Mixed elliptic motives

Alexander Goncharov

September 25, 1997

Contents

1 Introduction 1

2 Two basic examples: a survey 8

3 Mixed motives and motivic Lie algebras 14

4 Conjectures on the Galois group of the category of mixed elliptic motives 20

5 The complexes B(E, n)^{•} and B^{∗}(E, n)^{•} 32

6 The regulator integrals, Eisenstein - Kronecker series and a conjecture on L(Sym^{n}E, n + 1) 40

7 The complexes B(E; n)^{•} and motivic elliptic polylogarithms 49

8 Elliptic Chow polylogarithms and generalized Eisenstein - Kronecker series 57

1 Introduction

Summery. Let E be an elliptic curve over an arbitrary field k and H the motive H^{1}(E)(1). We define complexes B(E; n+2)^{•} and conjecture that they are quasiisomorphic to RHom_{MM_{k}}(ℚ, Sym^{n}H(1)). If k is a number field this together with Beilinson’s conjecture on regulators leads to a precise conjecture expressing the special values L(Sym^{n}E, n + 1) via the classical Eisenstein-Kronecker series. It can be considered as an elliptic analog of Zagier’s conjecture.

We give a simple motivic interpretation of the elliptic polylogarithms and show how it together with the motivic formalism implies that the complexes B(E; n + 2)^{•} should map naturally to RHom_{MM_{k}}(ℚ, Sym^{n}H(1)). When E degenerates to the nodal curve our complexes lead to the motivic complexes from [G1-2] reflecting the properties of the classical polylogarithms.

The complex B(E; 3)^{•} was constructed in [GL]. The groups similar to H^{1}B(E; n+ 2)^{•} were also discussed in [W], [W2], where it was conjectured that they inject into Ext_{MM_{k}}^{1}(ℚ, Sym^{n}H(1)).

We formulate several conjectures about the structure of the motivic Galois group of the category of mixed elliptic motives generalizing some conjectures about the mixed Tate motives ([G1-2]). In the end we define the generalized Eisenstein-Kronecker series which should be related to L(Sym^{n}E, n + m). For n = 1, m = 2 this was conjectured in [D3] and proved in [G3].

1. The trichotomy: special values of L-functions, motivic complexes and motivic Galois groups. Let E be an elliptic curve over a number field k and L(Sym^{n}E, s) the L-function of the n-th symmetric power of h^{1}(E). The seminal Beilinson conjecture relates the special values of the L-function of a motive X over a number field, considered up to a ℚ^{∗}-factor, with the volume of the image of certain pieces of the algebraic K-theory of X under the regulator maps. However for symmetric powers of elliptic curves one should be able to say more about the special values.

It is natural to adress the problem in the language of motives. Let MM_{k} (resp. MM_{X}) be the (hypothetical) abelian ℚ-category of all mixed motives over a field k (resp. all mixed motivic sheaves over a regular scheme X, [Be]). Let ℚ(−1) := h^{2}(P^{1}) be the Tate motive, ℚ(n) := ℚ(1)^{⊗n} for any integer n and M(n) := M ⊗ ℚ(n).

Let E be an elliptic curve over a field k. Then h^{1}(E) is a pure motive of weight 1. The cup product defines an isomorphism Λ^{2}h^{1}(E) ⟶ ℚ(−1). Set H := h^{1}(E)(1). It is a simple object of weight −1. The elliptic motives are the direct summands of the motives H^{⊗n}. They form a rigid abelian tensor category P_{E}.

Example. If E has no complex multiplication then P_{E} is equivalent to the category of finite dimensional GL_{2}-modules over ℚ. The objects S^{n}H(m) are simple and mutually non isomorphic. Any simple object in P_{E} is isomorphic to one of them.

The category M_{E} of mixed elliptic motives is the smallest abelian tensor subcategory of MM_{k} which contains H and is closed under extensions.

For an integer a let L^{∗}(Sym^{n}E, a) be the leading coefficient of the Taylor expansion of the L-function at s = a. For a given pair of integers n, m there are the following intimately related problems:

Problem A. Find explicit formulas for special values of the L-functions of pure elliptic motives, i.e.

L^{∗}(Sym^{n}E, n + m) E/k, k is a number field

Problem B. Construct explicitly elliptic motivic complexes

RHom_{MM_{k}}(ℚ(0), S^{n}H(m)), k is an arbitrary field  (1)

They may be non zero only if −n − 2m, the weight of the motive Sym^{n}H(m), is negative.

Problem C. Find a precise description of the Galois group of the category of mixed elliptic motives.

A weaker version of the Problem A concerns the special values of L-functions up to a ℚ^{∗}-factor. We will refer to it as Problem A*.

We are making sense of the motivic Ext’s in a usual way. Namely, let E^{(n)} be the kernel of the sum map

E^{n+1} ⟶ E,  (x_{1}, ..., x_{n+1}) ⟼ x_{1} + ... + x_{n+1}  (2)

For a group A living on E^{n+1} the notation A_{sgn} means the part alternating under the action of the group S_{n+1}. Beilinson’s description of Ext_{MM_{X}}^{i} (ℚ(0), ℚ(n)) implies (see lemma 3.4) that one should have

Ext_{MM_{k}}^{i}(ℚ(0), S^{n}H(m)) = gr_{n+m}^{γ}K_{n+2m−i}(E^{(n)})_{sgn} ⊗ ℚ  (3)

(Here γ is the γ-filtration on the K-groups). If k is a number field these groups are expected to be zero for i > 1. Beilinson’s regulator map

Ext_{MM_{k}}^{1}(ℚ(0), S^{n}H(m)) ⟶ Ext_{ℝ−MHS}^{1}(ℝ(0), S^{n}H(m))

is provided by the realization functor from MM_{k} to the category ℝ − MHS of mixed Hodge structures over ℝ.

Problem A is the deepest one. It is of arithmetic nature, while Problems B and C are geometric. We expect such a solution of Problem C that gives the desired answer for Problem B. This answer in the case of number fields should resolve Problem A*. The regulator map should be constructed first over ℂ, brining analysis into the picture and providing a key for Problem B.

In these problems one can consider the category of mixed motives generated by powers of any simple pure motive X. However the case of mixed elliptic motives seems to be especially interesting.

To smell the flavor of the problems let us look at similar questions for the category of mixed Tate motives over a field k (i.e. the rigid tensor subcategory of MM_{k} generated by the Tate motive ℚ(1) ).

Then Problem A is about special values ζ_{k}(n) of the Dedekind zeta function of a number field k. The ideal answer to Problem A* is given by Zagier’s conjecture [Z2].

In Problem B we are looking for complexes RHom_{MM_{k}}(ℚ(0), ℚ(n)) for an arbitrary field k. In [G1],[G2] we have constructed complexes B(ℚ(n); k)^{•}:

B_{n}(k) ⟶ B_{n−1}(k) ⊗ k^{∗} ⟶ ... ⟶ B_{2}(k) ⊗ Λ^{n−2}k^{∗} ⟶ Λ^{n}k^{∗}  (4)

which reflect the properties of classical n-logarithm function Li_{n}(z). Here B_{n}(k) is the quotient of ℤ[k^{∗}] along a certain subgroup R_{n}, which in the case k = ℂ is the subgroup of all functional equations for the classical n-logarithm function. It is placed in degree 1. These complexes ⊗_{ℚ} were conjectured to be quasiisomorphic to RHom_{MM_{k}}(ℚ, ℚ(n)).

A detailed exposition of this philosophy in the two simplest cases, for the motives ℚ(2) and H(1), is given in chapter 2.

According to the Tannakian formalism the category of mixed Tate motives a field k is equivalent to the category of finite dimensional representations of its Galois group. The Galois group is a semidirect product of 𝔾_{m} and a prounipotent algebraic group scheme U(k) over ℚ. Let L(k) be the Lie algebra of U(k). The action of 𝔾_{m} provides a grading L(k)_{•} = ⊕_{n≥1}L(k)_{−n} (it is negatively graded thanks to a weight argument).

The answer to the Problem C for mixed Tate motives which we have in mind is this. Let I(k)_{•} := ⊕_{n≥2}L(k)_{−n}. It is an ideal, and L(k)_{•}/I(k)_{•} = (k_{ℚ}^{∗})^{∨}. (Here V → V^{∨} is a duality between the inductive and projective limits of finite dimensional ℚ-vector space.)

The Freeness Conjecture. ([G1],[G2]) I(k)_{•} is a free graded pro-Lie algebra generated by the groups B_{n}(k)^{∨}, n ≥ 2, sitting in degree n.

For a more precise version see Conjecture 1.20 in [G2], where we proved that it is equivalent to the description (4) of the motivic complexes.

There are several other candidates for the motivic complexes, for example Bloch’s beautiful complexes of higher Chow groups [Bl] and their versions. However the complexes (4) are the smallest possible and the only ones directly related just to the classical polylogarithms. For a ”cycle” construction of the motivic Lie algebra see [BK].

In this paper I will formulate an elliptic analog of the freeness conjecture, see conjecture 1.2 below.

The Tannakian formalism tells us that there exists a pro-Lie algebra L(E) in the tensor category P_{E} such that the category of mixed elliptic motives is equivlent to the category of modules over L(E) in the category P_{E}. There is a different tensor structure ⊗′ on the category of pure elliptic motives. Assume that E has no complex multiplication. Then S^{n}H(m)⊗′ S^{n′}H(m′) = S^{n+n′}H(m+m′). Denote by P_{E}^{∗} the category of pure elliptic motives with this tensor structure.

Theorem 1.1 Let us assume for simplicity that E is not a CM curve. Then there exists a differential graded pro-Lie algebra L̃^{∗}(E) in the category P_{E}^{∗} such that for (m, n) ≠ (0, 0) one has

H_{P_{E}^{∗}}^{i}(L̃^{∗}(E))_{S^{n}H(m)^{∨}} = gr_{n+m}^{γ}K_{n+2m−i}(E^{(n)})_{sgn} ⊗ ℚ

Here V_{M} is the M-isotypical component of a simple object V in P_{E}^{∗}. The DG pro-Lie algebra L̃^{∗}(E) is constructed in s.4.4.

Conjecture 1.2 Assume that k = k̄.

a) L(E) has a unique quotient L^{∗}(E) which is a pro-Lie algebra with respect to the both tensor structures on the category of pure elliptic motives and H_{P_{E}^{∗}}^{•}(L^{∗}(E)) = H_{P_{E}}^{•}(L(E)).

b) There is an ideal I^{∗}(E) ⊂ L^{∗}(E) with an abelian quotient

L^{∗}(E)/I^{∗}(E) = (k^{∗})^{∨} ⊠ H ⊕ J^{∨} ⊠ H

which should be a free Lie algebra in the tensor category P_{E}^{∗}.

More about the Lie algebra L^{∗}(E) in s. 4.8. The conjectures about L^{∗}(E) imply a very specific structure for the elliptic motivic complexes.

Assuming the following vanishing conjecture (a version of the Beilinson-Soule conjecture; the left hand side is defined via (3))

Ext_{MM_{k}}^{i}(ℚ(0), S^{n}H(m)) = 0 for i < 0

one can show that H_{∂}^{j}(L̃^{∗}(E)) = 0 for j < 0. Here ∂ is the differential in L̃^{∗}(E).

Conjecture 1.3 Assume k = k̄.

a) H_{∂}^{j}(L̃^{∗}(E)) = 0 for j ≠ 0.

b) H_{∂}^{0}(L̃^{∗}(E)) is isomorphic to the pro-Lie algebra L^{∗}(E) anticipated in conjecture 1.2.

Remark. Let G be a reductive group. Then there is a new tensor structure the category of finite dimensional G-modules given by V_{λ} ⊗ V_{μ} ⟶^{=} V_{λ+μ}, where V_{λ} is the G-module with the highest weight λ. Notice that the category of all pure motives is equivalent to the category of finite dimensional modules over a pro-reductive group.

2. The complexes B(E, n)^{•}. In chapters 5-7 we study Problems A^{∗} and B in the case m = 1. We construct a complex B(E, n + 2)^{•} which is conjecturaly quasiisomorphic to RHom_{MM_{k}}(ℚ, Sym^{n}H(1)). When E is an elliptic curve over a number field k this leads to a precise conjecture on special values L(Sym^{n}E, n + 1).

This complex is an elliptic deformation of the complex (4): if E is a nodal curve and k = k̄, it is quasiisomorphic to (4).

Remark. A better notation for the group B_{n+2}(E) and the complex B(E, n+ 2)^{•} would be B_{S^{n}H(1)} and B(S^{n}H(1))^{•}.

Let us assume first that k = k̄. Let ℤ[X] be the free abelian group generated by a set X, {x} are the generators. We will define subgroups

R_{n}(E/k) ⊂ ℤ[E(k)], and set B_{n}(E) := \frac{ℤ[E(k)]}{R_{n}(E/k)}

Put formally B_{0}(E) = ℤ. By definition R_{1}(E) is generated by the elements {x} + {y} − {x + y} where x, y ∈ E(k). So B_{1}(E) = J(k), where J is the Jacobian of E.

Define a homomorphism δ : ℤ[E(k)] ⟶ ℤ[E(k)] ⊗ J(k) by the formula {a} ⟼ {a} ⊗ a. We prove in chapter 5 that δ(R_{n}(E)) ⊂ R_{n−1}(E) ⊗ J, so we get a homomorphism δ : B_{n}(E) ⟶ B_{n−1}(E) ⊗ J. Consider the following cohomological complex

B_{n}(E) ⟶ B_{n−1}(E) ⊗ J ⟶ ... ⟶ B_{1}(E) ⊗ Λ^{n−1}J ⟶ B_{0}(E) ⊗ Λ^{n}J (5)

where the differential is given by the formula

δ : {a} ⊗ b_{1} ∧ b_{2} ∧ ... ∧ b_{m} ⟼ {a} ⊗ a ∧ b_{1} ∧ b_{2} ∧ ... ∧ b_{m}

It is the complex B(E, n + 1)^{•}. We put it in degrees [1, n + 1]. The complex is acyclic in the last two terms. To emphasize dependency on the ground field k we use a notation B(E/k, n + 1)^{•}. If k is not an algebraically closed field we

postulate the Galois descent property: B(E/k, n)^{•} := (B(E/k̄, n)^{•})^{Gal(k̄/k)}. The complexes B(E/k, n)^{•} for n = 2, 3 were constructed in [GL].

Conjecture 1.4 a) There exists a canonical homomorphism

H^{i}(B(E/k, n + 2)_{ℚ}^{•}) ⟶ gr_{n+1}^{γ}K_{n+2−i}(E^{(n)})_{sgn} ⊗ ℚ  (6)

b) This homomorphism is an isomorphism.

Here “canonical” means compatibility with the regulator map, see below. The most nontrivial is the “surjectivity conjecture” from part b).

In [W] J.Wildeshaus, assuming standard conjectures on mixed motives, gave a conjectural inductive definition of groups similar to H^{1}B(E, n + 2)^{•} and formulated a conjecture similar to the part a) of conjecture (1.4) for i = 1. See also some new developments in this direction in the recent preprint [W2].

In chapter 7 we construct, assuming the standard conjectures, groups B_{n}(E). One has a surjective map B_{n}(E) ⟶ B_{n}(E) which is hope to be an isomorphism. These groups form a complex B(E; n)^{•} similar to the complex (5). We show that the motivic formalism implies that (for k = k̄) this complex maps to the standard cochain complex of the motivic Lie algebra L(E), providing a morphism of complexes (for an arbitrary field k)

B(E/k, n + 2)_{ℚ}^{•} ⟶ RHom_{MM_{k}}(ℚ(0), S^{n}H(1))  (7)

Conjecture 1.5 The morphism (7) is a quasiisomorpism.

This conjecture implies conjecture (1.4).

3. The Eisenstein-Kronecker series, groups B_{n+1}(E) and regulators. The group R_{n+1}(E/ℂ) is a subgroup of the functional equations for the elliptic n-logarithm studied by Bloch for n = 2 [Bl] and by Beilinson and Levin [BL] in general. Its real version is the Eisenstein - Kronecker series known from the XIX-th century [We]. These are the special functions we need in our conjecture on L(Sym^{n}E, n + 1). Let us say a few words about them.

Let E be an elliptic curve over ℂ. Choose a holomorphic differential ω on E(ℂ). Let Γ ∈ ℂ be the lattice of periods of ω. We will always normalize ω in such a way that Γ := ℤ ⊕ ℤτ where Imτ > 0. The differential ω defines via the Abel-Jacobi map an isomorphism E(ℂ) ⟶ ℂ/Γ. The intersection form on Γ = H_{1}(E(ℂ), ℤ) provides a pairing

(·, ·) : E(ℂ) × Γ ⟶ S^{1};  (z, γ) := exp(\frac{2πi(zγ̄ − z̄γ)}{τ − τ̄})  (8)

Consider the Eisenstein-Kronecker series

K_{i,j}(z; τ) = ∑_{γ∈Γ\0} \frac{(z, γ)}{γ^{i}γ̄^{j}}, i, j ≥ 1

There is a homomorphism K_{n} : ℤ[E(ℂ)] ⟶ Sym^{n−2}H_{B}^{1}(E(ℂ), ℂ) given by the formulas

{z} ⟼ ∑_{a+b=n} K_{a,b}(z) · (dz)^{a−1}(dz̄)^{b−1}

Theorem (6.2) below claims that it sends the subgroup R_{n}(E(ℂ)) to zero.

Suppose E is defined over ℝ. Choose a differential ω ∈ Ω^{1}(E_{/ℝ}) over ℝ. Then the lattice of periods Γ is invariant under the action of complex conjugation z ⟼ z̄ and the isomorphism E(ℂ) ⟶ ℂ/Γ is compatible with complex conjugation.

For any lattice Γ one has K̄_{i,j}(z; τ) = (−1)^{i+j}K_{j,i}(z; τ). If a lattice Γ and a divisor P on ℂ/Γ are invariant under complex conjugation, then K_{i,j}(P; τ) ∈ ℝ. In this situation (+ means invariants under the complex conjugation acting on E(ℂ) and ℝ(1)) we find a map

K_{n} : B_{n}[E(ℂ)]^{+} ⟶ Sym^{n−2}H^{1}(E(ℂ), ℝ(1))^{+}

Restricting K_{n} to Ker(B_{n}(E/ℂ) → B_{n−1}(E/ℂ) ⊗ J(ℂ)) and assuming conjecture (1.4) we should get the regulator map

gr_{n−1}^{γ}K_{n−1}(E^{(n−2)})_{sgn} ⊗ ℚ ⟶ Sym^{n−2}H^{1}(E(ℂ), ℝ(1))^{+}

This together with Beilinson’s conjecture on regulators implies a precise conjecture on L(Sym^{n}E, n + 1) for elliptic curves over number fields, see chapter 6.

4. The structure of the paper. In chapter 2 we recall explicit construction of the Bloch-Suslin complex ([S]) which essentially computes RHom_{MM_{k}}(ℚ(0), ℚ(2)) and its elliptic analog ([GL]) which does the job for RHom_{MM_{k}}(ℚ(0), H(1)). These constructions and results in the special case when k is a number field lead to explicit formulas for the special values of the Dedekind ζ-function and the L-function of elliptic curves at s = 2. In chapter 3 we spell out the basic formalism of motivic Galois groups and motivic Lie algebras. These chapters are expository.

In chapter 4 we prove theorem 1.1 and formaluate some conjectures on the structure of the motivic Lie algebra of the category of mixed elliptic motives. Let me mention three of them: conjecture 4.2 on the Ext groups in the category of mixed elliptic motives, conjecture 4.3 about the small motivic Lie algebra L^{∗}(E) and the freeness conjecture 4.10 for L^{∗}(E).

In chapter 5 we construct the complexes B(E, n+2)^{•} and B^{∗}(E, n+2)^{•} (they are canonically quasiisomorphic). In chapter 6 a conjecture on L(Sym^{n}E, n+1) is formulated. In chapter 7 we show that restriction of the Eisenstein-Kronecker series K_{n+1} to the 2n-th power of the augmentation ideal in ℤ[E] is a real period of an explicitely constructed mixed elliptic motive. This motivic interpretaion seems to be quite different (and simpler) then the one suggested in [BL]. We show how this plus the motivic formalism of chapter 3 allow us to deduce conjecture 1.4a) from standard conjectures on mixed motives. In chapter 8 we define the single valued and multivalued elliptic Chow polylogarithms and introduce the generalized Eisenstein-Kronecker series. We hope to develope further to the material of this chapter in future.

Acknowledgement. Chapter 5-6 of this paper were written during my stay in the IHES in the Summer of 1995 and circulated as [G5]. The paper was finished when I enjoed the hospitality of MPI(Bonn) and University Paris XI at Orsay. I am grateful to IHES and MPI and University Paris XI for hospitality and support. This work was partially supported by NSF grant DMS - 9500010.

I am grateful to A.Levin and V. Schechtman for useful discussions and to N. Schappacher for useful remarks about the text.

2 Two basic examples: a survey

1. The ℚ(1)-story: the Bloch-Suslin complex and ζ_{F} (2). i) Let R_{ℚ(2)}(k) is the subgroup of ℤ[k^{∗}] generated by the elements ∑_{i}(−1)^{i}{r(x_{1}, ..., x̂_{i}, ..., x_{5})}, where x_{i} runs through all 5-tuples of distinct points over k on the projective line, and r is the cross ratio. The Bloch-Suslin complex B(ℚ(2); k)^{•} for an arbitrary field k is defined as follows:

B_{ℚ(2)}(k) ⟶^{δ} Λ^{2}k^{∗}; B_{ℚ(2)}(k) := \frac{ℤ[k^{∗}]}{R_{ℚ(2)}(k)}; δ : \{x\} ⟼ (1 − x) ∧ x

We put B_{ℚ(2)}(k) in degree 1. Then one should have a canonical isomorphism in the derived category

B(ℚ(2); k)^{•} =^{qis} RHom_{MM_{k}}(ℚ(0), ℚ(2))

The main reason for this is the following results describing the relation of the cohomology of the complex B(ℚ(2); k)^{•} ⊗ ℚ with weight 2 motivic cohomology.

According to Matsumoto’s theorem H^{2}B(ℚ(2); k)^{•} = K_{2}(k).

