Zagier’s conjecture on L(E, 2)

A.B. Goncharov, A.M. Levin

Contents

1 Introduction 1

2 The group B_{2}(E) 9

3 An elliptic analog of the Bloch-Suslin complex 14

4 B_{2}(E), θ-functions and action of isogenies 30

1 Introduction

In this paper we introduce an elliptic analog of the Bloch-Suslin complex and prove that it (essentially) computes the weight two parts of the groups K_{2}(E) and K_{1}(E) for an elliptic curve E over an arbitrary field k. Combining this with the results of Bloch and Beilinson we proved Zagier’s conjecture on L(E, 2) for modular elliptic curves over ℚ.

1. The elliptic dilogarithm. 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|

The elliptic analog of the dilogarithm was defined and studied by Spencer Bloch in his seminal paper [Bl1].

The story goes as follows. Let E(ℂ) = ℂ^{∗}/q^{ℤ} be the complex points of an elliptic curve E. Here q := exp(2πiτ), Imτ > 0.

The function L_{2}(z) has a singularity of type |z| log |z| near z = 0. It satisfies the relation L_{2}(z) = −L_{2}(z^{−1}). So averaging L_{2}(z) over the action of the group ℤ on ℂ^{∗} generated by z ⟼ qz we get the convergent series:

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

This function can be extended by linearity to the set of all divisors on E(ℂ) setting L_{2,q}(P) := ∑_{i} n_{i}L_{2,q}(P_{i}) for a divisor P = ∑ n_{i}(P_{i}).

2. The results on L(E, 2). Let L(E, s) = L(h^{1}(E), s) be the Hasse-Weil L-function of an elliptic curve E over ℚ. We will always suppose that an elliptic curve E has at least one point over ℚ: zero for the addition law.

Let v be a valuation of a number field K, and h_{v} the corresponding canonical local height on E(K). As usual x ∼_{ℚ^{∗}} y means that x = qy for a certain q ∈ ℚ^{∗}. Let J = J(E) be the Jacobian of E.

Theorem 1.1 Let E be a modular elliptic curve over ℚ. Then there exists a ℚ-rational divisor P = ∑ n_{j}(P_{j}) over ℚ̄ which satisfy the conditions a)-c) listed below and such that

L(E, 2) ∼_{ℚ^{∗}} π · L_{2,q}(P)  (1)

The conditions on divisor P:

a)  ∑ n_{j}P_{j} ⊗ P_{j} ⊗ P_{j} = 0 in S^{3}J(ℚ̄)  (2)

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

∑ n_{j}h_{v}(P_{j}) · P_{j} = 0 in J(ℚ̄) ⊗ ℝ  (3)

c) For every prime p where E has a split multiplicative reduction one has an integrality condition on P, see (4) below.

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_{3}(x) := x^{3} − \frac{3}{2}x^{2} + \frac{1}{2}x be the third 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) 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_{3}(\frac{ν}{eN}) = 0  (4)

Remarks. 1. For a p-adic valuation v of the field K(P) one has (log p)^{−1}h_{v}(P_{j}) ∈ ℚ. So the condition b) in this case looks as follows

(log p)^{−1} ∑ n_{j}h_{v}(P_{j}) · P_{j} = 0 in J(K(P)) ⊗ ℚ  (5)

In particular the right hand side is a finite dimensional ℚ-vector space.

2. Lemma 1.5 below shows that, assuming (2), if the condition (3) is valid for all archimedean valuations but one then it is valid for all of them. In particular if P ∈ ℤ[E(ℚ)] we can omit (3) for the archimedean valuation.

The proof of theorem (1.1) is based on the results of S. Bloch [Bl1] on regulators on elliptic curves, a “weak” version of Beilinson’s conjecture for modular curves proved by A.A. Beilinson in [B2] and the results presented in s.2-3 below.

To prove the theorem we introduce for an elliptic curve E over an arbitrary field k a new complex (the elliptic motivic complex B(E; 3)) and prove that its cohomology essentially computes the weight 2 parts of K_{2}(E) and K_{1}(E) (see theorems (1.5) and (3.8)). This complex mirrors the properties of the elliptic dilogarithm. It is an elliptic deformation of the famous Bloch-Suslin complex which computes K_{3}^{ind}(F) ⊗ ℚ and K_{2}(F) for an arbitrary field F (see [DS], [S] and s.1.6).

In particular we replace the “arithmetical” condition b) by its refined “geometrical” version (see s. 1.4), which is equivalent to the condition b) for curves over number fields.

Our results imply

Theorem 1.2 Let E be an elliptic curve over ℚ. Then

i) For any element γ ∈ K_{2}(E) there exists a ℚ-rational divisor P on E satisfying the conditions a), b) from theorem (1.1) such that the value of the Bloch-Beilinson regulator map r_{2} : K_{2}(E) ⟶ ℝ on γ is ∼_{ℚ^{∗}} L_{2,q}(P)

ii) For any ℚ-rational divisor P on E satisfying the conditions a), b) there exists an element γ ∈ K_{2}(E) ⊗ ℚ such that r_{2}(γ) ∼_{ℚ^{∗}} L_{2,q}(P).

Theorem (1.2ii) implies immediately

Corollary 1.3 Let E be an elliptic curve over ℚ. Let us assume that the image of K_{2}(E)_{ℤ} ⊗ ℚ under the regulator map is L(E, 2) · ℚ. (This is a part of the Bloch-Beilinson conjecture).

Then for any ℚ-rational divisor P on E(ℚ̄) satisfying the conditions a)

- c) of theorem (1.1) one has

q · L(E, 2) = π · L_{2,q}(P)

where q is a rational number, perhaps equal to 0.

Remark. Corollary (1.3) has an analog for an elliptic curve over any number field. Its formulation is an easy exercise to the reader.

Unlike in Zagier’s conjecture on ζ-functions of number fields one can not expect P_{i} ∈ E(ℚ): the Mordell-Weil group of an elliptic curve over ℚ could be trivial.

The conditions a)-b) are obviously satisfied if P is (a multiple of) a torsion divisor. Moreover, if E is a curve with complex multiplication then L(E, 2) is the value of the elliptic dilogarithm on a torsion divisor ([Bl1]). However if E is not a CM curve this should not be true in general. Thus one has to consider the non-torsion divisors, and so it is necessary to use the conditions a)-b) in full strength.

The conditions a) and b) were guessed by D. Zagier several years ago after studying the results of the computer experiments with ℚ-rational points on some elliptic curves, which he did with H. Cohen.

3. A numerical example. E is given by equation y^{2} −y = x^{3} −x. The discriminant Δ = conductor = 37. So E has split multiplicative reduction at p = 37 with one irreducible component of the fiber of the Néron model. Therefore the integrality condition is empty.

Local nonarchimedean heights on E. Let P = [a/p^{2δ}, b/p^{3δ}] ∈ E(ℚ) where a, b are prime to p. If p is prime to Δ then h_{p}(P) = 0 if δ ≤ 0 and h_{p}(P) = δ · log p if δ > 0. The local height at p = 37 is given by h_{37}(P) = −1/6+2δ (see the formula for the local height in s. 4.3 of or [Sil]).

The Mordell-Weil group has rank one and is generated by the point P = [0, 0]. Consider the following integral points on E:

P = [0, 0], 2P = [1, 0], 3P = [1, 1], 4P = [2, 3], 6P = [6, −14]

and also

5P = [\frac{1}{4}, \frac{5}{8}], 10P = [\frac{161}{16}, \frac{2065}{64}]

There are no height conditions at p ≠ 37 for the integral points and there is just one at p = 2 for the points 5P and 10P.

Consider the divisor ∑ n_{k}(kP). Notice that S^{3}J(ℚ) = ℤ and the condition a) is ∑ n_{k} · k^{3} = 0. The height condition at p = 37 gives ∑ n_{k} · k = 0 provided that the coordinates of (kP) are prime to 37.

The divisor

P_{k} = (kP) − k(P) − \frac{k^{3} − k}{6} ((2P) − 2(P))

satisfies the conditions ∑ n_{k} · k = ∑ n_{k} · k^{3} = 0. Also P_{10} − 4 · P_{5} satisfies the height condition at p = 2.

The computer calculation (using PARI) shows

\frac{8π · L_{2,q}(P_{3})}{37 · L(E, 2)} = −8.0000..., \frac{8π · L_{2,q}(P_{4})}{37 · L(E, 2)} = −26.0000...,

\frac{8π · L_{2,q}(P_{6})}{37 · L(E, 2)} = −90.0000..., \frac{8π · L_{2,q}(P_{10} − 4 · P_{5})}{37 · L(E, 2)} = −248.0000...

4. The group B_{2}(E) and a refined version of conditions a) - b). Let E be an elliptic curve over an arbitrary field k and J := J(k) be the group of k-points of the Jacobian of E. Let ℤ[X] be the free abelian group generated by a set X. We will define in s. 2.1 a group B_{2}(E/k) = B_{2}(E) such that

a) one has an exact sequence

0 ⟶ k^{∗} ⟶ B_{2}(E/k) ⟶^{p} S^{2}J(k) ⟶ 0  (6)

b) one has a canonical (up to a choice of a sixth root of unity) surjective homomorphism

h : ℤ[E(k)\0] ⟶ B_{2}(E/k)  (7)

whose projection to S^{2}J(k) is given by the formula {a} ⟼ a · a. c) if K is a local field there is 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). Moreover the canonical local height h_{K} is given by the composition

ℤ[E(K)\0] ⟶^{h} B_{2}(E/K) ⟶^{H} ℝ

The group B_{2}(E) appears naturally as a version of the theory of biextensions. It is a “motivic” version of theta functions. Set {a}_{2} := h({a}) ∈ B_{2}(E).

