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Universal Mixed Elliptic Motives

In this paper we construct a Q-linear tannakian category MEM_1 of universal mixed elliptic motives over the moduli space M_{1,1} of elliptic curves. It contains MTM, the category of mixed Tate motives unramified over the integers. Each object of MEM_1 is an object of MTM endowed with an action of SL_2(Z) that is compatible with its structure. Universal mixed elliptic motives can be thought of as motivic local systems over M_{1,1} whose fiber over the tangential base point d/dq at the cusp is a mixed Tate motive. The basic structure of the tannakian fundamental group of MEM is determined and the lowest order terms of all relations are found (using computations of Francis Brown), including the arithmetic relations, which describe the"infinitesimal Galois action". We use the presentation to give a new and more conceptual proof of the Ihara-Takao congruences.


Introduction
Among the principal goals of the theory of motives is to construct a Q-linear tannakian category MM(T ) of mixed motivic sheaves over a smooth base T whose ext groups are the motivic cohomology groups of T : , where M is a motive over T , such as Q(n). This goal was partially achieved by Voevodsky in his remarkable work [55] in which he constructed a triangulated tensor category of mixed motives. Many obstructions remain to constructing tannakian categories of mixed motives including, most notably, the problem of establishing Beilinson-Soulé vanishing. One case where this goal has been achieved is that of mixed Tate motives over a ring of S integers in a number field. This was established by Levine [37] and Deligne-Goncharov [16] using the work of Voevodsky. The theory of mixed Tate motives can be regarded as the story of motives associated to genus 0 curves and their moduli spaces, [7].
In this paper we take a step towards extending this story to genus 1 curves and their moduli spaces. We construct a Q-linear tannakian category MEM 1 of universal mixed elliptic motives over M 1,1/Z , the moduli stack of elliptic curves, which contains MTM, the category of mixed Tate motives unramified over Z. The ring of functions on its tannakian fundamental group is a Hopf algebra in the category of ind-objects of MTM which encodes the periods of classical modular forms of level 1.
Each object V of MEM 1 consists of an object V of MTM whose Betti realization is endowed with an action of SL 2 (Z). This action is required to determine a compatible set of local systems (Betti, Q-de Rham, Hodge, ℓ-adic) over M 1,1 whose fibers over the canonical tangent vector t := ∂/∂q of M 1,1 at the cusp are the various realizations of V . An object of MEM 1 can be specialized to an elliptic curve E/T by pulling back its associated local systems along the the structure map T → M 1,1 to obtain a family of local systems over T .
This hybrid approach to constructing a tannakian category of mixed motives over M 1,1 is necessitated by the lack of understanding of motives associated to modular forms. In particular, it circumvents the problem of proving Beilinson-Soulé vanishing for the category of mixed motives generated by motives of classical modular forms. We expect the local systems (Betti, de Rham, Hodge, ℓ-adic) associated to an object of MEM 1 to be the set of compatible realizations of a mixed motivic sheaf over M 1,1/Q in the sense of Ayoub [2] or Arapura [1]. If this is the case, then their pullbacks to T should be the set of realizations of a mixed elliptic motives over T in the sense of Goncharov [20]. We have not pursued this as our interests lie in the tannakian aspects of the theory, especially in the determination of the fundamental group of MEM 1 and its relation to classical modular forms and mixed Tate motives.
The most basic object of MEM 1 is the local system H := R 1 π * Q associated to the universal elliptic curve π : E /Z → M 1,1/Z . The simple objects of MEM 1 are the Tate twists S m H(r) of the symmetric powers of H. The elliptic polylogarithms of Beilinson and Levin [6] produce basic objects of MEM 1 . These provide, for each n > 1, a non-trivial extension which corresponds to the Eisenstein series of weight 2n. The tannakian fundamental group of MEM 1 is an extension of GL 2 by a prounipotent group U MEM . Using work of Brown [9] and Pollack [44], we construct a (possibly incomplete) set of minimal relations between them and determine their leading quadratic terms. Each minimal relation corresponds to a Hecke eigenform, and each Hecke eigenform determines a countable set of minimal relations. If one assumes that the "regulator mapping" Ext 2 MEM1 (Q, S 2n H(r)) ⊗ R → H 2 D (M 1,1/R , S 2n H R (r)) is injective 1 , an analogue of Beilinson's conjecture [5,Conj. 8.4.1], then these relations generate all relations in u MEM

1
. This partial presentation is discussed in more detail later in the introduction.
One goal of this work is to illuminate the relationship between cusp forms of SL 2 (Z) and the depth filtration of the fundamental group of MTM. In this vein, in Section 29, we show that the relations in u MEM 1 coming from cusp forms imply the congruences between the generators of the infinitesimal Galois action on the unipotent fundamental group of P 1 − {0, 1, ∞} in depth 2 that were discovered by Ihara and Takao [32] and made explicit by Goncharov [21] and Schneps [46]. Our proof gives a conceptual explanation of how and why cusp forms impose relations on the depth graded quotients of mixed Tate motives.
We now give a more detailed, but still informal, discussion of universal mixed elliptic motives. The full definition is given in Section 6. Suppose that r and n are non-negative integers with r+n > 0. Denote the moduli stack over Spec Z of smooth elliptic curves with n marked points and r non-zero tangent vectors by M 1,n+ r/Z . For * ∈ {1, 1, 2} we construct a category MEM * of universal mixed elliptic motives over M 1, * /Z . Objects of MEM * will be called universal mixed elliptic motives of type * .
The normalization of the fiber E 0 over q = 0 is isomorphic to P 1 and has a natural coordinate w, unique up to the involution w ↔ w −1 , that takes the value 1 at the identity and defines an isomorphism of the smooth points E 0 of E 0 with G m/Z . There is therefore a map Spec Z((t)) → Spec Z((q, v)) → M 1,2/Z , q → t, v → t, where v = w − 1, which determines a tangential base point of M 1,2/Z and the tangential base point Spec Z((t)) → Spec Z((q, v)) → M 1, 1/Z of M 1, 1/Z . We shall denote any of these distinguished base points, including t, by v o . Informally, a mixed elliptic motive of type * ∈ {1, 1, 2} over Z is a "motivic local system" V of Q-vector spaces over M 1, * /Z with a weight filtration W • by sub local systems that satisfies: (i) each weight graded quotient of V is a sum of the simple local systems S n H(m), where H = R 1 π * Q(0) and π : E → M 1, * is the universal elliptic curve; (ii) the fiber V o of V over v o is an object of MTM; its weight filtration M • is the relative weight filtration associated to the weight filtration W • and the monodromy operator about the nodal cubic.
The fiber of H over v o will be denoted by H. It is isomorphic to Q(0) ⊕ Q(−1) as an object of MTM.
Objects of MEM * have compatible Betti, Hodge, ℓ-adic and Q-de Rham realizations. The Hodge realization of a mixed elliptic motive V is an admissible variation of mixed Hodge structure over M an 1, * whose fiber over v o is the associated limit mixed Hodge structure. The ℓ-adic realization of V is a lisse sheaf V ℓ over M 1, * /Z[1/ℓ] whose fiber over v o is the ℓ-adic realization of V o . The Q-de Rham realization of V is a bifiltered bundle (V, F • , W • ) over M 1, * /Q with a regular connection that underlies Deligne's canonical extension of V to M an 1,1 and its Hodge and weight filtrations. There are natural functors MTM → MEM 1 → MEM 2 → MEM 1 → MTM whose composite is the identity.
Along with the pure motives S n H(m), the simplest objects of MEM * include the "geometrically constant" motives. These are the objects of MTM that are pulled back along the structure map M 1, * /Z → Spec Z. Their Hodge realizations are constant variations of MHS over M an 1, * . More interesting objects of MEM * can be constructed from the unipotent fundamental group of punctured elliptic curves. Denote by p p p the local system over M 1,2 whose fiber over (E, x) is the Lie algebra p(E ′ , x) of the unipotent completion of π 1 (E ′ , x), where E ′ denotes E − {0}. The local system p p p is a pro-object of MEM 2 . This is an elliptic analogue of the result of Deligne and Goncharov [16] that the Lie algebra of the unipotent fundamental group of P 1 − {0, 1, ∞} is a pro-object of MTM.
The category MEM * , being a Q-linear neutral tannakian category, is the category of representations of an affine group scheme, unique up to inner automorphism, that we shall denote by π 1 (MEM * ). One goal of this paper is to determine the structure of π 1 (MEM * ). While we do not find an explicit presentation of its Lie algebra, we are able to give a good idea of its "shape". The first step in this direction is taken in Section 16 where we show that π 1 (MEM * ) is an extension of GL(H) by a prounipotent group and prove that there are natural isomorphisms π 1 (MEM 1 ) ∼ = π 1 (MEM 1 ) ⋉ Q(1) and π 1 (MEM 2 ) ∼ = π 1 (MEM 1 ) ⋉ π un 1 (E ′ t , v o ), where E t is the fiber of the universal elliptic curve over t. This reduces the problem of finding presentations of π 1 (MEM * ) to the case * = 1 and to the problem of determining how it acts on the Lie algebra of π un 1 (E ′ t , v o ). When trying to understand π 1 (MEM * ), it is convenient to define the geometric fundamental group π geom 1 (MEM * ) of MEM * to be the kernel of the natural surjection π 1 (MEM * ) → π 1 (MTM). Taking the fiber over the tangent vector v o corresponding to the Tate curve gives a splitting s t : π 1 (MTM) → π 1 (MEM * ) of this homomorphism, so that one has an isomorphism π 1 (MEM * ) ∼ = π 1 (MTM) ⋉ π geom 1 (MEM * ).
A key technical point (established in Section 23), which simplifies the problem of finding presentations of the Lie algebras of the π 1 (MEM * ), is that the de Rham realization of every object of MEM * has a canonical (i.e., unique natural) bigrading that splits the Hodge filtration and both weight filtrations. This generalizes the canonical grading of the de Rham realization of a mixed Tate motive that splits it weight and Hodge filtrations. This bigrading allows us to canonically identify the de Rham realization V DR of an object V of MEM * with its associated bigraded module Gr V DR = m,n Gr M 2m Gr W n V DR and reduces the problem of finding a presentation of the Lie algebra of π 1 (MEM * ) to the problem of finding a presentation of its associated bigraded Lie algebra. For this reason, we now specify the fiber functor to be the de Rham fiber functor. This allows us to identify the Lie algebra g MEM * of π 1 (MEM * ) with the completion of its associated bigraded Lie algebra. Denote its pronilpotent radical by u MEM * A fundamental result [16] of Deligne and Goncharov asserts that π 1 (MTM) is an extension of G m by a prounipotent group whose Lie algebra is freely generated (non canonically) by a set {σ σ σ 2m+1 : m ≥ 1}. In Part 3, we show that u MEM is generated (topologically) by the set {e j 0 · e 2n : n > 1, 0 ≤ j ≤ 2n − 2}, where e 0 ∈ sl(H) lowers sl(H)-weights by 2 and where u · v denotes the adjoint action of u on v.
There are two kinds of relations in u MEM 1 : namely, geometric relations: the relations between the geometric generators {e 2n : n ≥ 0}, arithmetic relations: which describe how the arithmetic generators z 2m+1 act on the geometric generators e 2n . In Section 25, we compute the quadratic leading terms of a minimal generating set of relations of u MEM 1 . There is one such geometric relation for each normalized Hecke eigen cusp form and each integer d ≥ 2, the degree of the relation as an expression in the e 2n , n ≥ 0. Each Eisenstein series G 2m determines a countable set of arithmetic relations which express [z 2m−1 , e 2n ] in terms of the geometric generators {e k : k ≥ 0}. We conjecture that regulator should be an isomorphism. (Cf. Conjecture 17.1(i).) This is an analogue of Beilinson's Conjecture [5]. If true, it would imply that the relations we have found generate all relations in u MEM * . In Part 3, we approach the problem of finding the geometric relations by studying the monodromy representation u geom 1 → Der p, where p denotes the Lie algebra of π un 1 (E t , v o ). Relations between the images of the e 2n in Der p give an upper bound on the relations that hold between the e 2n in u geom 1 . The problem of determining these relations can be studied by passing to the associated bigraded Lie algebras. Since p is free, it is isomorphic to the completion of its associated bigraded, which is the free Lie algebra L(H) ∼ = L(A, T ). The Q-de Rham story gives a formula for the images of the e 2n in Der L(A, T ) via the elliptic KZB equation [12,39,25]. The image of the generator e 2n of u geom 1 is 2ǫ 2n /(2n − 2)!, where ǫ 2n is the unique derivation 2 of L(H) satisfying ǫ 2n (T ) = − ad 2n T (A) and ǫ 2n ([T, A]) = 0, n > 1.
Pollack, in his undergraduate thesis [44], found all quadratic relations that hold between the {ǫ 2n : n ≥ 0} in Der L(A, T ) and showed that they correspond to cuspidal cocycles of SL 2 (Z) invariant under the "real Frobenius" operator. He also found relations of each degree d ≥ 3 that hold between the ǫ 2n modulo "depth 3", one for each invariant cuspidal cocycle of SL 2 (Z). His results are summarized in Section 24.
A lower bound on the relations in u MEM 1 is obtained from the computation of the cup products of the classes of the extensions (1.1) in the real Deligne cohomology groups H • D (M an 1,1 , S 2n H(r)). These are deduced from the fundamental work of Francis Brown [9] who computed periods of twice iterated integrals of Eisenstein series. His work is closely related to unpublished computations of Tomohide Terasoma [53] who computed cup products in Deligne cohomology of the classes in higher Chow groups of Kuga-Sato varieties that correspond to Eisenstein series, generalizing to higher weight Beilinson's computations [4] in weight 2. 3 It turns out that the upper and lower bounds on the quadratic heads of the relations in u MEM 1 coincide, as we show in Section 25, which allows us to determine the quadratic leading terms of all relations in u MEM 1 .
In related work, Enriquez [18] has defined an elliptic generalization of Drinfeld's braided monoidal categories [17] and defined an affine Q-group GRT ell for which the scheme of elliptic associators is a torsor. 4 It is a split extension where GRT is the affine Q-group, defined by Drinfeld [17], and where the projection to GRT is induced by the map (27.3). One expects there to be a homomorphism π 1 (MEM 1 ) → GRT ell with an appropriate choice of fiber functors, which projects the extension above to 1 → π geom 1 (MEM 1 ) → π 1 (MEM 1 ) → π 1 (MTM) → 1.
Assuming that it exists, this homomorphism will be surjective if and only if Enriquez's conjecture [18,Conj. 9.1] holds and the standard homomorphism π 1 (MTM) → GRT is surjective. It will be injective if and only if u MEM → Der p is injective. 5 The existence of this homomorphism will imply that the relations in u MEM 1 will also hold in the Lie algebra grt ell of GRT ell . This paper is in four parts. In the first part we define universal mixed elliptic motives. Before doing this, we give a detailed description of the local system H over M 1,1 and its fiber H over t in all of its manifestations. We also consider the basic structure of π 1 (MEM * ) that one gets from formal considerations.
The second part is devoted to proving that the elliptic polylogarithmic extensions (1.1) lie in MEM * and that they exhaust all extensions of Q by a simple object S m H(r) of MEM 1 when m > 0. For this we use the Hodge theory of the relative completion of SL 2 (Z), which is described in detail in [26]. Results from [27] allow us to relate extensions of admissible variations of mixed Hodge structure over M an 1, * to the Deligne-Beilinson cohomology groups H • D (M an 1, * , S m H(r)). We also use weighted and crystalline completion to study the ℓ-adic aspects. This part concludes with a discussion of the relationship between the groups Ext 2 MEM * (Q, S m H(r)) and certain of the conjectures of Beilinson and Bloch-Kato on regulators associated to modular curves.
In Part 3 we consider the problem of finding a presentation of the Lie algebra u MEM * . We first show that the e 2n and the z 2m+1 generate u MEM 1 . We then recall Pollack's relations and use Brown's period computations to show that they lift to relations in u MEM * . In Part 4 we use the results of Part 3 to begin the study of the relationship between universal mixed elliptic motives and mixed Tate motives. This is achieved 3 There is an inconsistency between the results of Brown and Terasoma. Our computations are consistent with Brown's. Nonetheless, we believe that Terasoma's results are basically correct. 4 Implicit in [18] is that this group depends on the choice of a free Lie algebra of rank 2 over Q together with an ordered pair of generators. Two natural choices are the Betti and de Rham realizations of the Lie algebra of π un 1 (E ′ t , vo) with a pair of generators which descends to a symplectic basis of its abelianization. Such a choice will correspond to the choice of a fiber functor. 5 Brown's result [8] implies that π 1 (MTM) → GRT is injective.
by specialization to the nodal cubic. As an application, we deduce the Ihara-Takao congruences from Pollack's relations. We also work out the precise relationship between the depth filtration on π 1 (MTM) and the pull back of the "elliptic weight filtration" W • of Der p.
1.1. History. This paper evolved slowly over the past 8 years and owes much to its antecedents. These include the paper [6] of Beilinson and Levin on the elliptic polylogarithm, the work of Deligne [14] on the fundamental group of P 1 − {0, 1, ∞}, and the paper of Deligne and Goncharov [16] on mixed Tate motives. The works of Calaque-Enriquez-Etingof [12] and Levin-Racinet [39] on the elliptic KZB connection proved to be useful, both in understanding the Q-de Rham picture and in establishing formulas that play a key role in Part 4. Finally, Nakamura's computation [40] of the action of the absolute Galois group G Q on the fundamental group of E ′ t was also a useful reference point and helped us understand the ℓ-adic picture. This work originated in informal computations done in the Spring of 2007 in which we found (under optimistic assumptions) the basic form (described in §21) of a presentation of the ℓ-adic crystalline completion of π 1 (M 1,1/Z[ℓ −1 ] , t). Standard conjectures on the ranks of Selmer groups and an optimistic assumption on the size of the image of cup products in Galois cohomology-which now appears likely to be true-led us to predict that each cusp form of SL 2 (Z) should determine a sequence of relations in the unipotent radical of the crystalline completion, and thus between the derivations ǫ 2n discussed above. The first author gave the problem of finding some of these relations to Aaron Pollack, who was then an undergraduate. He found all the quadratic relations between the ǫ 2n and also relations of each degree ≥ 3 that held modulo depth ≥ 3. This suggested that our optimistic assumptions were indeed true. However, the proof that Pollack's relations lifted to u MEM 1 had to wait until the period computations of Brown [9]. There is no ℓ-adic proof of this. It would be very interesting to have one.

