. The same argument can then be applied to each minor of P , and we eventually deduce by induction on the size of P that every entry φ j (s i ) vanishes for all s i , a contradiction. Let s 1 , . . . , s m ∈ S such that det(P ) = 0. Let e k be the column vector with a 1 in the kth row and zeros elsewhere. Let λ k , for 1 ≤ k ≤ m, denote the vector in C m such that λ T k P = e k . Set x k = (s 1 , . . . , s m ).λ k ∈ s∈S Cs. By construction it satisfies φ i (x j ) = δ ij .
Suppose that we have a non-trivial relation of minimal length n for all τ ∈ H where the ω ij ∈ {ω 1 , . . . , ω N }, a set of linearly independent 1-forms of the kind . Composition of paths for iterated integrals [9] states that where α, β are two composable paths, and αβ denotes the path α followed by β. By linearity, it holds more generally for α any linear combination of paths which can be composed with β. By the previous two lemmas, there exist elements Apply the composition of paths formula with β a path from τ to τ 0 , and α = x j . Since αβ is a linear combination of paths from points in the SL 2 (Z)-orbit of τ to τ 0 and since the above linear combination of iterated integrals vanishes along each such path, we deduce N relations of the form which have length ≤ n − 1. By minimality, the coefficients λ I such that i 1 = j vanish. Since this holds for all j = 1, . . . , N , we conclude that the λ I all vanish, a contradiction.

5.
A Lie algebra of geometric derivations 5.1. Geometric context*. The de Rham fundamental group π dR 1 (E × ∂/∂q , → 1 0 ) of the punctured first order smoothing of the Tate curve [21,17], with the basepoint given by (a choice of) unit tangent vector at the origin, is the de Rham realisation of a mixed Tate motive over Z [20] equipped with an extra filtration W . These filtrations admit a canonical splitting, hence an isomorphism where Π is the pro-unipotent affine group scheme whose Lie algebra is the completion of the bigraded Lie algebra Lie(a, b) discussed below.

5.2.
Derivations. Consider the free bigraded Lie algebra Lie(a, b) on two generators a, b. It has two gradings M, Proposition-Definition 5.1. There exists a distinguished family of derivations [34,31,30,21] for every n ≥ 0: which are uniquely determined by the property ε ∨ 2n+2 Θ = 0, the formula ε ∨ 2n+2 (a) = ad(a) 2n+2 (b) , and the fact that Their action on b is given by where i, j ≥ 0. These derivations were first written down by Tsunogai [34].
The Lie algebra Lie(a, b) admits a right action by SL dR where γ is given by (1.1). The infinitesimal version of this action gives rise to an action of sl 2 via the following derivations: The notation is consistent with (5.2) since ε ∨ 0 is indeed the case n = 0 of (5.2). Since they annihilate Θ, they generate a copy of sl 2 inside Der Θ Lie(a, b), via These facts are proved in [21].
Let us denote by ε 2n+2 the operators obtained from ε ∨ 2n+2 by conjugating by the element S, where (a, b) S = (−b, a). These are lowest weight vectors and satisfy
to be the bigraded Lie subalgebra generated by the ε ∨ 2n+2 for n ≥ 1, together with the action of sl 2 . By the previous lemma, it is the bigraded Lie subalgebra generated by the derivations ad(ε ∨ 0 ) i ε ∨ 2n+2 for all n ≥ 1. With this definition, which differs slightly from the version given in [6], u geom does not contain the element ε ∨ 2 = ε 2 , which plays a limited role, and commutes (as one may verify) with all elements of u geom . Therefore the Lie algebra generated by all derivations ε ∨ 2n+2 for n ≥ −1 is isomorphic to Definition 5.4. Let U geom denote the affine group scheme whose Lie algebra is the completion of u geom . It admits a right action of SL dR 2 (Z). 5.4. Relations. The elements ε 2n+2 satisfy many non-trivial relations, which were first studied by Pollack [30] following a suggestion of Hain (in Pollack's notation, the notations ε and ε ∨ are reversed). One shows that the elements are linearly independent when a, b ≥ 1 and a + b ≤ 4, but starting from M -degree ≤ −24, Pollack showed that there are relations , which are the first two in an infinite sequence of quadratic and higher order relations. There are several different proofs of these relations in the literature. We shall derive a new interpretation of these relations via the orthogonality of equivariant iterated integrals of Eisenstein series to cusp forms for the Petersson inner product (theorem 2.2). 5.5. Embeddings. One way to think of the Lie algebra u geom is as a quotient of a certain free Lie algebra generated by Eisenstein symbols (see the next section). Another is to embed it inside the free Lie algebra Lie (a, b).