The conditions a)-b) on a divisor ∑_{j} n_{j}(P_{j}) are equivalent to the following single one:

∑ n_{j}{P_{j}}_{2} ⊗ P_{j} = 0 in B_{2}(E(ℚ̄)) ⊗ J(ℚ̄)  (8)

More precisely,

Lemma 1.4 Let K be a number field and P_{j} ∈ E(K). Then

∑ n_{j}{P_{j}}_{2} ⊗ P_{j} = 0 in B_{2}(E(K)) ⊗ J(K) ⊗ ℚ

if and only if the following two conditions hold:

∑ n_{j}P_{j} ⊗ P_{j} ⊗ P_{j} = 0 in S^{3}J(K) ⊗ ℚ

and for any valuation v of the field K

∑ n_{j}h_{v}(P_{j}) · P_{j} = 0 in J(K) ⊗_{ℚ} ℝ

Proof. Multiplying the exact sequence (6) by J(K) ⊗ ℚ we get

0 ⟶ K^{∗}⊗J(K)⊗ℚ ⟶ B_{2}(E(K))⊗J(K)⊗ℚ ⟶^{p⊗id} S^{2}J(K)⊗J(K)⊗ℚ ⟶ 0

and use the fact that the local norms |·|_{v} separate all the elements in K^{∗}⊗ℚ.

5. The elliptic motivic complex. Let us suppose first that k is an algebraically closed field. In chapter 3 we define a subgroup R_{3}(E) ⊂ ℤ[E(k)]. When k = ℂ it is a subgroup of all functional equations for the elliptic dilogarithm. In particular the homomorphism

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

annihilates the subgroup R_{3}(E/ℂ). Consider the homomorphism (J := J(k))

δ_{3} : ℤ[E(k)] ⟶ B_{2}(E) ⊗ J, \{a\} ⟶ −\frac{1}{2}\{a\}_{2} ⊗ a

An important result ( theorem (3.3)) is that δ_{3}(R_{3}(E)) = 0 . Setting

B_{3}(E) := \frac{ℤ[E(k)]}{R_{3}(E)}

we get a homomorphism δ_{3} : B_{3}(E) ⟶ B_{2}(E) ⊗ J. Let us consider the following complex

B(E; 3) :  B_{3}(E) ⟶^{δ_{3}} B_{2}(E) ⊗ J ⟶ J ⊗ Λ^{2}J ⟶ Λ^{3}J  (9)

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. This is our elliptic motivic complex.

Let I_{E} be the augmentaion ideal of the group algebra ℤ[E], and I_{E}^{4} its fourth pour. Let B_{3}^{∗}(E) 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 homomorphism

δ_{3} : B_{3}^{∗}(E) ⟶ k^{∗} ⊗ J  (10)

which fits the following commutative diagram

0  0
↓  ↓
B_{3}^{∗}(E) ⟶ k^{∗} ⊗ J
↓  ↓
B_{3}(E) ⟶^{δ_{3}} 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. Let us denote by B^{∗}(E; 3) the complex (10). It is canonically quasiisomorphic to the complex B(E; 3).

The complex B^{∗}(E; 3) looks simpler then B(E; 3). However a definition of the differential in B^{∗}(E; 3) which does not use the embedding to B(E; 3) is rather awkward, see s. 4.6.

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

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

6. Relation with algebraic K-theory. Let k be an arbitrary field. Let K_{2} be the sheaf of K_{2} groups in the Zariski topology on E. One has canonical inclusion K_{2}(k) ↪ H^{0}(E, K_{2}) and surjective projection H^{1}(E, K_{2}) → k^{∗}.

Theorem 1.5 Let k = k̄. Then there is a sequence

T or(k^{∗}, J) ↪ \frac{H^{0}(E, K_{2})}{K_{2}(k)} ⟶ B_{3}^{∗}(E) ⟶ k^{∗}⊗J ⟶ Ker(H^{1}(E, K_{2}) → k^{∗}) → 0

It is exact in the term k^{∗} ⊗J and exact modulo 2-torsion in the other terms.

For an abelian group A(E) depending functorially on E let A(E)^{−} be the subgroup of skewinvariants under the involution x → −x of E. Recall that one has the γ-filtration on the Quillen K-groups. One can show that modulo 2-torsion

\frac{H^{0}(E, K_{2})}{K_{2}(k)} = gr_{2}^{γ}K_{2}(E)^{−}, Ker(H^{1}(E, K_{2}) → k^{∗}) = gr_{2}^{γ}K_{1}(E)^{−}

Recall that the Bloch-Suslin complex for an arbitrary field k is defined as follows:

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

Here R_{2}(k) is the subgroup 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.

One should compare theorem (1.5) with the following exact sequence provided by Suslin’s theorem on K_{3}^{ind}(k) ([S]) and Matsumoto’s theorem on K_{2}(k) ([M]) (see also a closely related results by Dupont and Sah [DS]):

0 ⟶ T̃or(k^{∗}, k^{∗}) ⟶ K_{3}^{ind}(k) ⟶ B_{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.

Remark. To guess an elliptic analog of the Steinberg relation (1 − x) ⊗ x ∈ ℂ^{∗} ⊗ ℂ^{∗} we might argue as follows. E(ℂ) = ℂ^{∗}/q^{ℤ}, so let us try to make sence out of ∑_{n∈ℤ}(1 − q^{n}x) ⊗ q^{n}x. Let p : ℂ^{∗} → E(ℂ). Projecting ℂ^{∗} ⊗ ℂ^{∗} to ℂ^{∗} ⊗ J(ℂ) we get ∏_{n∈ℤ}(1 − q^{n}x) ⊗ x, x ∈ E(ℂ). Regularizing the infinite product we obtain θ(x) ⊗ x where

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

Unfortunately this seems to make no sence: θ(x) is not a function on E(ℂ) with values in ℂ^{∗}, but a section of a line bundle. Only introducing the group B_{2}(E/ℂ) and realizing that θ(x) is a function on E(ℂ), but with values in the group B_{2}(E/ℂ), we find the elliptic analog of the Steinberg relation: θ(x) ⊗ x ∈ B_{2}(E/ℂ) ⊗ J(ℂ) (compare with s. 4.1, 4.2, 4.6). For an arbitrary base field {x}_{2} ∈ B_{2}(E) replaces θ(x).

In [W] J.Wildeshaus, assuming standard conjectures about mixed motives, gave a conjectural inductive definition of groups similar to B_{n}(E) and discuss an elliptic analog of weak version of Zagier’s conjecture.

Acknowledgement. This paper was essentially written when the first author enjoyed the hospitality of MPI(Bonn) and IHES during the Summer of 1995, and completed when the authors met in 1996 at the MPI (Bonn). The generous support of both institutions gratefully acknowledged.

A.G. was partially supported by the NSF Grant DMS-9500010 and A.L. by the Soros International Scientific Foundation.

It is our pleasure to thank Don Zagier and Maxim Kontsevich for interesting discussions. We are very grateful to the referee who made a lot of useful remarks and pointed out some errors.

2 The group B_{2}(E)

1. A construction of the group B_{2}(X). Let k be an arbitrary field. For any two degree zero line bundles L_{1} and L_{2} on a regular curve X

over k let us define, following Deligne ([De]), a k^{∗}-torsor [L_{1}, L_{2}].

Motivation. Let s_{i} be a section of a line bundle L_{i}. If div(s_{2}) = ∑ m_{i}P_{i} then < s_{1}, s_{2} > is s_{1}(div(s_{2})) ∈ ⊗L_{1}^{m_{i}}|_{p_{i}} = [L_{1}, L_{2}]. Such a tensor product turns out to be symmetric and does not depend on the choice of s_{2}. If L_{1} = O then [L_{1}, L_{2}] = k^{∗}.

For f ∈ k(X)^{∗} and a closed point x ∈ X_{1} set f̄(x) := Nm_{k(x)/k}f(x). We extend f̄ by linearity to the group of k-rational divisors on E.

Definition. The elements of [L_{1}, L_{2}] are pairs < s_{1}, s_{2} >, where s_{i} is a section of the line bundle L_{i} and the divisors div(s_{1}) and div(s_{2}) are disjoint. For a rational function f, g such that div(f) is disjoint from div(s_{2}) and div(g) is disjoint from div(s_{1}), one has ((s) := div(s)):

< f · s_{1}, s_{2} >= f̄((s_{2})) < s_{1}, s_{2} >, < s_{1}, g · s_{2} >= ḡ((s_{1})) < s_{1}, s_{2} >

There are two a priory different expansions

< f · s_{1}, g · s_{2} >= f̄((g))f̄(s_{2}) < s_{1}, g · s_{2} >= f̄((g))f̄((s_{2}))ḡ((s_{1})) < s_{1}, s_{2} > (11) and

< f ·s_{1}, g·s_{2} >= ḡ((f))ḡ((s_{1})) < f ·s_{1}, s_{2} >= ḡ((f))ḡ((s_{1}))f̄((s_{2})) < s_{1}, s_{2} >

(12)

The right hand sides coincide thanks to the Weil reciprocity f̄((g)) = ḡ((f)). So the k^{∗}-torsor [L_{1}, L_{2}] is well defined.