Acknowledgments:
The authors gratefully acknowledge the support of MSRI in 2008 where some of the initial work was done. The first author also gratefully acknowledges the generous support of the Friends of the Institute for Advanced Study and Duke University which allowed him to spend his sabbatical year 2014-15 at the IAS where part of this paper was written. The first author would like to thank Francis Brown for numerous stimulating discussion during which the material in Parts 3 and 4 evolved. The authors would also thank Tomohide Terasoma for his correspondence related to the computation of the cup product in Deligne cohomology of Kuga-Sato varieties. They would also like to thank Aaron Pollack for his contributions to this project during the early stages of this work.

Notation and Conventions
All moduli spaces in this paper will be regarded as stacks and their underlying analytic spaces will be regarded as orbifolds (i.e., stacks in the category of complex analytic varieties).
2.1. Path multiplication. We use the topologists' convention. The product αβ of two paths [0, 1] → X in a topological space is defined when α(1) = β(0). The product αβ traverses α first, then β. This is the opposite of the algebraists' convention, where paths are multiplied in the "functional order".

2.2.
The topological fundamental group. Suppose that X is an algebraic variety over a subring R of C. Denote the corresponding analytic variety by X an . For x ∈ X(C), we denote the topological fundamental group of X an by π top 1 (X, x). If X is a DM stack over R, the associated analytic space X an is an orbifold (i.e., a stack in the category of topological spaces). For each suitable base point x of X an , we denote the orbifold fundamental group of X an by π top 1 (X, x). Fundamental groups of stacks are defined in [42], for example. Typically in this paper, X will be the stack/orbifold quotient of a smooth variety by a finite group. In this case the orbifold fundamental group is easy to describe directly. See, [24,Sec. 3], for example.
Denote the orbifold quotient of a topological space X by a discrete group Γ by Γ\\X . If X is simply connected, then for each choice of a base point x o ∈ X , there is a natural isomorphism π top 1 (Γ\\X , x o ) ∼ = Γ. In addition, we can regard the projection p : X → Γ\\X as a base point. In this case Γ is naturally isomorphic to π top 1 (Γ\\X , p). For example π top 1 (SL 2 (Z)\\h, p) ∼ = SL 2 (Z). 2.3. Theétale fundamental group. Denote theétale fundamental group of a scheme X over a ring R by π 1 (X, x), where x is a geometric point of X. Theétale fundamental group of a DM stack X can also be defined -see [42]. We will also denote it by π 1 (X, x). Suppose that R is a field k with algebraic closure k. The fundamental group of Spec k with respect to the geometric point Spec k → Spec k is simply Gal(k/k). We denote it by G k . The structure morphism X → Spec k induces a homomorphism π 1 (X, x) → G k . One has the canonical short exact sequence 2.4. Topological versusétale fundamental groups. When X is a DM-stack over C and x ∈ X(C), there is a natural isomorphism 6 π 1 (X, x) ∼ = π top 1 (X, x) ∧ between theétale fundamental group of X and the profinite completion of the topological fundamental group of the associated analytic variety. When k is a subfield of C, the exact sequence (2.1) becomes Tannakian fundamental groups. Suppose that F is a field of characteristic zero. We will denote the tannakian fundamental group of an F -linear neutral tannakian category C with respect to a fiber functor ω : C → Vec F by π 1 (C, ω). It is an affine group scheme that is an inverse limit of affine algebraic groups. When the context is clear, we will omit ω from the notation.
We will denote the category of linear representations of a group G over a field F by Rep F (G) and the sub-category of finite dimensional representations by Rep fte f (G). If G is an affine F -group (scheme), we will write Rep(G) instead of Rep F (G). If F has characteristic zero, then Rep(G) is equivalent to the category of ind-objects of Rep fte (G). 6 Strictly speaking, our path multiplication convention implies that this is an anti-isomorphism.
If one prefers, one can replace theétale fundamental group by its opposite group. Subsequently, the term "algebraic group" will mean "affine algebraic group". In particular, we will not refer to elliptic curves as algebraic groups.
2.6. Cohomology of affine group schemes. The cohomology of an affine group scheme G over a field F of characteristic zero with coefficients in a G-module V is defined by where Rep(G) denotes the category of G-modules. The fact that every G-module is a direct limit of its finite dimensional submodules implies that if V is finite dimensional over F , then where Rep fte (G) denotes the category of finite dimensional G-modules. If G is an extension of a reductive group R by a prounipotent group U , and if U has Lie algebra u, then there are natural isomorphisms For background and more details, see [33,Chapts. 3,4] and [27].

2.7.
Free Lie algebras. The free Lie algebra generated by a vector space V will be denoted by L(V ). The free Lie algebra generated by a set X will be denoted by L(X ). It is isomorphic to the free Lie algebra generated by the vector space with basis X .

Mixed Tate Motives
Deligne and Goncharov [16], using work of Voevodsky [55] and Levine [38,37], constructed a Q-tannakian category of mixed Tate motives MTM(O K,S ) over a number field K, unramified outside a subset S of primes of O K . This category has simple objects Q(n) and has the property that Each object V of MTM(O K,S ) has a weight filtration, which we denote by M • . There are realization functors MTM(O K,S ) that take the object V to its Betti realization (V B , M • ), its de Rham realization (V DR , M • , F • ), and for each prime number ℓ, its ℓ-adic realization (V ℓ , M • ), which is a filtered G K := Gal(Q/K) module, unramified outside ℓ. There are comparison isomorphisms The Betti and de Rham realizations combine to give the Hodge realization of V , which is a mixed Hodge structure whose weight graded quotients are of Tate type. The weight filtration of V = Q(n) is The realizations of Q(n) are The Galois action on V ℓ is given by the nth power of the ℓ-adic cyclotomic character Hodge realization is the one dimensional Hodge structure of type (−n, −n). Denote the category MTM(Z) by MTM. The fiber functors The category MTM ss of semi-simple mixed Tate motives (i.e., those that are direct sum of copies of Q(m)'s) is neutral tannakian and has fundamental group isomorphic to G m ; the mixed Tate motive Q(m) corresponds to the mth power of the standard character of G m . The functor Gr M • : MTM → MTM ss is a retraction of MTM onto MTM ss .
For each of these fiber functors, the inclusion MTM ss → MTM induces a homomorphism π 1 (MTM, ω) → G m . Denote its kernel by K (or K ω when we want to identify the fiber functor). It is prounipotent. Since for every mixed Tate motive V , there is a natural isomorphism • induces a splitting of the homomorphism π 1 (MTM, ω DR ) → π 1 (MTM ss , ω DR ) = G m . The Lie algebra k of K is thus a G mmodule. It is (non-canonically) isomorphic to the completion of the free Lie algebra L(σ σ σ 3 , σ σ σ 5 , z 7 , σ σ σ 9 , . . . ) where G m acts on σ σ σ 2n+1 via the (2n + 1)st power of the standard character.
One important property of MTM is that the functor that takes MTM to the category MHS of Q-MHS is fully faithful. 7

Moduli Spaces of Elliptic Curves
Suppose that r and n are non-negative integers satisfying r + n > 0. Denote the moduli stack over Spec Z of smooth elliptic curves with n marked points and r non-zero tangent vectors by M 1,n+ r . Here the n marked points and the anchor points of the r tangent vectors are distinct. Taking each tangent vector to its anchor point defines a morphism M 1,n+ r → M 1,n+r that is a principal G r m -bundle. The Deligne-Mumford compactification [35] of M 1,n will be denoted by M 1,n . It is also defined over Spec Z. In this paper, we are primarily concerned with the cases (n, r) = (1, 0), (0, 1) and (2,0 The unique cusp of M 1,1 is the moduli point of the nodal cubic. Denote it by e o . The standard line bundle L over M 1,1 is the conormal bundle of the zero section of E → M 1,1 . Sections of L ⊗n over M 1,1 are modular forms of weight n, and those that vanish at the cusp e o are cusp forms of weight n.
The restriction of the relative tangent bundle of E to its identity section is the dualĽ of L. The moduli space M 1, 1 isĽ ′ , the restriction ofĽ to M 1,1 with its zero-section removed. This is isomorphic to the complement L ′ of the zero-section of L.
We will also identify M 1,1 with the identity section of E. With this convention, e o also denotes the identity of the nodal cubic in E and also the corresponding point of the partial compactificationĽ of M 1, 1 .
An explicit description of L ′ over M The point (u, v) corresponds to the plane cubic y 2 = 4x 3 − ux − v and the abelian differential dx/y. The G m -action is λ : (u, v) → (λ −4 u, λ −6 v). In this case, M 1,1 and M 1,1 are the quotient stacks Similarly, when 2 and 3 are invertible in A, M 1,2/A is the quotient of the scheme The point (u, v, x, y) corresponds to the point (x, y) on the cubic y 2 = 4x 3 − ux − v and the abelian differential dx/y. The action of λ multiplies dx/y by λ.
of M 1,1 at the moduli point of the nodal cubic is integral and its reduction mod p is non-zero for all prime numbers p. Denote it by t. Identify M 1,1 with the identity section of the completion M 1,2 of the universal elliptic curve over M 1,1 . With this identification, e o is the identity of the nodal cubic.
The fiber E 0 of the Tate curve over q = 0 is an integral model of the nodal cubic. Its normalization is P 1 Z . Let E 0 be E 0 with its double point removed. It is isomorphic to G m/Z . Let w be a parameter on E 0 whose pullback to the normalization of E 0 takes the values 0 and ∞ on the inverse image of the double point, and the value 1 at the identity. It is unique up to w → w −1 . It determines the tangent vector w o := ∂/∂w of E 0 at the identity e o of E 0 . It is integrally defined and non-zero at all primes.
We thus have the tangent vector v o := t + w o of M 1, 1 (and thus of M 1,2 and E as well) at the identity e o of E 0 which is non-zero at all primes.

4.3.
Moduli spaces as complex orbifolds. To fix notation and conventions, we give a quick review of the construction of M an 1,1 , E an and their Deligne-Mumford compactifications. All will be regarded as complex analytic orbifolds. This material is classical and very well-known. A detailed discussion of these constructions and an explanation of the notation can be found in [24] where the left-hand map takes τ to q := exp(2πiτ ) and where C 2 is included in   as the scalar matrices.

4.3.2.
The line bundle L an . The restriction of L an to M an 1,1 is the orbifold quotient of the trivial line bundle C × h → h by the SL 2 (Z) action It is an orbifold line bundle over M an 1,1 . Its restriction to the punctured q-disk D * is naturally isomorphic to the trivial bundle C × D * → D * , and therefore extends naturally to a (trivial) line bundle over D. The line bundle L an over M 1,1 is the unique extension of the line bundle above to M an 1,1 that restricts to this trivial bundle over the q-disk.