Let us write H = Qa ⊕ Qb. The natural map is injective, since a derivation δ is uniquely determined by δ(a), δ(b). Hence is also injective. In fact, more is true. It is straightforward to show that, as a consequence of the relation δ(Θ) = 0, the action of a derivation on either a or b defines a pair of injective linear maps denote the de Rham version of the relative completion of the fundamental group of the moduli scheme of elliptic curves equipped with a non-zero abelian differential, with base point the unit tangent vector at the cusp. The monodromy action defines a canonical homorphism After choosing suitable splittings of the weight filtrations, it gives rise to a morphism from the graded Lie algebra of G dR 1, → 1 to ε 2 Q ⊕ u geom . Using the fact that ε 2 is central, one deduces the existence of a homomorphism G dR 1,1 −→ U geom where G dR 1,1 is the de Rham relative completion of the fundamental group of M 1,1 . One knows [2,21] that the previous map factors through its quotient U E .
Definition 6.1. There is a unique morphism of Lie algebras satisfying for all n ≥ 1, which is equivariant for the respective actions of sl 2 on both sides. It respects the M and W gradings.
The map µ is surjective. The map µ induces a surjective (faithfully flat) homomorphism of affine group schemes µ : U E −→ −→ U geom .
Proof. It follows from the identity and the definition of µ applied to (4.2) that where for simplicity we write ε (m) 2n+2 for (ad(−ε 0 )) m ε 2n+2 . These elements vanish whenever m > 2n as a consequence of lemma 5.2.
The image of C E under the homomorphism µ is a non-abelian cocycle where SL 2 (Z) acts via the Betti action (3.4). The function J satisfies for all τ ∈ H, and G γ satisfies the cocycle equation for all g, h ∈ SL 2 (Z).

Action of complex conjugation
Complex conjugation acts via q → q on the unit disc which corresponds on the upper half plane to the involution τ → −τ .
It acts on U geom (C) by complex conjugation on the coefficients. It is important to note that it does not respect the cocycle relation (6.7), since and h is not in general equal to h since the (Betti) action of SL 2 (Z) on U geom (C) is via (3.4), which involves powers of 2πi (see (3.5)). To remedy this, we compose the action of complex conjugation with the element −1 ∈ G m (Q), where G m is the multiplicative group corresponding to the M -grading.
More precisely, consider the involution It follows that the involution s now preserves the cocycle relation: for all g, h ∈ SL 2 (Z), and hence sG ∈ Z 1 (SL 2 (Z), U geom (C)). It was shown in [2] that the space of cocycles is a torsor over a certain automorphism group, and the technical heart of this paper is to relate G and sG via this automorphism group. To state this, recall that Z sv ⊂ R denotes the ring of single-valued multiple zeta values.
The following theorem involves a considerably amount of the machinery constructed in [2], and may be taken as a black box.
The group Aut(U geom ) SL 2 denotes the SL 2 -invariant automorphisms of U geom , i.e., automorphisms φ : U geom ∼ → U geom satisfying φ(xg) = φ(x)g for all g ∈ SL dR 2 , where SL dR 2 denotes the de Rham action of SL 2 on U geom . Since SL 2 (Z) is Zariski-dense in SL 2 , an element φ ∈ Aut(U geom ) SL 2 commutes with the (Betti) image of SL 2 (Z) in SL dR 2 (C) under the comparison map (3.4). 7.1. Further properties. Before turning to the proof, we state a number of further properties satisfied by the elements b, φ of the theorem.
(ii). There exists a derivation which preserves u geom (Z sv ) and whose restriction to u geom (Z sv ) satisfies This is a highly non-trivial constraint. For example, it expresses the non-obvious fact that the derivation on the right hand side is uniquely determined by its action on a single element a (or b). Conversely, the fact that ψ ψ ψ in turn normalises the image of all geometric derivations u geom is equivalent to an infinite sequence of combinatorial constraints of the form [ψ ψ ψ, ε 2n+2 ] ∈ u geom (Z sv ) for every n ≥ 1. It is far from obvious that there exist any solutions to these equations and follows from the Tannakian theory implicit in the proof of theorem 7.2 (we know that 'motivic' derivations must satisfy a similar property).
(iii). The equation (7.2), in the case γ = T , is equivalent to the 'inertial relation' where the element N + ∈ u geom is the element of M -degree −2 given by and is the unipotent part of the logarithm of (T, G T ) ∈ SL 2 ⋉ U geom (C). The inertial relation ties together b 0 and δ: information about b 0 can be deduced from information about δ and vice-versa. The equation (7.2) is uniquely determined by its two instances γ = T and γ = S. In other words, (b, φ) are uniquely determined (up to twisting by a ∈ (U geom ) SL 2 via (b, φ) → (ab, aφa −1 )), by the inertial condition and the equation (iv). It follows as a consequence of §18 of [2] that to lowest order is canonical, where ζ sv (2n + 1) = 2ζ(2n + 1). The element φ is also known to lowest order by [2], theorem 16.9, via the inertial relation. In particular, The element ψ ψ ψ satisfies ψ ψ ψ(a) ≡ a and ψ ψ ψ(b) ≡ b modulo terms of degree ≥ 3 in b.