There is a canonical isomorphism of k^{∗}-torsors

[L_{1} ⊗ L_{2}, M] ⟶ [L_{1}, M] ⊗_{k^{∗}} [L_{2}, M]

and a similar additivity isomorphism for the second divisor. Further, the two possible natural isomorphisms

[L_{1} ⊗ L_{2}, M_{1} ⊗ M_{2}] ⟶ ⊗_{1≤i,j≤2}[L_{i}, M_{j}]

coincide.

Therefore for any element s ∈ S^{2}J_{X} one gets a k^{∗}-torsor [s] defined up to an isomorphism. Moreover [s_{1} + s_{2}] = [s_{1}] ⊗_{k^{∗}} [s_{2}].

The above facts just mean that the collection of k^{∗}-torsors [s] defines an extension of type (6). This is the definition of the group B_{2}(X).

To construct the homomorphism (7) we want to make sense of elements < s_{1}, s_{2} > where the divisors of the sections s_{i} may not be disjoint. The definition is suggested by the motivation given above. Namely

< s_{1}, s_{2} > ∈ [L_{1}, L_{2}] ⊗_{k^{∗}} V (div(s_{1}), div(s_{2}))  (13)

where V (L_{1}, L_{2}) is a k^{∗}-torsor defined as follows. Recall that if A → B is a homomorphism of abelian groups and T is an A-torsor, we can define a B-torsor f_{∗}T := T ⊗_{A} B where A acts on B via the homomorphism f. Thus for a k(x)^{∗}-torsor T_{x}^{∗}X we define a k^{∗}-torsor N(T_{x}^{∗}X) using the norm homomorphism Nm : k(x)^{∗} → k^{∗}.

For two arbitrary divisors l_{1} and l_{2} on X consider the following k^{∗}-torsor

V (l_{1}, l_{2}) := ⊗_{x∈X_{1}}N(T_{x}^{∗}X)^{⊗ord_{x}l_{1}·ord_{x}l_{2}}

Here ord_{x}l is the multiplicity of the divisor l at the point x. One has a canonical isomorphism V (l_{1} + l_{2}, m) ⟶ V (l_{1}, m) ⊗ V (l_{2}, m). If the divisors l_{1} and l_{2} are disjoint then V (l_{1}, l_{2}) = k^{∗}.

Any rational function f provides a canonical isomorphism

φ_{f} : V (l_{1}, l_{2}) ⟶ V ((f) + l_{1}, l_{2}) v ⟼ f̃(l_{2}) · v

In this formula f̃(x) ∈ N(T_{x}^{∗}X)^{⊗ord_{x}(f)} is the “leading term” of the function f at x. It is defined as follows. Choose a local parameter t at the point x. If f(t) = at^{k}+ higher order terms then f̃(x) = a(dt)^{k} ⊗ 1 ∈ N(T_{x}^{∗}X)^{k} = (T_{x}^{∗}X)^{k} ⊗ k^{∗}. So f̃(x)^{ord_{x}l_{2}} ∈ N(T_{x}^{∗}X)^{⊗ord_{x}(f)·ord_{x}l_{2}} and the formula above makes sense.

Let us recall the full version of the Weil reciprocity law. Let

∂_{x} : \{f, g\} ⟼ (−1)^{ord_{x}(f)·ord_{x}(g)} \frac{f(x)^{ord_{x}(g)}}{g(x)^{ord_{x}(f)}}  (14)

be the tame symbol. Then for any two rational functions f, g on a curve over an algebraically closed field k one has ∏_{x∈X} ∂_{x}(f, g) = 1.

There exists a canonical isomorphism of k^{∗}-torsors

S : [L_{1}, L_{2}] ⊗ V ((s_{1}), (s_{2})) ⟶ [L_{2}, L_{1}] ⊗ V ((s_{2}), (s_{1}))

given on generators by the formula

S : < s_{1}, s_{2} > ⟼ (−1)^{degL_{1}·degL_{2}+∑_{x∈X_{1}} ord_{x}s_{1}·ord_{x}s_{2}} < s_{2}, s_{1} >

The defining properties of the torsor [L_{1}, L_{2}] look as follows:

< f ·s_{1}, s_{2} >= f̄((s_{2})) < s_{1}, s_{2} >, < s_{1}, g ·s_{2} >= S(< s_{1}, (g) >) < s_{1}, s_{2} >

The formula

S < (f), (g) >=< (f), (g) >  (15)

is equivalent to the Weil reciprocity. Similar to (11), (12) and using the formula (15) we see that S is well defined.

2. The homomorphism (7). From now on X = E is an elliptic curve over k, so there is canonical isomorphism T_{x}^{∗}E = T_{0}^{∗}E. For c ∈ k^{∗} one obviously has < c · s_{1}, s_{2} >=< s_{1}, s_{2} >. Therefore we may consider < s_{1}, s_{2} > when s_{1}, s_{2} are divisors on E. Consider a map

{a} ⟼< (a) − (0), (a) − (0) >∈ (T_{0}^{∗}E)^{⊗2} ⊗_{k^{∗}} [a · a]

There is an almost canonical (up to a sixth root of unity) choice of an element in (T_{0}^{∗}E)^{⊗2}. Namely, the quotient of E by the involution x ⟼ −x is isomorphic to P^{1}. The image of 0 on E is the point ∞ on P^{1}. The images of the three nonzero 2-torsion points on E gives 3 distinguished points on P^{1}. Their ordering = choice of a level 2 structure on E. A choice of ordering gives a canonical coordinate t on 𝔸^{1} := P^{1}\∞ for which t = 0 is the first point and t = 1 is the second. This coordinate provides a vector in T_{∞}^{∗}P^{1} and so a vector in (T_{0}^{∗}E)^{2}. Therefore we have six different trivializations of (T_{0}^{∗}E)^{2} and thus a canonical one of (T_{0}^{∗}E)^{12}: their product. The sixth root of (16× this trivialization) is the (almost) canonical element in (T_{0}^{∗}E)^{2} we need.

Using this element in (T_{0}^{∗}E)^{2} we get a map (7). The composition ℤ[E(k)\0] ⟶ B_{2}(E) ⟶ S^{2}J is obviously given by {a} ⟼ a · a.

If E is written in the Weierstrass form y^{2} = (x − e_{1})(x − e_{2})(x − e_{3}) then e_{1}, e_{2}, e_{3} are the coordinates of the distinguished points and \frac{x−e_{i}}{e_{j}−e_{k}} is the canonical coordinate corresponding to the ordering e_{i}, e_{j}, e_{k}. Let Δ be the discriminant of E. Then Δ = 16·∏_{i<j}(e_{i}−e_{j})^{2}. The canonical trivialization is Δ^{1/6}(dx/2y)^{2}.

3. The canonical height. In this section we recall the construction of the canonical local heights via the biextension (compare with [Za] and [Bl2]). The canonical local height gives a homomorphism B_{2}(K) ⟶ ℚ (resp. to ℝ) when K is a nonarchimedean (resp. archimedean) local field.

The construction of the group B_{2}(E) provides us with a collection of k^{∗}-torsors T_{(x,y)} where (x, y) is a point of J ×J. The torsors T_{(0,y)} and T_{(x,0)} are trivialized: we have a distinguished element < ∅, (y) − (0) >∈ T_{(0,y)} and a similar one in T_{(x,0)}.

The collection of torsors T_{(x,y)} glue to a k^{∗}-bundle on J × J. It is isomorphic to the (rigidified) Poincaré line bundle minus zero section.

Let L be a degree zero line bundle on an elliptic curve over k. Let us denote by L^{∗} the complement of the zero section in L. It is a principal homogeneous space over a certain commutative algebraic group A(L) over k which is an extension

0 ⟶ 𝔾_{m} ⟶ A(L) ⟶ J ⟶ 0

The group A(L) is described as follows. For any a ∈ E let t_{a} : x ⟼ a + x be the shift by a. Then t_{a}^{∗}L is isomorphic to L (because L is of degree zero). The set of isomorphisms from L to t_{a}^{∗}L form a k^{∗}-torsor. These torsors glue together to a k^{∗}-torsor over E which is isomorphic to L^{∗}. On the other hand the collection of isomorphisms L ⟶ t_{a}^{∗}L form a group. This is the group A(L). It is commutative: the commutator provides a morphism from E × E to 𝔾_{m}, which has to be a constant map.

Now let K be a local field and E be an elliptic curve over K. We get a group extension

0 ⟶ K^{∗} ⟶ A(L)(K) ⟶ J(K) ⟶ 0

Let U(G) be the maximal compact subgroup of a locally compact commutative group G. One can show (use lemma 6.1, ch. 11 in [La]) that U(A(L)(K)) projects surjectively onto J(K) if K is archimedean or E has a good reduction over K. In the case of bad reduction the image is a subgroup of finite index.

There is canonical homomorphism

A(L)(K) ⟶ A(L)(K)/U(A(L)(K)) =: H

The quotient H is isomorphic to ℤ (resp. to a subgroup in ℚ which is an extension of ℤ by a finite group ) when K is nonarchimedean and E has good reduction (resp. bad reduction), and to ℝ if K = ℂ, ℝ.