4.3.3.
Eisenstein series. To fix notation and normalizations we recall some basic facts from the analytic theory of elliptic curves.
Suppose that k ≥ 1. The (normalized) Eisenstein series of weight 2k is defined by the series where σ k (n) = d|n d k . When k > 1 it converges absolutely to a modular form of weight 2k. When properly summed, it also converges when k = 1. In this case the logarithmic 1-form The ring of modular forms of SL 2 (Z) is the polynomial ring C[G 4 , G 6 ].
This has discriminant the normalized cusp form of weight 12: The following statement is easily verified. For a proof of the last statement, see [25,Prop. 19.1].
Proposition 4.1. The abelian differential dx/y corresponds to 2πidz and the differential xdx/y of the second kind corresponds to (2πi) −1 ℘ τ (z)dz. These differentials form a symplectic basis of H 1 DR (E τ ) as of M an 1, 1 , which also corresponds to the curve y 2 = 4x 3 − g 2 (τ )x − g 3 (τ ) with the abelian differential ξdx/y.
The universal elliptic curve π an : E an → M an 1,1 is the orbifold quotient of the projection C × h → h by Γ, which acts on h via the quotient map Γ → SL 2 (Z). The fiber of π an over the orbit of τ is E τ .
It can also be regarded as the orbifold quotient SL 2 (Z)\\E h of the universal framed elliptic curve where (m, n) : (z, τ ) → (z + mτ + n, τ ), by the natural action of SL 2 (Z). The universal elliptic curve E an → M an 1,1 is obtained by glueing in the Tate curve as described in [24, §5]. The restriction to the q-disk D of E an minus the double point of E 0 is the quotient of C * × D by the Z-action To relate this to the algebraic construction, note that M an 1,2+ 1 is the analytic variety Γ\ C * × C × h), where γ(ξ, z, τ ) = (cτ + d)ξ, γ(z, τ ) . The function It is invariant with respect to the-C * action λ·(ξ, z, τ ) = (λξ, z, τ ) on Γ\(C * ×C×h) and the C * -action on M 1,2+ 1 that multiplies the abelian differential by λ.
where B 3 denotes the braid group on 3 strings and SL 2 (Z) denotes the inverse image of SL 2 (Z) in the universal covering group SL 2 (R) of SL 2 (R). Again, this is well-known; details can be found in [24, §8].
Similarly, there are natural isomorphisms

The Local System H
Roughly speaking, the local system H over M 1,1 is the "motivic local system" R 1 π * Q associated to the universal elliptic curve π : E → M 1,1 . While we could work in Voevodsky's category of motivic sheaves [55], we will instead define H to be a set of compatible realizations: Betti, Q-de Rham, Hodge, and ℓ-adicétale. These are described below in detail.
In subsequent sections, we will abuse notation and denote the pull back of H to M 1,n+ r by H for all r, n ≥ 0 with r + n > 0.
5.1. Betti realization. The Betti realization of H is the orbifold local system H B := R 1 π an * Q over M an 1,1 associated to the universal elliptic curve π an : E an → M an 1,1 . Since Poincaré duality induces an isomorphism H 1 (E) ∼ = H 1 (E) for all elliptic curves E, H B is also isomorphic to the local system whose fiber over [E] ∈ M 1,1 is H 1 (E, Q).
Since h is contractible, there is a natural isomorphism The sections a and b thus trivialize the pullback of H B to h. The induced action of SL 2 (Z) on H 1 (E h , Z) is given by left multiplication: It corresponds to a right action of SL 2 (Z) on Qa ⊕ Qb.
Denote the dual basis of H 1 (E h , Z) byǎ,b. The action of SL 2 (Z) on this frame is given by γ : b −ǎ → b −ǎ γ. The basisǎ,b corresponds to the basis −b, a of H 1 (E τ ) under Poncaré duality. The corresponding action of SL 2 (Z) on this frame is: This defines a left action of SL 2 (Z) on H B × h, where H B = Qa ⊕ Qb should be thought of as the fiber of H B over p, and also over t. Define a holomorphic section w of H an h by where ω τ denotes the class of the holomorphic 1-form dz in H 1 (E τ ).
This bundle has a Hodge filtration The section w trivializes F 1 H an h . The sections a and w descend to give a framing Since ∇ 0 a = 0, the connection on H h is given by To define an extension H an of H an to a vector bundle over M Define a connection ∇ 0 on H DR by The connection is G m -invariant, defined over Q, has regular singularities along the discriminant divisor D = u 3 − 27v 2 = 0 and is holomorphic on its complement. It therefore descends to a rational connection over M 1,1/Q , with a regular singular point at the cusp e o .
Define a Hodge filtration This is a G m -invariant sub-bundle, and thus defined over M 1,1/Q .
The following statement follows from [25,Prop. 19.6], [25,Prop. 19.7], and Proposition 4.1 above. 8 that respects the Hodge filtration. When pulled back to M 1, 1/Q = A 2 − {0}, the section T corresponds to dx/y and the section S to xdx/y. After pulling back to the q-disk along the slice q → g 2 (q), g 3 (q) ∈ A 2 , we have T = w and S = a−2G 2 (q)w.

5.4.
Hodge realization. The Hodge realization of H is the polarized variation of Q-Hodge structure (PVHS) of weight 1 over M an 1,1 whose underlying local system is the Betti incarnation H B of H and whose associated flat vector bundle, with its Hodge filtration, is the flat vector bundle (H DR , ∇ 0 , F • ) ⊗ O M an 1,1 . It is naturally polarized by the cup product. Since it is of weight 1, its weight filtration is We compute the limit mixed Hodge structure on the fiber H := H t of H over t = ∂/∂q.
The fiber H DR of H DR over the cusp e o has Q-basis S and T . Proposition 5.2 implies that T = w and that, when q = 0, S = a − G 2 (0)w/2 = a + w/12.
Consequently, H DR = Qa ⊕ Qw. The residue of the connection at e o is the nilpotent operator The associated monodromy weight filtration of H DR (centered at the weight, 1, of It remains to determine the Betti structure H B on H associated to the tangent vector t = ∂/∂q. This is computed according to Schmid's prescription [45]. The complex vector space underlying H is H DR ⊗ C. Its integral lattice is id + log q a∂/∂w v. Since b = (log q)a − w /(2πi), the comparison map takes a ∈ H B to a ∈ H DR and b ∈ H B to lim q→0 id + log q a∂/∂w b = −(2πi) −1 w.
Since w spans F 1 H DR , it follows that the limit MHS is isomorphic to Z(0)⊕ Z(−1). The copy of Z(0) is spanned by a and the copy of The image of the positive generator of π top 1 (D * ) in π top 1 (M 1,1 , t) = SL 2 (Z) is Remark 5.3. For the uninitiated, it may seem strange that the pure Hodge structure on H 1 (E τ ) becomes a mixed Hodge structure in the limit. One way to come to terms with this is to think of the limit MHS H = H t as being the mixed Hodge structure on H 1 (E t ), where E t denotes a smoothing of the nodal cubic E 0 in the direction of t. 9 There is a continuous retraction E t → E 0 whose composition with the inclusion E * 0 ֒→ E t is the inclusion E * 0 ֒→ E 0 , where E * 0 denotes the non-singular locus of E 0 , which is isomorphic to C * . This sequence induces an exact sequence This MHS is not always split -if one replaces t by λ t, then the MHS on H 1 (E λ t ) is the extension of Z(−1) by Z(0) corresponding to λ ∈ C * ∼ = Ext 1 (Z(−1), Z(0)). To prove this, one replaces q by q/λ in the above computations of the limit. 5.5.Étale realization. The ℓ-adic realization of H is the lisse sheaf H ℓ := R 1 π * Q ℓ over the stack M 1,1/Q . Its fiber over the moduli point of an elliptic curve E over a number field K in Q is π 1 (E, 0) ⊗ Q ℓ (−1), endowed with the natural action of G K := Gal(Q/K). As in the Hodge case, its fiber H ℓ = H 1 et (E t/Q , Q ℓ ) over t = Spec Q((q 1/n : n ≥ 1)) is a split extension of Q ℓ (−1) by Q ℓ (0). Details can be found in Nakamura [40].

Its restriction
The notion of the fiber E v of E over a non-zero tangent vector v of the origin of the q-disk can be made precise. See, for example, see Appendix C of [25].
to the geometric fundamental group is G Q -equivariant and corresponds to the action of π top 1 (M 1,1 , t) on H B ⊗ Q ℓ under the comparison isomorphisms (v) For each prime number ℓ, the action π 1 (M 1,1/Q , t) → Aut H ℓ induced by the action of SL 2 (Z) on H B via the comparison isomorphism is G Qequivariant. This is the basic universal elliptic motive. Other universal elliptic motives will include S n H(r) := (S n H) ⊗ Q(r).

Universal Mixed Elliptic Motives
Suppose that * ∈ {1, 1, 2}. Denote the natural tangential base point of Let H be the structure described in the summary (5.6), pulled back to M 1, * .
Definition 6.1 (mixed elliptic motives). A universal mixed elliptic motive V over Z of type * consists of: (iii) a bifiltered vector bundle (V, W • , F • ) over M 1, * /Q whose fiber over e o is the Q-de Rham realization of (V, W • ); (iv) an integrable flat connection defined over Q with nilpotent residue along each component of the boundary divisor ∆ which preserves W • and satisfies Griffiths transversality: These are required to satisfy: (ii) each weight graded quotient Gr W m V of V is isomorphic to a direct sum of copies (with multiplicities) of S m+2r H(r).
Remark 6.2. The last condition implies that M • is the relative weight filtration associated to the nilpotent endomorphism log σ o (the positive generator of the fundamental group of the punctured q-disk) of the filtered vector space (V B , W • ). The isomorphism of the canonical extension of V B ⊗ O M an 1, * with the complexification of (V, W • , F • ) defines an admissible variation of MHS over M an 1, * whose limit MHS over v o is naturally isomorphic to the Hodge realization of (V, M • , W • ).
In addition, since V is an object of MTM, the associated Galois representation G Q → Aut(V ⊗ Q ℓ ) is unramified at all primes p = ℓ and is crystalline at ℓ. (ii) for all prime numbers ℓ, the diagram  Example 6.6 (Simple universal mixed elliptic motives). These are the Tate twists S m H(r) of symmetric powers of H. That they are universal elliptic motives follows from the discussion in the previous section that was summarized in Section 5.6. The mixed Tate motive underlying S m H(r) is The simple object S m H(r) of MEM * is pure of weight m − 2r. Not all objects of MEM * are pure or geometrically constant. The simplest non-trivial examples are extensions of Q by S 2n H(2n + 1) over M 1,2 for each n ≥ 1. These are the elliptic polylogarithms of Beilinson and Levin [6].
Example 6.8. An important and non-trivial example of a pro-object of MEM * is provided by the local system over M 1,2 whose fiber over [E, x] is the Lie algebra p(E ′ , x) of the unipotent completion 10 of π 1 (E ′ , x). Its restriction to M 1, 1 is the local system whose fiber over [E, v] is the Lie algebra of the unipotent completion of π 1 (E ′ , v).
Choose a parameter w on the fiber E 0 over q = 0 of the Tate curve that takes the value 1 at the identity and defines an isomorphism of the smooth points E 0 of E 0 with G m . It is unique up to the involution w → 1/w of E 0 . The tangent vector is integrally defined and is non-zero mod p for all prime numbers p. It will be used as a base point for both . Corollary 6.10. There is an object p p p of MEM 2 whose fiber over v o is p(E ′ t , w o ). 6.1. Variants: higher levels and more decorations. One can make a similar definition of mixed elliptic motives over M λ 1,n+ r/OK,S where r + n > 0, λ is a level given by a congruence subgroup of SL 2 (Z), and O K,S is the ring of S-integers of a number field K. One first has to choose an integrally defined tangent vector v o with everywhere good reduction. (That is, a tangent vector defined over O K,S that is non-zero mod ℘ for all prime ideals ℘ of O K,S .) Once r + n > 2, there are several possible choices of base point. All give equivalent categories of mixed elliptic motives. This will be proved in [11]. However, the cases (n, r) ∈ {(1, 0), (2, 0), (0, 1)} are the most relevant for our immediate purposes.
The category of mixed elliptic motives over M 1, 1 [N ] /Z[µ µ µ N ,1/N ] should be interesting. The coordinate ring of its tannakian fundamental group should be an ind-object of MTM(Z[µ µ µ N , 1/N ]) and there might be an interesting connection to Deligne's work [15].

Tannakian Considerations
The category MEM * , * ∈ {1, 1, 2}, is a rigid abelian tensor category over Q. The functor that takes an object of MEM * to its fiber (V, It is faithful, exact and preserves tensor products and unit objects. Consequently, each fiber functor ω : MTM → Vec F gives rise to a fiber functor ω • v * o : MEM * → Vec F that we shall also denote by ω. Proposition 7.1. The category MEM * is a neutral tannakian category over Q. For a faithful, exact functor ω from MEM * to a neutral tannakian category C that preserves tensor products, set π 1 (MEM * , ω) = Aut ⊗ ω. Its coordinate ring is an ind-object of C. When C = Vec F , where F is a field of characteristic zero, π 1 (MEM * , ω) is an affine F -group.
The standard fiber functors for both MTM and MEM * are: Other useful fiber functors include Gr W • ω B , Gr W • ω DR and Gr W • ω ℓ , which take values Vec Q and Vec Q ℓ , as appropriate.

The morphisms
One thus has a split extension which is split by v * o . One also has the group π 1 (MEM * , v * o ). It is an affine group scheme in MTM. More precisely: is a Hopf algebra in the category of ind-objects of MTM. The Betti, de Rham and ℓ-adic realizations of

respectively.
A universal mixed elliptic motive V is semi-simple if it is a direct sum of simple universal elliptic motives: The category MEM ss * of semi-simple universal elliptic motives is the full tannakian subcategory of MEM * generated by the S m H(r). For each of the standard fiber functors ω, we have The simple object H corresponds to the defining representation of GL(H ω ), and the simple object Q(−1) is Λ 2 H, and thus corresponds to the 1-dimensional representation det : GL(H ω ) → G m . Combining these, we see that S m H(r) corresponds to the mth symmetric power of the defining representation, twisted by det ⊗(−r) . Note that if V is a pure object V of MEM * of weight m, then the scalar λ id H matrix in GL(H ω ) acts on V ω as λ m id V . Proposition 7.3. For each of the standard fiber functors, the surjection is split and has prounipotent kernel.
Proof. Standard facts about variations of MHS imply that the functor Gr W . The splitting is the composite of this with any F -rational point of the "path torsor" Isom ⊗ (ω, Gr W • ω). That the kernel is prounipotent follows from the fact that one can only extend a pure universal elliptic motive (non-trivially) by one of lower weight. General tannakian theory implies that the Lie algebra of the kernel has negative weights and is thus pronilpotent. That said, we will give a concrete proof of this below.
Remark 7.4. Later we will need to know that the functor V → Gr W • Gr M • V that takes a mixed elliptic motive V to the associated bigraded of its fiber over v o is exact. This is proved in Appendix B.