(v). The weights of the coefficients of a, b, φ are determined by the M -filtration. In fact, much more is true: replacing b, φ with their 'motivic' versions as in [2], §18.3, the action of the de Rham motivic Galois group of MT (Z) on their coefficients is determined by its action on (b, φ). Restricting to the subgroup G m implies that the weights of the motivic multiple zeta values are induced by the M -grading on O(U geom ).
(vi). Since complex conjugation is an involution, the elements sv and hence (b, φ) satisfy an involution equation which we will not write down.
Remark 7.3. The element ψ ∈ Aut Π(C) is image of the single-valued element sv defined below, and could be computed independently via the periods of Π. In particular, it is compatible [2] with the Hain morphism Φ from the motivic fundamental group of the projective line minus three points P 1 \{0, 1, ∞}. The image of sv in its group of automorphisms was computed in [4] and involves single-valued multiple zeta values.
This provides yet another method to obtain information about b and φ, which we will not exploit here.
In conclusion, the elements b, φ are heavily constrained and it should in principle be possible to compute them explicitly to higher orders. 7.2. Proof of theorem 7.2. The proof follows closely the argument given in [2], §19 with some minor differences. We summarize the main ingredients, and refer to loc. cit. for further details.
(1) The objects G dR 1,1 , π dR 1 (E × ∂/∂q ,  (3) Let G dR H denote the group of tensor automorphisms of the fiber functor on the category H which sends (M B , M dR , c) to M dR . It is an affine group scheme over Q. Consider the composition of isomorphisms Since it is functorial in M , it defines a canonical element Thus for every object X in H, we deduce the existence of s ∈ Aut(X dR )(C) which is compatible with all morphisms in H, and computes the action of the real Frobenius F ∞ in the de Rham realisation.
(4) The element s is canonical, but it is convenient to modify it as follows. Here our presentation differs slightly from that of [2], §19. The action of G dR H on the de Rham component of the Lefschetz object Q(−1) = (Q, Q, 1 → 2πi) in H defines a morphism π : G dR H → Aut(Q) = G m . The image of s under π is −1 ∈ C × = G m (C). Now, the choice of splitting of the weight filtration M is equivalent to an action of the multiplicative group G m on the de Rham component of objects of H, i.e., a splitting of the homomorphism π : G dR H → G m . We now multiply s by the image of −1 ∈ G m (Q) under this splitting to obtain a modified element sv = (−1)s ∈ G dR H (C), which now acts by the identity on Q(−1). The element sv depends on the choice of splitting.
Note that since π 1 (E × ∂/∂q → 1 0 ) is mixed Tate, its M -filtration is canonically split in the de Rham realisation by F , and the action of sv upon its complex points is in fact canonical. The same applies for U geom , and so the statement of the theorem depends in no way on the choices of splittings.
(5) We therefore deduce the existence of an element sv ∈ Aut(G dR 1,1 )(C) × Aut(Π)(C) which is compatible with µ. It is the image of the element sv ∈ G dR H , which acts compatibly on both G dR 1,1 and Π(C). On the other hand, in [2] §10, we gave an explicit description of the group of automorphisms of G dR 1,1 . We showed that, for any choice of splitting G dR 1,1 ∼ = SL 2 ⋉ U dR 1,1 , any automorphism of G dR 1,1 which acts trivially on SL 2 defines a pair b ∈ U dR 1,1 (C) and φ ∈ Aut(U dR 1,1 ) SL2 (C) .
They are well-defined up to twisting by an element a ∈ (U dR 1,1 ) SL 2 (C). The (right) action of (b, φ) on (g, u) ∈ (SL 2 ⋉ U dR There exists, by the argument of [21] Appendix B, a splitting G dR compatible with the choice of M -splittings. We deduce that the image of sv under the monodromy homomorphism µ is represented by a pair b ∈ U geom (C) and φ ∈ Aut(U geom ) SL2 (C) , which are well-defined up to twisting by a ∈ (U geom ) SL 2 (C). It follows from the compatibility of sv with µ that the image of sv ∈ Aut(U geom )(C) is given by the automorphism b −1 φb. It is induced by the automorphism sv ∈ Aut(Π)(C), which we call ψ in the statement of the theorem.