Suslin proved ([S], see also [DS]) that

H^{1}B(ℚ(2); k)^{•} ⊗ ℚ = K_{3}^{ind}(k) ⊗ ℚ = gr_{2}^{γ}K_{3}(k) ⊗ ℚ

In fact, Suslin proved in [S] that the following sequence is exact:

0 ⟶ T̃or(k^{∗}, k^{∗}) ⟶ K_{3}^{ind}(k) ⟶ B_{ℚ(2)}(k) ⟶^{δ} Λ^{2}k^{∗} ⟶ K_{2}(k) ⟶ 0

Here T̃or(k^{∗}, k^{∗}) is a nontrivial extension of T or(k^{∗}, k^{∗}) by ℤ/2ℤ.

Finally, there exists a canonical morphism in the derived category

B(ℚ(2); k)^{•} ⊗ ℚ ⟶ τ_{≥1}RHom_{MM_{k}}(ℚ(0), ℚ(2))

inducing the isomorphism on H^{1} and H^{2}. The right hand side can be understood, for example, as the complex of Bloch’s higher Chow groups. The existence of such a morphism follows also from the motivic philosophy and the motivic construction of the dilogarithm in [BGSV]. According to the Beilinson-Soulé vanishing conjecture one should have

gr_{2}^{γ}K_{4+i}(k) ⊗ ℚ = 0 for i ≥ 0

One can reformulate this by saying that the canonical morphism

RHom_{MM_{k}}(ℚ(0), ℚ(2)) ⟶ τ_{≥1}RHom_{MM_{k}}(ℚ(0), ℚ(2))

is a quasiisomorphism

The Bloch-Suslin complex is definitely ”the smallest possible” representative of the motivic complex RHom_{MM_{k}}(ℚ(0), ℚ(2)). I think it is also ”the best possible”, but this is of cource matter of taste. It gives a solution of the Problem B for the motive ℚ(2) which we have in mind.

ii) Now let k = ℂ. Recall that the dilogarithm is the following multivalued analytic function of on ℂP^{1}\{0, 1, ∞}):

Li_{2}(z) = − ∫_{0}^{z} log(1 − t)\frac{dt}{t}

It has a single-valued version, the Bloch-Wigner function:

L_{2}(z) := ImLi_{2}(z) + arg(1 − z) · log |z|

Let

L̃_{2} : ℤ[ℂ^{∗}] ⟶ ℝ, {z} ⟼ L_{2}(z)

Then one can show that L̃_{2}(R_{ℚ(2)}(ℂ)) = 0, so we get a well defined homomorphism

L̃_{2} : B_{ℚ(2)}(ℂ) ⟶ ℝ

Theorem 2.1 The restriction of the homomorphism L̃_{2} to the subspace H^{1}B(ℚ(2); k)^{•}⊗ ℚ = K_{3}^{ind}(ℂ) ⊗ ℚ coinsides with the Borel regulator on K_{3}^{ind}(ℂ) ⊗ ℚ

A simple direct proof of this result see in [G1].

Finally, if k = F is a number field combining the two theorems above with the Borel theorem we get the solution of the Problem A* for ζ_{F} (2):

Theorem 2.2 a) There exists elements y_{1}, ..., y_{r} ∈ H^{1}B(ℚ(2); F)^{•} ⊗ ℚ such that

q · ζ_{F} (2) = π^{2(r_{1}+r_{2})}d_{F}^{−1/2}det(L̃_{2}(σ_{i}(y_{j})))

for certain q ∈ ℚ^{∗}.

b) For any y_{1}, ..., y_{r} ∈ H^{1}(B(ℚ(2); F)^{•}) ⊗ ℚ one has the formula above with q ∈ ℚ.

To solve the Problem A one should be able to determine effectively the constant q(y_{1} ∧ ... ∧ y_{r}). I do not know how to do this. So we see that the analysis (the dilogarithm) indeed suggested a key for the solution of the Problem B, which leads to a solution of the Problem A*. The relation with the Problem C was explained in details in [G2] and will be scketched briefly in section 4.

Remark. The standard notation for the groups B_{ℚ(n)}(k) is B_{n}(k).

2. The H(1)-story: the elliptic deformation of the Bloch-Suslin complex and L(E, 2) ([GL]) . Let us suppose that k is an algebraically closed field. Let I_{E} be the augmentaion ideal of the group algebra ℤ[E(k)], and I_{E}^{4} its fourth pour. Let B_{3}^{∗} be the quotient of I_{E}^{4} by the subgroup generated by the elements (f) ∗ (1 − f)^{−}, where ∗ is the convolution in the group algebra ℤ[E], f ∈ k(E)^{∗}, and g^{−}(t) := g(−t). Then there is a map

δ_{H(1)}^{∗} : B_{3}^{∗}(E) ⟶ k^{∗} ⊗ J  (9)

whose construction will be outlined below. The complex we get putting the left group in the degree 1 is called the complex B^{∗}(E, 3)^{•}. This is our candidate for RHom_{MM_{k}}(ℚ(0), H(1)).

If k = ℂ we can ”define” δ_{H(1)}^{∗} as follows:

∑ n_{i}{a_{i}} ⟼ ∑ n_{i}θ(a_{i}) ⊗ a_{i}

where

θ(z) = ∏_{n∈ℤ}(1 − q^{n}z) := q^{1/12}z^{− \frac{1}{2}} ∏_{j≥0}(1 − q^{j}z) ∏_{j>0}(1 − q^{j}z^{−1})

To make sense out of this formula for arbitrary field k we proceed as follows (see also s. 4.6 of [GL]).

i. The group B_{2}(E). Let E be an elliptic curve over an arbitrary field k. In s.2.1-2.2 of [GL] we defined a group B_{2}(E/k) = B_{2}(E) which fits in the following diagram

0

↑

0 ⟶ 𝔾_{m}(k) ⟶ B_{2}(E/k) ⟶^{p} S^{2}J(k) ⟶ 0

↑ θ

ℤ[E(k)\0]

such that p ∘ θ : {a} ⟼ a · a. here the horisontal sequence is exact, and the vertical is exact at least if k = k̄. The map θ is defined modulo 6-torsion.

Moreover, if K is a local field there exists a canonical homomorphism

h : B_{2}(E/K) ⟶ ℝ

whose restriction to the subgroup K^{∗} ⊂ B_{2}(E/K)) is given by x ⟼ log |x|, ( see s. 2.3). The canonical local height h_{K} is given by the composition

ℤ[E(K)\0] ⟶^{θ} B_{2}(E/K) ⟶^{h} ℝ

The group B_{2}(E) appears naturally from the theory of biextensions. Namely, let P be the rigidified Poincare line bundle over J × J: its fiber over (0, 0) is identified with k^{∗}. The restriction of P to x × J (resp. J × y) minus the zero section has canonical structure of a commutative algebraic group over k. It is the extension of J by 𝔾_{m} corresponding to x (resp y). This means that for every point (x, y) ∈ J × J we have a k^{∗} torsor T_{(x,y)} together with morphisms of k^{∗}-torsors

T_{(x_{1},y)} ⊗_{k^{∗}} T_{(x_{2},y)} ⟶ T_{(x_{1}+x_{2},y)} T_{(x,y_{1})} ⊗_{k^{∗}} T_{(x,y_{2})} ⟶ T_{(x,y_{1}+y_{2})}

providing the group structure ”in horisontal and vertical directions” such that the diagram

T_{(x_{1},y_{1})} × T_{(x_{2},y_{1})} × T_{(x_{1},y_{2})} × T_{(x_{2},y_{2})} ⟶ T_{(x_{1},y_{1}+y_{2})} × T_{(x_{2},y_{1}+y_{2})}
↓  ↓
T_{(x_{1}+x_{2},y_{1})} × T_{(x_{1}+x_{2},y_{2})}  ⟶  T_{(x_{1}+x_{2},y_{1}+y_{2})}

is commutative. The horisontal sequence in the diagram above is just a different way to spell these properties of the torsors T_{(x,y)}. Set {a}_{2} := θ({a}) ∈ B_{2}(E).

ii) The group B_{3}^{∗}(E). Consider the homomorphism (J := J(k))

δ_{H(1)} : ℤ[E(k)] ⟶ B_{2}(E) ⊗ J, \{a\} ⟶ −\frac{1}{2}\{a\}_{2} ⊗ a

Let R_{3} ⊂ ℤ[E(k)] be the subgroup generated by R_{3}^{∗} and the distribution relations

m · ({a} − m · ∑_{mb=a}{b}), a ∈ E(k), m = −1, 2

Then δ_{H(1)}(R_{3}(E)) = 0 by theorem (3.3) in [GL]. Setting

B_{3}(E) := \frac{ℤ[E(k)]}{R_{3}(E)}

we get a homomorphism δ_{H(1)} : B_{3}(E) ⟶ B_{2}(E) ⊗ J. There is a map i : B_{3}^{∗}(E) ⟶ B_{3}(E) provided by the inclusion i : I_{E}^{4} ↪ ℤ[E]. We define δ_{H(1)} as a morphism making commutative the diagram

B_{3}^{∗}(E) ⟶^{δ_{H(1)}^{∗}} k^{∗} ⊗ J
↓  ↓
B_{3}(E) ⟶^{δ_{H(1)}} B_{2}(E) ⊗ J

Let us consider the following complex

B(E; 3)^{•} :  B_{3}(E) ⟶^{δ_{H(1)}} B_{2}(E) ⊗ J ⟶ J ⊗ Λ^{2}J ⟶ Λ^{3}J  (10)

Here the middle arrow is {a}_{2} ⊗ b ⟼ a ⊗ a ∧ b and the last one is the canonical projection. The complex is placed in degrees [1, 4]. It is acyclic in the last two terms. It was proved in [GL] that we get the commutative diagram:

0  0
↓  ↓
B_{3}^{∗}(E) ⟶^{δ_{H(1)}^{∗}} k^{∗} ⊗ J
↓  ↓
B_{3}(E) ⟶^{δ_{H(1)}} B_{2}(E) ⊗ J ⟶ J ⊗ Λ^{2}J ⟶ Λ^{3}J
↓ ↓ ↓= ↓=
S^{3}J ⟶ S^{2}J ⊗ J ⟶ J ⊗ Λ^{2}J ⟶ Λ^{3}J
↓  ↓
0  0

where the vertical sequences are exact, and the bottom one is the Koszul complex, and thus also exact. So B^{∗}(E; 3)^{•} is canonically quasiisomorphic to B(E; 3)^{•}. If k ≠ k̄ we postulate the Galois descent:

B^{∗}(E/k; 3)_{ℚ}^{•} := B^{∗}(E/k̄; 3)_{ℚ}^{•}^{Gal(k̄/k)}

iii) The elliptic dilogarithm, algebraic K-theory and regulator on K_{2}(E/ℂ). Let E(ℂ) = ℂ^{∗}/q^{ℤ} be the complex points of an elliptic curve E. Here q := exp(2πiτ), Imτ > 0.

L_{2,q}(z) := ∑_{n∈ℤ} L_{2}(q^{n}z), L_{2,q}(z^{−1}) = −L_{2,q}(z)

(The series converges since L_{2}(z) has a singularity of type |z| log |z| near z = 0 and L_{2}(z) = −L_{2}(z^{−1})). When k = ℂ the group R_{3}(E) is the subgroup of all functional equations for the elliptic dilogarithm. In particular the homomorphism

L̃_{2,q} : ℤ[E(ℂ)] ⟶ ℝ, {a} ⟼ L_{2,q}(a)

annihilates the subgroup R_{3}(E/ℂ).

Let K(E)^{−} be the minus part of the action of the involution x → −x, x ∈ E.

Theorem 2.3 [GL] a) Let k = k̄. Then there is a sequence

0 ⟶ T or(k^{∗}, J) ⟶ gr_{2}^{γ}K_{2}(E)^{−} ⟶ B_{3}^{∗}(E) ⟶ k^{∗} ⊗ J ⟶ gr_{2}^{γ}K_{1}(E)^{−} → 0

It is exact modulo 2-torsion (and exact in k^{∗} ⊗ J).

b) Now let k = ℂ. Then the composition

gr_{2}^{γ}K_{2}(E)^{−} ⟶ B_{3}^{∗}(E) ⟶^{L̃_{2,q}} ℝ

coinsides with the Bloch - Beilinson regulator map.

Since we have the Galois descent on gr_{2}^{γ}K_{i}(E)_{ℚ}^{−}, we immediately get Corollary 2.4 Let E be a curve over an arbitrary field k. Then

H^{1}(B^{∗}(E; 3)^{•}) = gr_{2}^{γ}K_{2}(E)_{ℚ}^{−}, H^{2}(B^{∗}(E; 3)^{•}) = gr_{2}^{γ}K_{1}(E)_{ℚ}^{−}

Using the complex B(E, 3) instead of B^{∗}(E, 3) the following lemma makes the conditions on the divisor P effectively computable (see a numerical example in s.1.3 of [GL]) .

Lemma 2.5 Let E be an elliptic curve over a number field F. Then an F - rational divisor P = ∑ n_{j}(P_{j}) over ℚ̄ belongs to Ker(B_{3}(E) ⟶ B_{2}(E) ⊗ J) if and only if it satisfy the following conditions:

a)  ∑ n_{j}P_{j} ⊗ P_{j} ⊗ P_{j} = 0 in S^{3}J(ℚ̄)  (11)

b) For any valuation v of the field F (P) generated by the coordinates of the points P_{j}

∑ n_{j}h_{v}(P_{j}) · P_{j} = 0 in J(ℚ̄) ⊗ ℝ  (12)

In particular for any field k the group of k-rational divisors on E(ℚ̄) satisfying the conditions a)-b) above maps surjectively to gr_{2}^{γ}K_{2}(E_{/k})_{ℚ}^{−}, and when k is a number field this map is compatible with the regulator map in the obvious way.

Combining lemma 2.5 with Beilinson’s conjecture on regulators we come to an explicit formula for L(E, 2), which we formulate only for k = ℚ leaving the general case to the reader.

Conjecture 2.6 Let E be an elliptic curve over ℚ. Then for any ℚ-rational divisor P on E(ℚ̄) satisfying the conditions (11) and (12), and an integrality condition at each prime p where E has a split multiplicative reduction (see 47) one has

πL̃_{2,q}(P) = q · L(E, 2) for a certain q ∈ ℚ

Adding to the game Beilinson’s results on regulators for modular curves we proved this formula for modular elliptic curves over ℚ ([GL]):

Theorem 2.7 Let E be a modular elliptic curve over ℚ. Then there exists a ℚ-rational divisor P on E(ℚ̄) satisfying the conditions of conjecture above such that

πL̃_{2,q}(P) = q · L(E, 2) for a certain q ∈ ℚ^{∗}

So for the motive H(1) we see the same kind of relationship between the Problems A* and B. It remains to see the role of the motivic Galois group of the category of mixed elliptic motives, i.e. to understand

Why the motivic complex RHom_{MM_{k}}(ℚ(0), H(1)) has the shape (9), how it reflects the structure of the motivic Galois group, and what it tells us about the motivic Galois group?

In particular, how to define the group B_{3}^{∗}(E) it terms of the motivic Lie algebra of the category M_{E}? The answers are given in chapter 4 below.

3 Mixed motives and motivic Lie algebras

1. Categories of mixed motives. Let MM_{X} be the (hypothetical) abelian category of all mixed motivic sheaves over a regular scheme X. When X = Spec(k), k is an arbitrary field, we get the category MM_{k}. We will assume that it satisfies all the expected properties conjectured by Beilinson [Be]. In particular any object of MM_{X} has a canonical increasing weight filtration W_{•}; morphisms are stricktly compatible with W_{•}. We will ignore the fact that existence of such an abelian category is not known yet.

Let π : X ⟶ Spec(k) be the structure morphism. There are the Tate sheaves ℚ(n)_{X} := π^{∗}ℚ(n)_{Spec(k)} which we usually denote simply by ℚ(n). The basic conjecture is Beilinson’s description of Ext’s between them:

Conjecture 3.1

Ext_{MM(X)}^{i}(ℚ(0)_{X}, ℚ(n)_{X}) = gr_{n}^{γ}K_{2n−i}(X) ⊗ ℚ

Consider the category of pure motives over a field k. One can have in mind Grothendieck’s category of motives with morphisms given by the Chow correspondences modulo numerical equivalence. We will assume that it is a semisimple abelian category.

Now let P be a rigid tensor subcategory of the category of pure motives, and M_{P} the tensor category of mixed motives generated by P. This means that M_{P} is closed under extensions, contains P as a full subcategory and the weight graded quotients of any object of M_{P} belong to P.

Examples. 1. P is the category of pure Tate motives. M_{P} is the category of mixed Tate motives.

2. P is the category of pure elliptic motives. Then M_{P} is the category of mixed elliptic motives.

3. P is the category of all pure motives over k, so M_{P} is the category of all mixed motives.

There is a canonical fiber functor

Ψ : M_{P} ⟶ P, M ⟼ ⊕_{n∈ℤ}gr_{n}^{W} M

Let us axiomatise this situation. Namely, let P be an abelian semisimple rigid tensor ℚ-category . We will say that M_{P} is a mixed category over P if it is an abelian rigid tensor ℚ-category containing P as a full subcategory with the following properties:

1. Any object M of M_{P} carries a canonical finite filtration W_{•}M (the weight filtration).

2. Morphisms are strictly compatible with W_{•}.

3. The graded objects gr_{i}^{W} M belong to P.

4. Hom_{M_{P}} (M, N) are finite dimensional.

Remark. We can assume also that M_{P} is an F -category where F is an arbitrary field of characteristic zero. This is essential when E is a curve with complex multiplication by O_{F}, so M_{E} is an F -category. However to simplify a bit notations we will assume F = ℚ below.

Below we will assume that M_{P} is such a category, not necessarily of motivic origin.

2. The fundamental Lie algebra of a mixed category over P. Set H(M_{P}) := End(Ψ). It is a cocommutative Hopf algebra in the tensor category P.

Let L(M_{P}) be the Lie algebra of all derivations of the functor Ψ:

L(M_{P}) = Der(Ψ) = {F ∈ End(Ψ)|F_{X⊗Y} = F_{X} ⊗ id_{Y} + id_{X} ⊗ F_{Y} }

It is a Lie algebra in the tensor category P. Its universal enveloping algebra is isomorphic to the Hopf algebra H(M_{P}). The Tannakian formalism shows that the functor Ψ provides an equivalence between the category M_{P} and the category of L(M_{P})-modules in the category P.

A Lie coalgebra in a tensor category C is an object L together with a cobracket δ : L → Λ_{C}^{2}L such that the composition

L ⟶^{δ} Λ_{C}^{2}L ⟶^{δ⊗id−id⊗δ} Λ_{C}^{3}L

is zero. The standard complex of L is defined as follows:

C_{C}^{•}(L) := L ⟶^{δ} Λ_{C}^{2}L ⟶^{δ⊗id−id⊗δ} Λ_{C}^{3}L ⟶ Λ_{C}^{4}L ⟶ ...

Here L placed in degree 1, and the differential has degree +1. We define the cohomology groups of a Lie coalgebra L setting H_{C}^{•}(L) := H^{•}(C^{•}(L)).

There is a duality V → V^{∨}, (V^{∨})^{∨} = V between the inductive and projective limits of objects in the cateogory P. Set L(M_{P}) := L(M_{P})^{∨}. It is a Lie coalgebra in the tensor category P_{M}.

For a ℚ-vector space V and an object X of a category C there is an object V ⊠ X of C such that Hom_{C}(Y, V ⊗ X) = V ⊗ Hom_{C}(Y, X). Any pure motive W can be canonically decomposed into the isotypical components:

W = ⊕_{M}Hom_{M}(M, W) ⊠ M

where the sum is over the isomorphism classes of simple objects in M_{P}. The standard complex of L(M_{P}) splits into a direct sum of isotypical components

(L(M_{P}))_{M} ⟶^{δ} (Λ^{2}L(M_{P}))_{M} ⟶^{δ} (Λ^{3}L(M_{P}))_{M} ⟶^{δ} ...  (13)

Notice that L(M_{P})_{M} is zero unless the weight M is > 0. Therefore each of the complexes (13) has finite length.

3. The Galois group of a mixed category. Let Φ : M_{P} ⟶ V ect_{ℚ} be composition of the fiber functor Ψ with a fiber functor φ from P to the category of finite dimensional ℚ-vector spaces.

Then G(M_{P}) := Aut^{⊗}Φ is a proalgebraic group scheme over ℚ. It is the Galois group of the category M_{P}.

There are two canonical functors between the tensor categories M_{P} and P: the inclusion functor P ↪ M_{P}, and the functor

gr^{•} : M_{P} ⟶ P gr^{W} : X ⟼ ⊕_{n∈ℤ}gr_{n}^{W} X

Their composition is the identity functor on P. These functors obviously respect the fiber functor, and so lead to homomorphisms of groups G(M_{P}) ⟶ G(P) and G(P) ⟶ G(M_{P}). Thus the group G(M_{P}) is a semidirect product:

0 ⟶ U(M_{P}) ⟶ G(M_{P}) ⟶ G(P) ⟶ 0

Passing to Lie algebras we get:

0 ⟶ LieU(M_{P}) ⟶ LieG(M_{P}) ⟶ LieG(P) ⟶ 0

So LieU(M_{P}) is a pronilpotent Lie algebra in the category of G(P)-modules.

The category of finite dimensional G(M_{P})-modules is equivalent to the category of U(M_{P})-modules in the category finite dimensional G(P)-modules. Since the group scheme U(M_{P}) is prounipotent, it is equivalent to the category of LieU(M_{P})-modules in the category of G(P)-modules. One can think about it as of the category G(P)-modules equipped with an action of the Lie algebra LieU(M_{P}) such that the action LieU(M_{P}) ⊗ V ⟶ V is a morphism of G(P)-modules.