The restriction functor MEM
Since N an is a covering space of E an , there are neighbourhoods U of the zero section of N an and V of the zero section of E an such that V gets mapped biholomorphically onto U by the projection N → E and such that U is a deformation retract of N . Denote the complement of the zero section in U and V by U ′ and V ′ . Then the sequence of maps Proposition 7.5. There is a natural restriction functor MEM 2 → MEM 1 that is the identity on the fiber over the base point v o and takes the object V with monodromy representation ρ V : π top 1 (M 1,2 , v o ) → Aut V B to an object of MEM 1 with monodromy representation the composite The only task is to prove that if V is an object of MEM 2 , then its de Rham realization (V DR , ∇), a vector bundle with connection over M 1,2/Q , pulls back to a vector bundle with connection over M 1, 1/Q satisfying the required compatibilities with the other realizations.
Denote the zero section of E /Q by Z and its ideal sheaf in O E by m Z . Then There is a natural isomorphism O E ∼ = O Z ⊕ m Z that is induced by the inclusion Z ֒→ E and the projection E → M 1,1 ∼ = Z. This induces the eigenspace decomposition which is characterized by the properties: (i) it has regular singular points along Z, To complete the proof, we sketch a proof of the compatibility of the monodromy representations with the Betti realizations. More precisely, we explain why the diagram , and where ρ ∇ and ρ ∇ are the monodromy representations of ∇ and ∇. It suffices to consider the case where v = ∂/∂ξ. Let τ o ∈ h be a point that lies above P ∈ Z.
By standard ODE theory (see, for example, [56]), since ∇ has nilpotent residue at each point along Z, there is a polynomial whose regularized value p(τ o , 1, 0) at v := ∂/∂ξ ∈ T P M 1,2 is the identity of V P and with the property that all flat sections of V are of the form vp(τ, ξ, log ξ) for some v ∈ V P . The monodromy of ∇ about an element γ ∈ π 1 (M an 1,2 , v) is obtained by taking the analytic continuationp of p along γ and then taking its regularized value at v: On the other hand, ODE theory implies that the flat sections of ∇ over M 1, 1 are of the form p(τ, 1, log ξ). The regularized monodromy representation of ∇ on γ ∈ π 1 (M an 1, 1 , v) is computed fromp. Since we may assume that γ lies in the neighbourhood U ′ of Z, we have the claimed compatibility of monodromy representations.

Hodge Theoretic Considerations
For F = Q or R, define MHS F (M 1, * , H) to be the category of admissible variations of F -MHS over M 1, * whose weight graded quotients are sums of polarized variations of Hodge structure of the form S m H ⊗ A, where A is a Hodge structure. 12 It is neutral tannakian. Denote the forgetful functor that takes a variation to the 11 Examples of how this works in practice can be found in [25, §13]. 12 It is important to note that A is not necessarily of type (p, p).
F -vector space underlying its fiber over v o by ω o . When F = Q we will omit the subscript.
Theorem 8.1. The forgetful functor MEM * → MHS(M 1, * , H) that takes a universal mixed elliptic motive V to the associated variation of MHS V an over M an 1, * is fully faithful. Consequently, is surjective.
Proof. Since Hom C (A, B) is naturally isomorphic to Hom C (Q(0), Hom Q (A, B)) for all objects A and B of MEM * and C = MEM * and MHS(M 1, * , H), it suffices to show that Injectivity is easily proved and is left to the reader. We will prove surjectivity.
Since the Hodge realization functor MTM → MHS is fully faithful [16], this map is the Hodge realization of a morphism Q(0) → V in MTM.
Identify the morphism v : as v ℓ is fixed by both G Q and π 1 (M 1, * /Q , v o ).
To complete the construction of a morphism φ : GAGA implies that the natural morphism is an isomorphism. This implies: Suppose that k is a subfield of C and that X is a smooth variety defined over k with an action by a k-algebraic group G. If V is a G-invariant vector bundle over X with a rational, G-invariant connection ∇ : V → V ⊗ Ω 1 X ⊗ OX k(X), then for all P ∈ X(k) the square this implies that the square is also cartesian.

Part 2. Simple Extensions in MEM *
In this section we compute the groups Ext 1 (Q, S m H(r)) in the various categories of realizations of objects of MEM * and show that these compatible realizations can be assembled into an extension in MEM * . This allows use to compute Ext 1 MEM * (Q, S m H(r)) for all m and r.

Cohomology of M 1, *
Here we recall the basic facts we need. Full details in the Hodge case can be found in [26]. References in the ℓ-adic case are given below. 9.1. The cohomology of M an 1,1 . The first basic observation is that, since we are regarding M an 1,1 as the orbifold SL 2 (Z)\\h, there is a natural isomorphism where V denotes the local system over M an 1,1 that corresponds to V . Since SL 2 (Z) is virtually free, this implies that H j (M an 1,1 , V) vanishes for all j ≥ 2 whenever V is a local system of Q-modules.
Well-known results of Shimura, Manin, Drinfeld and Zucker (cf. [36,57]) imply that the cohomology groups H • (M an 1,1 , S n H) and their mixed Hodge structures can be expressed in terms of modular forms. Denote the space of (holomorphic) modular forms of SL 2 (Z) of weight w by M w and the subspace of cusp forms by M o w . Recall that these are trivial when w is odd.
In this section we regard H as a polarized variation of Hodge structure over M an 1,1 of weight 1. As explained in Section 4, its fiber over the tangent vector t is where the comparison isomorphism H B ⊗ C → H DR ⊗ C takes a to a and identifies w with −2πib. The Hodge filtration F 1 H DR is spanned by w.
This is a holomorphic 1-form on h with values in S 2n H h . Since it is SL 2 (Z) invariant and since the section w spans F 1 H h , Since this form is holomorphic, it defines a class in H 1 (M an 1,1 , S 2n H). When f is the Eisenstein series G 2n , we denote ω f by ψ 2n .
The following statement is a composite of well-known results. (Cf. [6], [57].) Theorem 9.2 (Shimura, Zucker, Manin-Drinfeld). For each n > 0, the group H 1 (M an 1,1 , S 2n H) has a natural MHS with Hodge numbers (2n + 1, 0), (0, 2n + 1) and (2n + 1, 2n This is a 2-dimensional R Hodge substructure of H 1 (M an 1,1 , S 2n H). Its complexification V f,C is spanned by ω f and its complex conjugate. As a real Hodge structure, M an 1,1 decomposes The copy of R(−2n − 1) is spanned by the class 2πi ψ 2n+2 . There is a similar decomposition of H 1 (M an 1,1 , S 2n H) as a Q-MHS. Suppose that f ∈ B 2n+2 . As above, let a p be the pth Fourier coefficient of f . Let m p (x) be the minimal polynomial of a p over Q. Then This is a Q Hodge substructure of H 1 (M an 1,1 , S 2n H Q ) with the property that where g ranges over the Hecke eigen cusp forms whose Fourier expansions are Galois conjugate to that of f . It has dimension 2 dim Q K f . There is a decomposition where f ranges over the Galois equivalence classes of Hecke eigen cusp forms of weight 2n + 2.
. This group is generated by the class of the form (4.2) corresponding to the Eisenstein series G 2 . 13 If m > 0, then the projection to M an 1,1 induces an isomorphism

This implies that
. One can show that the cup product is injective and has image in the middle factor H 1 (M an 1,1 , S 2n+2 H) ⊕2 (−1). 13 It is natural to extend the definition of the ψ 2n (n > 1) to the case n = 1 by defining

Relative Unipotent Completion
The next task is to describe the tannakian fundamental group of MHS(M 1, * , H). Computing it will allow us to bound the size of π 1 (MEM * , ω B ). To do this, we need to review some basic facts about relative completion of discrete groups and facts about the relative completion of modular groups and their relation to extensions of variations of MHS. References for this material include [22,26,27].
10.1. Relative unipotent completion. Suppose that Γ is a discrete group and that R is a reductive algebraic group over a field F of characteristic zero. Suppose that ρ : Γ → R(F ) is Zariski dense representation. Consider the full subcategory This category is neutral tannakian over F . The completion of Γ relative to ρ is the affine group scheme π 1 (R(Γ, ρ), ω) over F , where ω takes a module M to its underlying vector space. It is an extension is the Hopf algebra of matrix entries of the objects of R(Γ, ρ).

10.2.
Cohomological properties. Set G = π 1 (R(Γ, ρ), ω) and U = U(Γ, ρ). Denote the Lie algebra of U by u. It is pronilpotent. The action of G on u induced by the inner action of G on U induces an action of G on H • (u). Standard arguments imply that it factors through the quotient mapping G → R, so that The following basic property of relative completion is easily proved by adapting the proof of [28,Thm. 4.6] or [29,Thm. 8.1]. A proof in the de Rham case is given in [22].
and an injection Using the fact (cf. [27]) that, for all G-modules V , there are natural isomorphisms Hom F (A, B)) R , one can restate the previous result as follows. For all G-modules A and B, the natural map is an isomorphism in degree 1 and injective in degree 2.
10.3. Unipotent completion. Unipotent completion is the special case where R is the trivial group. In this case, R(Γ, ρ) is the category of unipotent representations of Γ. The unipotent completion of Γ over F will be denoted by Γ un /F . Suppose that E is an elliptic curve over C.
Denote the unipotent completion of π 1 (E ′ , b) by P(E, b) and its Lie algebra by p (E ′ , b). For all such E and b, the coordinate ring O(P(E, b)) and Lie algebra p(E ′ , b) of P(E, b) have natural mixed Hodge structures that are compatible with their algebraic structures. The weight filtration W • of p(E ′ , b) is (essentially) its lower central series: ) of the associated weight graded of p(E ′ , b) with the free Lie algebra generated by H 1 (E). These form a variation of MHS over M an 1,2 when b is a point of E ′ , and over M an 1, 1 when b is a tangential base point.
There is also a canonical MHS on the unipotent completion of π 1 (E ′ t , w o ). It is the limit MHS associated to the tangent vector v o of M 1,2 at the identity of the nodal cubic. It has a weight filtration W • as above, and a relative weight filtration M • . This limit MHS is computed explicitly in [25, §18].
Denote the commutator subalgebra of p = p(E ′ , b) by p ′ and its commutator subalgebra by p ′′ . Then there is a short exact sequence in the category of Lie algebras with mixed Hodge structure. There is a canonical isomorphism of mixed Hodge structures Consequently, these variations yield extensions of variations of MHS over M an 1,2 and M an 1, 1 for each m ≥ 0. But since the log of monodromy about the cusp of E ′ lies in p ′ and spans the copy of Q(1) in H 1 (p ′ ), the monodromy of V m about any loop in a fiber of M an 1, 1 → M an 1,1 is trivial. Consequently, the restriction of this extension to M an 1, 1 descends to a variation over M an 1,1 . By standard manipulations, each of these variations is equivalent to a variation Similar constructions can be made in the ℓ-adic case.
The following result follows from Proposition 9.3 when * = 1 and the vanishing of H 1 (M an 1, * , Q) when * = 1.
Proposition 10.2. The unipotent completion of π un 1 (M an 1, * , x) is trivial when * ∈ {1, 2} and is isomorphic to G a when * = 1. Denote the Lie algebra of G rel * (x) by g rel * (x) and the Lie algebra of U rel