Therefore we get a homomorphism A(L)(K) ⟶ ℤ(resp ℚ) for the nonarchimedean case and A(L)(K) ⟶ ℝ for the archimedean one.

For a given x the torsors T_{(x,y)} form a group T_{(x,·)} which is isomorphic to the group A(L|_{x×J}), and there is a similar statement for the torsors T_{(x,y)} for a given y. Applying the homomorphism A(L|_{x×J}) ⟶ H we will get a homomorphism of the group T_{(x,·)} to H. Similarly we have a homomorphism of the group T_{(·,y)} to H.

Consider the map

U(T_{(x_{1},·)}) × U(T_{(x_{2},·)}) ⟶ T_{(x_{1}+x_{2},·)}

induced by the multiplication T_{(x_{1},y)} × T_{(x_{2},y)} ⟶ T_{(x_{1}+x_{2},y)}. Its image is a subgroup. This follows from the commutativity of 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})}

It is therefore a compact subgroup, and so a maximal compact subgroup in T_{(x_{1}+x_{2},·)}. In particular the restrictions of the homomorphisms T_{(x,·)} ⟶ H and T_{(·,y)} ⟶ H to T_{(x,y)} coincide.

So we get a well defined homomorphism B_{2}(E(K)) ⟶ H. Now the composition

ℤ[E(K)\0] ⟶^{h} B_{2}(E(K)) ⟶ H  (16)

is the canonical Néron height on E(K). The restriction of the homomorphism (16) to the subgroup K^{∗} ⊂ B_{2}(E(K)) coincides with the logarithm of the norm homomorphism. In the nonarchimedean case H is a subgroup of log p · ℚ ⊂ ℝ.

Remark. The homomorphism h : ℤ[E(K)\0] ⟶ B_{2}(E(K)) was defined up to a sixth root of unity. This does not affect the definition of the height because the norm vanishes on roots of unity.

3 An elliptic analog of the Bloch-Suslin complex

1. The group B_{3}(E) and complex B(E; 3). We will assume in s.3.1 - 3.3 that k = k̄. We will always use notation J := J(k). Set g^{−}(t) := g(−t). Denote by ∗ the convolution in the group algebra ℤ[E(k)].

Definition 3.1 R_{3}(E) is the subgroup of ℤ[E(k)] generated by the elements (f) ∗ (1 − f)^{−}, f ∈ k(E)^{∗}, {0}, and the ”distribution relations”

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

Remarks. a) For m = −1 we get the elements {a} + {−a} ∈ R_{3}(E). If we remove them from the definition of R_{3}(E), we get the same group.

b) It would be more natural to add to the subgroup R_{3}(E) the distribution relations for all m ∈ ℤ\0: we should get the same group (compare with lemma (3.20)). But we will not need this.

Consider the homomorphism

β : ⊗^{2}k(E)^{∗} ⟶ ℤ[E(k)] β : f ⊗ g ⟼ f ∗ g^{−} := ∑ n_{i}m_{j}{a_{i} − b_{j}}

(the Bloch map), where (f) = ∑ n_{i}(a_{i}) and (g) = ∑ m_{j}(b_{j}).

Recall that

δ_{3} : ℤ[E(k)] ⟶ B_{2}(E) ⊗ J, \{a\} ⟼ −\frac{1}{2}\{a\}_{2} ⊗ a

and i : k^{∗} ↪ B_{2}(E) is the canonical embedding (see (6)). Let I_{E} be the augmentation ideal of the group algebra ℤ[E(k)] and p : I_{E} → J the canonical projection.

Recall that if k = k̄ the tame symbol provides a homomorphism

⊗^{2}k(E)^{∗} ⟶^{∂} k^{∗} ⊗ ℤ[E]

The Weil reciprocity law shows that its image belong to k^{∗} ⊗ I_{E}.

Theorem 3.2 The following diagram is commutative

⊗^{2}k(E)^{∗} ⟶^{∂} k^{∗} ⊗ I_{E}
↓ β  ↓ i ⊗ p
ℤ[E(k)] ⟶^{δ_{3}} B_{2}(E) ⊗ J

Proof. Let (f) = ∑ n_{i}(a_{i}), (g) = ∑ m_{j}(b_{j}). Then

δ_{3} ∘ β(f ∧ g) = −\frac{1}{2} · ∑_{i,j} m_{i}n_{j} < a_{i} − b_{j}, a_{i} − b_{j} > ⊗(a_{i} − b_{j})

The term ∑_{i,j} m_{i}n_{j} < (a_{i} − b_{j}), (a_{i} − b_{j}) > ⊗a_{i} equals to

∑_{i,j} m_{i}n_{j} < (a_{i})−(0), (a_{i})−(0) > ⊗a_{i}+∑_{i,j} m_{i}n_{j} < (b_{j})−(0), (b_{j})−(0) > ⊗a_{i}−

2 · ∑_{i,j} m_{i}n_{j} < (a_{i}) − (0), (b_{j}) − (0) > ⊗a_{i}

The first term here is zero because ∑_{j} m_{j} = 0. The second is zero because ∑_{i} n_{i}a_{i} = 0 in J. The last one can be written as

−2 · ∑_{i} m_{i} < (a_{i}) − (0), (g) > ⊗a_{i}

So the theorem follows from the definition of the tame symbol.

Theorem 3.3 δ_{3}(R_{3}(E)) = 0.

Proof. We will denote by {a}_{3} projection of the generator {a} onto the quotient B_{3}(E). The map δ_{3} kills the distribution relations:

δ_{3}(m({a}_{3} − m ∑_{mb=a}{b}_{3}))) = m({a}_{2} ⊗ a − m ∑_{mb=a}{b}_{2} ⊗ b) =

m({a}_{2} − ∑_{mb=a}{b}_{2}) ⊗ a = 0

The last equality is provided by corollary (4.3), which will be proved later, in s. 4.4. (The proof does not depend on any results or constructions in chapter 3).

The fact that δ_{3}(f ∗(1−f)) = 0 lies deeper and follows from the theorem (3.2). Theorem (3.3) is proved.

Set B_{3}(E) := ℤ[E(k)]/R_{3}(E). We get a complex

B(E, 3) := B_{3}(E) ⟶ B_{2}(E) ⊗ J ⟶ J ⊗ Λ^{2}J ⟶ Λ^{3}J

Let I_{E}^{k} be k-th power of the augmentation ideal.

Lemma 3.4 β(f ⊗ g) ∈ I_{E}^{4}. Moreover, β is surjective onto I_{E}^{4}.

Proof. A divisor ∑ n_{i}(a_{i}) is principal if and only if ∑ n_{i} = 0 and ∑ n_{i}a_{i} = 0 in J(E). So I_{E}^{2} coincides with the subgroup of ℤ[E] given by the divisors of functions. So the convolution of two principal divisors belongs to I_{E}^{4} and, moreover, generate it.

Let B_{3}^{∗}(E) be the quotient of I_{E}^{4} by the subgroup generated by the elements (f) ∗ (1 − f)^{−}.

Lemma 3.5 δ_{3}(I_{E}^{4}) ∈ k^{∗} ⊗ J ⊂ B_{2}(E) ⊗ J

Proof. It is easy to see that δ_{3}(I_{E}^{4}) ⊂ h(I_{E}^{3}) ⊗ J. Further, h(I_{E}^{3}) ⊂ k^{∗} because p ∘ h(I_{E}^{3}) = 0 (see the properties of the group B_{2}(E) listed in s. 1.4). The lemma follows.

So we get a complex

B^{∗}(E; 3) : B_{3}^{∗}(E) ⟶^{δ_{3}} k^{∗} ⊗ J  (17)

2. Relation with algebraic K-theory. Let us remind the long exact sequence of localization

K_{3}(k(E)) ⟶^{∂_{3}} ⊕_{x∈E_{1}}K_{2}(k(x)) ⟶ K_{2}(E) ⟶

K_{2}(k(E)) ⟶^{∂_{2}} ⊕_{x∈E_{1}}k(x)^{∗} ⟶ K_{1}(E) ⟶ k(E)^{∗} ⟶^{∂_{1}} ⊕_{x∈E_{1}}ℤ

The group K_{1}(E) has a subgroup k^{∗} which comes from the base. One can show that

H^{0}(E, K_{2}) = gr_{2}^{γ}K_{2}(E), H^{1}(E, K_{2}) = gr_{2}^{γ}K_{1}(E) = K_{1}(E)/k^{∗}

Lemma 3.6 Modulo 2-torsion one has

H^{0}(E, K_{2}) = K_{2}(k) ⊕ H^{0}(E, K_{2})^{−}
Ker(H^{1}(E, K_{2}) ⟶ k^{∗}) = H^{1}(E, K_{2})^{−} = K_{1}(E)^{−}

Proof. It follows easily using the transfer related to the projection E → P^{1} given by factorization along the involution x → −x.

Recall that H^{3}(B(E, 3)) = H^{4}(B(E, 3)) = 0.

Theorem 3.7 Let k = k̄. Then the commutative diagram from theorem (3.2) provides a morphism of complexes

\frac{K_{2}(k(E))}{k^{∗}·k(E)^{∗}} ⟶^{∂} k^{∗} ⊗ J
β̃ ↓  ↓ id
B_{3}^{∗}(E) ⟶^{δ_{3}} k^{∗} ⊗ J

where β̃ is surjective and Kerβ̃ = T or(k^{∗}, J) modulo 2-torsion.