Lemma 3.2 Let M be a pure object. Then

RHom_{LieU(M_{P})−mod}(φ(ℚ(0)), φ(M)) ⊠ M^{∨} = ((Λ^{•}L(M_{P}))_{M}, ∂)

4. A description of the fundamental Lie coalgebra. ( Compare with [BMS] and [BGSV]). Let A, B be simple objects of the category P. Let us say that an object M of M_{P} is n-framed by A, B if we are given nonzero homomorphisms

v_{A} : A → gr_{0}^{W} M, f_{B} : gr_{−n}^{W}M → B

Consider the finest equivalence relation on the set of all objects n-framed by A, B such that (M; v_{A}; f_{B}) and (M′; v′_{A}; f′_{B}) are equivalent if there is a morphism M → M′ respecting the frames. Denote by A(A, B) the set of equivalence classes of mixed motives n-framed by A, B.

Let us define an addition law setting

(M; v_{A}; f_{B}) + (M′; v′_{A}; f′_{B}) := (M ⊕ M′; (v_{A}, v′_{A}); f_{B} + f′_{B})

The neutral element is A⊕B with the obviuos framing. Indeed, the equivalence between (M ⊕ A ⊕ B; (v_{A}, id_{A}); (f_{B} + id_{B})) and (M; v_{A}; f_{B}) is provided by the natural morphisms M ⟵ M ⊕ A ↪ M ⊕ A ⊕ B.

The inversion is given by −(M; v_{A}; f_{B}) := (M; v_{A}; −f_{B}), and one also has (M; v_{A}; −f_{B}) = (M; −v_{A}; f_{B}). Let us prove the first claim. The (A, B)- framed object object W_{≤0}M/Kerf_{B} is equivalent to (M; v_{A}; f_{B}). So we may assume without loss of generality that W_{≤0}M = M, W_{<−n}M = 0, and moreover gr_{0}^{W} M = A, gr_{−n}^{W}M = B. There is a morphism

i_{−} : W_{<0}M ↪ M ⊕ M, m ⟼ (m, −m)

Then

M ⊕ M ⟶ \frac{M ⊕ M}{i_{−}(M)} =: N_{1}

is a morphism of (A, B)-framed objects. We get an extension

0 ⟶ W_{<0}N_{1} ⟶ N_{1} ⟶ A ⊕ A ⟶ 0

Let us consider the extension 0 ⟶ W_{<0}N_{1} ⟶ N ⟶ A ⟶ 0 induced by the morphism j_{−} : A → A ⊕ A, a ⟼ (a, −a). Then there is a well defined morphism α : A → N given by a ⟼ (n_{a}, −n_{a}) where n_{a} ∈ N is any element

projecting to a. So we get a morphism of (A, B)-framed objects A ⊕ B ⟶^{(α,β)} N, where β : B ↪ N is the natural inclusion.

It is easy to check that A(A, B) is an abelian group. Since M_{P} is a ℚcategory, it is a ℚ-vector space, and in fact an inductive limit of finite dimensional F -vector spaces.

If B is a simple object then A(ℚ(0), B) ⊠ B^{∨} is an object of P defined by the isomorphism class of B up to a unique isomorphism. Set

H′(M_{P}) := ⊕A(ℚ(0), B) ⊠ B^{∨}

The sum is over all isomorphism classes of simple objects B in P, but only finitely many terms are non zero.

The tensor product of mixed motives provides H′(M_{P}) with a structure of a commutative algebra in the tensor category P:

A(ℚ(0), B_{1}) ⊠ B_{1}^{∨} ⊗ A(ℚ(0), B_{2}) ⊠ B_{2}^{∨} → A(ℚ(0), B_{1} ⊗ B_{2}) ⊠ B_{1}^{∨} ⊗ B_{2}^{∨}

Let us define the coproduct. Let C be a simple pure object of weight −k. Let c ∈ Hom_{P}(gr_{−k}^{W}M, C) and c^{∗} ∈ Hom_{P}(C, gr_{−k}^{W}M). Let

v(c^{∗}) : ℚ(0) ⟶ C ⊗ C^{∨} ⟶^{c^{∗}⊗id} gr_{−k}^{W}M ⊗ C^{∨} = gr_{0}^{W} (M ⊗ C^{∨})

(The first arrow is the canonical morphism). Set

α(c) := (M; v_{ℚ(0)}; c) ∈ A(ℚ(0), C) ⊠ C^{∨}
α^{∗}(c^{∗}) := (M ⊗ C^{∨}; v(c_{i}^{∗}); f_{B} ⊗ id_{C^{∨}}) ∈ A(ℚ(0), C^{∨} ⊗ B) ⊠ C ⊗ B^{∨}

Define a natural map

ν_{C} : Hom_{P}(gr_{−k}^{W}M, C) ⊗ Hom_{P}(C, gr_{−k}^{W}M) ⟶  (14)
(A(ℚ(0), C) ⊠ C^{∨}) ⊗ (A(ℚ(0), C^{∨} ⊗ B) ⊠ C ⊗ B^{∨})

by setting

ν_{C} : c ⊗ c^{∗} ⟼ α(c) ⊗ α^{∗}(c^{∗})

There is a canonical element in (14) defined as follows. Let c_{1}, ..., c_{m} be a basis of the ℚ-vector space Hom_{P}(gr_{−k}^{W}M, C) and c_{1}^{∗}, ..., c_{m}^{∗} the dual basis of Hom_{P}(C, gr_{−k}^{W}M), i.e. c_{i} ∘ c_{j}^{∗} = δ_{i,j}Id_{C}. Then ∑_{i} c_{i} ⊗ c_{i}^{∗} does not depend on the choice of basis c_{i}. By definition

Δ : (M; v_{ℚ(0)}; f_{B}) ⊠ B^{∨} ⟼ ⊕_{C}ν_{C}(∑_{i} c_{i} ⊗ c_{i}^{∗}) ∈

⊕_{C}(A(ℚ(0), C) ⊠ C^{∨}) ⊗ (A(ℚ(0), C^{∨} ⊗ B) ⊠ C ⊗ B^{∨})

The sum is over all isomorphism classes of simple objects in P whose weights are between 0 and wt(B). It turnes out to be a finite sum.

Notice that V := ⊕V_{B} ⊠B^{∨} is a pro (resp. ind) object in P if and only if V_{B} is pro (resp. ind) ℚ-vector space for each isomorphism class of simple objects in P. There is a duality between the pro- and ind-objects: if V is a pro (resp. ind) object in P, then V^{∨} := ⊕V_{B}^{∨} ⊠ B is an ind (resp. pro) object in P, and the canonical map V → (V^{∨})^{∨} is an isomorphism.

For any simple object B of P the B-isotypical componenmt of H′(M_{P}) is an ind-object in the category P. This means that H′(M_{P}) itself is an ind-object in the category P. On the other hand H(M_{P}) is a pro-object in P.

Let E ∈ End(Ψ) and E_{M} the corresponding endomorphism of Ψ(M). There is a natural map φ : H′(M_{P}) → H(M_{P})^{∨} given by

φ((M; v_{ℚ(0)}; f_{B}) ⊠ B^{∨})(E_{M}) := f_{B}E_{M}v_{ℚ(0)}(ℚ(0));

Here we extended f_{B} to a morphism Ψ(M) → B postulating that its restriction to gr_{−l}^{W}M for l ≠ n is zero. We will always extend the framing morphisms in a similar way.

Theorem 3.3 a) H′(M_{P}) is dual to H(M_{P}).

b) Δ provides H′(M_{P}) with a structure of a commutative Hopf algebra in the category P which is canonically isomorphic to H(M_{P})^{∨}.

Proof. a) According to the Tannaka theory (which remains valid for fiber functors with values in the semisimple tensor categories) the fiber functor Ψ provides an equivalence between the category M_{P} and the category H(M_{P})- mod of finite dimensional H(M_{P})-modules in the category P. So we will work with the category H(M_{P})-mod.

Let us show that φ is injective. Suppose that φ((M; v_{ℚ(0)}; f_{B}) ⊠ B^{∨}) =

0. We may assume that gr_{p}^{W} M = 0 for p > 0 and p < −n and gr_{0}^{W} M = ℚ(0), gr_{−n}^{W}M = B. Consider the cyclic submodule M′ := H(M_{P}) · Ψ(ℚ(0)).

Then its weights are bigger then −n, since otherwise φ((M; v_{ℚ(0)}; f_{B}) ⊠ B^{∨}) ≠ 0. So we get a morphism M′ ⊕B → M which obviously respects the frames. On the other hand there is canonical projection M′ ⊕ B → ℚ(0) ⊕ B providing by the projection M′ → gr_{0}^{M} = ℚ(0). Thus (M; v_{ℚ(0)}; f_{B}) is equivalent to ℚ(0)⊕B (with the natural frame).

Now let us show that φ is surjective. Let f ∈ H(M_{P})_{B}^{∨} be a subobject isomorphic to B^{∨}. Denote by Kerf the corresponding subobject of H(M_{P})_{B} such that H(M_{P})_{B}/kerf = B. Consider the H(M_{P})-module: ℚ(0)⊕H(M_{P})_{B}/Kerf whith the action of H(M_{P}) provided by f, the nontrivial part of the action is a morphism H(M_{P})_{B} ⊗ ℚ(0) → H(M_{P})_{B}/Kerf, and the obvious framing. The map φ sends it to f.

The part b) follows easyly from the definitions and the part a). The theorem is proved.

4 Conjectures on the Galois group of the category of mixed elliptic motives

1. A conjecture on Ext’s in the category of mixed elliptic motives.

Lemma 4.1

RHom_{MM_{k}}(ℚ(0), Sym^{n}H(m)) = RHom_{MM_{E^{(n)}}} (ℚ(0), ℚ(n + m))[n]_{sgn} (15)

Proof. Let p : E^{(n)} ⟶ Spec(k) be the canonical projection. Then we should have the motivic Leray spectral sequence

E_{2}^{p,q} = Ext_{MM_{k}}^{p}(ℚ(0), R^{q}p_{∗}ℚ(n + m))

degenerating at E_{2} and abbuting to Ext_{MM_{E^{(n)}}}^{p+q} (ℚ(0), ℚ(n + m)). Since

H^{i}(E^{(n)})_{sgn} = 0 for i ≠ n; H^{n}(E^{(n)})_{sgn} = Sym^{n}h^{1}(E)

we get (15). Conjecture (3.1) tells us that the cohomology of the elliptic motivic complexes are given by the formula

R^{i}Hom_{MM_{k}}(ℚ(0), Sym^{n}H(m)) ⊗ ℚ = gr_{n+m}^{γ}K_{n+2m−i}(E^{(n)})_{sgn} ⊗ ℚ (16)

Let us choose a fiber functor φ : P_{E} → V ect_{ℚ}. Set H := φ(H) Then φ(ℚ(1)) corresponds to the one-dimensional representation det : GL_{2}(H) → 𝔾_{m} and S^{m}H(n) to the GL_{2}-module S^{m}H(n), both with the trivial LieU(M_{E})-action.

The following conjecture plays a crucial role. Our results are little steps towards its confirmation.

Conjecture 4.2 Let E be an elliptic curve over an algebraicaly closed field k The canonical morphism

RHom_{M_{E}}(ℚ(0), S^{n}H(m)) ⟶ RHom_{MM_{k}}(ℚ(0), S^{n}H(m))

is a quasiisomorphism. In particular the Ext groups in the category M_{E} should be isomorphic to the Ext groups (between the same objects) in the category MM_{k}.

Remark. This conjecture should not be true if k is not an algebraically closed field. A similar conjecture for the category of mixed Tate motives is expected to be valid for all fields k.

Question. Consider the mixed category M_{{X_{i}}} generated by given simple pure motives X_{i}. When could we expect that the Ext’s in the category M_{X} coinside with the Ext’s (between the same objects) in the category MM_{k}?

2. The main hero: a Lie coalgebra L^{∗}(E). Let us assume for simplicity that E is an ellliptic curve over a field k without complex multiplication. Then any simple object of the category P_{E} of pure elliptic motives is isomorphic to just one of the objects S^{n}H(m).

P_{E} is a rigid tensor category. Let us define a new tensor structure ⊗′ on this category setting

S^{n}H(m) ⊗′ S^{n′}H(m′) ⟶^{=} S^{n+n′}H(m + m′)

where the morphism is the usual tensor product followed by the projection to the S^{n+n′}H(m + m′) component. We get an abelian tensor category P_{E}^{∗} which will be called the small tensor category of elliptic motives.

Remark. P_{E}^{∗} is not a rigid tensor category since there is no dual object for S^{n}H(m) if n > 0.

The mixed Tate motives are contained in the category of mixed elliptic motives, and we have a canonical functor M_{T} (k) ⟶ M_{E}. Let π^{∗} : L(k) → L(E) be the corresponding morphism of Lie coalgebras.

Conjecture 4.3 There exists a Lie subcoalgebra L^{∗}(E) ⊂ L(E) which enjoys the following properties:

1. L^{∗}(E) is also a Lie coalgebra in the small tensor category P_{E}^{∗}.

2. The natural morphism of cohomology groups

H_{P_{E}^{∗}}^{•}(L^{∗}(E)) ⟶ H_{P_{E}^{∗}}^{•}(L(E))

is an isomorphism.

3. π^{∗} maps L(k)_{ℚ(n)^{∨}} isomorphically onto L^{∗}(E)_{ℚ(n)^{∨}}.

A construction of a Lie coalgebra in the tensor category P_{E}^{∗} which hypothetically satisfies all these properties is given in s. 4.5.

Remarks. 1. A Lie subcoalgebra of L(E) is usually not a Lie coalgebra in the tensor category P_{E}^{∗}.

2. H_{P_{E}^{∗}}^{•}(L^{∗}(E)) are quite different from H_{P_{E}}^{•}(L^{∗}(E)). The reason is clear from the following example:

(V ⊠ H) ⊗ (V ⊠ H) = (V ⊗ V) ⊠ S^{2}H ⊕ (V ⊗ V) ⊠ S^{2}H

(V ⊠ H) ⊗′ (V ⊠ H) = (V ⊗ V) ⊠ S^{2}H

3. Consider L^{∗}(E) as a Lie coalgebra in P_{E}. Let M_{E}^{∗} be the category of comodules over it in the category P_{E}. If L^{∗}(E) is a Lie coalgebra of L(E), then M_{E}^{∗} is a subcategory of M_{E}.

The weight consideration shows that the picture of nonzero isotypical components of L^{∗}(E) looks as follows:

n
↑
7  • • • • • • • •
6  • • • • • • •
5  • • • • • • •
4  • • • • • •
3  • • • • • •
2  • • • • •
1  • • • • •
0  • • • •
−3 −2 −1 0 1 2 3 4 ⟶ m

A boldface point with coordinates (m, n) corresponds to nonzero isotypical component L^{∗}(E)_{S^{n}H(m)^{∨}}. For example L^{∗}(E)_{ℚ(0)} = 0, L^{∗}(E)_{S^{2}H(−1)^{∨}} = 0.

3. Corollaries. According to conjectures (4.3) and (4.2) one has the following quasiisomorphisms:

(Λ_{P_{E}^{∗}}^{•}L^{∗}(E), ∂)_{S^{n}H(m)^{∨}} =^{4.3} RHom_{M_{E}}(ℚ(0), S^{n}H(m)) =^{4.2}  (17)
RHom_{MM_{k}}(ℚ(0), S^{n}H(m))  (18)

So we get the basic formula

H^{i}(Λ_{P_{E}^{∗}}^{•}L^{∗}(E))_{S^{n}H(m)^{∨}} = gr_{n+m}^{γ}K_{n+2m−i}(E^{(n)})_{sgn} ⊗ ℚ  (19)

Using this we immediately conclude that

L^{∗}(E)_{ℚ(1)^{∨}} = k^{∗} ⊠ ℚ(1)^{∨}, L^{∗}(E)_{H^{∨}} = J ⊠ H^{∨}

More generally, let J_{S^{2m+1}H(−m)} be the image of the subgroup of homologically trivial cycles in CH^{m}(E^{(2m−1)})_{sgn} under the Abel-Jacobi map. Then

L^{∗}(E)_{S^{2n+1}H(−n)^{∨}} = J_{S^{2n+1}H(−n)} ⊠ S^{2n+1}H(−n)^{∨}, n > 0

Indeed, if H is a pure motive of weight 1 then

C^{•}(L^{∗}(E))_{H} = L^{∗}(E)_{H} = H_{P^{∗}(E)}^{1}(L^{∗}(E))_{H} =^{2} H^{1}(L^{∗}(E))_{H}

Recall that B_{2}(k) is the Bloch group of a field k. Let B_{3}(k) be its analog for the classical trilogarithm introduced in [G1-2]).

Lemma 4.4 (17) together with the results of [G2], [GL] implies that the lower left corner of the diagram looks as folows (all groups are tensored by ℚ):

J_{S^{3}H(−1)} •  •  •  •  •
•  •  •  •  •
J B_{3}^{∗}(E)  •  •  •
0 k^{∗} B_{2}(k) B_{3}(k) •

Proof. The structure of the bottom row follows from 3) and the main results in [G2]. The H(1)^{∨}-isotypical component of the standard complex of L^{∗}(E) is

L^{∗}(E)_{H(1)^{∨}} ⟶ L^{∗}(E)_{ℚ(1)^{∨}} ⊗ L^{∗}(E)_{H^{∨}} = k^{∗} ⊗ J ⊠ H(1)^{∨}

The motivic interpretation of the group B_{3}^{∗}(E) as a group of H(1)-framed mixed elliptic motives (see theorem 7.15 below) leads to a morphism of complexes

B_{3}^{∗}(E) ⊠ H(1)^{∨} ⟶ k^{∗} ⊗ J ⊠ H(1)^{∨}
↓  ↓=
L^{∗}(E)_{H(1)^{∨}} ⟶ k^{∗} ⊗ J ⊠ H(1)^{∨}

Thanks to the main result of [GL] and property 2) this map is a quasiisomorphism. So L^{∗}(E)_{H(1)^{∨}} = B_{3}^{∗}(E) ⊠ H(1)^{∨}.

The Lie coalgebra L(E) has a much more complicated structure, see s.4.8 below for the simplest example: description of L(E)_{ℚ(1)^{∨}}.

Zero cycles on E^{(n)}. The only way how the object S^{n}H could appear as a direct summand of a tensor product of n simple objects of negative weights in the small tensor category P_{E}^{∗} is this:

S^{n}H = H ⊗′ ... ⊗′ H (n times)

Set

L^{∗}(E)_{S^{2}H^{∨}} := B_{S^{2}H}^{∗} ⊠ S^{2}H^{∨}  (20)

The S^{2}H^{∨}- isotypic component of the cochain complex Λ_{P_{E}^{∗}}^{•}L^{∗}(E) looks as follows:

(L^{∗}(E) ⟶ Λ_{P_{E}^{∗}}^{2}L^{∗}(E))_{S^{2}H^{∨}} = (B_{S^{2}H}^{∗}(E) ⟶^{δ_{S^{2}H}} Λ^{2}J) ⊠ S^{2}H^{∨}

Similarly the S^{n}H^{∨}-isotypic component of Λ_{P_{E}^{∗}}^{•}L^{∗}(E) is ( ... ⟶ Λ^{n}J)⊠S^{n}H^{∨}. Therefore we get

Ext^{n}(ℚ(0), S^{n}H) = CH^{n}(E^{(n)})_{sgn} = a quotient of Λ^{n}J (21)

The last equality indeed follows from the Bloch’s theorem on zero cycles on abelian varieties [Bl5].

Similarly we deduce that for m > 0 one should have

Ext_{M_{E}}^{n+m}(ℚ(0), S^{n}H(m)) = a quotient of Λ^{n}J ⊗_{ℚ} K_{m}^{M}(k)

4. The duality between DG Lie coalgebras and commutative DG algebras. Let L^{•} be a DG Lie coalgebra in a tensor category C, which is not supposed to be a rigid tensor category. This means that we have a complex (L^{•}, ∂) and a cobracket

δ : L^{•} → Λ_{C}^{2}(L^{•})

which is a morphism of complexes satisfying the co Jacobi identity. Let

C(L^{•}) := L^{•}[−1] ⊕ S_{C}^{2}(L^{•}[−1]) ⊕ S_{C}^{3}(L^{•}[−1]) ⊕ ...

be the free super commutative algebra (without unit) generated by L^{•}[−1]. There are two cohomological differentials on C(L^{•}): the first is provided by the differential in L^{•}, and the second comes via the Leibniz rule from the cobracket δ. Their sum is a differential providing a structure of a commutative DG algebra on C(L^{•}). For example, if L^{•} = L is concentrated in degree 0, then S_{C}^{n}(L[−1]) = Λ_{C}^{n}L[−n] and we get the standard cochain complex of the Lie coalgebra:

C(L) := L[−1] ⊕ Λ_{C}^{2}(L)[−2] ⊕ Λ_{C}^{3}(L)[−3] ⊕ ...

Let L^{•} be a DG Lie coalgebra and A^{•} is a DG commutative algebra. Define MC(Hom_{k}(L^{•}[−1], A^{•})) as the set of all degree zero elements in ω ∈ Hom_{k}(L^{•}[−1], A^{•}) satisfying the Maurer-Cartan equation dω + \frac{1}{2}ω ∧ ω = 0. Here ω ∧ ω is the composition

L^{•}[−1] → S_{C}^{2}(L^{•}[−1]) → S_{C}^{2}A^{•} → A^{•}

If we think of ω as of element in (L^{•}[−1])^{∨} ⊗ A^{•}, we get the usual formula for ω ∧ ω. Then one has

MC(Hom_{k}(L^{•}[−1], A^{•})) = Hom_{DGCom}(C(L), A)

We get a functor

C : DGCoLie → DGCom

There is a functor in the opposite direction:

F : DGCom → DGCoLie

satisfying

Hom_{DGcoLie}(L^{•}, F (A^{•})) = MC(Hom_{k}(L^{•}, A^{•}[1]))

Namely, F (A^{•}) is the cofree Lie supercoalgebra F(A^{•}[1]) generated by the complex A^{•}[1]. There is a canonical projection p : F(A^{•}[1]) ⟶ A^{•}[1]. According to the universality property of the cofree Lie coalgebras to define the differential on F(A^{•}[1]) it is sufficient to define its projection F(A^{•}[1]) → A^{•}[1]. By definition it is the sum of the differential in A^{•}[1] and the product Λ^{2}(A^{•}[1]) = S^{2}(A^{•})[2] → A^{•}[1].