Relative completion of
) that vanish at the identity. Its bracket is induced by the skew symmetrized coproduct of O(G rel * (x)). The theorem implies that g rel * (x) is an inverse limit of Lie algebras with mixed Hodge structure and that u rel * (x) is a pronilpotent Lie algebra in the category of mixed Hodge structures. These mixed Hodge structures have the property that and similarly for their Lie algebras. The coordinate ring of G rel * and each of the Lie algebras has two weight filtrations, W • and M • . Both are compatible with the algebraic operations.
Since π top 1 (M 1,1 S m H) ∼ = SL 2 (Z) has virtual cohomological dimension 1, Proposition 10.1 implies that H 2 (u 1 ) = 0. This implies that u 1 is topologically freely generated by H 1 (u). (See the discussion in Section 18.) Proposition 10.1 also implies that there are natural SL(H)-module isomorphisms Both are isomorphisms of MHS. Recall from [26, §14] that the exact sequence The projection onto G a is induced by the Hurewicz homomorphism and is obtained by integrating the 1-form (4.2).
Applying completion to the extension . The corresponding sequence of Lie algebras 0 → p → g rel 2 → g rel 1 → 0 is exact in the category of pro-objects of MHS Q .
10.5. Variations of MHS over M an 1, * . The category MHS of Q-MHS is neutral tannakian. Let ω B be the fiber functor that takes a MHS to its underlying Q-vector Suppose that G is an affine group scheme over Q whose coordinate ring is a Hopf algebra in the category of ind-object of MHS. A Hodge representation of G is a G-module V , endowed with a MHS for which the corresponding coaction is a morphism of MHS. The action of π 1 (MHS) on O(G) respects its Hopf algebra structure. We can thus form the semi-direct product π 1 (MHS) ⋉ G. The Hodge representations of G are precisely the representations of π 1 (MHS) ⋉ G.
A proof of the following result is sketched in [27]. A more detailed proof of a more general result will appear in a joint paper with Gregory Pearlstein.
where ω B x denotes the fiber functor that takes a variation to the Q-vector space underlying its fiber over x.
This result applies to tangential base points as well. This follows as the weight filtration W • can be recovered from the relative weight filtration M • and the monodromy logarithm N . (Cf. [27,Lem. 4.3].) It implies that a Hodge representation of G rel * must also respect the weight filtration W • .
Proposition 10.6. If V is an admissible variation of mixed Hodge structure over M an 1, * whose monodromy representation factors through the relative completion of • V is a trivial local system -that is, if and only if V is a unipotent local system.
If V is a unipotent local system, then its monodromy representation factors through the unipotent completion of π 1 (M an 1, * , x). By Proposition 10.2, this is trivial when * = 1, 2. So V is a trivial local in this case, and therefore trivial as a variation of MHS as well. When * = 1, the discriminant map M an 1, 1 → C * induces an isomorphism on unipotent fundamental groups. So the last statement follows from the classification of unipotent variations of MHS in [31].
10.6. Density results. The forgetful functor induces a homomorphism G * → π 1 (MEM * , ω B ). There is thus a natural homomorphism Proof. Since MEM * → MHS(M 1, * , H) takes geometrically constant MEMs to constant variations of MHS, the surjection π 1 (MHS(M 1, * , H)) → π 1 (MEM * , ω B ) commutes with their projections to π 1 (MHS) and to π 1 (MTM, ω B ). So one has the commutative diagram The surjectivity of the middle map, which follows from Theorem 8.1, and the splitting π 1 (MEM, ω B ) ∼ = π 1 (MTM, ω B ) ⋉ π geom 1 (MEM, ω B ), imply the surjectivity of the left hand map. The last assertion follows as the canonical homomorphism Recall that K = K B is the kernel of the natural homomorphism π 1 (MTM, ω B ) → G m . It is prounipotent. Denote the kernel of π 1 (MEM * , ω B ) → π 1 (MTM, ω B ) by U MEM * . Since the commutative diagram (in which all fiber functors are ω B ) has exact rows and columns, we have: that is compatible with products. The conjugation action of G * on G rel * induces an action of π 1 (MHS F ) on H • (G rel * , V ). So H • (G rel * , V ) has a natural F -MHS. An explicit construction of this MHS is given in [27].
The spectral sequence of the extension 1 → G rel * → G * → π 1 (MHS F ) → 1 and the fact that Ext j MHSF (A, B) vanishes for all MHS A and B when j > 1, implies that there is an exact sequence (11.1) where Γ denotes the functor Hom MHS (F, ). For all objects V of MHS F (M 1, * , H), there is a natural homomorphism . In the case of modular curves, we get the best possible result.
is an isomorphism of MHS. Consequently, the homomorphism (11.2) is an isomorphism.
Plugging this into the exact sequence (11.1) yields the following useful computational tool. When m = 2n > 0, the extension corresponds to the Eisenstein series G 2n+2 . 14 In degree 2, we consider only extensions of real variations of MHS, and so, only real Deligne-Beilinson cohomology. The reason for doing this should become apparent in Section 21. We will compute the answer only for * ∈ {1, 1} as we shall not need the case * = 2.  ). This vanishes when m is odd. So suppose that m = 2n. The decomposition (9.1) implies that if r ≥ 2n + 2, where n > 0 when * = 1 and n ≥ 0 when * = 1, we have and that this ext group vanishes in the remaining cases.
11.2. The F ∞ action. In this section, we take * ∈ {1, 1}. Since M 1, * and the universal elliptic curve over it are defined over Z, complex conjugation (aka "real Frobenius") F ∞ ∈ Gal(C/R) acts on E an → M an 1, * . This implies that F ∞ acts on H R , and thus on H • (M an 1, * , S 2n H R ) as well. It extends to a C-linear involution of H • (M an 1, * , S 2n H C ). As is well known, this can be computed in terms of modular symbols. (Cf. [26,Lem. 17.7].) The action on the class of the Eisenstein series is given by: Lemma 11.6. The real Frobenius operator F ∞ multiplies the class of ψ 2n+2 in H 1 (M an 1, * , S 2n H C ) by −1. Proof. Since F ∞ commutes with the action of the Hecke operators, F ∞ multiplies the eigenform ψ 2n+2 by ±1. To determine this multiple we use the residue map The (Betti) copy of Q(r − 2n − 1) in H 1 (M an 1, * , S 2n H Q (r)) is spanned by (2πi) r−2n−1 ψ 2n+2 .

Weighted and Crystalline Completion
In this section we review the definitions of the ℓ-adic weighted and crystalline completion functors and make some initial observations about the crystalline completion of π 1 (M 1,/ * /Q , v o ). More details can be found in [28,29].
Fix a prime number ℓ. Suppose that Γ is a profinite group and that R is a reductive algebraic group over Q ℓ , endowed with a central cocharacter c : G m → R. Suppose that ρ : Γ → R(Q ℓ ) is a continuous, Zariski dense representation. Say that a representation V of R has weight m if its pullback to G m via c is of weight m. Since c is central, Schur's Lemma implies that every irreducible representation of R has a weight.
Consider the full subcategory R cts Q ℓ (Γ, ρ, c) of the category of continuous Γ-modules Rep cts Q ℓ (Γ) consisting of those Γ-modules M that are finite dimensional and admit a (necessarily unique) filtration where the action of Γ on each graded quotient Gr W m M factors through a rational representation R → Aut(Gr W m M ) of weight m via ρ. Note that every irreducible R-module is an object of R cts Q ℓ (Γ, ρ, c). This category is neutral tannakian over Q ℓ . The weighted completion of Γ relative to ρ is the affine group scheme π 1 (R cts Q ℓ (Γ, ρ, c), ω) over Q ℓ , where ω is the fiber functor that takes a module M to its underlying vector space. It is an extension There is a natural homomorphism ρ : Γ → π 1 (R cts Q ℓ (Γ, ρ, c), ω)(Q ℓ ) whose composition with the canonical quotient map π 1 (R cts Q ℓ (Γ, ρ, c), ω) → R is ρ. It is Zariski dense.
Levi's Theorem implies that the central cocharacter c : G m → R can be lifted to a cocharacterc : G m → π 1 (R cts Q ℓ (Γ, ρ, c), ω). So every object V of R cts Q ℓ (Γ, ρ, c) is a G m -module viac. Denote the subspace of V on which G m acts via the nth power of the standard character by V n . The inclusion V n ֒→ V induces an isomorphism V n ∼ = Gr W n V .
of the weight filtration that is natural with respect to morphisms of R cts Q ℓ (Γ, ρ, c). It is compatible with duals and tensor products.
Here we recall from [28] the setup for the weighted completion of The defining representation of G m thus corresponds to Q ℓ (1). Since this has weight −2, we define the central cocharacter to be the homomorphism c : G m → G m defined by z → z −2 . The weighted completion of Γ with respect to χ ℓ and c is A wtd ℓ = π 1 (R cts Q ℓ (π 1 (Spec Z[1/l], Q), ρ, c), ω).
12.1. Crystalline completion. Suppose that K ⊂ Q is a number field. Set G K = Gal(Q/K). We continue with the notation of the previous paragraph. Suppose in addition that there is a surjection Γ → G K with a distinguished splitting s : G K → Γ. Define an object V of R cts Q ℓ (Γ, ρ, c) to be crystalline (with respect to s) if the representation ρ • s : G K → V is unramified at all primes that do not lie over ℓ and that it is crystalline at all primes that lie over ℓ. 15 Let R cris Q ℓ (Γ, ρ, c) be the full subcategory of R cts Q ℓ (Γ, ρ, c) whose objects are the crystalline objects. It is tannakian. The ℓ-adic crystalline completion of (Γ, s) is defined to be π 1 (R cris Q ℓ (Γ, ρ, c), ω). It is a quotient of the weighted completion π 1 (R cts Q ℓ (Γ, ρ, c), ω). As in the case of weighted completion, every object of R cts Q ℓ (Γ, ρ, c) has a natural weight filtration W • , the functors Gr W n are exact. Each liftc of the central cocharacter c to π 1 (R cts Q ℓ (Γ, ρ, c), ω) defines a natural splitting of the weight filtration of each object of R cts Q ℓ (Γ, ρ, c).

12.2.
Crystalline completion of G Q . The setup is similar to that in Example 12.3. Let Γ = G Q , R = G m and ρ : G Q → G m (Q ℓ ) the ℓ-adic cyclotomic 15 In some applications it is natural relax this definition and allow crystalline representations to be ramified at a set S of primes. We will not need to do that here as M 1, * /Z has good reduction at all primes.
character χ ℓ . The central cocharacter c : G m → G m is defined by z → z −2 . We also take K = Q and the homomorphism Γ → G Q to be the identity. Define There is a canonical surjection A wtd Define c : G m → GL(H) to be the cocharacter λ → λ id H . It is central and gives H ℓ weight 1. Define , ω). To define the ℓ-adic crystalline completion of π 1 (M 1, * /Q , v o ), we take s to be the section of , ω). Both groups are extensions of GL(H ℓ ) by a prounipotent group: Proposition 12.5. There are homomorphisms the last two of which are surjective, such that the diagram commutes. The sequence G rel * ⊗Q ℓ → G cris,ℓ → A cris ℓ → 1 is exact and the morphism G cris,ℓ → A cris ℓ is split.
Proof. The first homomorphism is induced by the forgetful functor . The existence of the middle surjection follows from the fact that is a full subcategory of R cts Q ℓ . The existence of the right-hand square follows from the commutativity of the diagram The last statement follows from the right exactness of completion and from the fact that one has functors R cris Q ℓ (G Q , χ ℓ , c) → R cris Q ℓ → R cris Q ℓ (G Q , χ ℓ , c) whose composite is the identity.

Extensions of ℓ-adic sheaves over M 1, * /Q
Our goal in this section is to prove the ℓ-adic analogues of the results in Section 11. The work [43] of Martin Olsson, which we use as a "black box", is key. These results are not needed in the sequel. They are included to show how the computations in this paper are related to certain fundamental questions in number theory.
The ℓ-adic analogues of the groups Ext • MHS(M1, * ,H) (R, S m H(r)) are the groups H • (G cris,ℓ * , S m H ℓ (r)). In this section, we compute these cohomology groups in degree 1 and bound them in degree 2.
For convenience, we denote the category R cris (π 1 (M 1, * /Z[1/ℓ] , v o ), ρ ℓ , c) by MEM ℓ * , so that MEM ℓ * is equivalent to Rep(G cris,ℓ * ) and In this and subsequent sections, we will use the following notation. Denote the category of G Q modules that are unramified at all p = ℓ and crystalline at ℓ by C ℓ . For an object V of C ℓ , define This group is typically denoted by H • f (G Q , V ). We have chosen a non-standard notation to avoid a notational conflict -f will often denotes a cusp form of SL 2 (Z). By [28,Thm. 4.6], there is a natural isomorphism degenerates at E 2 as the kernel has virtual cohomological dimension 1 and because the extension is split. This implies that ) m > 0. When m = 0, the image of (13.1) equals the image of , Q ℓ (r)).
So the fiber of E ′ 2n+1 over t splits as a sum of extensions of Q ℓ by Q ℓ (j), with 1 ≤ j ≤ 2n+1. Nakamura [40,Thms. 3.3 & 3.5] shows that all but the last of these is trivial, and the last one is a non-zero multiple of the generator of H 1 sm (G Q , Q ℓ (2n + 1)). 16 This implies that the restriction of the E ′ 2n+1 to t is crystalline, so that (13.1) is surjective.
For convenience, we denote the category R cris (π 1 (M 1, The following technical computation will be needed in the next section.

A bound on
Proof. We will prove the theorem when * = 1. The proof of the case * = 1 is similar and is left to the reader. We begin by establishing the vanishing of H 2 (G cris,ℓ 1 , S m H ℓ (r)) when r < m + 2 and when m is odd. There are at least two ways to do this. Both exploit the vanishing of H 1 (G cris,ℓ 1 , S m H ℓ (r)) when r < m + 1 or m is odd. We will use Lie algebra cohomology. The other approach uses Yoneda extensions.
Denote the Lie algebra of U cris,ℓ * by u cris,ℓ * . Since and since there is a natural isomorphism , then m is even and r ≥ m+2.

Consequently, the group of GL(H)-invariant 2-cochains
is non-zero only when m is even and r ≥ m + 2. It follows that H 2 (u cris,ℓ 1 , S m H ℓ (r)) vanishes when m is odd and when r < 2m + 2.
There are natural homomorphisms ) General properties of weighted completion (cf. [28,Thm. 4.6]) imply that the righthand map is injective. Our next task is to show that the second mapping is injective. We do this using Yoneda extensions.
Suppose that the 2-extension in MEM ℓ 1 represents a class in the kernel of H 2 (G cris,ℓ 1 , S 2n H ℓ (r)) → H 2 (G wtd,ℓ 1 , S 2n H ℓ (r)) By Yoneda's criterion, there is a G wtd,ℓ 1 -module V and a commutative diagram with exact rows in Rep(G wtd,ℓ 1 ). We view V as a G Q -module via the homomorphism where the first homomorphism is the section induced by the base point t. To prove injectivity, it suffices to show that V is a crystalline G Q -module as the fact that it is a G wtd ℓ -module implies that it is unramified away from ℓ. The action of G wtd ℓ on V determines compatible weight filtrations M • on E, F and V .
Let B be the kernel of F → Q ℓ . By the pruning lemma below (Lemma 13.4), we may assume that each W • graded quotient of B is a sum of copies of S m H ℓ (r) where r > m or, equivalently, that B = M −2 B. In this case B is an extension for some d ≥ 0. This implies that E is also an extension The pushout of the extension Since F is crystalline, this is an extension of crystalline G Q -modules. Proposition 11.3 of [30] implies that V is also a crystalline G Q -module and therefore a G cris,ℓ 1 module. This completes the proof of injectivity. The final task is to show that the image of ). When n = 0, this is clear. To prove this when n > 0, we use the fact that G rel 1 ⊗ Q ℓ is a crystalline representation of G Q , or more accurately, that its coordinate ring O(G rel 1 ) ⊗ Q ℓ is a crystalline G Q -module. (That it is unramified away from ℓ is proved in [30], the full statement is proved in [43].) Let A Mod ℓ to be the Zariski closure of the image of G Q in Aut(G rel 1/Q ℓ ). 17 Representations of G Q that factor through G Q → A Mod ℓ (Q ℓ ) are crystalline. Denote by C cris,ℓ 1 the tannakian category of finite dimensional representations of G Q ⋉ G rel 1/Q ℓ . Its tannakian fundamental group (with respect the underlying Q ℓ vector space) is a split extension The splitting is induced by the base point t. There is a commutative diagram Now suppose that n > 0. Since H 2 (G rel 1 , S 2n H(r)) vanishes for all n and r, since H 0 (G rel 1/Q ℓ , S 2n H ℓ (r)) = 0 when n > 0, and since there are G Q -module isomorphisms ) and . Applying H 2 to the diagram (13.3) implies that the image of (13.2) is contained in H 1 sm (G Q , H 1 et (M 1,1/Q , S 2n H ℓ (r))). Our final task is to prove the Pruning Lemma.
Lemma 13.4 (Pruning Lemma). Suppose that * ∈ {1, 1}. Let G be either G wtd,ℓ * or G cris,ℓ * . If is an extension of G-modules, then there is a G-sub module A of B such that the extension (13.4) is pulled back from an extension Proof. All modules in this proof will be G-modules and all extension groups will be extensions of G-modules.