(6) Now let us apply the element sv to the 'canonical cocycle'. The Betti component G B 1,1 of the relative completion G 1,1 admits a natural map This is one of the defining properties of relative completion. We deduce a map where c denotes the comparison isomorphism comp B,dR for short. The image of γ ∈ SL 2 (Z) is (cγ, C γ ), where cγ is the image of γ under (3.4) and C γ ∈ U dR 1,1 (C) is called the canonical cocycle. Its image under µ is precisely G γ ∈ U geom (C). Since γ is Betti-rational, the action of F ∞ , corresponds, via the comparison isomorphism, to complex conjugation on coefficients of c(γ). This action is computed by the element s. The element sv computes complex conjugation composed with the map −1 ∈ G m , which was called s (7.1). Since the affine ring of SL 2 is pure Tate, the latter action is trivial on SL 2 and hence sv acts trivially on γ (this is the reason for preferring sv over s, which does not). It follows that the (right) action of sv satisfies the equation Writing sv in terms of (b, φ) above, we deduce that (7) It remains to compute the coefficients of sv. For this we need the fact that the motivic fundamental group of the punctured Tate elliptic curve (or rather, the image of G 1,1 under the monodromy homomorphism µ) is a mixed Tate motive over Z [20]. It then follows from the results of [4] that, for any mixed Tate motive M over Z, the coefficients of sv in M dR (C) are single-valued multiple zeta values.
The idea of this proof applies in a general setting. It may be possible to circumvent the final step (7), which appeals to deep results about the category of mixed Tate motives over Z by a direct argument following the procedure outlined in remark 7.3.

Equivariant iterated Eisenstein integrals
We can now define modular-equivariant iterated integrals of Eisenstein series.
It is well-defined up to right multiplication by an element a ∈ (U geom ) SL2 (Z sv ).
Theorem 8.2. The series J eqv defines a real analytic function which is equivariant for the action of SL 2 (Z): Proof. Using the monodromy equation (6.6), we compute Proof. The differential equation follows from the definition 8.1 together with the observation that J −1 satisfies the equation dJ −1 = J −1 ω. This follows from differentiating the equation J −1 J = 1, which implies that (dJ −1 )J − J −1 ωJ = 0, and by multiplying on the right by J −1 . The formula for the value of J eqv at → 1 ∞ is a consequence of the definition, the fact that J( → 1 ∞ ) = 1, and the fact that φ, s are group homomorphisms, and therefore preserve the identity in U geom (Z sv ).
In particular, the holomorphic component of the differential equation is canonical, but the anti-holomorphic part depends on the choice of φ. The right-hand side is J eqv sω to leading order, and so dJ eqv ≡ −ωJ eqv + J eqv sω modulo lower order terms. Since the holomorphic components of these two differential equations agree, it follows that F = K −1 A, for some A : H → U geom (C) which satisfies A(γτ ) γ = A(τ ) and ∂A ∂τ = 0 .
Its modular equivariance follows from the equivariance of φ and the fact that s preserves SL 2 (Z). Consider any coefficient a : H → V 2n ⊗C of A. It defines a holomorphic section a ∈ Γ(SL 2 (Z)\ \H; V 2n ), where V 2n denotes the vector bundle associated to V 2n , which is analytic at the cusp. Such sections correspond to modular forms of weight 2n. On the other hand, from A = KF we see that the coefficients of A are linear combinations of iterated integrals of modular forms and their complex conjugates. By corollary 4.5, A is constant. We deduce that Multiplying by s(b) on the left and changing the constant A, this equivalent to The previous result can also be deduced from the fact that complex conjugation, and hence the elements s and sv in the proof of theorem 7.2, are involutions. This implies a cumbersome identity involving b, φ and s which we omit.

Definition of a class of real-analytic modular forms
Having defined a modular-equivariant function J eqv , we can extract real-analytic modular forms from its coefficients, which in turn generate the algebra MI E . 9.1. Coefficients of J eqv . For every τ ∈ H, we view J eqv (τ ) ∈ U geom (C) as a homomorphism O(U geom ) → C. For every SL 2 -equivariant linear map Since J eqv is equivariant, it follows that for all γ ∈ SL 2 (Z):
can be uniquely written in the form The proposition follows from applying the invertible change of variables In the de Rham basis of V dR 2n , this corresponds to where the second map is the multiplication on O(U geom ). The inclusion of an isotypical factor in an SL 2 -equivariant decompositioň of the left hand side implies that the product of coefficients c, c ′ can be decomposed as a linear combination of coefficients of the form Since, furthermore, the transformation (9.4) is linear with coefficients in Q[L ± , log q, log q], it follows that every product c r,s .c ′ r ′ ,s ′ of modular components ( §9.3) can in turn be expressed as a linear combination, with coefficients in Q[L ± , log q, log q], of c ′′ r+r ′ ,s+s ′ . In fact, the coefficients must lie in Q[L ± ], because any modular form satisfying (1.1) is translation-invariant, and so the coefficients in this formula must lie in the T -invariant subspace of Q[L ± , log q, log q], which is exactly Q[L ± ] (see [1], lemma 2.2).