The functors C and F are obviously adjoint. So there are canonical morphisms

A^{•} ⟶ C ∘ F (A^{•}) L^{•} ⟶ F ∘ C(L^{•})

Theorem 4.5 These morphisms are quasiisomorphisms.

This theorem was proved by Quillen [Q] when C is the category of ℚ-vector spaces, but it is true for an arbitrary tensor category C.

The functor F has another description via the bar construction. Namely, let B(A^{•}) be the reduced bar complex of a DG commutative algebra A^{•}. It has a structure of a Hopf algebra. Let B_{0}(A^{•}) be the augmentation ideal of B(A^{•}). Then the space B_{0}(A^{•})/(B_{0}(A^{•}) · B_{0}(A^{•})) of the indecomposable elements has a natural structure of a DG Lie coalgebra (a good reference for the Bar construction is ch. 2 of [BK]).

5. A cycle construction of L^{∗}(E) (Compare with [Bl4]). Let sgn be the one dimensional alternating representation of G_{n} where a generator of each factor ℤ/2ℤ acts by −1 and the restriction to S_{n} is the alternating representation. The group G_{n} acts naturally on E^{n}.

The idea of the construction. Let Γ_{X}(n) be a motivic complex on a regular scheme X, i.e., it is a complex of ℚ-vector spaces quasiisomorphic to RHom_{MM_{X}} (ℚ(0), ℚ(n)). We will need also a canonical morphism of complexes Γ_{X}(m) × Γ_{Y} (n) → Γ_{X×Y} (m + n). We will take below the complex of Bloch’s Higher Chow groups on X as a concrete version of Γ_{X}(n) to work with. Then

N^{∗}(E) := ⊕_{2m+n>0,n≥0}Γ_{E^{n}}(m + n)_{sgn} ⊠ S^{n}H(m)^{∨}

has a natural structure of a commutative DG algebra (without unit) in the small tensor category of pure elliptic motives. Namely, the product is provided by the natural morphism of complexes

Γ_{E^{n}}(m + n)_{sgn} × Γ_{E^{n′}} (m′ + n′)_{sgn} ⟶ Γ_{E^{n+n′}} (m + m′ + n + n′)_{sgn}

(take the external product of the complexes on E^{n+n′} and project it onto the signum part with respect to the action of the group G_{n+n′}). It remains to apply the functor F!

Now let us spell out the details for the complexes of Bloch’s Higher Chow groups. Let Z^{n}(E^{m}, k) be the ℚ-vector space of codimension n cycles with ℚ coefficients on E^{m} × (ℙ^{1}\{1})^{k} which are skewsymmetric with respect to the action of G_{k} on (ℙ^{1}\{1})^{k} and meet the faces of the coordinate cube defined by setting some of the coordinates equal to 0 or ∞ properly. Set Z^{m+n}(E^{n}, c)_{sgn} := (Z^{m+n}(E^{n}, c) ⊗ sgn)^{G_{n}}. We define a complex Z^{m+n}(E^{n}, ∗)_{sgn} placing

Z^{m+n}(E^{n}, c)_{sgn} in degree 2n + m − c

The differential is the alternating sum of the face maps:

Z^{m+n}(E^{n}, c)_{sgn} ⟶ Z^{m+n}(E^{n}, c − 1)_{sgn}

A ⊗′-product of RHom’s

RHom_{MM_{k}}(ℚ(0), S^{n}H(m)) ⊗′ RHom_{MM_{k}}(ℚ(0), S^{n′}H(m′)) ⟶ (22)

RHom_{MM_{k}}(ℚ(0), S^{n+n′}H(m + m′))

is provided by the usual tensor product

RHom(ℚ, A) ⊗ RHom(ℚ, B) ⟶ RHom(ℚ, A ⊗ B)

and the canonical projection S^{n}H(m) ⊗ S^{n′}H(m′) → S^{n+n′}H(m + m′).

Lemma 4.6 a) One has a quasiisomorphism

Z^{m+n}(E^{n}, ∗)_{sgn} = RHom_{MM_{k}}(ℚ(0), S^{n}H(m))

b) The product of cycles followed by the projection to the G_{n+n′}-alternating part provides a morphism of complexes

Z^{m+n}(E^{n}, ∗)_{sgn} ⊗ Z^{m′+n′}(E^{n′}, ∗′)_{sgn} ⟶ Z^{m+n+m′+n′}(E^{n+n′}, ∗ + ∗′)_{sgn}

which coincides in the derived category with the ⊗′-product of RHom’s (22).

Proof. Follows from lemma (4.1) and the results of Bloch [Bl2], [BK].

Set

N^{∗}(E) := ⊕_{2m+n>0}Z^{m+n}(E^{n}, ∗)_{sgn} ⊠ S^{n}H(m)^{∨}

There is a commutative product on N^{∗}(E) given by

(Z^{m+n}(E^{n}, ∗)_{sgn} ⊠ S^{n}H(m)^{∨}) ⊗ (Z^{m′+n′}(E^{n′}, ∗′)_{sgn} ⊠ S^{n′}H(m′)^{∨})
(Z^{m+n+m′+n′}(E^{n+n′}, ∗ + ∗′)_{sgn} ⊠ S^{n+n′}H(m + m′)^{∨})

Proposition 4.7 N^{∗}(E) is a commutative DG algebra (without unit) in the small tensor category of pure elliptic motives.

Thus setting L̃^{∗}(E) := F N^{∗}(E) and using theorem 4.5 we get a proof of theorem 1.1.

Conjecture 4.8 H^{i}(F N^{∗}(E)) = 0 if i ≠ 0.

H^{0}(F N^{∗}(E)) is a Lie coalgebra in the tensor category P_{E}^{∗} which is our candidate for L^{∗}(E).

Proposition 4.9 Assuming the conjecture (4.8) one has for (n, m) ≠ (0, 0):

(H_{P_{E}^{∗}}^{∗}(H^{0}(L̃^{∗}(E))))_{S^{n}H(m)^{∨}} = RHom_{MM_{k}}(ℚ(0), S^{n}H(m)) ⊠ S^{n}H(m)^{∨}

Proof. This folows immediately from theorem 4.5 and lemma 4.6.

6. The Freeness conjecture for mixed elliptic motives. The Lie coalgebras in this paper are Ind-objects in the cateogory of pure motives. Denote by L(E), L^{∗}(E), I^{∗}(E) the Lie algebras dual to the Lie coalgebras L(E), L^{∗}(E), I^{∗}(E). Set

I^{∗}(E) := ⊕_{n+m>1}L^{∗}(E)_{S^{n}H(m)}

It is clear from the picture above that I^{∗}(E) is an ideal in L^{∗}(E) and

L^{∗}(E)/I^{∗}(E) = (k^{∗})^{∨} ⊠ ℚ(1) ⊕ J^{∨} ⊠ H  (23)

is an abelian Lie algebra.

Conjecture 4.10 I^{∗}(E) is a free Lie algebra in the tensor category P_{E}^{∗}.

A Lie algebra L in P_{E}^{∗} is free if and only if H_{P_{E}^{∗}}^{i}(L) = 0 for i > 1.

Remark. According to the property 3) this conjecture implies the Freeness conjecture for mixed Tate motives.

7. The vector space L(E)_{ℚ(1)^{∨}}. It follows from 19 that one should have

L(E)_{S^{2n+1}H(−n)^{∨}} = J_{S^{2n+1}H(−n)} = CH^{n+1}(E^{2n+1})_{sgn}

Let M be a simple object of P_{E}. According to H.Weyl’s theorem Hom_{P_{E}}(M, ⊗^{m}H) is an irreducible G_{m}-module. Denote it by ρ_{M}^{(m)} (we will omit (m) sometimes). Set

CH^{2n+2}(E^{4n+2})_{ρ_{ℚ(1)^{∨}}} := (CH^{2n+2}(E^{4n+2}) ⊗ ρ_{ℚ(1)^{∨}})_{G_{4n+2}} ⊂ S^{2}(J_{S^{2n+1}H(−n)})

For any integer n ≥ 0 one can define an abelian group B_{ℚ(1)}^{(n)} together with the following exact sequence:

0 ⟶ k_{ℚ}^{∗} ⟶ B_{ℚ(1)}^{(n)} ⟶ CH^{2n+2}(E^{4n+2})_{ρ_{ℚ(1)^{∨}}} ⟶ 0  (24)

For n = 0 this is the group B_{2}(E) we discussed in chapter 2. In general the extension 24 comes in a similar way from the biextension related to codimension n + 1 cycles in E^{2n+1} (see [Bl3]).

Let us take the sum of all these extensions:

0 ⟶ k^{∗} ⊗ ℚ[t] ⟶ ⊕_{n≥0}B_{ℚ(1)}^{(n)} ⟶ ⊕_{n≥0}CH^{2n+2}(E^{4n+2})_{ρ_{ℚ(1)^{∨}}} ⊠ ℚ(1)^{∨} ⟶ 0

There is a homomorphism

α : k^{∗} ⊗ ℚ[t] ⟶ k^{∗}, a ⊗ t^{n} ⟼ a

Theorem 4.11

L_{ℚ(1)^{∨}} = \frac{⊕_{n≥0}B_{ℚ(1)}^{(n)}}{kerα}

Using the identification

⊕_{n≥0}(Λ^{2}L(E)_{S^{2n+1}H(−n)^{∨}})_{ℚ(1)^{∨}} = ⊕_{n≥0}CH^{2n+2}(E^{4n+2})_{ρ_{ℚ(1)^{∨}}} ⊠ ℚ(1)^{∨}

we get a canonical homomorphism, provided by (24)

L(E)_{ℚ(1)^{∨}} ⟶ ⊕_{n≥0}(Λ^{2}L(E)_{S^{2n+1}H(−n)^{∨}})_{ℚ(1)^{∨}}

This homomorphism gives the cobracket δ. Its kernel is isomorphic to k^{∗}. Notice how simple is L^{∗}(E)_{ℚ(1)^{∨}} = k_{ℚ}^{∗} and how complicated is L(E)_{ℚ(1)^{∨}}!

8. Reflections on elliptic motivic complexes. Suppose that E is defined over an algebraically closed field k. I conjecture that:

a) For any m, n such that n ≥ 0, n + 2m > 0 there exists an abelian group B_{S^{n}H(m)}^{∗} together with homomorphisms

δ_{1} : B_{S^{n}H(m)}^{∗} ⟶ B_{S^{n−1}H(m)}^{∗} ⊗ J, n + 2m > 0
δ_{2} : B_{S^{n}H(m)}^{∗} ⟶ B_{S^{n}H(m−1)}^{∗} ⊗ k^{∗} m ≥ 1

For m ≥ 1 one should have

B_{S^{2m+1}H(−m)}^{∗} = J_{S^{2m+1}H(−m)} and B_{ℚ(m)}^{∗} = B_{m}(k)

where B_{m}(k) are the groups introduced in [G1-2]. For instance B_{H}^{∗} = J and B_{ℚ(1)}^{∗} = k^{∗}.

b) One should get a bicomplex Γ_{S^{n}H(m)}^{∗} of the following shape.

For m > 0:

B_{S^{n}H(m)}^{∗}  ⟶  B_{S^{n}H(m−1)}^{∗} ∧ k^{∗}  ⟶ ... ⟶  B_{S^{n}H(1)}^{∗} ∧ Λ^{m−1}k^{∗}
↓  ↓  ...  ↓
B_{S^{n−1}H(m)}^{∗} ∧ J ⟶ B_{S^{n−1}H(m−1)}^{∗} ∧ J ∧ k^{∗} ⟶ ... ⟶ B_{S^{n−1}H(1)}^{∗} ∧ J ∧ Λ^{m−1}k^{∗}
↓  ↓  ...  ↓
...  ...  ...  ...
↓  ↓  ...  ↓
B_{ℚ(m)}^{∗} ∧ Λ^{n}J ⟶ B_{ℚ(m−1)}^{∗} ∧ Λ^{n}J ∧ k^{∗} ⟶ ... ⟶  Λ^{n}J ∧ Λ^{m}k^{∗}

Here the horizontal differentials are b ∧ x ⟼ δ_{1}(b) ∧ x and the vertical ones are b ∧ x ⟼ δ_{2}(b) ∧ x.

For m = 0:

B_{S^{n}H}^{∗} ⟶ B_{S^{n−1}H}^{∗} ∧ J ⟶ ... ⟶ B_{S^{2}H}^{∗} ∧ Λ^{n−2}J ⟶ Λ^{n}J

For m < 0, set m′ := −m. Then

B_{S^{n}H(−m′)}^{∗} ⟶ B_{S^{n−1}H(−m′)}^{∗} ⊗ J ⟶ ... ⟶ B_{S^{2m′+1}H(−m′)}^{∗} ⊗ ∧^{n−2m′−1}J

c) The total complex associated with the bicomplex Γ_{S^{n}H(m)}^{∗} (abusing notations I will denote it also by Γ_{S^{n}H(m)}^{∗}) after tensoring with ℚ should be quasiisomorphic to RHom_{MM_{k}}(ℚ(0), S^{n}H(m)).

Example. Γ_{H(m)}^{∗} is the total complex associated with the bicomplex

B_{H(m)}^{∗} → B_{H(m−1)}^{∗} ∧ k^{∗} → ... → B_{H(2)}^{∗} ∧ Λ^{m−2}k^{∗} → B_{3}^{∗}(E) ∧ Λ^{m−1}k^{∗}
↓ ↓ ... ↓ ↓
B_{ℚ(m)}^{∗} ∧ J → B_{ℚ(m−1)}^{∗} ∧ J ∧ k^{∗} → ... → B_{ℚ(2)}^{∗} ∧ J ∧ Λ^{m−2}k^{∗} →  J ∧ Λ^{m}k^{∗}

d) There should be a variation P_{n,m} of mixed elliptic motives framed by ℚ(0) and S^{n}H(m) over a certain finite dimensional variety X(n, m) over k. The groups B_{S^{n}H(m)}^{∗} should come from it as follows. The variation P_{m,n} provides a homomorphism

l̃_{n,m} : ℚ[X(n, m)(k)] → A(ℚ(0), S^{n}H(m))

where l_{n,m}({x}) for x ∈ X_{n,m} is the class of the framed mixed motive P_{n,m}(x). (P_{n,m}(x) is the fiber at x of the variation P_{n,m}). Passing to the coalgebra Lie we get a homomorphism

l_{n,m} : ℚ[X(n, m)(k)] → L(E)_{S^{n}H(m)^{∨}}

Set B_{S^{n}H(m)} := Im(l_{n,m}). The group B_{S^{n}H(m)}^{∗} is the largest ℚ-subspace of B_{S^{n}H(m)} such that the restriction of the coproduct δ to the group B_{S^{n}H(m)}^{∗} has non zero components only in

L_{S^{n−1}H(m)^{∨}} ∧ J ⊠ H^{∨} ⊕ L_{S^{n}H(m−1)^{∨}} ∧ k^{∗} ⊠ ℚ(1)^{∨}, if m > 1

and in

L_{S^{n−1}H(m)^{∨}} ∧ J ⊠ H^{∨}, if m ≤ 1

By definition δ_{1} and δ_{2} are the components of the restriction of δ to B_{S^{n}H(m)}^{∗} ⊠ S^{n}H(m)^{∨}. The restriction of δ to B_{S^{n}H(m)} ⊠ S^{n}H(m)^{∨} may be more complicated.

The periods of the ℝ-Hodge realization of the variation P_{n,m} are the new transcendental functions denoted P_{n,m} which is needed to get the special values L(S^{n}H, m + n).

Example. P_{n,1} is the (motivic) elliptic polylogarithm sheaf on E\0, and P_{n,1} are the Eisenstein-Kronecker series from 1.5. The group B_{S^{n}H(1)} (resp. B_{S^{n}H(1)}^{∗}) should coincide with the group

B_{n+2}(E) = \frac{ℚ[E(k]}{R_{n+2}(E)}, B_{n+2}^{∗}(E) = \frac{I_{E}^{2n+2}}{R_{n+2}(E)}

which will be defined later on. Here I_{E} is the augmentation ideal of the group algebra ℚ[E(k)].

8. The structure of L^{∗}(E) and elliptic motivic complexes. Let us define the ℚ-vector spaces C_{S^{n}H(m)}^{∗} by setting

H^{1}(I^{∗}(E)) = ⊕C_{S^{n}H(m)}^{∗} ⊠ S^{n}H(m)^{∨}

Let δ^{∗} be the the cobracket in the Lie coalgebra L^{∗}(E). Conjecture (4.10) means that

δ^{∗} : C_{S^{n}H(m)}^{∗} ⟶ C_{S^{n}H(m−1)}^{∗} ∧ k^{∗} ⊕ C_{S^{n−1}H(m)}^{∗} ∧ J  (25)

Lemma 4.12 The S^{n}H(m)^{∨}-isotypical component of the Serre-Hochshild spectral sequence for the ideal I^{∗}(E) ⊂ L^{∗}(E) computing cohomology of L^{∗}(E) collapces to the total complex associated with a bicomplex ( which should be tensored ⊠S^{n}H(m)^{∨}):

C_{S^{n}H(m)}^{∗}  →  C_{S^{n}H(m−1)}^{∗} ⊗ k^{∗}  → ... →  C_{S^{n}H(−[ \frac{n−1}{2} ])}^{∗} ⊗ S^{m+[ \frac{n−1}{2} ]}k^{∗}
↓  ↓  ↓
C_{S^{n−1}H(m)}^{∗} ⊗ J → C_{S^{n−1}H(m)}^{∗} ⊗ J ⊗ k^{∗} → ... → C_{S^{n−1}H(−[ \frac{n−2}{2} ])}^{∗} ⊗ J ⊗ S^{m+[ \frac{n−2}{2} ]}k^{∗}
↓  ↓  ↓
...  ...  ...
↓  ↓  ↓
C_{H(m)}^{∗} ⊗ Λ^{n−1}J → C_{H(m−1)}^{∗} ⊗ Λ^{n−1}J ⊗ k^{∗} → ... →  C_{H(1)}^{∗} ⊗ Λ^{n−1}J ⊗ Λ^{m−1}k^{∗}
↓ ↓ ↓
C_{ℚ(m)}^{∗} ⊗ Λ^{n}J → C_{ℚ(m−1)}^{∗} ⊗ Λ^{n}J ⊗ k^{∗} → ... →  Λ^{n}J ⊗ Λ^{m}k^{∗}

The length of the rows decreases when we are going down. Here is how it looks for n = 4, m = 1:

C_{S^{4}H(1)}^{∗}  →  C_{S^{4}H}^{∗} ⊗ k^{∗}  → C_{S^{4}H(−1)}^{∗} ⊗ Λ^{2}k^{∗}
↓  ↓  ↓
C_{S^{3}H(1)}^{∗} ⊗ J → C_{S^{3}H}^{∗} ⊗ J ⊗ k^{∗} → C_{S^{3}H(−1)}^{∗} ⊗ J ⊗ Λ^{2}k^{∗}
↓  ↓
C_{S^{2}H(1)}^{∗} ⊗ Λ^{2}J → C_{S^{2}H}^{∗} ⊗ Λ^{2}J ⊗ k^{∗}
↓

C_{H(1)}^{∗} ⊗ Λ^{3}J

↓
k^{∗} ⊗ Λ^{2}J

Proof. The E_{1} term of this spectral sequence is

E_{1}^{p,q} = C^{p}(k^{∗} ⊕ J, H_{P_{E}^{∗}}^{q}(I^{∗}(E)))_{S^{n}H(m)^{∨}}

Since H_{P_{E}^{∗}}^{q}(I^{∗}(E)) is zero for q > 1 the only non zero rows are E_{1}^{•,0} and E_{1}^{•,1}.

Moreover E_{1}^{p,0} is zero unless p = n + m, and

E_{1}^{m+n,0} = Λ^{n}J ⊗ Λ^{m}k^{∗}

Further, E_{1}^{p,1} is non zero only if 0 ≤ p < m + n. If so, then

E_{1}^{p,1} = ⊕_{a+b=p} C_{S^{n−a}H(m−b)}^{∗} ⊗ Λ^{a}J ⊗ Λ^{b}k^{∗}

So the E_{1} term of the spectral sequence gives us precisely the groups in the bicomplex. The differential d_{1} : E_{1} → E_{1} provides all the differentials exept the last two in the right bottom corner targeting to Λ^{n}J ⊗ Λ^{m}k^{∗}. These two we get from the differential d_{2}. The lemma is proved.

The structure of the quotient L^{∗}(E)/[I^{∗}(E), I^{∗}(E)]. There is an exact sequence of Lie algebras

0 → \frac{I^{∗}(E)}{[I^{∗}(E), I^{∗}(E)]} → \frac{L^{∗}(E)}{[I^{∗}(E), I^{∗}(E)]} → \frac{L^{∗}(E)}{I^{∗}(E)} → 0  (26)

The action of L^{∗}(E) on I^{∗}(E) leads to the action of L^{∗}(E)/I^{∗}(E) on I^{∗}(E)/[I^{∗}(E), I^{∗}(E)]. The Lie algebra structure on L^{∗}(E)/[I^{∗}(E), I^{∗}(E)] is determined by this action.

The inclusion L^{∗}(E)_{ℚ(1)^{∨}} ⊕ L^{∗}(E)_{H(1)^{∨}} ↪ L^{∗}(E) provides a canonical splitting s : L^{∗}(E)/I^{∗}(E) → L^{∗}(E)/[I^{∗}(E), I^{∗}(E)] as an extension of ℚ-vector spaces. Recall that L^{∗}(E)/I^{∗}(E) = (k_{ℚ}^{∗})^{∨} ⊠ ℚ(1) ⊕ J_{ℚ}^{∨} ⊠ H is an abelian Lie algebra. So to define the Lie algebra structure on L^{∗}(E)/[I^{∗}(E), I^{∗}(E)] we need to know the ℚ-vector spaces C_{S^{n}H(m)}^{∗} and the homomorphisms (25). That is precisely what the following two conjectures are doing.