This theorem and lemma (3.6) implies immediately

Theorem 3.8 Let k = k̄. Then there are an embedding

i : T or(k^{∗}, J(k)) ↪ \frac{H^{0}(E, K_{2})}{K_{2}(k)}

and canonical isomorphisms

\frac{H^{0}(E, K_{2})}{T or(k^{∗}, J(k)) + K_{2}(k)} = H^{1}B^{∗}(E; 3) modulo 2-torsion  (18)
Ker(H^{1}(E, K_{2}) ⟶ k^{∗}) = H^{2}B^{∗}(E; 3)  (19)

For an arbitrary field k (18) and (19) are isomorphisms modulo torsion.

Proof of theorem (3.7). We will first prove that we have a morphism of complexes and T or(k^{∗}, J) ⊂ Kerβ̃, and then that T or(k^{∗}, J) = Kerβ̃ modulo 2-torsion.

Recall that the tame symbol homomorphism ∂ maps K_{2}(k(E)) to k^{∗}⊗I_{E}. Therefore the complex computing the groups H^{0}(E, K_{2}) and Ker(H^{1}(E, K_{2}) ⟶

k^{∗}) looks as follows

K_{2}(k(E)) ⟶^{∂} k^{∗} ⊗ I_{E}

Notice that ∂({c, f}) = c ⊗ (f), so it has a subcomplex

k^{∗} · k(E)^{∗} ⟶^{∂} k^{∗} ⊗ I_{E}^{2}  (20)

where · is the product in K-theory. One obviously has Ker∂ = K_{2}(k), and factorising by K_{2}(k) we get the identity map k^{∗} ⊗ I_{E}^{2} → k^{∗} ⊗ I_{E}^{2}.

The map β : ⊗^{2}k(E)^{∗} → I_{E}^{4} followed by the natural projection I_{E}^{4} → B_{3}^{∗}(E) leads to surjective map

β̃ : ⊗^{2}k(E)^{∗} ⟶ B_{3}^{∗}(E)

Notice that β(k^{∗} ⊗ k(E)^{∗}) = 0 and β(f ⊗ (1 − f)) = 0 by the definition of the group B_{3}^{∗}(E). So we get the desired morphism of complexes.

Tensoring the exact sequence 0 → I_{E}^{2} → I_{E} → J → 0 by k^{∗} and using the fact that I_{E} is a free abelian group, we get an exact sequence

0 ⟶ T or(k^{∗}, J) ⟶ k^{∗} ⊗ I_{E}^{2} ⟶ k^{∗} ⊗ I_{E} ⟶^{id⊗p} k^{∗} ⊗ J ⟶ 0  (21)

So we get the following commutative diagram, where the vertical sequences are complexes, the complex on the right is the exact sequence (21), and α is injective:

0 ↓ T or(k^{∗}, J)

↙  ↓
k^{∗} ⊗ I_{E}^{2} ⟶^{=} k^{∗} ⊗ I_{E}^{2}
α ↓  ↓
\frac{K_{2}(k(E))}{K_{2}(k)} ⟶^{∂} k^{∗} ⊗ I_{E}
β̃ ↓  ↓
B_{3}^{∗}(E) ⟶^{δ_{3}} k^{∗} ⊗ J

↓ 0

From this diagram we see that T or(k^{∗}, J) ⊂ Kerβ̃. Notice that

Coker(k^{∗} ⊗ I_{E}^{2} ⟶^{α} \frac{K_{2}(k(E))}{K_{2}(k)}) = \frac{⊗^{2}I_{E}^{2}}{\{(1 − f) ∗ (f)^{−}\}}

Theorem 3.9 The map β̄ : \frac{⊗^{2}I_{E}^{2}}{\{(1−f)∗(f)^{−}\}} ⟶ B_{3}^{∗}(E) is an isomorphism modulo 2-torsion.

Proof of theorem (3.9). It consists of three steps of quite different nature.

Step 1. Set (t_{a}f)(x) := f(x − a).

Proposition 3.10 Let f and g be rational functions on E. Then

\{f, g\} − \{t_{a}f, t_{a}g\} = 0 in \frac{K_{2}(k(E))}{k^{∗} · (k(E))^{∗}}

Let L/K be an extension of fields. Then one has a natural map p^{∗} : K_{2}(K) → K_{2}(L) and the transfer map p_{∗} : K_{2}(L) → K_{2}(K). We need the following result ([BT]).

Lemma 3.11 Let L/K be a degree 2 extension of fields. Then K_{2}(L) is generated by symbols {k, l} with k ∈ K, l ∈ L and p_{∗}({k, l}) = {k, Nm_{L/K}l}.

In particular p^{∗}p_{∗} = Id+σ where σ is a nontrivial element in Gal(L/K), and thus modulo 2-torsion any Galois invariant element in K_{2}(L) belongs to p^{∗}K_{2}(K).

Lemma 3.12 Assume k = k̄. Then any rational function f on an elliptic curve E over k can be decomposed into a product of functions with divisors of the following kind: (a) − 2(b) + (−a + 2b).

Proof of the lemma 3.12. Any function f can be decomposed into a product of the functions with divisors of the following kind: (a) − (b) − (c) + (−a + b + c).

Indeed, let (f) = ∑ n_{a}(a). We will use induction on ∑ |n_{a}|. Since ∑ n_{a} = 0, replacing if needed f by f^{−1} we can find points a, b, c such that n_{a} > 0, n_{b} < 0, n_{c} < 0 . Then (f) − [(a) − (b) − (c) + (−a + b + c)] is a principal divisor with smaller ∑ |n_{a}|.

Further (a) − (b) − (c) + (−a + b + c) = [(a) − 2(d) + (−a + b + c)] − [(b) − 2(d) + (c) for 2d = b + c.

Proof of the proposition (3.10). According to the lemma we can assume that (f) = (b) − 2(0) + (−b), (g) = (c) − 2(d) + (−c + 2d). Therefore (t_{a}f) = (b + a) − 2(a) + (−b + a), (t_{a}g) = (c + a) − 2(d + a) + (−c + 2d + a).

The quotient of E under the involution σ_{a} : x → a − x is isomorphic to ℙ^{1}. The symbol {t_{a}f, t_{a}g} + {σ_{a}t_{a}f, σ_{a}t_{a}g} is σ_{a}-invariant, so it comes from K_{2}(k(ℙ^{1})). It is known that K_{2}(k(ℙ^{1})) is generated by k(ℙ^{1})^{∗} ⊗ k^{∗}. So

{f, g} − {t_{a}f, t_{a}g} ∼ {f, g} + {σ_{a}t_{a}f, σ_{a}t_{a}g} = {f, g · σ_{0}g} ∼ 0

because σ_{a}t_{a} = σ_{0}, σ_{0}f = f, and the symbol {f, g · σ_{0}g} is σ_{0}-invariant.

Step 2. Let A be an abelian group. Let S_{k}(A) ⊂ S^{k}I_{A} be the subgroup generated by the elements

((x_{1})∗X_{1})∘...∘((x_{k})∗X_{k})−X_{1}∘...∘X_{k}, x_{1}+...+x_{k} = 0, x_{i} ∈ A, X_{i} ∈ I_{A}

This subgroup clearly belongs to the kernel of the convolution map

S^{k}I_{A} → I_{A}^{k}, X_{1} ∘ ... ∘ X_{k} ⟼ X_{1} ∗ ... ∗ X_{k}

So we get a homomorphism α_{k} : S^{k}I_{A}/S_{k}(A) ⟶ I_{A}^{k}.

Proposition 3.13 For any abelian group A the homomorphism α_{k} is injective.

Proof. We may assume that A is finitely generated. Therefore the group ring ℤ[A] looks as follows:

ℤ[A] = ℤ[t_{1}, ..., t_{a}, t_{1}^{−1}, ..., t_{a}^{−1}] × ∏_{a+1≤j≤m} \frac{ℤ[t_{j}]}{(t_{j}^{N_{j}} − 1)}  (22)

Under this isomorphism the augmentation ideal I_{A} goes to the maximal ideal (t_{1} − 1, ..., t_{m} − 1). The subgroup S_{k}(A) is generated by the elements

(t_{i_{1}} − 1)f_{1} ∘ ... ∘ (t_{i_{k−1}} − 1)f_{k−1} ∘ (t_{i_{k}} − 1)f_{k} −  (23)
(t_{i_{1}} − 1) ∘ ... ∘ (t_{i_{k−1}} − 1) ∘ (t_{i_{k}} − 1)f_{1}...f_{k}

So any element of the quotient S^{k}I_{A}/S_{k}(A) can be written as

(t_{i_{1}} − 1) ∘ ... ∘ (t_{i_{k−1}} − 1) ∘ (t_{i_{k}} − 1)f  (24)

The homomorphism α_{k} sends it to (t_{i_{1}} − 1)...(t_{i_{k−1}} − 1)(t_{i_{k}} − 1)f.

Let us suppose first that a > 0, i.e. A is an infinite group. We will use the induction on both m and k. The case m = 1 is trivial: if (t − 1)^{k}f = 0 then (t − 1) ∘ ... ∘ (t − 1)f = 0, even in the case f(t) ∈ _{\frac{ℤ[t_{j}]}{(t_{j}^{N_{j}} −1)}}.