The Elliptic Polylogarithm
Recall from Corollary 6.10 that there is an object p p p of MEM 2 whose corresponding local system has fiber the Lie algebra p(E ′ , x) of π un 1 (E ′ , x) over the point [E, x] of M an 1,2 . Denote by p p p ′ the sub-local system whose fiber over [E, x] is the commutator subalgebra  By Proposition 7.5, p p p, p p p ′ and Pol ell 2 restrict to objects of MEM 1 , which we will also denote by p p p 1 , p p p ′ 1 and Pol ell 1 . The fiber of p p p 1 over the point The monodromy of the local system Pol ell 1 over M 1, 1 about the fiber of the G m -bundle M 1, 1 → M 1,1 is the unipotent automorphism of p(E ′ , v) induced by the element of π 1 (E ′ , v) that rotates the tangent vector once about the identity. Its logarithm spans a copy of Q(1) in Der p(E ′ , v) and induces a map  1)). This has infinite order when m ≥ 3 is odd, which implies that the (2n + 1)st sub extension of Pol ell 2 (the "(2n + 1)st elliptic polylog") is non-trivial in Ext 1 (H(1), S 2n+1 H(2n + 2)) ∼ = Ext 1 (Q, S 2n+2 H(2n + 2)) ⊕ Ext 1 (Q, S 2n H(2n + 1)).
Since the first summand is trivial, this implies that the second component of E 2n+1 is non-trivial. Finally, one can prove the non-triviality using ℓ-adic methods. Since p is a mixed elliptic motive, one can check non-triviality of an extension in any realization. Nakamura [40,Thms. 3.3 & 3.5] computed the ℓ-adic Galois action on p and showed that the extension above is given by a non-zero rational multiple of the Soulé character.

Simple Extensions in MEM *
We can now assemble the results of this part to give a computation of the extension groups of Q by S m H(r) in MEM * .  Remark 15.2. Since the fiber H of H over t is Q(0) ⊕ Q(−1), the fiber V over t of a generator of Ext 1 MHS(M1,1) (Q, S 2n H(2n + 1)) is an extension of Q by Q(1) ⊕ Q(2) ⊕ · · · ⊕ Q(2n + 1).
and thus determines elements It is easy to see that λ j = 0 when j < 2n + 1; the proof is in the next paragraph. The proof of the theorem implies that the class of λ 2n+1 is a non-zero rational multiple of the class of ζ(2n + 1). And indeed, this is exactly what one sees in the computations of Brown [9] and Nakamura [40,Thms. 3.3 & 3.5].
We know that N := log σ o acts on the limit V as a morphism of type (−1, −1). Since the residue of the form ψ 2n at q = 0 is non-zero, The map is non-zero and therefore injective. Suppose that 1 ≤ j ≤ 2n. Since j < 2n + 1, N : S 2n H → S 2n H maps Q(j) onto Q(j + 1). It thus maps the extension λ j of Q by Q(j) isomorphically into an extension of Q(1) by Q(j + 1) that lies inside S 2n H(2n + 1). But all such extensions are split, so λ j = 0.

Structure Theorems
The results of the preceding sections allow us to determine how the fundamental groups of the various categories MEM * are related.
Let ω be any fiber functor that is the composition of the fiber functor MEM * → MTM with any fiber functor MTM → Vec F . Denote the unipotent completion of The inclusion E ′ t ֒→ M 1,2 induces a restriction functor MEM 2 → Rep(P) and therefore a homomorphism P → π 1 (MEM 2 , ω).
The inclusion of a formal punctured disk D ′ ֒→ E ′ t induces a homomorphism Q(1) = π un 1 (D ′ , w o ) → P. We will denote this subgroup of P by G a (1). It is also the unipotent fundamental group of the fiber of M 1, 1 → M 1,1 . As above, restriction to the fiber induces a homomorphism G a (1) → π 1 (MEM 1 , ω) such that the diagram commutes.
Theorem 16.1. There are exact sequences Proof. It suffices to prove the result when ω = ω B . We first show that P is a normal subgroup of π 1 (MEM 2 , ω B ). Since the Lie algebra of P is an object of MEM 2 , there is a natural action of π 1 (MEM 2 , ω B ) on P. The composite is the inner action. It is injective as P has trivial center, which shows that P is a subgroup of π 1 (MEM 2 , ω B ).
To see that it is normal, consider the diagram Since the right-hand vertical map is surjective, and since P is a normal subgroup of G 2 , it follows that P is a normal subgroup of π 1 (MEM 2 , ω B ). We can thus consider the quotient group π 1 (MEM 2 , ω B )/P. Its representations are objects of MEM 2 that have trivial monodromy on each fiber of M 1,2 → M 1,1 , and are thus constant on each fiber. This implies that its representations are precisely the pullbacks of objects of MEM 1 along the projection. This implies that π 1 (MEM 2 , ω B )/P is isomorphic to π 1 (MEM 1 , ω B ).

Motivic Remarks
Suppose that * ∈ {1, 1}. We expect that the ext groups of the category MEM * to be faithfully represented in the various categories of realizations. To be more precise, set MHS * := MHS(M an 1, * , H) and MEM ℓ * = Rep(G cris,ℓ * ). The realization functors real MHS : MEM * → MHS * and real ℓ : MEM * → MEM ℓ * induce "regulator mappings" that are compatible with products. We have already seen that, in degree 1, the first is an isomorphism and the second is an isomorphism after tensoring with Q ℓ . We can also map to real Deligne cohomology In degree 1, this map is an isomorphism after tensoring with R. The story is more interesting in degree 2, where we have regulator mappings  The goal of this part is to find a presentation of π 1 (MEM * , ω DR ), or more accurately, a presentation of the Lie algebra u MEM * of its prounipotent radical. While we do not achieve this goal, we are able to come close, thanks to computations of Brown [9] and Pollack [44]. Specifically, we are able to determine the "quadratic heads" of all relations in u MEM * . If the variant Conjecture 17.1(i) of standard conjecture holds, then the quadratic head of every non-trivial minimal relation is non-trivial.
In Section 23, we show that the de Rham realization of every object of MEM * has a canonical bigrading which splits the Hodge filtration and both weight filtrations. This is an important tool. It means that u MEM * is the completion of a bigraded Lie algebra in the category of sl(H)-modules.

Presentations of Graded Lie algebras
Before attempting the problem of finding a presentation of π 1 (MEM * , ω), we first consider a more abstract setting. Suppose that G is an affine group scheme over a field of characteristic zero that is an extension of a connected, reductive group S by a prounipotent group U. Suppose that the Lie algebra g of G is graded: where g 0 is isomorphic to the Lie algebra s of S and u = n<0 g n is the Lie algebra of U. For simplicity, we will suppose that each g n is finite dimensional. The splitting g = s ⋉ u gives a Levi decomposition G ∼ = S ⋉ U of G. To give a presentation of G, it suffices to give a presentation of the associated graded Lie algebra u • of u in the category of S-modules.
The first step in finding a minimal such presentation of u • is to choose an Sinvariant splitting of the graded surjection This induces a graded S-equivariant surjection L(H 1 (u • )) → u • that induces the identity on H 1 .
Set f = L(H 1 (u • )). The ideal of relations r := ker φ is graded submodule of the commutator subalgebra [f, f]. The Lie algebra version of Hopf's theorem implies that, since φ induces an isomorphism on H 1 , there is a natural isomorphism So, loosely speaking, every pronilpotent Lie algebra in Rep F (R) has a minimal presentation of the form u ∼ = L(H 1 (u)) ∧ / H 2 (u) . Of course, one has to determine the mapping ψ. One part is easy to describe. Namely, the quadratic part.
Denote the lower central series of a Lie algebra h by The bracket f ⊗ f → f induces an isomorphism Examining the Chevalley-Eilenberg cochain complex of u • , one sees immediately that: is the dual of the cup product Equivalently, there is an exact sequence where ∆ denotes the coproduct; i.e., the graded dual of the cup product.
The final observation is that the canonical isomorphism implies that (provided that every irreducible S-module is absolutely irreducible) there is a natural isomorphism whereŠ denotes the set of isomorphism classes of irreducible S-modules and {V α } an S-module in the class α.
If we fix a Cartan subalgebra h of s and a set of positive roots, we can decompose s as s = n + ⊕ h ⊕ n − where n + (resp. n − ) is the nilpotent subalgebra consisting of positive (resp. negative) root vectors. So we can think of u • as being generated as an n − module by its highest weight vectors. Applying this to the isomorphism (18.1) gives the following result.
Proposition 18.2. The space of highest weight vectors of the generating space of a minimal presentation of u • projects isomorphically to The space of highest weight vectors in a minimal space r/[r, f] of relations projects isomorphically to Combined with Lemma 18.1, we see that the space r/(r∩L 3 f) of quadratic leading terms of the space of minimal relations is dual to the cup product.
commutes, where ∆ denotes the dual of the cup product.

Splittings
A fundamental fact about motives is that (say) the complex vector space underlying a mixed Hodge structure is naturally (though not canonically) isomorphic to the direct sum of its weight graded quotients. In Appendix B we show that we can simultaneously split the weight filtrations M • and W • on the various realizations of all objects of MEM * , as well as all G * and G cris,ℓ * modules. This will simplify the problem of writing down the presentations as it allows us to work in the category of bigraded Lie algebras with sl(H)-action, instead of in the much more slippery category of pronilpotent Lie algebras.
Specifically, let G be any of the groups π 1 (MEM * , ω), where * ∈ {1, 1, 2}. Then every G-module V has a natural bigrading It is natural in the sense that it is preserved by G-module homomorphisms. It is compatible with ⊗ and Hom. One can also choose the bigradings to be compatible with the functors MEM 2 → MEM 1 → MEM 1 .
Remark 19.1. One consequence of the proof is that the functors Gr M • Gr W • are exact. If one knew this in advance, one could use the fiber functor Gr M • Gr W • to prove the result. Analogous results hold for G * -modules and G cris,ℓ -modules. One can arrange for their splittings to be compatible with those of π 1 (MEM * , ω) via the homomorphisms G * → π 1 (MEM * , ω B ) and G cris,ℓ * → π 1 (MEM * , ω ℓ ) ⊗ Q ℓ .
Remark 19.2. In Section 23, we will construct a canonical Q-de Rham bigrading of each object of MEM * which splits both weight filtrations and also the Hodge filtration. This bigrading generalizes, and reduced to, the canonical grading of the Q-DR realization of a mixed Tate motive which splits the weight filtration and the Hodge filtration. where a = 2πiA and T = −b. This is the Q-de Rham basis used in [25]. In this part, we will be mainly working in homotopy, and thus with H(1) = H 1 (E t ). Note that SL(H) and SL(Ȟ) are naturally isomorphic, as are their Lie algebras. We will identify them.
Below we will take ω to be either ω B or ω DR . When we use ω B , we will use the Q Betti basis a, b ofȞ; when we use ω DR , we will use the Q − DR basis A, T ofȞ. This decomposition corresponds to a Levi decomposition Since a prounipotent group is determined by its Lie algebra, to give a presentation of π 1 (MEM * , ω), it suffices to give a presentation of u = u MEM * in the category of GL(H)-modules. To do this, it suffices to give a presentation of the bigraded Lie algebra  We may assume that these maps are compatible with the projections u MEM . We now fix a basis of H 1 (Gr u MEM * ). Its image under the section will be a generating set of Gr u MEM * . These generators will not be unique as they will depend on the choice of the section. More will be said about this in Remark 20.3.
For the rest of this section * ∈ {1, 1}. Denote by V n the 1-dimensional G mmodule Q on which G m acts by the nth power of the standard character. Let be the basis vector on which the motivic zeta value ζ m (2m + 1) defined in [8] takes the value 1. Let be the corresponding element of H 1 (Gr u MEM * ). The projection u MEM * → k takes σ σ σ 2m+1 to z 2m+1 . Let e 2n ∈ Ext 1 MEM * (Q, S 2n−2 H(2n − 1)) * ⊗ S 2n−2 H(2n − 1) be the element whose value on the class ψ 2n of the normalized Eisenstein series is the highest weight vector 2πib 2n−2 ∈ Q(1) of S 2n−2 H(2n − 1). Remark 20.4. The results of Section 29 imply that there is a canonical Q-de Rham choice of the homomorphism Namely, it is the homomorphism that takes z ∈ Gr M −4m−2 k to its component (denoted by z (4m+2) in Section 29) of the image of z in the sl(H) invariants of is identified with its W -graded quotient using the canonical Q-DR splitting of W constructed in Section 23.
Notation 20.5. Denote the adjoint action of the enveloping algebra of g MEM * on u MEM * by f · u. In particular, if x ∈ g MEM * and u ∈ u MEM * , then x j · u = ad j x (u). With this notation, S 2n−2 H(2n − 1) = span{e j 0 · e 2n : 0 ≤ j ≤ 2n − 2}. One also has the relation e 2n−1 0 · e 2n = 0. Note that Their images under the chosen section of L(H 1 (Gr u MEM * )) → Gr u MEM * will be a generating set, so that, for example, where f 1 := L(e j 0 · e 2n+2 , z 2m+1 : n ≥ 0, m ≥ 1, 0 ≤ j ≤ 2n) and f 1 := f 1 /(e 2 ). This is free. Denote the Lie subalgebra of f * generated by the e j 0 · e 2n+2 , j, n ≥ 0, by f geom * . The Lie algebra u geom * is a quotient of f geom * . We will refer to the e 2n with n ≥ 0 as geometric generators as they generate Gr u geom * . The z 2m+1 's will be called arithmetic generators as they come from lifts of elements of k. , it is useful to ponder their shape and structure. As above, * ∈ {1, 1}. In this and subsequent sections, we denote π 1 (MEM * , ω B ) by G MEM * and π 1 (MEM * , ω B ) geom by G geom * . Relations between the e 2n 's will be called geometric relations. Since k is free, there are no relations between the σ σ σ 2m+1 's and hence no relations between the z 2m+1 's modulo the e 2n 's. Since u geom * Remark 20.6 implies that the geometric relations of M -weight −2d have degree d in the {e 2j : j ≥ 0}. It also implies that the right hand side of the arithmetic relation (21.1) has degree 2m + 2 in the {e 2j : j ≥ 0} as it has M -weight −4m − 4.
A minimal space of geometric relations is a subspace of the sl(H) highest weight vectors in r geom * that injects into H 0 n − , r * /[r * , f * ] under the canonical projection Since k is free, [r * , f * ] ⊆ r geom * , which implies that any minimal space of geometric relations is isomorphic to H 0 (n − , r geom As remarked above, these have degree d in the e 2m 's and W -weight −2n− 2d. They occur as highest weight vectors of copies of S 2n H(d) in the relations.
21.1. Conjectural size and shape of the relations. In this section we explain how standard conjectures in number theory, if true, would constrain the size and form of a minimal set of relations of Gr u MEM * . In Section 25 we will give an unconditional proof that a set of minimal relations is of this form and give a partial computation of it. Corollary 18.3 implies that the highest weight vectors in the space of leading quadratic terms of a set of minimal relations of u MEM * of M -weight d and sl(H)-weight 2n is dual to the image of the map obtained by composing the cup product with a GL(H)-invariant projection Since k is free and acts trivially on is injective. The image is the set of quadratic leading terms of the geometric relations. The quadratic parts of geometric relations of M -weight −2d have degree d as expressions in the {e 2m : m ≥ 0}, but are quadratic in f * . 18 To better understand the relations we restrict to the case * = 1. Since u MEM where e 2 is central, there is no loss of generality. The generator e 2m+2 of f 1 (Betti or de Rham, according to taste) is dual to a generator of H 1 (G MEM * , S 2m H(2m + 1)) and is a highest weight vector of the unique copy of S 2m H in H 1 (f 1 ). When m = 0, the generator z 2m+1 of f 1 is dual to a generator of H 1 (G MEM 1 , Q(2m + 1)). The projection (21.3) is dual to an inclusion and thus corresponds to a line of highest weight vectors in H 1 (f 1 ) ⊗2 of sl(H)-weight 2n. The line lies in Gr M −2d as the highest weight vector of S 2n H(2n + d) has Mweight −2d. Since the cup product is anti-commutative, the image of S 2n H(2n + d) To get an idea of the shape of the relations and to see how they might be computed, we assume Conjecture 17.1(i) for the rest of this section. This means that we identify H Recall that V f denotes the two-dimensional real Hodge structure associated to a Hecke eigen cusp form f and that B 2n+2 denotes the set of normalized Hecke eigen cusp forms of weight 2n+ 2. Each V f,R decomposes as the sum of two 1-dimensional 18 For example, [e j 0 · e 2a , e k 0 · e 2b ], where a, b > 1 and j, k ≥ 0, is a quadratic element of f 1 , but has degree d = j + k + 2 in the em's.
eigenspaces V ± f of the de Rham Frobenius F ∞ . The Petersson inner product induces an isomorphism Hom(V ± f , R) ∼ = V ∓ f . Conjecture 17.1(i) and Proposition 11.7 imply that there is an isomorphism The copy of R when d is even corresponds to the Eisenstein series G 2n+2 .
Proposition 21.1. When * = 1, Conjecture 17.1(i) implies that the cup product (21.2) is surjective for all n and d.
Proof. The interpretation of Brown's period computations [9] in terms of Deligne-Beilinson cohomology [27, §7] implies that the image of the cup product (21.2) under the regulator reg R , after tensoring with R, is isomorphic to H 2 D (M 1,1 , S 2n H(2n + d)).
Corollary 21.2. If we assume the Conjecture 17.1(i), then each minimal relation is determined by its leading quadratic term. That is, the projections So, granted Conjecture 17.1(i), for each f ∈ B 2n+2 and each d ≥ 2, there is a map where ǫ is + when d is even and − when d is odd. 19 The map V ǫ f ֒→ Gr M −2d H 0 (n − , r 1 /(r 1 ∩ L 3 f 1 )) obtained by composing (21.4) with the projection is dual to the cup product. It is non-zero and therefore injective. The last statement of Proposition 21.1 implies that the image of this last map lies in the geometric part That is, the leading quadratic terms of each cuspidal relation are geometric. More is true: Proof. As pointed out above, there are no relations between the z 2m−1 's modulo the ideal generated by the e 2n 's. Since u geom When d ≥ 2 is even, write d = 2m + 2, where m ≥ 0. The dual of the composition of the cup product with the projection associated to the Eisenstein series G 2n+2 will contain the leading quadratic terms of the arithmetic relation (21.1) associated to the Eisenstein series G 2m+2 . Determining this map is a joint project with Francis Brown.