Conjecture 4.13 i) There is a canonical map B_{S^{n}H(m)}^{∗} ↪ C_{S^{n}H(m)}^{∗} such that δ^{∗}|_{B_{S^{n}H(m)}^{∗}} = δ_{1} + δ_{2}.

ii) B_{S^{n}H(m)}^{∗} = C_{S^{n}H(m)}^{∗} if n = 0, 1. Otherwise one has an exact sequence

0 → B_{S^{n}H(m)}^{∗} → C_{S^{n}H(m)}^{∗} → B_{S^{n}H(−[ \frac{n−1}{2} ])}^{∗} ⊗ S^{m+[ \frac{n−1}{2} ]}k^{∗} → 0 (27)

Set C̄_{S^{n}H(m)}^{∗} := C_{S^{n}H(m)}^{∗}/B_{S^{n}H(m)}^{∗}. Then δ^{∗} induces a map

C̄_{S^{n}H(m)}^{∗} ⟶ C̄_{S^{n}H(m−1)}^{∗} ⊗ k^{∗} ⊕ C̄_{S^{n−1}H(m)}^{∗} ⊗ J  (28)

Now we can formulate the second part of conjecture (4.10).

Conjecture 4.14 The second component of the map (28) is zero, and the first component coinsides with the homomorphism

B_{S^{n}H(−[ \frac{n−1}{2} ])}^{∗} ⊗ S^{m+[ \frac{n−1}{2} ]}k^{∗} ⟶ B_{S^{n}H(−[ \frac{n−1}{2} ])}^{∗} ⊗ S^{m+[ \frac{n−1}{2} ]−1}k^{∗} ⊗ k^{∗} (29)

given by the identity × the Koszul differential.

Theorem 4.15 Assume all the conjectures of this section. Then the complex Γ_{S^{n}H(m)}^{∗} is quasiisomorphic to RHom_{M_{E}}(ℚ(0), S^{n}H(m))

Proof. The complex RHom_{M_{E}}(ℚ(0), S^{n}H(m)) is quasiisomorphic to the standard complex C^{•}(L(E)) of the Lie coalgebra L(E). According to the property 1) there is a morphism of complexes

C_{P_{E}^{∗}}^{•}(L^{∗}(E)) ⟶ C_{P_{E}}^{•}(L(E))

which is a quasiisomorphism thanks to the property 2).

The part ii) of conjecture (4.13) just means that there is a canonical embedding of complexes

j : Γ_{S^{n}H(m)}^{∗} ↪ C_{P_{E}^{∗}}^{•}(L^{∗}(E))

We are going to show that it is a quasiisomorphism.

Using the part i) of conjecture (4.13) we see that the quotient of the bicomplex we got from the spectral sequence along the bicomplex Γ_{S^{n}H(m)}^{∗} looks as follows. The two bottom rows become zero, and each of the remaining rows is B_{?}^{∗} times a Koszul complex. For instance in the case n = 4, m = 1 the quotient is get

B_{S^{4}H(−1)}^{∗} ⊗ S^{2}k^{∗} → B_{S^{4}H(−1)}^{∗} ⊗ k^{∗} ⊗ k^{∗} → B_{S^{3}H(−1)}^{∗} ⊗ Λ^{2}k^{∗}
0 ↓ 0 ↓ 0 ↓
B_{S^{3}H(−1)}^{∗} ⊗ J ⊗ S^{2}k^{∗} → B_{S^{3}H(−1)}^{∗} ⊗ J ⊗ k^{∗} ⊗ k^{∗} → B_{S^{3}H(−1)}^{∗} ⊗ J ⊗ Λ^{2}k^{∗}
0 ↓  0 ↓
B_{S^{2}H}^{∗} ⊗ Λ^{2}J ⊗ k^{∗} →  B_{S^{2}H}^{∗} ⊗ Λ^{2}J ⊗ k^{∗}

5 The complexes B(E, n)^{•} and B^{∗}(E, n)^{•}

1. An auxillary complex. Let A be an abelian group. Consider the following complex

ℚ[A] ⟶^{δ} ℚ[A] ⊗ A_{ℚ} ⟶^{δ} ℚ[A] ⊗ Λ^{2}A_{ℚ} ⟶^{δ} ℚ[A] ⊗ Λ^{3}A_{ℚ} ⟶^{δ} ...  (30)

The differential is defined by the formula

δ : {a} ⊗ b_{1} ∧ b_{2} ∧ ... ∧ b_{m} ⟼ {a} ⊗ a ∧ b_{1} ∧ b_{2} ∧ ... ∧ b_{m}  (31)

It is infinite if A_{ℚ} := A ⊗ ℚ is infinite dimensional. Let I_{A}^{k} be the k-th degree of the augmentation ideal I_{A} ⊂ ℤ[A].

Lemma 5.1 δ(I_{A}^{k}) ⊂ I_{A}^{(k−1)} ⊗ A.

Proof. I_{A}^{k} is generated by the elements

({a_{1}} − {0}) ∗ ({a_{1}} − {0}) ∗ ... ∗ ({a_{k}} − {0})

Clearly

δ({a_{1}} − {0}) ∗ ... ∗ ({a_{k}} − {0}) = ∑_{i} ∏_{j≠i}({a_{j}} − {0}) ∗ {a_{i}} ⊗ a_{i}

So the complex (30) has the “diagonal” filtration by subcomplexes

I_{A}^{n} ⟶^{δ} I_{A}^{n−1} ⊗ A_{ℚ} ⟶^{δ} I_{A}^{n−2} ⊗ Λ^{2}A_{ℚ} ⟶^{δ} I_{A}^{n−3} ⊗ Λ^{3}A_{ℚ} ⟶^{δ} ...

Each graded quotient is isomorphic to the Koszul complex

S^{n}A_{ℚ} ⟶ S^{n−1}A_{ℚ} ⊗ A_{ℚ} ⟶ ... ⟶ A_{ℚ} ⊗ Λ^{n−1}A_{ℚ} ⟶ Λ^{m}A_{ℚ}

2. The groups B_{n}(E). Recall that B_{0}(E) := ℤ and B_{1}(E) = J(k). We will usually write J for J(k). The group B_{2}(E) is the group discussed in s. 2.1. In particular if k = k̄ it is a quotient of ℤ[E(k)].

Recall that

E^{(n−1)} = {(x_{1}, ..., x_{n}) ⊂ E^{n}| ∑ x_{i} = 0}

The group S_{n} acts naturally on E^{(n−1)}. Let p_{i} : E^{(n−1)} → E be the projection to the i-th factor. We will use the “coordinate notations” denoting p_{i}^{∗}f by f(x_{i}) etc. Let us define the following diagram:

K_{n}^{M}(k(E^{(n−1)}))
μ̄_{n} ↗
S^{n}(k(E)^{∗})
β_{n} ↘
I_{E}^{2n}

by setting

μ̄_{n} : f_{1}∘...∘f_{n} ⟼ ∑_{σ∈S_{n}}(−1)^{|σ|}{p_{σ(1)}^{∗}f_{1}, ..., p_{σ(n)}^{∗}f_{n}} = ∑_{σ∈S_{n}}{p_{1}^{∗}f_{σ(1)}, ..., p_{n}^{∗}f_{σ(n)}}

For n ≥ 2 set

β_{n} : S^{n}k(E)^{∗} ⟶ I_{E(k)}^{2n} f_{1} ∘ ... ∘ f_{n} ⟼ (f_{1}) ∗ (f_{2}) ∗ ... ∗ (f_{n})

Definition 5.2 Let k = k̄ and n ≥ 3. Then R_{n}(E/k) is the subgroup of ℤ[E] generated by β_{n−1}(Kerμ̄_{n−1}) and “distribution relations” :

{a}_{n} − m^{n−2} · ∑_{mb=a}{b}_{n}, a ∈ E(k),  m = −1, 2  (32)

Example R_{3}(E) is generated by the elements (1 − f) ∗ (f)^{−}, where {x}^{−} := {−x}, and (33).

Remark. It would be more natural to add to the subgroup R_{n}(E/k) the distribution relations for all m ∈ ℤ, m ≠ 0. However we will get the same group, and for our purposes the definitioon above is technically more convinient.

Definition 5.3 Let k = k̄.

B_{n}(E) := \frac{ℤ[E(k)]}{R_{n}(E/k)}

Theorem 5.4 δ(R_{n}(E)) ⊂ R_{n−1}(E) ⊗ J.

The proof consists of two independent parts. For a more dificult one see proof of the theorem (5.9) below. The easy one follows from

Lemma 5.5 For any m ∈ ℤ, m ≠ 0, one has

δ_{n}({a}_{n} − m^{n−2} · ∑_{mb=a}{b}_{n}) = 0 in the group B_{n−1}(E) ⊗ J  (33)

Proof. For n = 2 this is done in [GL]. The general case follows by induction:

δ({a}_{n} − m^{n−2} · ∑_{mb=a}{b}_{n}) = {a}_{n−1} ⊗ a − m^{n−3} · ∑_{mb=a}{b}_{n−3} ⊗ mb =

({a}_{n−1} − m^{n−3} · ∑_{mb=a}{b}_{n−1}) ⊗ a = 0

So we get a homomorphism δ : B_{n}(E) ⟶ B_{n−1}(E) ⊗ J and thus the following complex B(S^{n−2}H(1))^{•}:

B_{n}(E) ⟶^{δ} B_{n−1}(E) ⊗ J ⟶^{δ} ... ⟶^{δ} B_{2}(E) ⊗ Λ^{n−2}J ⟶^{δ} J ⊗ Λ^{n−1}J ⟶^{δ} Λ^{n}J

(34)

Here the very left group sits in degree one. The differential is defined by the formula (31) and has degree +1.

If k is not an algebraically closed we postulate the Galois descent:

B(E/k, n + 2)_{ℚ}^{•} := (B(E/k̄, n + 2)_{ℚ}^{•})^{Gal(k̄/k)}

Let us also define the groups B_{n}^{∗}(E) for k̄ = k:

B_{n}^{∗}(E) := Im(I^{2n−2} ⟶ B_{n}(E))

Here the map is induced by the natural inclusion I^{2n−2} ↪ ℤ[E].

It follows from the lemma (5.1) that δ_{n}(B_{n}^{∗}(E)) ⊂ B_{n−1}^{∗}(E) ⊗ J. So we can consider the following subcomplex B^{∗}(E, n)^{•} of the complex (34).

B_{n}^{∗}(E) ⟶^{δ} B_{n−1}^{∗}(E) ⊗ J ⟶^{δ} ... ⟶^{δ} k^{∗} ⊗ Λ^{n−2}J

Proposition 5.6 The canonical morphism of complexes

B^{∗}(E, n + 2)_{ℚ}^{•} ⟶ B(E, n + 2)_{ℚ}^{•}

is a quasiisomorphism.

Proof. This morphism is injective by the definition of the groups B_{n}^{∗}(E). It follows immediately from the lemma below that the quotient is isomorphic to the Koszul complex

S^{n}J_{ℚ} ⟶ S^{n−1}J_{ℚ} ⊗ J_{ℚ} ⟶ ... ⟶ S^{2}J_{ℚ} ⊗ Λ^{n−2}J_{ℚ} ⟶ J_{ℚ} ⊗ Λ^{n−1}J_{ℚ} ⟶ Λ^{n}J_{ℚ}

Lemma 5.7

B_{n}(E)/B_{n}^{∗}(E) ⊗ ℚ = S^{n}J_{ℚ}

Proof. We may assume that k is closed. We need to study the quotient of the group ℚ[E(k)]/I^{2n−2} along the subgroup generated by the distribution relations. Notice that

ℚ[E(k)]/I^{2n−2} = ℚ ⊕ J_{ℚ} ⊕ S^{2}J_{ℚ} ⊕ ... ⊕ S^{2n−3}J_{ℚ}

where the isomorphism is given by i = (i_{0}, ..., i_{2n−3}) where

i_{m} : ℚ[E(k)] ⟶ S^{m}J_{ℚ}; {a} ⟼ a^{m}

Let us denote by DR_{n} the subgroup generated by the distribution relations (33). Then the homomorphism

in̂ = (i_{0}, ..., î_{n}, ..., i_{2n−3}) : DR_{n} ⟶ ℚ ⊕ J_{ℚ} ⊕ ... ⊕ Ŝ^{n}J_{ℚ} ⊕ ... ⊕ S^{2n−3}J_{ℚ}

is surjective. Indeed,

i_{k}({a}_{n} − m^{n−2} · ∑_{mb=a}{b}_{n}) = (1 − m^{n−k}) · a^{k}

In particular i_{n}(DR_{n}) = 0. The lemma and hence the proposition are proved.

3. The complex B^{∗}(E/k, n + 1)_{ℚ}^{•} is an elliptic deformation of the complex B(Spec(k), n)_{ℚ}^{•} . Let me remind that the complex B(Spec(k), n)^{•} looks as follows:

B_{n}(k) ⟶ B_{n−1}(k) ⊗ k^{∗} ⟶ ... ⟶ B_{2}(k) ⊗ Λ^{n−2}k^{∗} ⟶ Λ^{n}k^{∗}

where B_{n}(k) is the quotient of ℤ[k^{∗}] along a certain subgroup R_{n}.

There is an important difference between these two complexes. The complex B(Spec(k), n)_{ℚ}^{•} is defined directly in terms of a field k, while to define the complex B(E/k, n + 1)_{ℚ}^{•} we have to go to the algebraic closure of k and then postulate the Galois descent property. So in general they can only be quasiisomorphic.

Suppose k = k̄. When E degenerates to (P^{1}, {0} ∪ {∞}) the complex B(E/k, n + 1)_{ℚ}^{•} degenerates to a complex quasiisomorphic to B(Spec(k), n)_{ℚ}^{•} (for n = 2 see s.3.4 in [GL]).

In a sence the elliptic situation is simpler. For example our definition of the group R_{n}(E) is not inductive and more explicit then the definition of the group R_{n}(k) in [G2]. In fact when E degenerates to (P^{1}, {0} ∪ {∞}) our definition suggests a new way of defining of the groups B_{n}(k) from [G1]-[G2].

4. Proof of the theorem (5.4). Let σ ∈ S^{n} be a permutation and a_{i} ∈ E. Consider the following codimension p cycle in E^{(n−1)}:

X(σ; a_{1}, ..., a_{p}) := {(x_{1}, ..., x_{n}) ⊂ E^{(n−1)}|x_{σ(1)} = a_{1}, ..., x_{σ(p)} = a_{p}}

It is a product of elliptic curves E. Let us define a homomorphism

μ_{n;p} : Λ^{p}ℤ[E] ⊗ S^{n−p}k(E)^{∗} ⟶ ∐_{X∈(E^{(n−1)})^{p}} Λ^{n−p}(k(X))^{∗}

by setting

{a_{1}}∧...∧{a_{p}}⊗f_{p+1}∘...∘f_{n} ⟼ ∑_{σ∈S_{n}}(−1)^{|σ|}{f_{p+1}(x_{σ(p+1)}), ..., f_{n}(x_{σ(n)})}|_{X(σ;a_{1},...,a_{p})}

Denote by D̃_{(n)}^{p} the image of the homomorphism μ_{n;p−1}.

The elements of type f_{1}(x_{1}) ∧ ... ∧ f_{m}(x_{m}) in Λ^{m}k(E^{(m−1)})^{∗} and their linear combinations may be called the decomposable elements; this suggests the notation.

Let F be an arbitrary field with a discrete valuation v and the residue class F̄_{v}. The group of units U has a natural homomorphism U ⟶ F_{v}^{∗} , u ↦ ū. Choose a uniformizer π ∈ F^{∗}, ord_{v}π = 1. There is canonical homomorphism

Λ^{n}(F^{∗}) ⟶^{∂_{v}} Λ^{n−1}(F_{v}^{∗})

uniquely defined by the properties

∂_{v}(u_{1} ∧ ... ∧ u_{n}) = 0; ∂_{v}(π ∧ u_{2} ∧ ... ∧ u_{n}) = (ū_{2}, ..., ū_{n})

Consider the following complex on E^{(n−1)}:

Λ^{n}(k(E^{(n−1)})) ⟶^{∂} ∐_{X∈(E^{(n−1)})^{1}} Λ^{n−1}(k(X)) ⟶^{∂} ... ⟶^{∂} ∐_{X∈(E^{(n−1)})^{n−1}} k(X)^{∗}

(35)

Here ∂ := ∑_{X} ∂_{v(X)} where v(X) is the valuation corresponding to the irreducible divisor X.

Lemma 5.8 The groups D̃_{(n)}^{p} form a subcomplex in the complex (35)

Proof. Clear from the definitions.

Let us define for p < n a homomorphism

β̃_{n}^{(p+1)} : D̃_{(n)}^{p+1} ⟶ I_{E}^{2(n−p+1)} ⊗ Λ^{p}J

We define β̃_{n}^{(p+1)} first on space of decomposable elements on the subvariety x_{1} = a_{1}, ..., x_{p} = a_{p} by the formula

β̃_{n}^{(p+1)} : f_{p+1}(x_{p+1}) ∧ ... ∧ f_{n}(x_{n})|_{x_{1}=a_{1},...,x_{p}=a_{p}} ⟼
(f_{p+1}) ∗ ... ∗ (f_{n}) ∗ (a_{1} + ... + a_{p}) ⊗ a_{1} ∧ ... ∧ a_{p}

It extends uniquely to the space of all decomposable elements assuming the skewsymmetry with respect to the action S_{n}. In particulary it is defined on D̃_{(n)}^{p}.

If p = n − 1 then D̃_{(n)}^{n} = k^{∗} ⊗ ℤ[E^{(n−1)}]^{sgn} and we have a homomorphism

β̃_{n}^{(n)} : k^{∗}⊗ℤ[E^{(n−1)}]^{sgn} ⟶ k^{∗}⊗Λ^{n−1}J; x⊗(a_{1}, ..., a_{n}) ⟼ x⊗a_{1}∧...∧a_{n−1}

Finally, one can define a homomorphism

I_{E}^{4} ⊗ Λ^{n−2}J ⟶^{δ} k^{∗} ⊗ Λ^{n−1}J

Namely, there is a homomorphism

B_{3}(E) ⊗ Λ^{n−2}J ⟶^{δ} B_{H(1)}(E) ⊗ Λ^{n−1}J {a}_{3} ⊗ b_{1} ∧ ... ⟼ {a}_{2} ⊗ a ∧ b_{1} ∧ ...

Let B_{n}^{(k)}(E) be the subgroup of B_{n}(E) generated by k-th degree of the augmentation ideal I_{E}^{k}. The restriction of this homomorphism to B_{3}^{(4)}(E) ⊗ Λ^{n−2}J ⊂ B_{3}(E) ⊗ Λ^{n−2}J lands in k^{∗} ⊗ Λ^{n−1}J. Indeed, the composition

I_{E}^{4} ⊗ Λ^{n−2}J ⟶^{δ} B_{2}(E) ⊗ Λ^{n−1}J ⟶ S^{2}J ⊗ Λ^{n−1}J

is equal to zero.

Theorem 5.9 The maps β̃_{n}^{(i)} provide a homomorphism of complexes

D̃_{(n)}^{1} ⟶^{∂}  D̃_{(n)}^{2}  ⟶^{∂} ... ⟶^{∂}  D̃_{(n−1)}^{n}  ⟶^{∂}  D̃_{(n)}^{n}
↓ β̃_{n}^{(1)}  ↓ β̃_{n}^{(2)}  ↓ β̃_{n}^{(n−1)}  ↓ β̃_{n}^{(n)}
I_{E}^{2n}  ⟶^{δ} I_{E}^{2n−2} ⊗ J ⟶^{δ} ... ⟶^{δ} I_{E}^{4} ⊗ Λ^{n−2}J ⟶^{δ} k^{∗} ⊗ Λ^{n−1}J

Proof. We will do in details the commutativity of the left square, at the same time proving the theorem (5.4). The commutativity of the other squares except the very right one is completely similar.

The commutativity of the right square is a more subtle statement. For n = 2 it was already proved in [GL], s.3. The general case n > 2 is similar.

Consider an element

f_{1}(x_{1}) ∧ ... ∧ f_{n}(x_{n}) ∈ Λ^{n}k(E^{(n−1)})  (36)

Let v_{a}(f) be the order of zero of the function f(t) at t = a.

The part of the coboundary (in the complex D̃_{(n)}^{•}) of the element (36) sitting on the divisor x_{1} = a_{1} is equal to

∑_{a_{1}∈E(k)} v_{a_{1}}(f_{1}) · f_{2}(x_{2}) ∧ ... ∧ f_{n}(x_{n}), x_{2} + ... + x_{n} = −a_{1}  (37)

Let t_{a} be the shift by a on the group E(k), so t_{a}{b} = {a + b} and t_{a}f(x) = f(x − a) (sic). Then (t_{a}f) = (f) ∗ (a). Then it can be written as

∑_{a_{1}∈E(k)} v_{a_{1}}(f_{1}) · f_{2}(y_{2}) ∧ ... ∧ t_{a_{1}}f_{n}(y_{n}), y_{2} + ... + y_{n} = 0

Applying the homomorphism β̃_{n}^{(2)} to the element (37) we get

∑_{a_{1}∈E(k)} v_{a_{1}}(f_{1}) · (f_{2}) ∗ ... ∗ (f_{n}) ∗ (a_{1}) ⊗ a_{1}  (38)

From the other hand writing

(f_{1}) ∗ ... ∗ (f_{n}) = ∑_{a_{i}∈E(k)} v_{a_{1}}(f_{1}) · ... · v_{a_{n}}(f_{n}) · {a_{1} + ... + a_{n}}

we get

δ((f_{1})∗...∗(f_{n})) = ∑_{a_{i}∈E(k)} v_{a_{1}}(f_{1})·...·v_{a_{n}}(f_{n})·{a_{1}+...+a_{n}}⊗(a_{1}+...+a_{n}) (39)

Collecting the terms with a_{1} in the right factor we get just what needed: the formula (38). Taking into consideration the coboundaries of the element (36) sitting on the divisors x_{p} = a_{p} we will get the other terms (with a_{p}) in (39).

To prove the commutativity of the right square we replace it by a bigger diagram

D̃_{(n−1)}^{n}  ⟶^{∂}  D̃_{(n)}^{n}
↓ β̃_{n}^{(n−1)}  ↓ β̃_{n}^{(n)}
B_{3}(E) ⊗ Λ^{n−2}J ⟶^{δ} B_{2}(E) ⊗ Λ^{n−1}J

and then prove its commutativity in a way similar to the proof of theorem 4.5 in [GL]. The theorem is proved.