Consider an element

P = ∑_{j}(t_{i_{1}(j)} − 1) ∘ ... ∘ (t_{i_{k}(j)} − 1)f_{j} ∈ Kerα_{k}  (25)

Any element f of the right hand side of (22) can be written as f′(t_{2}, ..., t_{m})+ (t_{1} − 1)f′′(t_{1}, ..., t_{m}), So writing f_{j} = f′_{j} + (t_{1} − 1)f′′_{j} and setting P′ := ∑_{j}(t_{i_{1}(j)} − 1) ∘ ... ∘ (t_{i_{k}(j)} − 1)f′_{j} we get P = P′ + (t_{1} − 1) ∘ Q′′.

Further, let P′_{I} be the sum of all the terms of P′ where non of the indices i_{l}(j) equal to 1. Then P′ = P′_{I} + (t_{1} − 1) ∘ Q′_{II}. The restriction of P′_{I} to the divisor t_{1} = 1 in Spec ℤ[A] coincides with the restriction of P. Therefore it belongs to Kerα_{k} and thus is zero by the induction asumption for (k, m−1). Since by the definition P′_{I} does not depend on t_{1}, this implies P′_{I} = 0.

Thus P = (t_{1} − 1) ∘ Q for some Q ∈ S^{k−1}I_{A}/S_{k−1}(A). Therefore α_{k−1}(Q) = 0 since t_{1} − 1 is not a divisor of zero (we have assumed a > 0). Thus Q = 0 by the induction assumption for (k − 1, m).

Now let A be a finite group. Decomposing f(t) in (24) into a sum of monomials (t_{1} − 1)^{a_{1}}...(t_{q} − 1)^{a_{q}} we can write an element (24) as a sum

P = ∑(t_{1} − 1)^{b_{1}} ∘ ... ∘ (t_{k} − 1)^{b_{k}}(t_{k+1} − 1)^{b_{k+1}}...(t_{k+l} − 1)^{b_{k+l}}  (26)

We will treat for a moment this sum as element of S^{k}ℤ[t_{1}, ..., t_{m}], not S^{k}I_{A}/S_{k}(A). Let

α̃_{k}(P) := ∑(t_{1} − 1)^{b_{1}}... · (t_{k+l} − 1)^{b_{k+l}} ∈ ℤ[t_{1}, ..., t_{m}]

be its image in ℤ[t_{1}, ..., t_{m}] under the product map. If α̃_{k}(P) = 0 then clearly P = 0 modulo the relations (23) with f_{i} ∈ ℤ[t_{1}, ..., t_{m}].

Since α_{k}(P) = 0, α̃_{k}(P) is a linear combination of monomials of type (t_{i}^{N_{i}} −1)·Q(t). We are going to show that one can find another presentation P′ for the element in S^{k}I_{A}/S_{k}(A) given by P such that α̃_{k}(P′) = α̃_{k}(P) − (t_{i}^{N_{i}} − 1) · Q(t).

Let us factorize t_{i}^{N_{i}} − 1 = (t_{i} − 1) · p_{i}(t_{i} − 1), where p_{i}(u) = ∑ c_{j}u^{j} is a polynomial in one variable. Notice that c_{0} ≠ 0. We must have a monomial in the sum P which goes under the map α̃_{k} to c_{0} · (t_{i} − 1) · Q(t). It can be written (modulo S_{k}(A)) as c_{0} · (t_{i} − 1) ∘ R. Therefore the terms in (26) which are maped by α̃_{k} to c_{j−1}(t_{i} − 1)^{j} · Q(t), j > 1, can be written as c_{j−1}(t_{i} − 1)^{j} ∘ R. Therefore the sum of these terms in (26) can be written as (t_{i}^{N_{i}} − 1) ∘ R and thus represent a zero element in S^{k}I_{A}/S_{k}(A). The proposition is proved.

Step 3. Let us write β as a composition β = i ∘ α:

⊗^{2}I_{E}^{2} ⟶^{i} ⊗^{2}I_{E}^{2} ⟶^{α} I_{E}^{4}

i : A ⊗ B ⟼ A ⊗ B^{−}; α : A ⊗ B ⟼ A ∗ B

Let S_{E} ⊂ ⊗^{2}I_{E}^{2} be the subgroup generated by the elements (f) ⊗ (1 − f)^{−}, f ∈ k(E)^{∗}. One has

⊗^{2}I_{E}^{2} = \frac{⊗^{2}k(E)^{∗}}{k^{∗} ⊗ k(E)^{∗} + k(E)^{∗} ⊗ k^{∗}} , \frac{⊗^{2}I_{E}^{2}}{S_{E}} = \frac{K_{2}(k(E))}{k^{∗} · k(E)^{∗}}

Let A_{+} (resp A_{−}) be the coinvariants of the involution x → −x on E acting on a group A functorially depending on E (resp A ⊗_{ℤ} λ, where λ is the standard ℤ-line where the involution acts by inverting the sign).

Lemma 3.14 (Kerα)_{+} ⊂ S_{E}.

Proof. Let x ∈ (Kerα)_{+}. Then ∂(x) = 0, so x defines an element of (H^{0}(E, K_{2})/K_{2}(k))_{+}. But this group is zero modulo 2-torsion by lemma (3.6).

Lemma 3.15 Λ^{2}I_{E}^{2} ⊂ S_{E}.

Proof. A⊗B+B⊗A belongs to the subgroup generated by the Steinberg relations (f) ⊗ (1 − f). Thus A ⊗ B^{−} + B ⊗ A^{−} ∈ S_{E}. The lemma above states that B ⊗ A^{−} + B^{−} ⊗ A ∈ S_{E}. So A ⊗ B^{−} − B^{−} ⊗ A ∈ S_{E}.

Let α′ : S^{2}I_{E}^{2} → I_{E}^{4}, A ∘ B ⟼ A ∗ B. To prove the theorem we need to show that (Kerα′)_{−} ⊂ S_{E}.

Let A_{i} ∈ I_{E}. The element

< A_{1}, A_{2}, A_{3}, A_{4} >:= (A_{1} ∗ A_{2}) ∘ (A_{3} ∗ A_{4}) − (A_{1} ∗ A_{3}) ∘ (A_{2} ∗ A_{4})

clearly belongs to Kerα, and thus to Ker∂.

Proposition 3.16 < A_{1}, A_{2}, A_{3}, A_{4} >∈ S_{E}.

To prove this we need the following lemma.

Lemma 3.17 If A_{i} is a principal divisor for some 1 ≤ i ≤ 4, then < A_{1}, A_{2}, A_{3}, A_{4} >∈ S_{E}. In particular we get a well defined homomorphism

< ·, ·, ·, · >: ⊗^{4}J → S^{2}I_{E}^{2}/S_{E}  (27)

Proof. Let us show that if B_{0}, B_{1} ∈ I_{E}, then < B_{0} ∗ B_{1}, A_{2}, A_{3}, A_{4} >∈ S_{E} (the other cases are similar). We will write A =^{S_{E}} B if A − B ∈ S_{E}. It follows from the proposition (3.10) that for A, B ∈ I_{E}^{2} and X ∈ ℤ[E(k)] one

has (X ∗ A) ∘ B =^{S_{E}} A ∘ (X ∗ B). So

(B_{0}∗B_{1}∗A_{2})∘(A_{3}∗A_{4}) =^{S_{E}} (B_{0}∗B_{1})∘(A_{2}∗A_{3}∗A_{4}) =^{S_{E}} (B_{0}∗B_{1}∗A_{3})∘(A_{2}∗A_{4})

Since (a) − (0) + (b) − (0) − ((a + b) − (0)) ∈ I_{E}^{2}, we get (27). The lemma is proved.

Proof of the proposition (3.16). Let (A) be the image of A ∈ I_{E} in the Jacobian. Using (A^{−}) = −(A) and the previous lemma we get

< A_{1}, A_{2}, A_{3}, A_{4} > − < A_{1}^{−}, A_{2}^{−}, A_{3}^{−}, A_{4}^{−} >∈ S_{E}

On the other hand according to the lemma (3.6)

< A_{1}, A_{2}, A_{3}, A_{4} > + < A_{1}^{−}, A_{2}^{−}, A_{3}^{−}, A_{4}^{−} >∈ S_{E}

So 2 < A_{1}, A_{2}, A_{3}, A_{4} >∈ S_{E}. Using lemma (3.17) and the divisibility (by 2) of J(k) we conclude that < A_{1}, A_{2}, A_{3}, A_{4} >∈ S_{E}.

Proposition 3.18 Ker(S^{2}I_{E}^{2} ⟶^{α′} I_{E}^{4}) is generated by the elements

((x) ∗ A) ∘ ((−x) ∗ B) − A ∘ B, A, B ∈ I_{E}^{2}

and < A_{1}, A_{2}, A_{3}, A_{4} >, A_{i} ∈ I_{E}.

Proof. Let T ⊂ I_{E}^{2} be the subgroup generated by the elements < A_{1}, A_{2}, A_{3}, A_{4} >. There is a surjective homomorphism

S^{4}I_{E} ⟶ S^{2}(I_{E}^{2})/T, A_{1} ⊗ ... ⊗ A_{4} ⟼ (A_{1} ∗ A_{2}) ∘ (A_{3} ∗ A_{4})

The proposition follows immediately from the proposition (3.13). Theorem (3.9) follows from proposition (3.18) (3.13) and (3.17).