In particular, for any r, s, r ′ , s ′ ≥ 0, Elements in MI E k are generated by the real and imaginary parts of iterated integrals of Eisenstein series of length less than or equal to k. The length filtration is compatible with the multiplicative structure defined above.
For example, in length zero L 0 O(U geom ) ∼ = Q. Therefore In length one, we shall show in §14 that where the sum is over indices r, s satisfying r + s = 2n ≥ 2. 3 Note that the M -degree of L is compatible with the Hodge-theoretic weight of the function log |q|, which is the single-valued period of a family of Kummer extensions, and has weight 2. The motivic Lefschetz period which is associated to 2πi also has weight 2, so these definitions are forced upon us from the fact that τ and τ have M -degree zero.
Remark 10.5. Conjecturally, the weight filtration is a grading on Z sv . One way to exploit this is to work with motivic periods instead of real numbers, since motivic single-valued multiple zeta values are indeed weight-graded. This amounts to replacing b, φ with their 'motivic' versions, defined in [2]. Then the weight grading on the motivic periods is precisely dual to the M -grading on O(U geom ). It remains to verify that this grading is not disturbed when extracting coefficients and modular components §9.3. To see this, note that the comparison isomorphism (3.3) must be replaced with the universal comparison isomorphism in which one replaces 2πi by its motivic version (2πi) m . It is Tate, and weight-graded of degree 2. The motivic version of the change of variables (9.4), in which one replaces 2πi by (2πi) m , is therefore homogeneous of total degree 0, which implies that deg M c r,s = deg M c. It follows that the M -grading on O(U geom ) induces an M -grading on all coefficient functions. This leads to a class of 'motivic' modular forms which are formal expansions (1.3) whose coefficients are motivic periods (in this case, motivic single-valued multiple zeta values).
Their image under the period homomorphism are genuine modular forms in M. This will be discussed elsewhere. Note that the M -grading can be used to determine the powers of L in the right-hand side of (10.1).
The remark implies, in particular, that the M -filtration is well-defined (independent of the choice of representatives b, φ in theorem 7.2) since it is induced by the M filtration on O(U geom ). Since the generators of u geom have strictly negative M -degrees, and since Proof. It is enough to show that the dimension of each SL 2 -isotypical component of gr M m O(U geom ), or equivalently gr M −m u geom , is finite-dimensional. Since the latter is a quotient of u E , it is enough to prove it for the free Lie algebra on e 2n+2 V dR 2n . Note in passing that since gr M e 2n+2 = −2, the space gr M −m u E only involves Lie words of bounded length in the e 2n+2 V dR 2n . Now the inclusion of an isotypical component has M -degree −2(n 1 + . . . + n r − ℓ). This follows from the fact that the map δ k dR defined in (3.1) has M -degree 2k, so its dual has M -degree −2k. The integer k is given by n 1 + . . . + n r − ℓ. The integer ℓ is constrained by the modular weight: 2ℓ = w. The statement follows from the fact that the number of strictly positive integers n 1 , . . . , n r whose sum is bounded above is finite, and that the subring of Z sv of bounded weight is finite-dimensional. 10.7. Compatibilities. The above structures are all mutually compatible: for example, the length filtration and M -filtration are compatible with the algebra structure.
We leave the precise formulation to the reader.
11. Differential structure of MI E The ordinary differential equations satisfied by J eqv are equivalent to a system of differential equations involving the operators ∂ and ∂ satisfied by its modular components. These in turn give rise to inhomogeneous Laplace eigenvalue equations.
Proposition 11.1. Let F, A, B : H → V 2n ⊗ C be real analytic. Then the equation is equivalent to the following system of equations: for all r + s = 2n, and r, s ≥ 0. In a similar manner, is equivalent to the following system of equations: The proof is a straightforward computation ([1] §7).
11.2. Differential structure of MI E . Recall that the graded Q-vector space generated by Eisenstein series is denoted It is placed in M -degree zero. where we define c r,s to be zero if r or s is negative. Similarly, which follows from the differential equation dI E = −Ω E I E . The coproduct dual to the multiplication law in U E is SL 2 -equivariant, and therefore induces a coproduct on coefficient functions c :V dR 2n → O(U E ) of the form ∆c = k≥0 δ k dR (c ′ ⊗ c ′′ ) , using a version of Sweedler's notation. It follows that We conclude from the formula (4.3) for Ω E that where A : H → V 2n ⊗ C is a Q-linear combination of terms of the form where c ′′ is a coefficient function of strictly smaller length than c. The theorem follows on applying proposition 11.1 and lemma 11.2.
Since MI E is generated by coefficient functions we deduce the Corollary 11.4. The space MI E has the following differential structure: Proof. The first part is immediate from the previous theorem. The statement about the M -filtration follows since the coproduct ∆ respects the M -grading on O(U E ), the Eisenstein series G 2m+2 lie in M -degree zero, and the fact that in lemma 11.2, the M -degree of L exactly matches the M -degree of δ k dR .