Consider the Gersten complex for the Milnor K-theory on E^{(n−1)} ([NS]):

K_{n}^{M}(k(E^{(n−1)})) ⟶^{∂} ∐_{X∈(E^{(n−1)})^{1}} K_{n−1}^{M}(k(X)) ⟶^{∂} ... ⟶^{∂} ∐_{X∈(E^{(n−1)})^{n−1}} k(X)^{∗}

The Gersten complex is obtained from (35) by factorisation along the subgroup generated by the Steinberg relations. Let

D_{(n)}^{•} : D_{(n)}^{1} ⟶ D_{(n)}^{2} ⟶ ... ⟶ D_{(n)}^{n}

be the image of the complex D̃_{(n)}^{•} in the Gersten complex. In other words

D_{(n)}^{p} = D̃_{(n)}^{p}/St_{(n)}^{p}

where St_{(n)}^{p} is the intersection of D̃_{(n)}^{p} with the subgroup generated by the Steinberg relations in ∐_{X∈(E^{(n−1)})^{p−1}} Λ^{n−p+1}k(X)^{∗}.

Lemma 5.10 β_{n}^{(p)}(St_{(n)}^{p}) = 0 in B_{n+2−p}^{∗}(E) ⊗ Λ^{p−1}J.

Proof. Consider the subvariety X(id; a_{1}, ..., a_{p−1}) in E^{(n−1)} and its projection

p : X(id; a_{1}, ..., a_{p−1}) ⟶ E^{n−p} p : (x_{1}, ..., x_{n}) ⟼ (x_{p}, ..., x_{n−1})

The subgroup of Steinberg relations is generated by

{f(x_{p}, ..., x_{n−1}), (1 − f)(x_{p}, ..., x_{n−1}), g_{1}(x_{p}, ..., x_{n−1}), ...}

Notice that x_{n} = −a_{1} − ... − a_{p−1} − x_{p} − ... − x_{n−1}. This means that an element ∑_{j}{f_{p}^{(j)}(x_{p}), ..., f_{n}^{(j)}(x_{n})} of the subgroup of Steinberg relations can be written as

p^{∗} ∑_{j}{f_{p}^{(j)}(y_{p}), ..., t_{−a_{1}−...−a_{p}}f_{n}^{(j)}(y_{n})}

on

E^{(n−p)} = {(y_{p}, ..., y_{n})|y_{p} + ... + y_{n} = 0}

Now the lemma follows immediately.

Theorem 5.11 There is a canonical homomorphism of complexes

D_{(n)}^{1}  ⟶  D_{(n)}^{2}  ⟶ ... ⟶  D_{(n)}^{n}
↓ β̄_{n}^{(1)}  ↓ β̄_{n}^{(2)}  ↓ β̄_{n}^{(n)}
B_{n+1}^{∗}(E) ⟶^{δ} B_{n}^{∗}(E) ⊗ J ⟶^{δ} ... ⟶^{δ} k^{∗} ⊗ Λ^{n−1}J

Proof. Follows immediately from lemma (5.10) and theorem (5.9).

6 The regulator integrals, Eisenstein - Kronecker series and a conjecture on L(Sym^{n}E, n + 1)

1. Beilinson’s conjecture in the case of L(Sym^{n}h^{1}(E), n + 1). Let E be an elliptic curve over a number field k. According to the general Beilinson conjecture on regulators the special value L(Sym^{n}h^{1}(E), n+1) should be equal, up to standard factors, to the (co)volume of a certain lattice obtained as follows.

The n-th Deligne complex ℝ(n)_{D} on a regular variety X can be defined as the total complex associated with the following bicomplex:

A_{X}^{0}(n − 1) ⟶^{d} … ⟶^{d} A_{X}^{n}(n − 1) ⟶^{d} A_{X}^{n+1}(n − 1) ⟶^{d} …
↑ π_{n}  ↑ π_{n}
Ω_{X}^{n}  ⟶^{∂}  Ω_{X}^{n+1}  ⟶^{∂}

(40)

Here (A_{X}^{•}, d) is the C^{∞}-De Rham complex, π_{n} : A_{X}^{m} ⊗ ℂ ⟶ A_{X}^{m}(n − 1) is the projection induced by π_{n} : ℂ ⟶ ℝ(n − 1), z ⟼ z + (−1)^{(n−1)}z̄, the group A_{X}^{0}(n − 1) is placed in degree 1 and (Ω_{X}^{•}, ∂) is the De Rham complex of holomorphic forms with logarithmic singularities at infinity.

One has the regulator map

r_{Be} : H_{M}^{n}(E^{(n−1)}, ℚ(n))_{sgn} ⟶ H_{D}^{n}(E^{(n−1)} ⊗_{ℚ} ℝ, ℝ(n))_{sgn}  (41)

The right hand side is a group of purely topological origin:

H_{D}^{n}(E^{(n−1)} ⊗_{ℚ} ℝ, ℝ(n))_{sgn} = H_{B}^{n−1}(E^{(n−1)} ⊗_{ℚ} ℂ, ℝ(n − 1))_{sgn}^{+} =

(⊕_{σ:k↪ℂ}H_{B}^{n−1}(E_{σ}^{(n−1)}(ℂ), ℝ(n − 1))_{sgn})^{+} =
(⊕_{σ:k↪ℂ}Sym^{n−1}H_{B}^{1}(E_{σ}(ℂ), ℝ(1)))^{+}

Here + means invariants of the complex conjugation acting on σ’s and on the coefficients ℝ(n − 1).

The image of the regulator map (41) is conjectured to be a lattice. H_{D}^{n}(E^{(n−1)}⊗_{ℚ}

ℝ, ℚ(n))_{sgn} gives another lattice in H_{D}^{n}(E^{(n−1)} ⊗_{ℚ} ℝ, ℝ(n))_{sgn}. The covolume

of the lattice Imre_{Be} measured with respect to the second lattice should coincide (up to standard factors) with the special value of our L-function at s = n + 1.

2. The Eisenstein-Kronecker series. Let me recall their definition:

K_{i,j}(z; τ) = ∑_{γ∈Γ\0} \frac{(z, γ)}{γ^{i}γ̄^{j}}, i, j ≥ 1

For the relation with the elliptic polylogarithms see [BL], [Z].

Lemma 6.1 a) For any lattice Γ one has K̄_{i,j}(z; τ) = (−1)^{i+j}K_{j,i}(z; τ)

b) Suppose that the lattice Γ and a divisor P on ℂ/Γ are invariant under the complex conjugation. Then K_{i,j}(P; τ) ∈ ℝ.

Proof. Clear.

Consider for each n > 2 a homomorphism

K_{n} : ℤ[E(ℂ)] ⟶ Sym^{n−2}H^{1}(E(ℂ), ℂ)

{z} ⟼ ∑_{a+b=n} K_{a,b}(z; τ)(dz)^{a−1}(dz̄)^{b−1}

Theorem 6.2 K_{n}(R_{n}(E(ℂ))) = 0.

We will prove it in s 6.3 below. So we get a homomorphism

K_{n} : B_{n}[E(ℂ)] ⟶ Sym^{n−2}H^{1}(E(ℂ), ℂ)

This means that R_{n}(E(ℂ)) is a subgroup of functional equations for the Eisenstein-Kronecker series.

Lemma 6.3 Suppose that E is defined over ℝ and the lattice Γ was defined using a real differential ω. Then

K_{n} : B_{n}[E(ℂ)]^{+} ⟶ Sym^{n−2}H^{1}(E(ℂ), ℝ(1))^{+}

Here + means invariants of the complex involution acting on both ℝ(1) and E(ℂ).

Proof. Follows from lemma (6.1).

3. Computation of the regulator integral. The main result of this section is due to Deninger (see [D1], s.6). Our presentation is technically simpler since working with distributions we avoid the convergence problems.

For any functions f_{1}, ..., f_{n} on a manifold X consider the following (n − 1)- form with values in ℝ(n − 1) := (2πi)^{n−1}ℝ

r_{n}(f_{1}, ..., f_{n}) :=  (42)

Alt_{n} ∑_{j≥0} C_{j} · log |f_{1}|d log |f_{2}| ∧ ... ∧ d log |f_{2j+1}| ∧ di arg f_{2j+2} ∧ ... ∧ di arg f_{n}

Here C_{j} := \frac{1}{(2j+1)!(n−2j−1)!} and Alt_{n} is the operation of alternation of f_{i}’s. Then

dr_{n}(f_{1}, ..., f_{n}) = π_{n}(d log f_{1} ∧ ... ∧ d log f_{n})

This just means that the pair

(r_{n}(f_{1}, ..., f_{n}), d log f_{1} ∧ ... ∧ d log f_{n})

is an n-cycle in the Deligne complex (40). It is the product in the real Deligne cohomology of the 1-cocycles (log |f_{i}|, d log f_{i}). Set

ω_{p,q} := (∑ dz_{1}^{(−)} ∧...∧ dz_{p+q}^{(−)})^{(p,q)}  (43)

where dz_{i}^{(−)} means either dz_{i} or d̄z_{i} and the sum is taken over all possible terms. For example

ω_{2,0} = dz_{1} ∧ dz_{2}, ω_{1,1} = dz_{1} ∧ dz̄_{2} + dz̄_{1} ∧ dz_{2}, ω_{0,2} = dz̄_{1} ∧ dz̄_{2}

The forms π_{n}ω_{p,q} for p ≥ q, p + q = n − 1, form a basis over ℝ in H_{B}^{n−1}(E_{/ℝ}^{(n−1)}, ℝ(n − 1))_{sgn}.

We can represent elements in H_{B}^{n−1}(E_{/ℝ}^{(n−1)}, ℝ(n−1))_{sgn} by their cup product with forms π_{n}ω_{p,q}:

\frac{1}{(2πi)^{(n−1)}} ∫_{E^{(n−1)}(ℂ)} ∑_{σ∈S_{n}}(−1)^{|σ|}r_{n}(p_{σ(1)}^{∗}f_{1}, ..., p_{σ(n)}^{∗}f_{n}) ∧ π_{n}ω_{p,q}

Theorem 6.4

∫_{E^{(n−1)}(ℂ)} ∑_{σ∈S_{n}}(−1)^{|σ|}r_{n}(p_{σ(1)}^{∗}f_{1}, ..., p_{σ(n)}^{∗}f_{n})∧ω_{p,q} = c_{n} \frac{Imτ}{π} ·K_{p+1,q+1}(f_{1}∗...∗f_{n})

where c_{n} ∈ ℚ^{∗}.

The constant c_{n} can be obtained from the proof below.

Proof. It consists of several reductions of the integral.

Step 1. The form ω_{p,q} is skew invarant under the action of the group S_{n}. So the integral is equal to

n! · ∫_{E^{(n−1)}(ℂ)} r_{n}(p_{1}^{∗}f_{1}, ..., p_{n}^{∗}f_{n}) ∧ ω_{p,q}

Step 2. Let

α_{n}(f_{1}, ..., f_{n}) := ∑_{i=1}^{n}(−1)^{i}log|f_{i}|dlog|f_{1}| ∧ ...dl̂og|f_{i}|... ∧ dlog|f_{n}|  (44)

Lemma 6.5 For any functions f_{1}, ..., f_{n}

∫_{E^{(n−1)}(ℂ)} r_{n}(f_{1}, ..., f_{n}) ∧ ω_{p,q} = b_{n} · ∫_{E^{(n−1)}(ℂ)} α_{n}(f_{1}, ..., f_{n}) ∧ ω_{p,q}

where b_{n} ∈ ℚ^{∗} is a (computable) constant.

Proof. One always have either d log f_{i} ∧ ω_{p,q} = 0 or d log f̄_{i} ∧ ω_{p,q} = 0. So we can replace everywhere di arg f by ±d log |f|.

Step 3.

∫_{E^{(n−1)}(ℂ)} α_{n}(f_{1}, ..., f_{n})∧ω_{p,q} = n·∫_{E^{(n−1)}(ℂ)} log |f_{n}|d log |f_{1}|∧...∧d log |f_{n−1}|∧ω_{p,q}

(45)

Indeed, log |f_{1}|d log |f_{2}| + log |f_{2}|d log |f_{1}| = d(log |f_{1}| · log |f_{2}|) and so by the Stokes theorem

∫_{E^{(n−1)}(ℂ)} d(log |f_{1}| · log |f_{2}|)d log |f_{3}| ∧ ... ∧ d log |f_{n}| ∧ ω_{p,q} = 0

Step 4. Let us compute the right integral in (45).

Lemma 6.6

log |f(z)| = −\frac{Imτ}{2π} ∑_{γ∈Γ\0} v_{a}(f)\frac{(z − a, γ)}{|γ|^{2}} + C_{f}  (46)

where C_{f} is a certain constant.

Proof. One can get a proof applying ∂ ¯∂ to the both parts of (45). The constant C_{f} can be computed from the decomposition of f on the product of theta functions using the formula in s. 18 ch. VIII of [We]. According to step 5 it does not play any role in our considerations.

Step 5. By the Stokes formula

∫_{E^{(n−1)}(ℂ)} C_{f} · d log |f_{1}| ∧ ... ∧ d log |f_{n−1}| ∧ ω_{p,q} = 0

we see that one can neglect the constants C_{f}.

Step 6. We may suppose that in (45) the form ω_{p,q} is written in variables z_{1}, ..., z_{n−1}. Then for each summand in ω_{p,q} one can replace d log |f_{i}| by 1/2 · ∂ log f_{i} or 1/2· ¯∂log f_{i} depending whether in this summand appeared dz̄_{i} or dz_{i}.

For example for the summand dz_{1} ∧...∧dz_{p} ∧dz̄_{p+1} ∧...∧dz̄_{n−1} we will have the integral

\frac{n}{2^{n−1}} · ∫_{E^{(n−1)}(ℂ)} log |f_{n}(z_{n})|¯∂log f_{1}(z_{1}) ∧ ... ∧ ¯∂log f_{p}(z_{p})∧

∂ log f_{p+1}(z_{p+1}) ∧ ... ∧ ∂ log f_{n−1}(z_{n−1}) ∧ dz_{1} ∧ ... ∧ dz_{p} ∧ dz̄_{p+1} ∧ ... ∧ dz̄_{n−1}

Differentiating the distributions we get

∂ log f(z̄) = ∑_{γ∈Γ\0} v_{a}(f)\frac{(z − a, γ) · γ̄}{|γ|^{2}}

¯∂log f(z) = − ∑_{γ∈Γ\0} v_{a}(f)\frac{(z − a, γ) · γ}{|γ|^{2}}

The condition z_{1} + ... + z_{n} = 0 just mean that we compute the value of the convolution of one variable distributions:

log |f_{n}(z)| ∗ \frac{¯∂log f_{2}(z̄)}{∂z̄} ∗ ... ∗ \frac{¯∂log f_{p}(z̄)}{∂z̄} ∗ \frac{∂ log f_{p}(z)}{∂z} ∗ ... ∗ \frac{∂ log f_{n−1}(z)}{∂z}

at 0. Using the fact that the Fourier transform sends the convolution to the product and the formulas above we get the theorem.

Recall that we have a homomorphism K_{n+1} : ℤ[E(ℂ)] ⟶ Sym^{n−1}H_{B}^{1}(E(ℂ), ℂ) which is constructed as follows

{z} ⟼ ∑_{i=0}^{n−1} K_{a,b}(z) · (dz)^{a−1}(dz̄)^{b−1} ∈ Sym^{n−1}H_{B}^{1}(E(ℂ), ℂ)

Theorem (6.2) claims that it sends the subgroup R_{n+1}(E(ℂ)) to zero.

4. Proof of the theorem (6.2). For any function f on a manifold one has d log |f| ∧ d log |1 − f|. So a non zero term in the integral (6.5) could be

∫_{E^{(n−1)}(ℂ)}(log |f|d log |1−f|−log |1−f|d log |f|)∧d log |f_{3}|∧...∧d log |f_{n}|∧ω_{p,q}

One always has

(log |f|d log |1 − f| − log |1 − f|d log |f|) ∧ ω_{p,q} =
±i · (log |f|darg(1 − f) − log |1 − f|dargf) ∧ ω_{p,q}

Further one has, even in the sence of the distribution,

dL_{2}(f) = log |f|darg(1 − f) − log |1 − f|dargf

So we can rewrite the integral as

±i · ∫_{E^{(n−1)}(ℂ)} d(L_{2}(f) ∧ d log |f_{3}| ∧ ... ∧ d log |f_{n}| ∧ ω_{p,q}) = 0

It is zero by the Stokes formula.

Now using theorem (6.4) relating Eisenstein-Kronecker series to the regulator integral we come to the proof of the theorem.

Theorem (6.2) and lemma (6.1) imply

Theorem 6.7 The Eisenstein-Kronecker map K_{n+1} provides a homomorphism

K_{n+1} : B_{n+1}[E(ℚ̄)]^{Gal(ℚ̄/F)} ⟶ (⊕_{σ:F ↪ℂ}Sym^{n−1}H_{B}^{1}(E_{σ}(ℂ), ℝ(1)))^{+}

Combining these results with conjecture (1.4), our construction of the elliptic motivic complexes presented in s. 4.1 and Beilinson’s conjecture on regulators we come to conjecture (6.8) about L-function of Sym^{n−1}h^{1}(E) at s = n for an arbitrary elliptic curve E over a number field.

5. A conjecture on L(Sym^{n}E, n + 1) In this section we will assume for simplicity that E be an elliptic curve over ℚ. We will left to reader as an easy exercise to generalise all the discussion to the case of an elliptic curve over an arbitrary field F.

Conjecture (1.4) together with Beilinson’s conjecture on regulators imply a precise conjecture on L(Sym^{n}E, n + 1)

For any divisor P = ∑ n_{s}(P_{s}) on E(ℂ) put K_{i,j}(P) := ∑ n_{s}K_{i,j}(P_{s}).

The integrality condition . Suppose E has a split multiplicative reduction at p with N-gon as a special fibre. Let L be a finite extention of ℚ_{p} of degree n = ef and O_{L} the ring of integers in L. Let E^{0} be the connected component of the Néron model of E over O_{L}. Let us fix an isomorphism E_{F_{p^{f}}}^{0} = 𝔾_{m}/F_{p^{f}}. It provides a bijection between ℤ/eNℤ and the components of E_{F_{p^{f}}}. For a divisor P such that all its points are defined over L denote by d(P; ν) the degree of the restriction of the flat extension of a divisor P to the ν’th component of the (eN)-gon.

Let B_{n+1}(x) be the (n + 1)-th Bernoulli polynomial. The integrality condition at p is the following condition on a divisor P, provided by the work of Schappaher and Scholl ([SS]). For a certain (and hence for any, see s. 3.3 in [GL]) extention L of ℚ_{p} such that all points of the divisor P are defined over L one has ([L : ℚ_{p}] = ef):

∑_{ν∈ℤ/(eN)ℤ} d(P; ν)B_{n+1}(\frac{ν}{eN}) = 0  (47)

Let C_{n} be the conductor of the system of the l-adic representations related to Sym^{n}h^{1}(E). Set

β_{2l+1} = C_{2l+1}^{−(l+1)} · (Imτ)^{(l+1)(l+2)}, β_{2l} = C_{2l}^{− \frac{2l+1}{2}} · π^{−2l}(Imτ)^{(l+1)^{2}}

Conjecture 6.8 a) For any elliptic curve over ℚ there exist [\frac{n}{2}] + 1 ℚ-rational divisors P_{a} on E(ℚ̄) such that

L(Sym^{n}h^{1}(E), n + 1) ∼_{ℚ^{∗}} β_{n}det|K_{b,n+2−b}(P_{a}; τ)|  (48)

(1 ≤ a, b ≤ [\frac{n}{2}] + 1), and the divisors P_{a} satisfy the following two conditions:

i)  δ(P_{a}) = 0 in B_{n+1}(E) ⊗ J(ℚ̄)_{ℚ}  (49)

ii) the integrality condition: at each prime p where E has a split multiplicative reduction with special fibre a Neron N-gon

∑_{ν∈ℤ/(eN)ℤ} d(P; ν)B_{n+1}(\frac{ν}{eN}) = 0  (50)

b) For any [\frac{n}{2}] + 1 ℚ-rational divisors P_{a} on E(ℚ̄), satisfying the conditions above the right hand side of (48) is equal to q · L(S^{n}h^{1}(E), n + 1) where q is a rational number, perhaps equal to 0.

In [W] J.Wildeshaus, assuming standard conjectures about mixed motives, formulated a conjecture similar to the part b) of the conjecture (6.8) ( an elliptic analog of the weak version of Zagier’s conjecture).

For n = 2 the formula (48) was proved for modular elliptic curves over ℚ in [GL]. Formula (48), even without precise conditions on the divisors P_{a}, is the most nontrivial part of the conjecture for n > 2 ( see also s.8 in [G4]). An efficient way to formulate the conditions on the divisors P_{a} without referring to the definition of the subgroups R_{n}(E) is given in the chapter 7.

When E degenerates to the nodal curve, the conjecture on L(Sym^{2}E, 3) leads to Zagier’s conjecture [Za2] at s = 3, which was proved in [G1]-[G2]. This gives a credit for the conjecture (6.8).

The key condition (49) is obviously satisfied if P_{a} are (multiples of) torsion divisors. The determinants from (48) for torsion divisors where considered by Deninger ([De2], s.5) (and inspired by the Eisenstein symbol of Beilinson [Be]). They work well for CM elliptic curves. However Mestre and Schappacher [SM] deduced from a result of Serre that for a given non CM elliptic curve over ℚ for all n > n_{0} the determinant is always zero for any ℚ-rational torsion divisors P_{a}. So to get the L-values one has to consider the non torsion divisors.

6. A more explicit form of the conditions on the divisors P_{a}. Let P = ∑ n_{i}P_{i} and k(P) be the field generated by the points P_{i}. Let us denote by h_{v} the canonical local height related to a valuation v of a number field K. Let K_{v} be the completion of a number field K corresponding to v. Recall the height homomorphism h_{v} : B_{2}(E(K_{v})) ⟶ ℝ. If v is a non archimedean valuation then the target of this homomorphism is (log p) · ℚ.