The Monodromy Representation
Any relation that holds in g MEM Restricting to this smaller algebra will simplify the computations in subsequent sections. For a basis v 1 , v 2 ofȞ and each n ≥ 0, define the derivation Here we are identifying L(Ȟ) with its image in Der L(Ȟ) under the inclusion ad : L(Ȟ) ֒→ Der L(Ȟ). Each ǫ 2n (v 1 , v 2 ) annihilates θ and is thus in Der 0 L(Ȟ), [25,Prop. 21.2]. 20 It is useful to note that if c 1 , c 2 ∈ C, then Recall the notation of Section 19.1. Set Pollack [44] found relations between the ǫ 2n . 22 The relevance of Pollack's computations to bounding the relations in Gr u MEM • comes from the following result.

The Canonical De Rham Splitting
The Q-vector spaces V DR that underlies a universal mixed elliptic motive V in MEM * is naturally isomorphic to its associated graded module • V DR with respect to the weight filtration M • . The canonical way to realize this isomorphism is via the identification 20 These derivations occur in the work [54] of Tsunogai on the action of the absolute Galois group on the fundamental group of a once punctured elliptic curve. They also occur in the paper of Calaque et al [12, §3.1]. 21 One can show that there is a unique copy of S 2n−2 H(2n − 1) in Gr W −2n Der 0 L(H). It has highest weight vector ǫ 2n . 22 Note that in [44], ǫ 2n denotes our ǫ 2n (a, −b).
induced by the inclusion F m V DR ∩ M 2m V DR ֒→ M 2m V DR . This grading splits both the Hodge and weight filtrations. As shown in Appendix B, this isomorphism can be lifted to a natural isomorphism with its associated bigraded module which splits the Hodge filtration and both weight filtrations. The purpose of this section is to show that this lift is canonical.
Theorem 23.1. For every object V of MEM * , there is a unique isomorphism where V DR is the Q-de Rham realization of the fiber V ∈ MTM of V over v o with the following properties: (i) The isomorphism is natural with respect to morphisms in MEM * and compatible with tensor products and duals. (ii) This bigrading of V DR refines the standard grading V DR ∼ = p F p W 2p V DR of the de Rham realization of a mixed Tate motive. That is, for each p ∈ Z, this isomorphism restricts to an isomorphism Proof. Denote π 1 (MEM * , ω DR ) by G * , its prounipotent radical by U * and its Lie algebra by g * . Results from Section 7 imply that there are natural homomorphisms G 1 → G 2 → G 1 . This means that every object of MEM * can be viewed as an object of MEM 1 . So it suffices to prove the result for * = 1. Applying Proposition B.1 to G 1 implies that such compatible bigradings which split the Hodge filtration and both weight filtrations exist.
To prove uniqueness, observe that such bigradings correspond to a lift of the central cocharacter G m → GL(H) that gives the Q-DR splitting of H to a cocharacter G m → G 1 . This cocharacter lies in F 0 W 0 M 0 G 1 and is unique up to conjugation by an element of F 0 W 0 M 0 U 1 . But since Gr u 1 is generated the e j 0 · e 2n and the z 2m+1 with n, m > 0, all of which lie in M −2 , and since F 0 ∩ M −2 = 0, F 0 W 0 M 0 U 1 = 0. Uniqueness follows.
Denote the Q-DR realization of the Lie algebra of the unipotent fundamental group of (E ′ t , w o ) by p.  [25, §15].) In order to exploit the formulas derived in [25] from the elliptic KZB connection, we need to show that these bigradings agree.
is the identity.
Proof. The isomorphism φ : p ⊗ C → L(A, T ) ∧ ⊗ C constructed from the KZB connection induces the isomorphism  It follows that all c n vanish, so that φ • ψ −1 (A) = A and φ • ψ −1 is the identity.
By identifying g MEM 1 with its associated bigraded Lie algebra via the de Rham splitting, we can regard each e 2n as an element of g MEM and that, in the notation of [26], the connection on V is of the form ∇ 0 + Ω, where ∇ 0 denotes the connection on Gr W n V that is described in Section 5.2 and Ω = n ψ 2n (ϕ ϕ ϕ 2n ) ∈ Ω 1 End V DR is the image of e 2n under the monodromy homomorphism Gr u DR 1 → End Gr V DR .

Pollack's Relations
In this section we recall Pollack's relations [44], which give an upper bound on the leading quadratic terms of these relations. If the cup product (21.2) is surjective, as would follow from Conjecture 17.1(i), this will give an upper bound on the space of minimal relations in u MEM 1 .
Pollack found all highest weight relations of degree d = 2 that hold between the ǫ 2n and all highest weight relations between the ǫ 2n of higher degree d > 2 that hold modulo a certain filtration that we now define.
Suppose that n is a Lie algebra. Denote its commutator subalgebra by n ′ . Define the filtration P • of n by where L m denotes the mth terms of the lower central series. The filtration P • of L(Ȟ) induces a filtration 23 on Der L(Ȟ) by It has the property that [P j n, P k n] ⊆ P j+k n.
So if δ acts trivially on H 1 (L(Ȟ)), then it acts trivially on H 1 L(Ȟ) ′ . Since the image of u MEM 1 in sl(H) is trivial, its image in Der 0 L(Ȟ) acts trivially on H 1 (L(Ȟ)) and therefore lies in P 1 Der 0 L(Ȟ). It follows that L m u MEM 1 is mapped into P m Der 0 L(Ȟ).
Regard S 2m H(2m + 1) be the copy of S 2a H in Gr u MEM 1 generated by e 2m+2 . If a ≥ b > 0, then the image of the bracket is The determination of the highest weight vectors in L 2 f 1 /L 3 of degree d is an exercise in the representation theory of sl 2 .
23 In Section 28, we will see that this is the filtration induced by the natural elliptic depth filtration of Der p.
spans the space of sl(H) highest weight vectors in the subspace Denote the standard cochain complex of SL 2 (Z) with coefficients in the leftmodule V by C • (SL 2 (Z), V ) and its differential by δ. Set where U = ST . We will identify a cuspidal cocycle with its value on S. The real Frobenius operator F ∞ acts on cuspidal cocycles via the formula , S m H) − be the eigenspace decomposition. The + (resp. −) eigenspace consists of polynomials with even (resp. odd) degree in a.
The map Z 1 cusp (SL 2 (Z), S 2n H) → H 1 cusp (M 1,1 , S 2n H) that takes a cuspidal cocycle to its cohomology class has kernel is spanned by the unique cuspidal coboundary δ(a 2n ) = b 2n − a 2n . There is thus a short exact sequence It is equivariant with respect to the F ∞ action. The class map has a section given by modular symbols.
Recall that the modular symbol of a cusp form f of SL 2 (Z) of weight 2n + 2 is the homogeneous polynomial if and only if where ǫ is + when d is even and − when d is odd.
Remark 24.4. The second statement can be rewritten to focus on the cocycle: This has sl(H)-weight 2N . There is one non-trivial such relation for each k > 1.

Pollack's Relations are Motivic
In this section we use Brown's period computations [9] to prove that there are relations in Gr u MEM 1 that project to Pollack's relations in Der L(H) /P 3 . If, in addition, we assume Conjecture 17.1, then we show that these lifts of Pollack's relations generate all relations in u MEM . We continue with the notation of the previous four sections. ⊗ C with is the even or odd part of the modular symbol of f , where ǫ is the sign of (−1) d . The Pollack relation corresponding to the trivial cuspidal cocycle lifts to the arithmetic relation The "tail" of r f,d is well defined only modulo the ideal generated by the r g,e where g has lower weight and e < d. Note also that if the Fourier coefficients of f are rational, then the coefficients of the quadratic part of r f,d a rational multiplies of a fixed complex number.
The arithmetic relations look a little nicer if we set The image of ǫ ǫ ǫ 2k in the derivation algebra is ǫ 2k . With this normalization, the arithmetic relations become When m = 2, these become This agrees with Pollack's computations of [z 3 , ǫ 2n ] for n ≤ 6 in Der 0 L(H) if one takes z 3 to 1/12 times his derivationz 3 , as w 4 1,n goes to his E 4 2n in the derivation algebra.
Baumard and Schneps [3] used combinatorial methods to prove that Pollack's cubic relations between the ǫ 2n in Der 0 L(H) /P 3 lift to relations in Der 0 L(H). Brown gives a stronger version of the second statement in [10]. Since we have proved that Pollack's relations lift to relations in u MEM , it follows that all of his relations lift to relations in the derivation algebra. Denote by P • the filtration of Aut p induced by the filtration P • of p. It is obtained by exponentiating the filtration P • of Der p defined above. It restricts to a filtration P • of S * . One has the following factorizations In the rest of the proof, ǫ d ∈ {+, −} is the sign of (−1) d and ǫ ′ d is the sign of (−1) d+1 .
Recall from [9] that when n > 0, the Haberlund-Petersson inner product induces a non-singular pairing under which the sequence (24.1) is dual to the sequence 0 → H 1 cusp (M an 1,1 , S 2n H) → H 1 (M an 1,1 , S 2n H) → S 2n H/ im e 0 → 0. The duality (Lemma 18.1) between cup products and quadratic heads of the relations implies that the Pollack relations of degree d with SL 2 highest weight 2n correspond to a surjection where [ ] * denotes dual, and that the restriction of r S to the image of the cup product j+k=n+d−2 H 1 S 1 /P 3 , S 2j H(2j + 1) ⊗ H 1 S 1 /P 3 , S 2k H(2k + 1) of the cup product with the projection to the F ∞ invariant part of the real Deligne The corresponding result holds for * = 1, 2 as well as the extra relations are all quadratic and their dual cohomology classes account for the additional cohomology. Details are left to the reader.
From the discussion in Section 18, we know that the map which takes a minimal relation to its quadratic head, is injective if and only if the cup product is surjective for all n and d. Since, by Theorem 25.3, the image of the cup product under  The indeterminacy in the tails of the relations r f,d and the indeterminacy in the matric Massey products correspond. 26.5. Motivic Sheaves. Ayoub [2], using the work of Voevodsky [55] has constructed a category of motivic sheaves. Arapura [1], using the work of Nori, has also constructed a category of motivic sheaves. These constructions are not known to be equivalent. This makes sense when * ∈ { 1, 2} as M 1, * /Z is a scheme in these cases. When * = 1, define a motivic sheaf over M 1,1/Z to be a G m -invariant motivic sheaf on M 1, 1/Z that is trivial on G m orbits.
An affirmative answer to this question will imply that the specialization of a universal mixed elliptic motive (as a set of compatible realizations) to a point [E] ∈ M 1,1 (F ) corresponding to an elliptic curve over F will actually be the set of realizations of a motive over F . Universal mixed elliptic motives are related to the study of the unipotent fundamental group of P 1 − {0, 1, ∞} via degeneration to the nodal cubic E 0 . This is because the nodal cubic can be identified with P 1 /Z with 0 and ∞ identified. The corresponding group is E 0 = G m/Z and the associated punctured elliptic curve E ′ 0 is G m − {id} = P 1 − {0, 1, ∞}. Unless otherwise stated, in this part we will work with Q-de Rham realizations and its canonical bigrading constructed in Section 23.