By proposition 2.1, an element ξ ∈ M r,s is uniquely determined by ∂ξ and ∂ξ, up to a possible multiple of L −r in the case r = s. When ξ ∈ MI E , this constant is an element of Z sv , whose M -weight can be determined from the M -grading.
Remark 11.5. By the independence of iterated integrals (corollary 4.5), the sums on the right-hand side in the previous corollary are direct, and so we may write and similarly for ∂. This is because E does not contain any constant functions.
Remark 11.6. Given an element f m,n ∈ MI E k of modular weights m, n ≥ 0, we can use the splitting of remark 11.5 to define functions f r,s ∈ MI E k for all m + n = r + s, and r, s ≥ 0 via is then a vector-valued modular form, and we see in this manner that coefficient functions can be reconstructed from their individual modular components. where A r,s , B r,s are understood to be zero if any subscript r or s is negative.
The statement follows from the definition of the Laplacian (2.7). 11.4. Laplace operator structure for MI E . Corollary 11.8. Every element F ∈ MI E k of modular weights (r, s) satisfies an inhomogeneous Laplace equation of the following form: where the eigenvalue is minus the total modular weight.
Proof. This follows from the previous lemma, and equations (11.6) and (11.7) (or by direct application of the definition of the Laplace operator, using these same two equations, and the Leibniz rule).
The sum in the right-hand side is direct, by corollary 4.5. It could be written

Algebraic structure of MI E
In this section we delve more deeply into the algebraic structure of MI E . Since the space MI E is generated from the coefficients of O(U geom ), its structure is closely related to that of the geometric Lie algebra u geom . Although the precise structure of the latter is not completely known, we can use the relationship with MI E to transfer information back and forth between modular forms and derivations in u geom . There is a canonical Z sv -linear isomorphism of algebras and furthermore, every irreducible SL 2 -submodule arises in this way. Taking the coefficient c v (J eqv ) and extracting the term (c v ) 2n,0 in the manner of proposition 9.1 defines a modular form in MI E of modular weights (2n, 0). It is given explicitly by (12.2) This extends to a Z sv -linear map which respects the M and L filtrations. It depends on the choice of (b, φ). By definition of MI E , every modular form of weights (2n, 0) arises in this way, and χ is surjective. To prove injectivity, it is enough to show that the associated graded of χ with respect to length filtration is injective. For this, consider lowest weight vectors v 1 , . . . , v n in L k O(U geom ) which are linearly independent in gr L k O(U geom ). They define linearly disjoint SL 2 -submodules c v1 , . . . , c vn of O(U geom ). The dual of the monodromy homomorphism (6.1) defines an embedding By the linear independence of iterated integrals (corollary 4.5), the corresponding modular forms χ(v 1 ), . . . , χ(v n ) are linearly independent modulo iterated integrals of length ≤ k − 1, since by the remarks following theorem 8.2, their leading terms are real and imaginary parts of iterated integrals of independent Eisenstein series.
We next show that χ is a homomorphism. Let v 1 , v 2 ∈ lw(O(U geom )) of weights m, n. Then c v1v2 is defined via the commuting diagram where the map along the top is dual to δ 0 dR : V dR m ⊗ V dR n → V dR m+n the vertical maps in the bottom square are given by the homomorphism J eqv (τ ) : O(U geom ) → C, and m denotes multiplication. It follows that From the definition of χ, we obtain χ(v 1 v 2 ) = χ(v 1 )χ(v 2 ). Since χ is an isomorphism, this also implies that lw(MI E ) ⊂ MI E is stable under multiplication.
Finally, the isomorphism (12.1) is obtained by replacing χ with its associated graded for the length filtration. It is well-defined since modifying J eqv by J eqv a, for a ∈ (U geom ) SL2 (Z sv ) changes χ by terms of lower length. This is because multiplication is trivial on the associated graded for the lower central series.
Note that the action of sl 2 on O(U geom ) does not correspond to the action of the differential operators ∂, ∂ on MI E . Remark 12.2. In fact, since u geom has no SL 2 -invariant generators it follows that (u geom ) SL2 ⊂ [u geom , u geom ] and hence the map χ defined in the proof is well-defined (independent of the choice of (b, φ)) on the two-step quotients L k /L k+2 lw(O(U geom )).

12.2.
Orthogonality to cusp forms. It follows from theorem 2.2 that the composite where S is the complex vector space of holomorphic cusp forms, is the zero map. This sheds some light on the structure of O(U geom ), and makes it clear that the generators of u geom have infinitely many relations coming from every cusp form.