Let us consider a homomorphism

d_{m} : B_{n+1}(E) ⟶ B_{m}(E) ⊗ S^{n+1−m}J_{ℚ}, {a}_{n+1} ⟼ {a}_{n+1−m} ⊗ a · ... · a

We will need the following pairs of homomorphisms. If m = 2:

i) A homomorphism

p_{2} ⊗ id : B_{2}(E) ⊗ S^{n−1}J_{ℚ} ⟶ S^{2}J_{ℚ} ⊗ S^{n−1}J_{ℚ}

where p_{2} : {a}_{2} ⟼ a · a.

For any valuation v of we have

ii) The height homomorphism

h_{v} ⊗ id : B_{2}(E) ⊗ S^{n−1}J_{ℚ} ⟶ ℝ ⊗_{ℚ} S^{n−1}J_{ℚ}

where v is any valuation of the field K(P).

For m > 2:

iii) The Bernoulli homomorphism, defined for any bad prime where E has a split multiplicative reduction with the Néron N-gon:

Ber_{m} : B_{m}(E) ⊗ S^{n+1−m}J_{ℚ} ⟶ S^{n+1−m}J_{ℚ},

\{a\}_{m} ⊗ b_{1} · ... · b_{n+1−m} ⟼ ∑_{ν} d(a, ν)B_{m}(\frac{ν}{N}) · b_{1} · ... · b_{n+1−m}

iv) The Eisenstein-Kronecker homomorphism

K_{m} ⊗ id : B_{m}(E) ⊗ S^{n+1−m}J_{ℚ} ⟶ Sym^{n−2}H^{1}(E(ℂ), ℝ(1))^{+} ⊗ S^{n+1−m}J_{ℚ}

Remarks. 1. In formulas above J_{ℚ} means J(ℚ̄)_{ℚ}. However for a given divisor P we land in Gal(ℚ̄/ℚ)-invariant part of powers of J(k(P))_{ℚ}.

2. Let us consider the Eisenstein-Kronecker homomorphism only on the kernel of the Bernoulli homomorphism. Then Beilinson’s conjecture on regulators means that it should land in

(the regulator lattice in Sym^{n−2}H^{1}(E(ℂ), ℝ(1))^{+}) ⊗ S^{n+1−m}J_{ℚ}

So by the Mordell-Weil theorem if k(P) is a given number field then the target group is a finite dimensional ℚ-vector space.

3. If v is a p-adic valuation of the field k(P), then (log p)^{−1} · h_{v}(P_{i}) ∈ ℚ, and so the target of the height homomorphism is a finite dimensional ℚ-vector space.

4. If the a divisor P is in the kernel of the height homomorphisms for all archimedean valuations but one then thanks to the product formula it is sent to zero by all of them. In particulary if k(P) = ℚ we can forget the archimedian valuation.

Composing each of these homomorphism with the appropriate map d_{m} we get homomorphisms

(p_{2} ⊗ id) ∘ d_{2}, (h_{v} ⊗ id) ∘ d_{2}, Ber_{m} ∘ d_{m}, (K_{m} ⊗ id) ∘ d_{m} (51)

here m > 2.

Proposition 6.9 The condition δ(P) = 0 in the group B_{n}(E)⊗J_{ℚ} implies that all of the homomorphisms (51) are equal to zero.

Proof.Clear.

The height condition is the crucial one. If it is satisfied, then for a given field k(P_{a}) the other conditions should give only a finite number of conditions on the divisors P_{a}.

If a ℚ-rational divisor P is sent to zero by all of the homomorphisms (51) then this essentially means that δ(P) = 0 in the group B_{n}(E) ⊗ J(ℚ̄)_{ℚ}. To see this we write the homomorphism d_{m} as a composition of homomorphisms δ ⊗id

B_{n+1}(E) ⟶ B_{n}(E)⊗J_{ℚ} ⟶ (B_{n−1}(E)⊗J_{ℚ})⊗J_{ℚ} ⟶ ... ⟶ B_{m}(E)⊗J_{ℚ}^{⊗n+1−m}

(52)

followed by the projection

B_{m}(E) ⊗ J_{ℚ}^{⊗n+1−m} ⟶ B_{m}(E) ⊗ S^{n+1−m}J_{ℚ}

Let us spell the details in the first interesting case: n = 3.

Proposition 6.10 Let us assume that the Bloch-Beilinson regulator r_{Be} : K_{2}(E)_{ℤ} ⟶ ℝ is injective. Then if for n = 3 a ℚ-rational divisor P belongs to the kernel of homomorphisms (51) then δ(P) = 0 in the group B_{3}(E) ⊗ J_{ℚ}

Proof. Consider the homomorphism

d_{2} : B_{4}(E) ⟶ B_{2}(E) ⊗ S^{2}J,  {a}_{4} ⟼ {a}_{2} ⊗ a · a  (53)

Suppose an element P ∈ B_{4}(E) is in the kernel of the homomorphism (p_{2} ⊗ id) ∘ d_{2}. Then P ∈ B_{4}^{∗}(E) and f(P) ∈ (ℚ̄^{∗} ⊗ S^{2}J(ℚ̄))^{Gal(ℚ̄/ℚ}.

The image of the divisor P under this homomorphism belongs to the subgroup k(P)^{∗} ⊗ S^{2}J(k(P)) . The hight condition is a way to say that it is equal to zero.

Let us write the map d_{2} as a composition

B_{4}(E) ⟶ B_{3}(E) ⊗ J ⟶ (B_{2}(E) ⊗ J) ⊗ J ⟶ B_{2}(E) ⊗ S^{2}J

Notice that

Ker(B_{3}(E)⊗J ⟶ (B_{2}(E)⊗J)⊗J)⊗ℚ = Ker(B_{3}(E) ⟶ B_{2}(E)⊗J)⊗J ⊗ℚ

Consider the homomorphism

B_{3}(E(ℂ)) ⊗ J(ℂ) ⟶ J(ℂ) ⊗ ℝ,  b ⊗ {a}_{3} ⟼ K_{2,1}(a) ⊗ b (54)

There is a homomorphism K_{2}(E(ℂ)) ⟶ B_{3}(E(ℂ)) such that the following diagram is commutative (see [GL]):

K_{2}(E(ℂ)) ⟶^{r_{Be}} ℝ
↓  ↓ Id
B_{3}(E(ℂ)) ⟶^{K_{3}} ℝ

So assuming the injectivity of the regulator K_{2}(E)_{ℤ} ⟶ ℝ we see that (54) should be injective on Ker(B_{3}(E) ⟶ B_{2}(E) ⊗ J) ⊗ J ⊗ ℚ.

7 The complexes B(E; n)^{•} and motivic elliptic polylogarithms

1. In this chapter k = k̄, and all abelian groups are tensored by ℚ, so we work with the corresponding ℚ-vector spaces. For instence J := J ⊗ ℚ etc.

Let

ℚ[E(k)] ⟶^{δ} ℚ[E(k)] ⊗ J, {a} ⟼ {a} ⊗ a

Theorem 7.1 Let us assume standard conjectures on mixed motives. Then there exist canonical homomorphisms l_{n} : ℚ[E(k)] → L(E)_{S^{n−2}H(1)^{∨}} such that the following digram is commutative:

ℚ[E(k)]  ⟶^{δ}  ℚ[E(k)] ⊗ J
l_{n−1} ↓  ↓ l_{n−2} ⊗ id
L(E)_{S^{n−2}H(1)^{∨}}  ⟶^{∂} L(E)_{S^{n−3}H(1)^{∨}} ⊗ J

Proof. The proof is based on the existence and basic properties of the motivic elliptic polylogarithms of Beilinson and Levin [BL]. For any nonzero point a ∈ E(k) let G_{a}^{(1)} be an element of Ext_{M_{E}}^{1}(ℚ(0), H) wich corresponds to a ∈ J under the isomorphism Ext_{M_{E}}^{1}(ℚ(0), H) = J. Set G_{a}^{(m)} := S^{m}(G_{a}^{(1)}). The motivic elliptic (n − 1)-logarithm at a is a mixed elliptic motive El_{n−1}(a) which provides a certain extension class in Ext_{M_{E}}^{1}(H, G_{a}^{(n−1)}(1)). In particular its weight graded quotients are

H, ℚ(1), H(1), ..., S^{(n−1)}H(1)

Therefore it has canonical framing and so defines an element of A(H, S^{(n−1)}H(1)). After tensoring it by H and twisting by ℚ(−1) we can introduce a natural framing by ℚ and S^{(n−2)}H(1) (since S^{(n−1)}H ⊗ H = S^{(n)}H ⊕ S^{(n−2)}H). Therefore we picked up an element l_{n−1}(a) ∈ A(ℚ, S^{(n−2)}H(1)).

The commutativity of the diagram follows from the properties of the elliptic polylogarithms ([BL]). The crucial point is this. Since W_{≤−3}El_{n−1}(a) is a symmetric power of the motive G_{a}^{(1)}, The projection to L(E) of the elements in A(S^{(k)}H(1), S^{(n−1)}H(1)) coming from the canonical framing by S^{(k)}H(1) and S^{(n−1)}H(1)) of the motive El_{n−1}(a) are zero provided 0 ≤ k < n − 2. So projecting the coproduct of El_{n−1}(a) to Λ^{2}L(E) the only nonzero contribution we get is given by the component of the coproduct coming from A(H, S^{(n−2)}H(1)) ⊗ A(S^{(n−2)}H(1), S^{(n−1)}H(1)). The fact that it is equal to l_{n−2}(a) ∧ a follows immediately from the basic property of the elliptic polylogarithm motive (see [BL]).

Definition 7.2

R_{n}(E) = Kerl_{n−1}, B_{n}(E) = \frac{ℚ[E(k)]}{R_{n}(E)}

Theorem (7.1) implies that δ provides a well defined homomorphism δ : B_{n}(E) ⟶ B_{n−1}(E) ⊗ J. So we get a complex B(E; n)^{•}:

B_{n}(E) ⟶ B_{n−1}(E) ⊗ J ⟶ ... ⟶ B_{2}(E) ⊗ Λ^{n−2}J ⟶ J ⊗ Λ^{n−1}J ⟶ Λ^{n}J Set

r_{n}(J) := Ker(J ⊗ Λ^{n−1}J ⟶ Λ^{n}J)

The theorem (7.1) and this definition immediately imply that there exists canonical homomorphism of complexes

B_{n}(E)  ⟶  B_{n−1}(E) ⊗ J  ⟶ ... ⟶ B_{2}(E) ⊗ Λ^{n−2}J ⟶ r_{n}(J)
l_{n−1} ↓  l_{n−2} ⊗ id ↓  ↓ l_{1} ⊗ id  ||
L(E)_{S^{n−2}H(1)^{∨}} ⟶ L(E)_{S^{n−3}H(1)^{∨}} ⊗ J ⟶ ... ⟶ L(E)_{ℚ(1)^{∨}} ⊗ Λ^{n−2}J ⟶ r_{n}(J)

Lemma 7.3 The bottom complex is a subcomplex of the S^{n−2}H(1)^{∨}-isotypical component of the standard cochain complex of the Lie coalgebra L(E).

So if k = k̄, we get a canonical injective morphism of the complexes

B(E; n + 1)^{•} ⟶ (Λ^{•}L(E), ∂)_{S^{n−1}H(1)^{∨}}  (55)

Let K_{n−1}^{M} be the sheaf of Milnor K-groups. Combining this morphism with the canonical morphism from the right hand side to RHom_{MM_{k}}(ℚ(0), S^{n−1}H(1)) provided by the inclusion functor M_{E} → MM_{k} we get

Corollary 7.4 Let us assume the formalism of mixed motives. Then a) there exists canonical homomorphisms

H^{i}(B(E; n + 1)_{ℚ}^{•}) ⟶ gr_{n}^{γ}K_{n+1−i}(E^{(n−1)})_{sgn} ⊗ ℚ  (56)

b) The homomorphism for i = 1 is injective.

Indeed, thanks to (55) this is true if k = k̄. The general case follows since we have the descent property both for rational K-theory and, (by definition), for the complexes B(E; n + 1)_{ℚ}^{•}.

Remark. We do not expect a morphism of complexes (55) exist if k is not algebraicly closed. The reason is this. If k is not closed we have postulated the Galois descent for the complexes B(E; n + 1)^{•}. On the other hand the standard complex of the Lie algebra L(E/k) should not satisfy the Galois decent.

I hope a stronger result should be valid:

Conjecture 7.5 a)

gr_{n}^{γ}K_{n+1−i}(E^{(n−1)})_{sgn} ⊗ ℚ = H^{i−1}(E^{(n−1)}, K_{n−1}^{M})_{sgn} ⊗ ℚ  (57)

b) There exists a canonical isomorpism in the derived category

B(E; n + 1)_{ℚ}^{•} = RHom_{MM_{k}}(ℚ(0), S^{n}H(1))

in particular

H^{i}(B(E, n + 1)_{ℚ}^{•}) = gr_{n}^{γ}K_{n+1−i}(E^{(n−1)})_{sgn} ⊗ ℚ  (58)

c) Let k = k̄. Then the homomorphism of complexes (55) is a quasiisomorphism.

If the conjecture (4.2) is correct, then b) is equivalent to c).

According to the lemma (7.3) this conjecture implies conjecture (1.4).

Part a) of the conjecture is trivial for n = 2.

Lemma 7.6 For n = 3 and k = k̄ the part a) of the conjecture (7.5) follows from the rigidity conjecture for K_{3}^{ind}(k).

Proof. The statement boils down to Kerβ = Imα in the sgn-part of the diagram

⟶ B_{2}(k(E^{(2)})) ⊗ k(E^{(2)})^{∗} ⟶  Λ^{3}k(E^{(2)})^{∗}
↓ α  ↓
∐_{Y ∈(E^{(2)})^{1}} B_{2}(k(Y)) ⟶^{β} ∐_{Y ∈(E^{(2)})^{1}} Λ^{2}k(Y)^{∗}
↓  ↓
0  ...

Let β = ∐_{Y} β_{Y}. Then Kerβ_{Y} = K_{3}^{ind}(k(Y))_{ℚ}. By the rigidity conjecture any point y ∈ Y provides an isomorphism K_{3}^{ind}(k)_{ℚ} = K_{3}^{ind}(k(Y))_{ℚ}. So Kerβ/Imα is a subgroup of

Coker(K_{3}^{ind}(k)_{ℚ} ⊗ k(E^{(2)})^{∗} ⟶ ∐_{Y ∈(E^{(2)})^{1}} K_{3}^{ind}(k)_{ℚ}) = CH^{1}(E^{(2)})_{sgn} ⊗ ℚ = 0

The lemma is proved.

2. Motivic realization of elliptic polylogarithms. Let f = (f_{1}, ..., f_{n}) be an n - tuple of rational functions on E.

Motivation. Consider the following multivalued analytic function on

{n − tuples of rational functions on E(ℂ)} × H_{n−1}(E^{(n−1)}(ℂ))_{sgn_{n}}

P (E^{(n−1)}; f; γ) := ∫_{γ} Alt_{(x_{1},...,x_{n})} log f_{1}(x_{1}))d log f_{2}(x_{2})∧...∧d log f_{n}(x_{n}) (59)

where γ is a cycle representing a nontrivial class in H_{n−1}(E^{(n−1)}(ℂ))_{sgn_{n}}. The subscript sgn_{n} means the skewsymmetric part with respect to the permutations. (A better way to define this function is given by formula (64) below.) We will show that this function is a period of a mixed elliptic motive.

Choose a coordinate z on ℙ^{1}. Let Δ_{n} be the coordinate cube in (ℙ^{1}; 0∪∞)^{n}, i.e. union of 2n hyperplanes z_{i} = 0, z_{i} = ∞. Notice that (ℙ^{1})^{n}\Δ_{n} = (𝔾_{m})^{n}.

Let us define a codimension n cycle

Z(E^{(n−1)}; f) ⊂ E^{(n−1)} × (ℙ^{1})^{n}

as follows:

Z(E^{(n−1)}; f) := Alt_{(x_{1},...,x_{n})}{x_{1}, ..., x_{n}; f_{1}(x_{1}), ..., f_{n}(x_{n})} ∪ E^{(n−1)} × {1, ..., 1}

Here we use the coordinate system z_{1}, ..., z_{n}, i.e. z_{i} = f_{i}(x_{i}). The group S_{n} acts on (𝔾_{m})^{n} by permutations. So we get an action of the group S_{n}×S_{n} on E^{(n−1)}× (𝔾_{m})^{n}. In this chapter we mark by the subscript sgn the skewsymmetric part under this action. Consider the following mixed motive

h(E^{(n−1)}; f) := H^{n}(E^{(n−1)} × (𝔾_{m})^{n}, Z^{0}(E^{(n−1)}; f))(n)_{sgn}

where Z^{0}(E^{(n−1)}; f) := Z(E^{(n−1)}; f)\(Z(E^{(n−1)}; f) ∩ Δ_{n}).

More precisely, h(E^{(n−1)}; f)(−n) := R^{n}p_{!}F(E; f)_{sgn} where F(E; f) is the following mixed motivic sheaf on E^{(n−1)} × (ℙ^{1})^{n}. Take the constant sheaf on the complement to Z(E^{(n−1)}; f) ∪ E^{(n−1)} × Δ_{n} in E^{(n−1)} × (ℙ^{1})^{n}; extend it by j_{∗} to the divisor E^{(n−1)} × Δ_{n} and then by j_{!} to Z(E^{(n−1)}; f).

Lemma 7.7 h(E^{(n−1)}; f) is a mixed elliptic motive framed by ℚ(0) and Sym^{n−1}H(1).

Proof. i) ℚ(0)-component of the frame. Let us prove that

gr_{2n}^{W}H^{n}(E^{(n−1)} × (𝔾_{m})^{n}, Z^{0}(E^{(n−1)}; f))_{sgn} = ℚ(−n)

Notice that H^{n}(𝔾_{m})_{sgn}^{n} = H^{n}(𝔾_{m})^{n} and H^{i}(𝔾_{m})_{sgn}^{n} = 0 for i < n. So the projection E^{(n−1)} × (𝔾_{m})^{n} ⟶ (𝔾_{m})^{n} induces an isomorphism

ℚ(−n) = gr_{2n}^{W}H^{n}((𝔾_{m})^{n})_{sgn} ⟶^{∼} gr_{2n}^{W}H^{n}(E^{(n−1)} × (𝔾_{m})^{n})_{sgn}

The canonical morphism

H^{n}(E^{(n−1)} × (𝔾_{m})^{n}, Z^{0}(E^{(n−1)}; f))_{sgn} ⟶ H^{n}(E^{(n−1)} × (𝔾_{m})^{n})_{sgn}

induces an isomorphism after taking gr_{2n}^{W}. Indeed, there is an exact sequence

H^{n−1}Z^{0}(E^{(n−1)}; f) ⟶ H^{n}(E^{(n−1)} × (𝔾_{m})^{n}, Z^{0}(E^{(n−1)}; f))_{sgn} ⟶
H^{n}(E^{(n−1)} × (𝔾_{m})^{n})_{sgn} ⟶ H^{n}Z^{0}(E^{(n−1)}; f)

and

gr_{2n}^{W}H^{n−1}Z^{0}(E^{(n−1)}; f) = gr_{2n−1}^{W}H^{n−1}Z^{0}(E^{(n−1)}; f) = 0

since Z^{0}(E^{(n−1)}; f) is an open regular variety of dimension n − 1.

ii) Sym^{n−1}H(1)-component of the frame. One has

H^{n}(E^{(n−1)} × (ℙ^{1})^{n}, Z(E^{(n−1)}; f))_{sgn} = Sym^{n−1}h^{1}(E)  (60)

Indeed, H^{n}(E^{(n−1)} × (ℙ^{1})^{n})_{sgn} = 0. So there is an exact sequence

H^{n−1}(E^{(n−1)} × (ℙ^{1})^{n})_{sgn} ⟶^{α} H^{n−1}(Z(E^{(n−1)}; f))_{sgn} ⟶^{β}

H^{n}(E^{(n−1)} × (ℙ^{1})^{n}, Z(E^{(n−1)}; f))_{sgn} ⟶ 0

Further,

H^{n−1}(E^{(n−1)} × (ℙ^{1})^{n})_{sgn} = Sym^{n−1}h^{1}(E)
H^{n−1}(Z(E^{(n−1)}; f))_{sgn} = Sym^{n−1}h^{1}(E) ⊕ Sym^{n−1}h^{1}(E)

and α is injective.

The restriction map induces an isomorphism

gr_{n−1}^{W}H^{n}(E^{(n−1)} × (ℙ^{1})^{n}, Z(E^{(n−1)}; f))_{sgn} ⟶  (61)
gr_{n−1}^{W}H^{n}(E^{(n−1)} × (𝔾_{m})^{n}, Z^{0}(E^{(n−1)}; f))_{sgn}

because (Z(E^{(n−1)}; f) is regular of dimension n − 1).

W_{n−1}H^{n−1}Z^{0}(E^{(n−1)}; f) = W_{n−1}H^{n−1}Z(E^{(n−1)}; f)

Combining (60) and (61) we get the Sym^{n−1}H(1)-component of the frame. The lemma is proved.

Finally, we show that h(E^{(n−1)}; f) is a mixed elliptic motive by induction using the following basic observation: intersection of the cycle Z(E^{(n−1)}; f_{1}, ..., f_{n}) with any codimension 1 face of the cube Δ_{n} is a sum of cycles of form Z(E^{(n−2)}; g_{1}, ..., g_{n−1}). For example

Z(E^{(n−1)}; f_{1}, ..., f_{n}) ∩ {z_{1} = 0} = ∑_{x∈E} m_{x}(f_{1}) · Z(E^{(n−2)}; f_{2}, ..., f_{n})

where m_{x}(f) is the multiplicity of zero at x. The lemma is proved.

Remark 7.8 One can apply the same construction to n arbitrary functions f_{1}(z_{1}, ..., z_{n}), ..., f_{n}(z_{1}, ..., z_{n}) on E^{(n−1)}. However it is not clear whether the motive h(E^{(n−1)}; f_{1}, ..., f_{n}) is a mixed elliptic motive in general.