Proposition 3.19 Assume that k = k̄. Then a) There is an injective homomorphism of complexes B^{∗}(E; 3) → B(E; 3). b)The quotient B(E; 3)/B^{∗}(E; 3) is isomorphic to the Koszul complex

S^{3}J ⟶ S^{2}J ⊗ J ⟶ J ⊗ Λ^{2}J ⟶ Λ^{3}J  (28)

In particular the complexes B(E; 3) and B^{∗}(E; 3) are quasiisomorphic.

We will need

Lemma 3.20 Suppose that

D := m ∑_{i}({a_{i}} − m ∑_{mb_{i}=a_{i}}{b_{i}}) ∈ I_{E}^{4}

Then D belongs to the subgroup generated by the elements (f) ∗ (1 − f)^{−}.

Let [m] : E → E be the isogeny of multiplication by m.

Lemma 3.21 Let f, g ∈ k(E)^{∗}, k = k̄. Then

[m]^{∗}\{f, g\} = m\{f, g\} in \frac{K_{2}(k(E)}{k^{∗} · k(E)}

Proof of the lemma (3.21). One has an exact sequence

0 ⟶ \frac{H^{0}(E, K_{2})}{K_{2}(k) + T or(k^{∗}, J)} ⟶ \frac{K_{2}(k(E))}{k^{∗} · k(E)} ⟶ k^{∗} ⊗ J ⟶ 0

The operator [m]_{∗} acts on the left and right ( different from 0) groups here by multiplication by m (see [BL], ch. 5,6 where this was proved rationally; that proof works integrally). Further, [m]_{∗} has no Jordan blocks since [m]_{∗}[m]^{∗} = m^{2}. The lemma follows.

Proof of the lemma (3.20). There is an isomorphism

j : ℤ[E(k)]/I_{E}^{4} = ℤ ⊕ J ⊕ S^{2}J ⊕ S^{3}J, {a} ⟼ (1, a, a · a, a · a · a)

One has

j(m({a} − m ∑_{mb=a}{b})) = m((1 − m^{3}), (1 − m^{2})a, (1 − m)a · a, 0) (29)

Using this we see that ∑_{i}{a_{i}} ∈ I_{E}^{4}. Thus ∑_{i}{a_{i}} = ∑(f_{j}) ∗ (g_{j})^{−}. It is easy to see that β([m]^{∗} ∑_{i}{f_{i}, g_{i}} − m{f_{i}, g_{i}}) = D. The lemma is proved.

Proof of the proposition (3.19). a) We need only to show that B_{3}^{∗}(E) injects to B_{3}(E). This boils down to the lemma (3.20) above, since in our definition of R_{3}(E) we used only the distribution relations for m = −1, 2.

b) By definition B_{3}(E)/B_{3}^{∗}(E) is isomorphic to the quotient of ℤ[E(k)]/I_{E}^{4} modulo (the image of) the distribution relations.

Using the computaion (29) and divisibility of J(k) we see that the map j maps the subgroup generated by the distribution relations for any given |m| > 1 surjectivly onto 2ℤ⊕J ⊕J^{2}. Therefore B_{3}(E)/B_{3}^{∗}(E) = S^{3}J. Let us recall that B_{2}(E)/k^{∗} = S^{2}J. So the terms of the quotient B(E; 3)/B^{∗}(E; 3) are the same as in (28). It is easy to see that the differentials coincide. The proposition is proved.

3. Zagier’s conjecture on L(E, 2) for modular elliptic curves over ℚ. Let us recall that for a curve X over ℝ one has H_{D}^{2}(X/ℝ, ℝ(2)) = H^{1}(X/ℝ, ℝ(1)). Let X̄ := X ⊗ℂ. The cup product with ω ∈ Ω^{1}(X̄) provides an isomorphism of vector spaces over ℝ:

H^{1}(X/ℝ, ℝ(1)) ⟶ H^{0}(X̄, Ω^{1})^{∨}

So we will present elements of H_{D}^{2}(X/ℝ, ℝ(2)) as functionals on H^{0}(X̄, Ω^{1}). Bloch constructed the regulator map

r_{D} : K_{2}(E) ⟶ H_{D}^{2}(E, ℝ(2))

If we represent an element of K_{2}(E) as ∑_{i}{f_{i}, g_{i}} (with all the tame symbols vanish) then Beilinson’s construction of the regulator looks as follows:

< r_{D} ∑_{i}\{f_{i}, g_{i}\}, ω >= \frac{1}{2πi} ∑_{i} ∫_{E(ℂ)} log |f_{i}|d arg(g_{i}) ∧ ω

Let f and g be rational functions on E such that

(f) = ∑ n_{i}(a_{i}), (g) = ∑ m_{j}(b_{j})

Let Γ = H_{1}(E(ℂ), ℤ). We may assume that Γ = {ℤ ⊕ ℤ · τ} ⊂ ℂ and z is the coordinate in ℂ. Let us briefly recall how the regulator integral < r_{D}{f, g}, dz > is computed by means of the elliptic dilogarithm ([Bl1], see also [RSS]). The intersection form on Γ provides a pairing

(·, ·) : E(ℂ) × Γ ⟶ S^{1};  (z, γ) := exp(\frac{2πi(zγ̄ − z̄γ)}{τ − τ̄})  (30)

Let

K_{2,1}(z; τ) := \frac{(Imτ)^{2}}{π} ∑_{γ∈Γ\0} \frac{(u, γ)}{γ^{2}γ̄}, z = exp(2πiu)

Then one has

\frac{1}{2πi} ∫_{E(ℂ)} log |f|d arg g ∧ dz = \frac{1}{iπ} ∑_{a,b∈E(ℂ)} v_{a}(f)v_{b}(g)K_{2,1}(a − b; τ)

To prove this one may use that

∫_{E(ℂ)} log |f|di arg g ∧ dz = − ∫_{E(ℂ)} log |f|d log |g| ∧ dz

together with the following lemma and the fact that the Fourier transform sends the convolution to the product.

Lemma 3.22

log |f(z)| = −\frac{Imτ}{2π} ∑_{γ∈Γ\0} v_{a}(f)\frac{(z − a, γ)}{|γ|^{2}} + C_{f}  (31)

where C_{f} is a certain constant.

Proof. One can get a proof applying ∂ ¯∂ to the both parts of (31). 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]. It does not play any role in our considerations since ∫_{E(ℂ)} C_{f} · d log |g| ∧ ω = 0 by the Stokes formula.

The relation between the Eisenstein-Kronecker series K_{2,1}(z) and the elliptic dilogarithm is the following ( [Bl1], [Z]): K_{2,1}(z; τ) = L_{2,q}(z)−iJ_{q}(z), where the function J_{q}(z) is defined as follows. Let us average the function J(z) := log |z| log |1 − z| over the action of the group ℤ generated by the shift z ⟼ qz regularizing divergencies. We will get

J_{q}(z) := ∑_{n=0}^{∞} J(q^{n}z) − ∑_{n=1}^{∞} J(q^{n}z^{−1}) + \frac{1}{3}(log |q|)^{2} · B_{3}(\frac{log |z|}{log |q|})

Here B_{3}(x) is the third Bernoulli polynomial. The function J_{q}(z) is invariant under the shift z ⟼ qz and satisfies J_{q}(z) = −J_{q}(z^{−1}).

It follows from the main result of Beilinson in [B2], see also [SS2], that for a modular elliptic curve E over ℚ there always exists an element in K_{2}(E)_{ℤ} whose regulator gives (up to a standard nonzero factor) L(E, 2). So we get the formula

L(E, 2) ∼_{ℚ^{∗}} π · ∑_{i} ∑_{a,b∈E(ℂ)} v_{a}(f^{(i)})v_{b}(g^{(i)})L_{2,q}(a − b)

Finally, the results of the present section implies that the element

∑_{i} ∑_{a,b∈E(ℂ)} v_{a}(f^{(i)})v_{b}(g^{(i)}){a − b}_{3} ∈ I_{E(_{ℚ̄})}^{4} must satisfy all the conditions

of the theorem(1.1).

Theorem (1.2) follows from the surjectivity of the map β and the arguments above.

The integrality condition ([BG], [SS]). Let E be an elliptic curve over ℚ. Choose a minimal regular model E_{ℤ} of E over ℤ. One has the exact sequence

K_{2}(E_{ℤ}) ⟶ K_{2}(E_{ℚ}) ⟶^{∂} ⊕_{p}K′_{1}(E_{p})  (32)

The group K′_{1}(E_{p}) ⊗ ℚ is not zero if and only if E_{p} has a split multiplicative reduction with special fibre a Néron N-gon. In this case K′_{1}(E_{p}) ⊗ ℚ = ℚ.

Consider an element ∑_{i}{f_{i}, g_{i}} ∈ K_{2}(ℚ(E)) which has zero tame symbol at all points. It defines an element of H^{0}(E, K_{2}). Suppose first that the closure of the support of the divisors f_{i}, g_{i} is contained on the smooth part of E_{ℤ}. Then Schappaher and Scholl ([SS]) proved that the image of this element under the map ∂ is computed by the following formula:

∂(∑_{i}\{f_{i}, g_{i}\} = ±\frac{1}{3N} ∑_{ν∈ℤ/Nℤ} d((f) ∗ (g^{−}); ν)B_{3}(\frac{ν}{N}) · Φ

here Φ is a generator in K′_{1}(E_{p}) ⊗ ℚ.