More precisely, consider the linear map P : The last map is well-defined because L 1 MI E ∼ = Z sv ⊕gr L 1 MI E is split. The injectivity of the first map follows from proposition 2.1. Since gr L 1 MI E is generated by real analytic Eisenstein series E r,s , the holomorphic projections can be computed [2], §9 using the Rankin-Selberg method, and give special values of L-functions of cusp forms. One can show by this method that the last map in the above sequence is surjective after tensoring with C. Via theorem 12.1 one can deduce the dimension of gr L 2 lw(O(U geom )), and indeed a description of the relations in its dual, gr L 2 u geom . This provides a modular interpretation of Pollack's quadratic relations [30], §5.4. Note that multiple zeta values play no role in this calculation.
12.3. Linearized double shuffle equations. A connection between u geom and linearised double shuffle equations was described in [6]. We briefly recall the statement.
The evaluation map (5.6), for x = a, provides an embedding The tensor coalgebra T c (a, b) is graded by the degree in b. Consider the linear map for r ≥ 1. The space pls r was defined in [6] to be a graded vector space of homogeneous rational functions f in the variables x 1 , . . . , x r , for all r ≥ 2 with the property which satisfy the linearised double shuffle equations. For r = 2, these equations are given by f ( The space pls = ⊕ r≥1 pls r is stable under a graded version of the Ihara Lie bracket, and is equipped with a compatible action of sl 2 .
Theorem 12.3. The linear map ρ, when restricted to u geom , defines an injective map of bigraded Lie algebras ρ : u geom −→ pls which commutes with the action of sl 2 on both sides.
By theorem 12.1 we deduce a canonical surjection gr • L lw(O(P ls)) ⊗ Z sv −→ gr • L lw(MI E ) , where P ls denotes the affine group scheme corresponding to the graded Lie algebra pls. It is natural to ask if the previous map is an isomorphism.
The b-degree on u geom coincides with the grading r on pls. In [6], we proved: denotes its Poincaré series, then Note that in [6], we included ε 0 in the definition of u geom , which led to a marginally different formula for u 1 (s). of solutions to linearised double shuffle equations in polynomials (i.e., with no poles) is related to the structure of depth-graded motivic multiple zeta values [22,5]. It follows from theorem 12.1 that the structure of MI E is intimately connected to the structure of depth-graded multiple zeta values. It would be interesting to compare with [13]. 13. Representation of elements in MI E .
There are several ways in which one can represent functions in MI E .
13.1. Iterated integrals of Eisenstein series. By definition 6.1, there is a surjective map u E → u geom . Dually, it gives rise to an injective map of affine rings: It is a morphism of Hopf algebras. Recall that the coproduct on the tensor coalgebra on the right-hand side is simply the deconcatenation coproduct. The map (13.1) gives a convenient way to write coordinates on O(U geom ). Elements of the tensor coalgebra T c ( n≥1 E 2n+2V dR 2n ), and hence indefinite iterated integrals, can be denoted using the bar notation, which in the present setting gives where n i ≥ 4 are even, and 0 ≤ i k ≤ n k − 2. The notation is well-defined by the independence of iterated integrals, proposition 4.2. We can write: Proposition 13.1. Every element in MI E of length ≤ ℓ can be uniquely written as a modular (in the sense of (1.2)) linear combination of symbols Proof. This follows from definition 8.1, proposition 4.2, and the change of variables (9.4). The uniqueness follows from corollary 4.5.
Although this representation is unique, it is cumbersome. It has the disadvantage that the M -filtration is obscured.
Such a morphism can be represented as a linear combination of elements of the form: where n 1 , . . . , n ℓ > 2 are even, and for some n ≥ 0, is the inclusion of aV dR n -isotypical component. The map π can equivalently be represented by the dual projection π ∨ : V dR n1−2 ⊗ . . . ⊗ V dR n ℓ −2 → V dR n . Note that these symbols are not independent for trivial reasons, since for any λ ∈ Q. We shall fix once and for all a basis of maps for all n, n 1 , . . . , n ℓ , which are decomposable in the following sense. All equivariant embeddingsV dR m+n−k →V dR m ⊗V dR n will be normalised to be the duals which we can simply denote by for some integers k 1 , . . . , k ℓ−1 ≥ 0. It follows by Schur's lemma and induction on the length ℓ that every π can be uniquely written as a linear combination of ι k . where the subscripts r, s ≥ 0 indicate the modular weights.
Proof. This follows from the construction of MI E by evaluating coefficients of c : V dR 2n → O(U geom ) on J eqv . These coefficients are uniquely represented as linear combinations of symbols of the form (13.2). Modifying J eqv by right-multiplication by an element a ∈ (U geom ) SL 2 (Z sv ) changes this representation by symbols of length ≤ ℓ − 1, so its projection (13.5) onto length ℓ is well-defined. The uniqueness follows essentially from corollary 4.5: two functions with the same representative (13.5) differ by coefficients of length ≤ ℓ − 1, and hence an element of MI E ℓ−1 .