Degeneration to the Nodal Cubic
In this section we recall some basic formulas from [25, Part 3] which were deduced from the elliptic KZB connection [12,39]. Denote the natural parameter on P 1 Z by w, so that P 1 Z − {0, 1, ∞} = Spec Z[w, 1/w, 1/(w − 1)]. Denote the tangent vector ∂/∂w at w = 1 by w o . We can identify the nodal cubic E 0 (the fiber of the universal elliptic curve over q = 0) with P 1 with 0 and ∞ identified.
w o ) on unipotent fundamental groups. The induced Lie algebra homomorphism is not a morphism in MEM 1 , but it is a morphism of MTM and is thus π 1 (MTM)equivariant.
To write down formulas, we now identify the de Rham realizations of k and Lie π un 1 (P 1 − {0, 1, ∞}, w o ) with the completions of their M • graded quotients L(σ σ σ 3 , σ σ σ 5 , σ σ σ 7 , σ σ σ 9 , . . . ) and L(X 0 , respectively via the canonical Q-de Rham splitting of mixed Tate motives. Likewise, we identify p := Lie π un 1 (E ′ t , w o ) DR with the completion of its associated (M • , W • )bigraded module via the Q-de Rham splitting of mixed elliptic motives constructed in Section 23. To fix notation, recall from Section 19.1 thať where T spans the copy of Q(0) DR and A spans the copy of Q(1) DR . The associated bigraded of p is naturally isomorphic to L(Ȟ DR ) = L(A, T ).
In [25, §18] it is shown that, after identification with associated graded Lie algebras as above, the map induced by (27.1) is given by where · denotes the adjoint action of Q A, T = U L(A, T ) on L(A, T ). This homomorphism is injective and k-equivariant. 26 Recall from Section 5 that σ o denotes the positive integral generator of the fundamental group π 1 (D * , t) of the q-disk, which is isomorphic to Q(1). It acts on π un 1 (E ′ t , w o ) and thus on p. The image of σ o in Aut p lies in the prounipotent subgroup W −2 Aut p and thus has a logarithm. Set In [25, §13] it is shown that, after identification with the associated bigraded module, where B 2m denotes the 2mth Bernoulli number. It commutes with the action of k as it spans a copy of Q(1).

Depth Filtrations
As in the classical case, there is a natural depth filtration in the elliptic case. The classical and elliptic depth filtrations are quite closely related via the weight filtration W • . Unfortunately, the link between them is not as straightforward as one might hope, as we shall see.
In this section we set E = E t and U = P 1 − {0, 1, ∞}. We will denote the Lie algebra of π un 1 (X, w o ) by p(X), when X = U , G m , E, and E ′ . Recall that the depth filtration D • of the Lie algebra π un 1 (U, w o ) is defined by where L d denotes the dth term of the lower central series. In the elliptic case, define 27 These filtrations are motivic and thus preserved by k. The first is a filtration of p(U ) in MTM and the second is a filtration of p(E ′ ) in MEM 1 . They can thus be described on the associated graded modules: D d L(X 0 , X 1 ) is the ideal spanned by the Lie monomials whose degree in X 1 is ≥ d; and D d L(A, T ) is the ideal spanned by the Lie monomials in which θ := [T, A] occurs at least d times.
Remark 28.1. Note that Gr M • p(E ′ ) is naturally isomorphic to the T -adic completion of L(A, T ).

The morphisms
preserve the base point w o and thus induce homomorphisms That the vertical homomorphisms are injective follows from the fact that a Lie subalgebra of a free Lie algebra is free. It follows that p(U ) → p(E ′ ) preserves D • .
Proposition 28.2. The inclusion p(U ) ֒→ p(E ′ ) is strictly compatible with the depth filtrations. That is, 27 This is the filtration P • defined in Section 24.
Proof. Since the filtrations are motivic, we can work with the associated gradeds. The Lie algebra D 1 L(X 0 , X 1 ) is freely generated by the set {X m 0 · X 1 : m ≥ 0}. Since it is free, it follows that for all m ≥ 1, D m L(X 0 , X 1 ) = L m D 1 L(X 0 , X 1 ). This remains true after completion when we replace "generated" by "topologically generated". The Lie algebra D 1 L(A, T ) is also free. Its abelianization is naturally isomorphic to the free SymȞ-module generated by θ. The map (27.2) induces the inclusion which implies the result.
The weight filtration W • of p(E ′ ) restricts to a filtration of p(U ): The three filtrations D • , M • and W • are related by the "convolution formula". 28 Proof. Identify p(U ) with L(X 0 , X 1 ) ∧ and p(E ′ ) with L(A, T ) ∧ . We first show that if m = n + d, then . Suppose that w is homogeneous element of L(X 0 , X 1 ) of bidegree (a, b) in X 0 and X 1 . Then w ∈ W −2n ∩ D d if and only if b ≥ d and a + b ≥ n. These conditions imply that a + 2b ≥ m. On the other hand, the corresponding word in L(A, T ) ∧ lies in W −a−2b L(A, T ) ∧ , which is contained in W −m when a + 2b ≥ m. It follows that To prove the reverse inclusion, define a filtration G • on L(X 0 , X 1 ) ∧ by The first part implies that the identity L(X 0 , X 1 ) ∧ , G • → L(X 0 , X 1 ) ∧ , W • is filtration preserving. To prove G • = W • , it suffices to show that the identity induces an isomorphism on associated gradeds. An easy linear algebra argument implies that Gr G −m L(X 0 , X 1 ) ∼ = n+d=m Gr M −2n Gr d D L(X 0 , X 1 ). 28 The convolution of two filtrations F • and G • of a vector space V is the filtration F * G defined by (F * G) n V : This implies that Gr G −m L(X 0 , X 1 ) is spanned by the monomials w in X 0 and X 1 of bidegree (a, b) with a + 2b = m. Since these also span Gr W −m L(X 0 , X 1 ), the identify induces an isomorphism on associated gradeds.
Remark 28.4. One might suspect that one can invert this formula to get a formula for D • in terms of M • and W • . However, while it is true that The point being that the right hand side is not the convolution of W • and M • .
Remark 28.5. The proposition does not hold if U is replaced by E ′ . For example, It is also a pro-object of MTM. The image of the natural action of k on p(E ′ ) lands in Der N p(E ′ ), so that we have the commutative diagram The kernel of the restriction map res is spanned by ǫ 0 . Since this has weight 0, the restriction map is an isomorphism after applying W −1 . The depth filtrations of Der p(E ′ ) and Der p(U ) induce depth filtrations on Der N p(E ′ ) and Der E p(U ), respectively.
The following useful result was proved by Pollack [44, 4.5]. Note that it is important that δ(θ) = 0. The derivation δ = ad A is not in  We can now prove that the elliptic and classical depth filtrations agree.
Proposition 28.8. The restriction mapping Der N p(E ′ ) → Der E p(U ) preserves the depth filtration and is strict with respect to it.
This implies that the restriction mapping Der N p(E ′ ) → Der E p(U ) respects the depth filtrations. It remains to show that the restriction mapping is strictly compatible with the depth filtrations. Suppose that the restriction of δ ∈ Der N L(A, T ) to L(X 0 , X 1 ) ∧ is in D d Der L(X 0 , X 1 ). That is, that δ(X 0 ) ∈ D d L(X 0 , X 1 ) ∧ . Then equation (28.1) implies that δ(A) ∈ D d L(A, T ). Applying Lemma 28.7, we see that δ ∈ D d Der 0 L(A, T ), as required.
So the depth filtration on k can be computed by pulling back the depth filtration either from Der p(U ) (the "classical" case) or from Der p(E ′ ) (the "elliptic" case). Next, we use this to show that the depth filtration on k is closely related to the elliptic weight filtration W • . This is motivic as it is the kernel of the morphism Der p(U ) → p(U ) that takes the derivation δ to δ(log γ 1 ). So it is a pro-object of MTM that is filtered by W • . The image of the canonical homomorphism k → Der p(U ) is contained in Der 0 p(U ). Choose any element γ 0 of π 1 (U, v o ) that, together with γ 1 , generates π 1 (U, v o ). Define a linear map φ : p(U ) → Der 0 p(U ) by f → δ f , where δ f (log γ 0 ) = [log γ 0 , f ]. This has kernel Q log γ 0 . Note that φ is not motivic, or even a morphism of MHS. However, its restriction to M −4 p(U ) is strict with respect to the depth filtration and both weight filtrations as we now show.
Since log γ 0 is not in M −4 p(U ) and since M −2n is the nth term of the lower central series of p(U ), the restriction Similarly, when m ≥ 3, δ f ∈ W −m Der p(U ) if and only if f ∈ W −m p(U ). In other words, (28.2) is strict with respect to W • as well.
It is well known that the image of k → M −4 Der 0 p(U ) is contained in the image of φ. Since M −4 p(U ) → Der 0 p(U ) is injective, this lifts to a (non-motivic) map k → p(U ). The result now follows from Proposition 28.3 and the fact that p(U ) → Der 0 p(U ) is strict with respect to the depth filtration and the two weight filtrations.
Although we cannot "invert" the convolution formula, we nonetheless are able to express the depth graded quotients of k in terms of the classical and elliptic weight filtrations.

The Infinitesimal Galois action and Ihara-Takao Congruences
The Ihara-Takao congruence (Theorem 29.6 below) was proved numerically by Ihara and Takao in [32] and proved again using modular symbols by Schneps [46]. In this section, we use Pollack's relations to give a more conceptual explanation of the Ihara-Tako congruence, which makes it clearer why cusp forms impose relations in the depth graded quotients of k. We begin with a discussion of the infinitesimal Galois action φ E : k → Der p(E ′ ). We continue with the notation of the previous section. In particular, U denotes P 1 − {0, 1, ∞}.
The derivations ǫ 2n are highest weight vectors. As we will see below, to "first order", the images of the z 2m+1 in Der 0 L(Ȟ) are lowest weight vectors.

Its Betti counterpart iš
These are lowest weight vectors for sl(H). 30 We will identify k with the completion of Gr M Proof. Denote the normalizer of the image of a Lie algebra homomorphism a → Der p(E ′ ) by N (a). Since g MEM After identification with associated bigradeds, we see that, modulo geometric derivations, the image of k is contained in [Der 0 L(A, T )] sl(H) .
Since the sl(H)-invariants have the property that their M -and W -weights are equal by (19.2), σ σ σ (r) 2m+1 must be geometric when r = 4m + 2. The second assertion follows immediately from the fact that . 30 The equalityǫ B 2n = (2πi) 2n−1ǫDR 2n reflects fact thatǫ 2n spans a copy of Q(2n−1) in Gr W
The next result is an immediate consequence of the fact that the odd W -graded quotients of u geom 1 vanish.

Appendix A. Relative Weight Filtrations
Universal mixed elliptic motives are mixed Tate motives with additional structure. This extra structure includes a second weight filtration W • and a nilpotent endomorphism N of V that preserves W • . Every universal mixed elliptic motive V has two weight filtrations: its weight filtration M • as an object of MTM and the second weight filtration W • . One of axioms of a universal mixed elliptic motive is that M • be the relative weight filtration of the nilpotent endomorphism N of the filtered vector space (V, W • ). In this section we review Deligne's definition [13] of the relative weight filtration of a nilpotent endomorphism of a filtered vector space. More information about relative weight filtrations can be found in [51]. A concise exposition is given in Section 7 of [23].
A.1. The weight filtration of a nilpotent endomorphism. There is a natural weight filtration of a vector space associated to a nilpotent endomorphism N of it. (ii) for each k ∈ Z, N k : Gr The filtration W (N ) • of V is called the weight filtration of N . Note that W (N ) • is centered at 0. When V is a motive of weight m, it is natural to reindex the weight filtration of a nilpotent endomorphism N of V so that it is centered at m. The shifted filtration Example A.2. Let H = Cw ⊕ Ca, regarded as a vector space of weight 1. Let N be the nilpotent endomorphism a ∂ ∂w of H. It induces a nilpotent endomorphism of V = S n H, the space of homogeneous polynomials in a and w of degree n, which we regard as a vector space of weight n. The shifted monodromy weight filtration M • of V is obtained by giving a weight 0 and w weight 2. The monomial a n−j w j has weight 2j. Then M k V = span of the monomials a n−j w j of weight ≤ k.
A.2. The weight filtration of a nilpotent endomorphism of a filtered vector space. Now suppose that N is a nilpotent endomorphism of a filtered finite dimensional vector space V over a field of characteristic zero. That is, V has a filtration 0 ⊆ · · · ⊆ W m−1 V ⊆ W m V ⊆ W m+1 V ⊆ · · · ⊆ V which is stable under N .
Since N preserves the weight filtration, it induces a nilpotent endomorphism This result will follow from the following lemma. Set diag(t 1 , t 2 ) := t 1 0 0 t 2 .
Define a homomorphism c : G m → GL 2 by c(t) = diag (1, t). This is a section of det : GL 2 → G m .
Lemma B.2. Suppose that F is a field of characteristic 0 and that is a commutative diagram of affine F -groups with exact rows and where K and U are prounipotent and σ is a section. If s A : G m → A is a section of A → G m such that ρ G • σ • s A = c, then there exists a section s G : GL 2 → G that extends σ • s A in the sense that the diagram Proof. Denote ρ −1 G im c by G c . This is an extension of G m by U . Choose a splitting s : GL 2 → G of ρ G . Then s induces a splitting s ′ : G m → G c . Since ρ G • σ • s A = c, the image of σ lies in G c . Levi's theorem implies that there is an element u of U such that σ = us ′ u −1 . Now define s G = usu −1 . Levi's Theorem applied to