The functions f_{1}, ..., f_{n} on the E define a map

f : E^{(n−1)} ⟶ (ℙ^{1})^{n}, (x_{1}, ..., x_{n}) ⟼ (f_{1}(x_{1}), ..., f_{n}(x_{1}))

Let E_{f}^{(n−1)} be the image of this map and Ẽ_{f}^{(n−1)} := E_{f}^{(n−1)}\(E_{f}^{(n−1)} ∩ Δ). Consider the following motive:

h̃(E^{(n−1)}; f) := H^{n}((𝔾_{m})^{n}, Ẽ_{f}^{(n−1)})(n)_{sgn}  (62)

Lemma 7.9 Suppose that f_{∗} : H_{n−1}(E^{(n−1)})_{sgn} → H_{n−1}(E_{f}^{(n−1)})_{sgn} is a nonzero map. Then

a) h̃(E^{(n−1)}; f) is a (ℚ(0), Sym^{n−1}H(1))-framed mixed elliptic motive.

b) h̃(E^{(n−1)}; f) = h(E^{(n−1)}; f) as framed motives.

c) If f_{∗} = 0 then h(E^{(n−1)}; f) = 0.

Proof. a) is very similar to the proof of the lemma above. For instance the Sym^{n−1}h^{1}(E) part of the framing comes from isomorphism

W_{n−1}H^{n}((ℙ^{1})^{n}\Δ_{n}, Ẽ_{f}^{(n−1)}) ⟶ Sym^{n−1}h^{1}(E)

Namely, H^{i}(P^{n})_{sgn} = 0, so there is an isomorphism

H^{n−1}(E_{f}^{(n−1)})_{sgn} ⟶ H^{n}((ℙ^{1})^{n}, E_{f}^{(n−1)})_{sgn}

Combining it with f^{∗} : H^{n−1}(E_{f}^{(n−1)})_{sgn} ⟶ H^{n−1}(E^{(n−1)})_{sgn} we get a morphism

H^{n}((ℙ^{1})^{n}, E_{f}^{(n−1)})_{sgn} ⟶ H^{n−1}(E^{(n−1)})_{sgn} = Sym^{n−1}h^{1}(E)  (63)

b) The projection

E^{(n−1)} × (ℙ^{1})^{n} ⟶ (ℙ^{1})^{n}

induces a morphism respecting the frames.

c) is clear from the construction. The lemma is proved.

The period corresponding to this framing is exactly the function (59). Indeed, consider the differential form

ω_{Δ_{n}} := d log(z_{1}) ∧ ... ∧ d log(z_{n})

in (ℂℙ^{1})^{n}\Δ_{n}. Let Γ be a relative n-chain in ℂP^{n} which bounds an (n−1)-cycle γ, [γ] ∈ H_{n−1}((ℂℙ^{1})^{n}, E_{f}^{(n−1)})_{sgn}. Then

L_{n}(E^{(n−1)}; f; γ) = ∫_{Γ} ω_{Δ_{n}}  (64)

The Stokes formula shows that the integrals (64) and (59) coinside.

Proposition 7.10 The ℝ-valued period of the Hodge realization of h(E^{(n−1)}; f) is given by

L(E^{(n−1)}; f) := ∫_{E^{(n−1)}(ℂ)} Alt_{x_{1},...,x_{n}}r_{n}(f_{1}, ..., f_{n}) ∧ ω_{p,q}

This integral coincides with the one computed by means of the Eisenstein - Kronecker series, as was explained before.

Recall that A(ℚ, Sym^{n−1}H(1)) is the group of mixed elliptic motives framed by ℚ(0) and Sym^{n−1}H(1).

Lemma 7.11 For any λ_{i} ∈ k^{∗} one has the equality of framed motives h(E^{(n−1)}; f_{1}, ..., f_{n}) = h(E^{(n−1)}; λ_{1} · f_{1}, ..., λ_{n} · f_{n})

Proof. The action of an element g = (λ_{1}, ..., λ_{n}) ∈ 𝔾_{m}^{n} on P^{n}\Δ_{n} = 𝔾_{m}^{n} provides a morphism of motives h(E^{(n−1)}; λ_{1}·f_{1}, ..., λ_{n}·f_{n}) ⟶ h(E^{(n−1)}; f_{1}, ..., f_{n}) which obviously preserves the framing.

Theorem 7.12 There is a well defined homomorphism of abelian groups

m_{n}^{∗} : S^{n}k(E)^{∗} ⟶ A(ℚ(0), Sym^{n−1}H(1)) f_{1} ∘ ... ∘ f_{n} ⟼ h(E^{(n−1)}; f)

It is zero if one of the functions f_{i} is a constant; one has m_{n}^{∗}(Kerμ_{n}) = 0.

Proof.

Below a generalization of the construction above is given. Let D^{0} be the group of degree zero divisors on E. For any d := (d_{1}, ..., d_{n}) let us construct a mixed elliptic motive h(E^{(n−1)}; d) := h(E^{(n−1)}; d_{1}, ..., d_{n}) framed by ℚ(0) and Sym^{n−1}H(1).

Let P be the rigidified Poincaré line bundle over J × J. For any two degree zero divisors d_{1}, d_{2} with disjoint support there is an element

< d_{1}, d_{2} >∈ P_{[d_{1}],[d_{2}]}

where [d_{i}] ∈ J is the class of a degree zero divisor d_{i}.

Consider the following (n − 1)-cycle

Z(E^{(n−1)}; d) ⊂ E^{(n−1)} × P^{n}  (65)

c(E^{(n−1)}; d) := Alt_{(x_{1},...,x_{n})}(x_{1}, ..., x_{n}; < d_{1}, (x_{1}) − (0) >; ...; < d_{n}, (x_{n}) − (0) >)

Here x_{1} + ... + x_{n} = 0 and (x_{1}, ..., x_{n}) belongs to Zariski open part of E^{(n−1)} where the divisors d_{i} and (x_{i}) − (0) are disjoint. and set

h(E^{(n−1)}; d) := H^{n}(E^{(n−1)} × P^{n}, Z(E^{(n−1)}; d))(n)

Theorem 7.13 a) h(E^{(n−1)}; d) is a mixed elliptic motive framed by ℚ(0) and Sym^{n−1}H(1).

b) There is a well defined homomorphism of abelian groups

S^{n}D^{0} ⟶ A(ℚ(0), Sym^{n−1}H(1)) d_{1} ∘ ... ∘ d_{n} ⟼ h(E^{(n−1)}; d)

Proof. Restriction to a fiber of the Poincare line bundle provides an isomorphism

gr_{2}^{W} H^{1}(P) ⟶ H^{1}(𝔾_{m}) = ℚ(−1)

and thus we get a first part of the framing:

ℚ(−n) ⟶ gr_{2n}^{W}H^{n}(P^{n}) ⟶^{π^{∗}} gr_{2n}^{W}H^{n}(E^{(n−1)} × P^{n})

where π : E^{(n−1)} × P^{n} → P^{n} is the natural projection.

The second part of the framing comes from the fundamental cycle of E^{(n−1)} just as before. The rest is straitforward.

3. Motivic construction of the complex B^{∗}(E; 3). Recall the convolution map β_{2} : S^{2}k(E)^{∗} ⟶ B_{3}^{∗}(E), f_{1} ∘ f_{2} ⟼ (f_{1}) ∗ (f_{2}). We are going to show that the diagram

S^{2}k(E)^{∗}
↓ β_{2}  ↘^{m_{2}^{∗}}
B_{3}^{∗}(E) ⟶^{l_{2}^{∗}} L^{∗}(E)_{H(1)}

provides a well defined homomorphism

l_{2}^{∗} : B_{3}^{∗}(E) ⟶ L^{∗}(E)_{H(1)}

(Here L^{∗}(E)_{H(1)^{∨}} = L^{∗}(E)_{H(1)} ⊠ H(1)^{∨}, so L^{∗}(E)_{H(1)} is a ℚ-vector space.) Consider the map

μ_{2} : S^{2}k(E)^{∗} ⟶ \frac{K_{2}(K(E)_{−}}{\{k^{∗}, k(E)^{∗}\}_{−}} , f_{1} ∘f_{2} ⟼ \{f_{1}(x), f_{2}(−x)\}−\{f_{1}(−x), f_{2}(x)\}

According to theorem 3.9 in [GL] one has

Theorem 7.14 μ_{2}(Kerβ_{2}) = 0

It remains to use that l_{2}^{∗}(f ∗ (1 − f)) = 0 by theorem 7.12.

Theorem 7.15 We get a commutative diagram

B_{3}^{∗}(E)_{ℚ}  ⟶^{δ_{H(1)}^{∗}}  (k^{∗} ⊗ J)_{ℚ}
l_{2}^{∗} ↓  ↓=
L^{∗}(E)_{H(1)} ⟶^{δ} L^{∗}(E)_{ℚ(1)} ⊗ L^{∗}(E)_{H}

To prove this theorem we need only to compute δh(E; f_{1}, f_{2}), which is left to the reader.

8 Elliptic Chow polylogarithms and generalized Eisenstein - Kronecker series

1. Elliptic Chow polylogarithms. The single valued version. Let C be a codimension n cycle in E^{k} ×(ℙ^{1})^{l}, k+l = 2n−1, skewinvariant under the action of G_{k} × G_{l}.

Recall the forms π_{n}ω_{p,q} (see (43)), which for p ≥ q, p+q = n−1 form a basis over ℝ in H_{B}^{n−1}(E_{/ℝ}^{n−1}, ℝ(n − 1))_{sgn_{n}}. We represent elements in H_{B}^{n−1}(E_{/ℝ}^{n−1}, ℝ(n − 1))_{sgn_{n}} by their cup product with the forms π_{n}ω_{p,q}:

The single valued elliptic Chow polylogarithm is a function

P_{k,l} : Z^{n}(E^{k}, l) ⟶ H_{B}^{n−1}(E_{/ℝ}^{n−1}, ℝ(n − 1))_{sgn_{n}} k + l = 2n − 1

defined as follows. Let π : C → (ℙ^{1})^{l} and p : C → E^{k}. If k > 0:

< P_{k,l}(C), ω_{p,q} >:= ∫_{C} π^{∗}r_{k−1}(z_{1}, ..., z_{k}) ∧ p^{∗}ω_{p,q} p ≥ q, p + q = n − 1,

Here we integrate over the nonsingular part of the complex points of the cycle C. The integral is always convergent (see [G6]). For example

P_{2,1}(C) := Alt_{(x_{2},x_{3})} ∫_{C} π_{1}^{∗} log |z_{1}|π_{2}^{∗}ω ∧ π_{3}^{∗}ω̄

The multivalued elliptic Chow polylogarithm, denoted P_{k,l}(C), is defined as follows. Let (x_{1}, ..., x_{k}, z_{k+1}, ..., z_{k+l}) be the coordinates on E^{k} × (ℙ^{1})^{l}. Assume l ≠ 0. Let π_{i} (resp. p_{j}) be the projection of C to the i-th coordinate ℙ^{1} (resp. j-th factor E) in E^{k} × (ℙ^{1})^{l}.

i). Assume k ≤ n. Then

P_{k,l}(C) :=

Alt_{(G_{k}×G_{l})} ∫_{p_{1}^{∗}γ×...×p_{k}^{∗}γ×π_{k+1}δ×...×π_{k+l}δ} log z_{n+1}d log z_{n+2} ∧ ... ∧ d log z_{2n−1}

ii). Assume n < k < 2n − 1. Then

P_{k,l}(C) :=

Alt_{(G_{k}×G_{l})} ∫_{p_{1}^{∗}γ×...×p_{n}^{∗}γ} p_{n+1}^{∗}ω ∧ ... ∧ p_{k}^{∗}ω ∧ (log z_{k+1}d log z_{k+2} ∧ ... ∧ d log z_{2n−1})

Example 1. The multivalued elliptic Chow dilogarithms:

P_{0,3}(C) := Alt_{(G_{3})} ∫_{π_{1}^{∗}δ} log z_{2}d log z_{3}

P_{1,2}(C) := Alt_{(G_{1}×G_{2})} ∫_{p_{1}^{∗}γ} log z_{2}d log z_{3}

P_{2,1}(C) := Alt_{(G_{2}×G_{1})} ∫_{p_{1}^{∗}γ} p_{2}^{∗}ω · log z_{3}

Example 2. The multivalued elliptic Chow trilogarithms:

P_{0,5}(C) := Alt_{(G_{5})} ∫_{π_{1}^{∗}δ×π_{2}^{∗}δ} log z_{3}d log z_{4} ∧ d log z_{5}

P_{1,4}(C) := Alt_{(G_{1}×G_{4})} ∫_{p_{1}^{∗}γ×π_{2}^{∗}δ} log z_{3}d log z_{4} ∧ d log z_{5}

P_{2,3}(C) := Alt_{(G_{2}×G_{3})} ∫_{p_{1}^{∗}γ×p_{2}^{∗}γ} log z_{3}d log z_{4} ∧ d log z_{5}

P_{3,2}(C) := Alt_{(G_{3}×G_{2})} ∫_{p_{1}^{∗}γ×p_{2}^{∗}γ} p_{3}^{∗}ω ∧ log z_{4} ∧ d log z_{5}

P_{4,1}(C) := Alt_{(G_{4}×G_{1})} ∫_{p_{1}^{∗}γ×p_{2}^{∗}γ} p_{3}^{∗}ω ∧ p_{4}^{∗}ω · log z_{5}

The multivalued elliptic Chow polylogarithms are periods of mixed motives, which are easy to write down.

2. Some interesting cycles. Let 𝕃_{n}(a) be the codimension n cycle in (ℙ^{1})^{2n−1} responsible for the classical n-logarithm (see [Bl6] and [BK]):

𝕃_{n}(a) := {x_{1}, ..., x_{k−1}, 1−x_{1}, 1−x_{2}/x_{1}, ..., 1−x_{k−1}/x_{k−2}, 1−a/x_{k−1}} ∈ Z^{n}(2n−1)

Consider the following cycle in Z^{n}(E^{(n−k−1)}, k + n):

Alt_{(x_{1},...,x_{n−k})}(x_{1}, ..., x_{n−k}, 𝕃_{k}(f_{1}(x_{1})), f_{2}(x_{2}), ..., f_{n−k}(x_{n−k})  (66)

Notice that we are using this E^{(n−k−1)}, not E^{n−k−1}.

Examples of cycles.

in Z^{2}(E, 2) : Alt_{(x_{1},x_{2})}{x_{1}, f_{1}(x_{1}), f_{2}(x_{2})}
in Z^{2}(3) : {z_{1}, 1 − z_{1}, 1 − a/z_{1}}
in Z^{3}(E^{(2)}, 3) : Alt_{(x_{1},x_{2},x_{3})}{x_{1}, x_{2}, f_{1}(x_{1}), f_{2}(x_{2}), f(x_{3})}
in Z^{3}(E, 4) : Alt_{(x_{1},x_{2})}{x_{1}, 1 − x_{1}, 1 − f_{1}(x_{1})/x_{1}, f_{2}(x_{2})}
in Z^{3}(5) : {z_{1}, z_{2}, 1 − z_{1}, 1 − z_{2}/z_{1}, 1 − a/z_{2}}

I think the single valued elliptic Chow polylogarithm on these cycles should provide the new transcendental functions needed for L(Sym^{n−k−1}E, n).

3. The generalized Eisenstein-Kronecker series and L(Sym^{n}E, n + m) for m ≥ 1. Conjecture 2.1 in [G2] for the field k(E^{(n)}) tells us that H_{M}^{n+1}(Speck(E^{(n)}), ℚ(n + m)) is generated by the sums of the elements of form

∑_{i}{f_{0}^{(i)}(x)}_{m} ⊗ f_{1}^{(i)}(x) ∧ ... ∧ f_{n}^{(i)}(x) in B_{m}(k(E^{(n)})) ⊗ Λ^{n}k(E^{(n)})^{∗}

satisfying the condition

∑_{i}{f_{0}^{(i)}(x)}_{m−1}∧f_{0}^{(i)}(x)∧f_{1}^{(i)}(x)∧...∧f_{n}^{(i)}(x) in B_{m−1}(k(E^{(n)}))⊗Λ^{n+1}k(E^{(n)})^{∗}

(67)

Definition 8.1 D_{(n,m)}^{0,1} is the subgroup of B_{m}(k(E^{(n)})) ⊗ Λ^{n}k(E^{(n)})^{∗} generated by the elements

Alt_{(x_{0},...,x_{n})}{f(x_{0})}_{m} ⊗ g_{1}(x_{1}) ∧ ... ∧ g_{n}(x_{n})  (68)

where f, g_{k} ∈ k(E)^{∗}, x_{0} + ... + x_{n} = 0.

Denote by X the element (68). Set

∂(X) := Alt_{(x_{0},...,x_{n})}{f_{0}(x_{0})}_{m−1} ∧ f_{0}(x_{0}) ∧ g_{1}(x_{1}) ∧ ... ∧ g_{n}(x_{n})

It belongs to the group B_{m−1}(ℂ(E^{(n)})) ⊗ Λ^{n+1}ℂ(E^{(n)})^{∗}.

r(X) := ∑_{i=1}^{n}(−1)^{i} ∑_{a∈E(k)} v_{a}g_{i}·Alt_{(x_{0},...,x_{n})}{f_{0}(x_{0})}_{m}⊗g_{1}(x_{1})∧...ĝ_{i}...∧g_{n}(x_{n})_{|_{x_{i}=a}}

Here v_{a}g is the valuation of g at the point a. Let d_{n,m} := ∂ + r. r(X) lies in the sum of the groups B_{m}(ℂ(X_{i,a}))^{∗} ⊗ Λ^{n−1}ℂ(X_{i,a})^{∗} where X_{i,a} is the divisor x_{i} = a.

Conjecture 8.2 a) There exists a map

Kerd_{n,m} ⟶ H_{M}^{n+1}(E^{(n)}, ℚ(n + m))_{sgn}

which in the case k = ℂ commutes with the regulator map.

b) One might hope that this map is surjective.

Part a) of this conjecture can be deduced from standard conjectures. If m = 1 this is exactly conjecture discussed in chapter 5. If n = 1 the conjecture follows from the conjecture 2.1 in [G2], see also conjecture 8 in [G4] and n = 1, m = 2 it is proved in [G3]. In general the main argument for the hope expressed in the part b) is simplicity of the ansatz used to define the elements (68).

Element (68) provides a cycle of type (66), as was explaned above. The value of the regulator on the element which lies in Kerd_{n,m} should coinside with the value of the elliptic Chow polylogarithm P_{n,m} on the corresponding cycle. Thus to get the generalized Eisenstein-Kronecker series responsible for the special values L(Sym^{n}E, n + m) we evaluate the Chow polylogarithm P_{n,m} on the cycle (66) using the Fourier decomposition of L_{k}(f(x)) and then the same method as in chapter 6. This boils down to computation of the following regulator integral (p + q = n)

r({f}_{m} ⊗g_{1} ∧...∧g_{n}; ω̄_{p,q}) := ∫_{E^{n}(ℂ)} log |f|^{m−2} ·α_{2}(1−f, f)∧α_{n}(g_{1}, ..., g_{n})∧ω_{p,q}

Set

r({f}_{m} ⊗ g_{1} ∧ ... ∧ g_{n}) :=

∑_{p+q=n} r({f}_{m} ⊗ g_{1} ∧ ... ∧ g_{n}; ω_{q,p})(dz)^{p}(dz̄)^{q} ∈ Sym^{n}H^{1}(E(ℂ), ℂ)

Let

(ℤ[E] ⊗ ℤ[E] ⊗ ℤ[E])_{E}= (ℤ[E × E × E])_{E}

be the abelian group generated by the elements {x, y, z} where x, y, z ∈ E(k), subject to the relations {x, y, z} = {x + c, y + c, z + c} for any c ∈ E(k). Define

β_{n,m} : {f}_{m} ⊗ g_{1} ∘ ... ∘ g_{n} ⟼ (ℤ[E] ⊗ ℤ[E] ⊗ ℤ[E])_{E}
{f}_{m} ⊗ g_{1} ∘ ... ∘ g_{n} ⟼ (1 − f) ⊗ (f) ⊗ (g_{1}) ∗ ... ∗ (g_{n})

Consider the folowing functions where p + q = n, m ≥ 2:

K_{n,m}^{p,q}(x, y, z) := (\frac{Imτ}{π})^{m}×

∑_{γ_{0}+...+γ_{m}=0}^{′} \frac{(x, γ_{0})(y, γ_{1} + ... + γ_{m−1})(z, γ_{m}) · (γ̄_{m}(γ_{0} − γ_{1}) + γ_{m}(γ̄_{0} − γ̄_{1})) · γ_{m}^{p−1} · γ̄_{m}^{q−1}}{|γ_{0}|^{2} · ... · |γ_{m−1}|^{2}|γ_{m}|^{2n}}

I will call them the generalized Eisenstein-Kronecker series. For n = 1 this is the functions K_{m+1}(x, y, z) defined in [G4], see also [G3]. For n = 1, m = 2 this function was considered by Deninger [D3].

Conjecture 8.3 There exists a variation of mixed elliptic motives over (E × E × E)/E such that its real periods are given by the generalized Eisenstein-Kronecker series.

Define a homomorphism

K_{n,m} : (ℤ[(E × E × E)(ℂ)])_{E}⟶ Sym^{n}H^{1}(E(ℂ), ℂ)

{x, y, z} ⟼ ∑_{p+q=n} K_{n,m}^{p,q}(x, y, z)(dz)^{p}(dz̄)^{q}

Theorem 8.4 Assume

Alt_{(x_{0},...,x_{n})}({f(x_{0})}_{m−1} ⊗ f(x_{0}) ∧ g_{1}(x_{1}) ∧ ... ∧ g_{n}(x_{n})) = 0

in the group B_{m−1}(ℂ(E^{(n)})) ⊗ Λ^{n+1}ℂ(E^{(n)})^{∗}. Then

r({f}_{m} ⊗ g_{1} ∧ ... ∧ g_{n}) = K_{n,m} ∘ β_{n,m}({f}_{m} ⊗ g_{1} ∘ ... ∘ g_{n})

The proof is completely similar to the proof of theorems 6.4 and theorems 3.4 and 4.7 in [G3], and thus is omitted.

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Max-Planck-Institute fur mathematik, Bonn, 53115, Germany; sasha@mpimbonn.mpg.de

Department of Mathematics, Brown University, Providence, RI 02912, USA.