By enumerating the symbols (13.5) we can deduce a very crude upper bound for the dimension of the space of functions in gr M m MI E of given M -degree and fixed modular weights (r, s), via (13.6) deg M (π; [E n1 | . . . |E n ℓ ]) r,s = n 1 + . . . + n ℓ − r − s , and so it suffices to count the number of such symbols in each degree. See also the proof of proposition 10.7. Now let us fix a choice of elements b, φ and hence J eqv as in theorem 7.2. In this case, since every function in MI E is obtained by extracting a coefficient of J eqv , and every coefficient represented as a linear combination of symbols, we can in fact assign to every f ∈ MI E ℓ a Z sv -linear combination of terms (13.5) of lengths ≤ ℓ: where r + s = n and c r,s is understood to be zero if s is negative.
Proof. This follows from the proof of theorem 11.3. Note that the differential equations with respect to the operator ∂ are considerably more complicated, since they involve b and φ.

Examples
We discuss some simple examples of functions in MI E . First of all, in length zero L 0 O(U geom ) ∼ = Q, and there is only the empty iterated integral, hence: 14.1. Length one. Since the generators ε ∨ 2n of u geom generate a copy of the irreducible representation of sl 2 of dimension 2n − 2, they are linearly independent and Furthermore, L 1 O(U geom ) = Q⊕gr L 1 O(U geom ) is canonically split by the augmentation map O(U geom ) → Q given by the identity element in U geom . Therefore every coefficient in length one is of the form It picks out the coefficients of ad(ε 0 ) i ε 2n+2 for 0 ≤ i ≤ 2n in J eqv . The modular components are precisely the real analytic Eisenstein series E r,s for r + s = 2n, whose symbols (13.2) are E r,s −→ −(id; E r+s+2 ) r,s From proposition 13.4 one retrieves the differential equations (2.11). Now since E 2n+2 , which is dual to ε 2n+2 , has degree +2, it follows that c has M -degree 2 and hence E r,s ∈ M 2 MI E 1 . The real analytic Eisenstein series are discussed in further detail in [1] §4. Their constant parts (2.13) satisfy E 0 r,s ∈ L Q + L −r−s ζ(r + s + 1)Q .
Remark 14.1. It is natural to consider a formal expansion in which one replaces the ζ(r + s + 1) in the coefficient of L −r−s in E r,s with the corresponding motivic multiple zeta value ζ m (r + s + 1) (all other coefficients in the expansion (1.3) of E r,s are rational numbers). The corresponding object is not a modular form, but its image under the period homomorphism is. Since motivic multiple zeta values are graded for the weight, this object of homogeneous M -degree 2.
14.2. Length two. Coefficients of O(U geom ) in length two are linear combinations of the following symbols, which we denote by Recall that J eqv is ambiguous up to right multiplication by some a ∈ (U geom ) SL2 (Z sv ). The only copy of a trivial representation of SL 2 in length 2 occurs when a = b and k = a − 2 , and in this case, the functions we shall consider below will be well-defined up to addition of a constant, in all other cases, they are uniquely defined by their symbol (remark 12.2). With this caveat, we can denote functions in length 2 as a linear combination of modular components of the following symbols, where r + s = a + b − 2k − 4, Since there are no cusp forms of weight < 12, or since there are no relations in u geom in low degrees, the following functions correspond to coefficients in O(U geom ):  where r + s + 2k = n 1 + n 2 for n 1 , n 2 and k as in the examples above. The differential equations for the functions 2 −k F 4,4,(k) r,s were written out in detail in [1], §9.3. The first non-trivial relation in the Lie algebra u geom is (5.5). Dually, it implies that the following linear combination of symbols corresponds to a function in MI

Further questions and topics
For reasons of space, we have not addressed the following topics.
(1) The action of a Tannaka or Galois group of automorphisms [2], which acts on the pair (b, φ). In a similar direction, the action of the motivic Galois group of the category of mixed Tate motives over Z on u geom was studied in [6], and is known to lowest orders. A related question is to study 'motivic' versions of the equivariant iterated integrals studied here. (2) In the physics literature (see references in [1]) it is conjectured that the functions in M arising from modular graph functions are determined by their constant part. For this one may need to assume the transcendence conjecture for multiple zeta values, or work with motivic periods as above. Failing this, it would be interesting to know how many coefficients in the q-expansion are required to determine an element of MI E uniquely. (3) The functions in MI E provide kernels for the Rankin-Selberg method. In lengths zero and one, these are known by the unfolding technique. It would be interesting to extend this to higher lengths.
The functions defined in this paper are constructed out of the action of the real Frobenius F ∞ . It would be interesting to study p-adic versions, which should be constructed in a similar manner using the Frobenius at a finite prime.