In general one should extend ℚ to ℚ((f_{i}), (g_{i})), which is the field of the definition of the divisors (f_{i}), (g_{i}). After this we get precisely the condition (4).

Let us explane why the expression ∑_{ν∈ℤ/(eN)ℤ} d(P; ν)B_{3}(\frac{ν}{eN}) does not depent on the field L. Let L and L′ are two extentions of ℚ_{p} such that all the points are defined over them. We can assume that L ⊂ L′ (by taking the composit). Denote by n_{1} = e_{1}f_{1} degree of this extension. Looking at the Tate uniformization we see that points which intersect ν-th component of special fiber corresponding to the field L intersect e_{1}ν-th component of special fiber corresponding to the field L′.

4. Main results from the motivic point of view. Let MM_{X} be the (hypothetical) abelian category of all mixed motivic sheaves over a regular scheme X over a field k. Let ℚ(−1) := h^{2}(P^{1}), ℚ(n) := ℚ(1)^{⊗n} and H := h^{1}(E)(1).

The motivic refinement of our results is the following

Conjecture 3.23 There exists a canonical quasiisomorphism in the derived category

B(E, 3) ⊗ ℚ = RHom_{MM_{k}}(ℚ(0), H(1))

Let us explain the relationship of this conjecture with our results.

Let π : E ⟶ Spec(k) be the structure morphism. There are the Tate sheaves ℚ(n)_{E} := π^{∗}ℚ(n). Beilinson’s description of Ext groups between the Tate sheaves over E gives us

Conjecture 3.24

Ext_{MM(E)}^{i}(ℚ(0)_{E}, ℚ(2)_{E}) = gr_{2}^{γ}K_{4−i}(E) ⊗ ℚ

Lemma 3.25

RHom_{MM_{k}}(ℚ(0), H(1)) = RHom_{MM_{E}}(ℚ(0), ℚ(2))^{−}  (33)

Indeed, let p : E ⟶ 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(2))

degenerating at E_{2} and abutting to Ext_{MM_{E}}^{p+q} (ℚ(0), ℚ(2)). Noting that

h^{0}(E)^{−} = h^{2}(E)^{−} = 0; h^{1}(E)^{−} = h^{1}(E)

we get (33). Conjecture (3.24) together with this lemma tell us that the cohomology of the elliptic motivic complexes are given by the formula

R^{i}Hom_{MM_{k}}(ℚ(0), H(1)) ⊗ ℚ = gr_{2}^{γ}K_{3−i}(E)^{−} ⊗ ℚ  (34)

Conjecture (3.23) follows from theorem (3.7) and conjecture (3.24), see [G2]. A very interesting and important is the following:

Problem. To construct explicitely the general elliptic motivic complexes

RHom_{MM_{k}}(ℚ(0), Sym^{n}H(m))

For m = 1 it is considered in [G2].

5. Degeneration to the nodal curve (compare with [Bl1], [DS]). Let k be an algebraicly closed field. Denote by I the group of rational functions such that f(0) = f(∞) = 0. Let I_{k^{∗}} be the augmentation ideal of ℤ[k^{∗}].

For an element f ⊗ g ∈ (1 + I) ⊗ k(t)^{∗} consider the Bloch map

β(f ⊗ g) := ∑_{x,y∈k^{∗}} v_{x}(f)v_{y}(g){y/x} + v_{∞}(g) · ((f) + (f^{−})) ∈ I_{k^{∗}}^{2}

Set

ℤ[k^{∗}] ⟶^{δ} k^{∗} ⊗ k^{∗},  {x} ⟼ (1 − x) ⊗ x, {1} ⟼ 0  (35)

Let p : I_{k^{∗}} → k^{∗} be the natural projection {x} → x.

Theorem 3.26 The following diagram is commutative:

(1 + I) ⊗ k(t)^{∗} ⟶^{∂} I_{k^{∗}} ⊗ k^{∗}
β ↓  ↓ p ⊗ id
I_{k^{∗}}^{2}  ⟶^{δ} k^{∗} ⊗ k^{∗}

The proof is a direct calculation similar to (but simpler then) the proof of theorem (3.2).

Let S(k) be the subgroup generated by the elements (1 − f) ⊗ f where f ∈ I and the subgroup (1 + I) ⊗ k^{∗}. Set B_{2}^{∗}(k) := I_{k^{∗}}^{2}/β(S(k)).

Lemma 3.27 Kerβ ⊂ S(k)

Proof. An easy analog of theorem (3.10) for the nodal curve claims that the homomorphism

q : k(t)^{∗} ⟶ (1 + I) ⊗ k(t)^{∗}, f(t) ⟼ f(x) ⊗ (x − 1)

is surjective. It is clear that β ∘ q is injective. The lemma is proved.

This lemma implies that β̄ : (1 + I) ⊗ k(t)^{∗} → B_{2}^{∗}(k) is an isomorphism. So we get a morphism of complexes

\frac{(1+I)⊗k(t)^{∗}}{S(k)} ⟶^{∂} k^{∗} ⊗ k^{∗}
β̄ ↓  ↓ id
B_{2}^{∗}(k) ⟶^{δ} k^{∗} ⊗ k^{∗}

Such that β̄ is surjective and Kerβ̄ = T or(k^{∗}, k^{∗}), similar to the theorem (3.8). So we see that when E degenerates to a nodal curve the complex B^{∗}(E/k, 3) degenerates to the complex B_{2}^{∗}(k) → k^{∗} ⊗ k^{∗}.

Let S_{1}(k) be the subgroup generated by the elements (1 − f) ⊗ f where f ∈ I.

Theorem 3.28 Let k = k̄. The identity map on ℤ[k^{∗}] provides a homomorphism of groups B_{2}^{∗}(k) → B_{2}(k). Its kernel is isomorphic to S^{2}k^{∗}. This map provides a quasiisomorphism of complexes

B_{2}^{∗}(k) ⟶ k^{∗} ⊗ k^{∗}
↓  ↓
B_{2}(k) ⟶ k^{∗} ∧ k^{∗}

One can show that the group B_{2}^{∗}(k) is isomorphic to a group defined by S. Lichtenbaum in [Li].

It is known ([Lev]) that

K_{2}(ℙ_{\{0,∞\}}^{1}, \{0, ∞\}) = \frac{(1 + I) ⊗ k(t)^{∗}}{S_{1}(k)}

Using this and the results of this subsection we can get a proof of Suslin’s theorem for k = k̄.

5. On an elliptic analog of the 5-term relation for the elliptic dilogarithm. Notice that

B_{2}(F) = Coker(ℤ[M_{0,5}(k)] ⟶^{∂} ℤ[𝔾_{m}(k)])

Here M_{0,5} is the configuration space of 5 distict points on the projective line. So one may ask how to present the group B_{3}(E) in a similar form:

B_{3}(E) =^{?} Coker(ℤ[X(k)] ⟶^{∂} ℤ[E(k)])

where X is a finite dimensional variety. In the our definition we have an infinite dimensional X (more precisely it is an inductive limit of finite dimensional varieties).

Here is a guess. Let us realize E as a cubic in P^{2}. Let p be a point in P^{2} and l_{1}, l_{2}, l_{3} any three lines through this point. Set A_{i} := l_{i} ∩ E. Let R̃_{3}^{∗}(E) ⊂ I_{E}^{4} be the subgroup generated by the elements

{p; l_{1}, l_{2}, l_{3}} := A_{1} ∗ A_{2}^{−} + A_{2} ∗ A_{3}^{−} + A_{3} ∗ A_{1}^{−}  (36)

and those linear combinations of the elements {a}_{3} + {−a}_{3} which lie in I_{E}^{4}. ( Probably they belong to the subgroup generated (36)).

Lemma 3.29 R̃_{3}^{∗}(E) ⊂ R_{3}^{∗}(E).

Proof. Let f_{i} be a linear homogeneous equation of the line l_{i}. Since the lines l_{1}, l_{2}, l_{3} intersect in a point, these equations are linearly dependent, so we may choose them in such a way that f_{1} + f_{2} = f_{3}. Thus f_{1}/f_{3} ∧ f_{2}/f_{3} = f_{1}/f_{3} ∧ (1 − (f_{1}/f_{3})). Applying the map β to this Steinberg relation and using the relations {a}_{3} + {−a}_{3} = 0 we get the element {p; l_{1}, l_{2}, l_{3}}.

One obviously has ∑_{j=1}^{4}(−1)^{j}{p; l_{1}, ...l̂_{j}, ..., l_{4}} = 0, so we can assume that the line l_{3} is, say, a vertical line.

Conjecture 3.30 R̃_{3}^{∗}(E) = R_{3}^{∗}(E).

4 B_{2}(E), θ-functions and action of isogenies

1. Elliptic curves over ℂ. Let us represent E as a quotient ℂ/ℤ + ℤτ. Below ξ denotes the coordinate on ℂ. The canonical trivialization of (T_{0}^{∗}E)^{12} is −16 ∏(e_{j} − e_{i})(dξ)^{12}, which is equal to Δ(τ)(dξ)^{12}, where

Δ(τ) = (2πi)^{12}η^{24}(τ), η(τ) = q^{\frac{1}{24}} ∏_{n=1}^{∞}(1 − q^{n}) (q = exp(2πiτ))

So the trivialization of (T_{0}^{∗}E)^{2} is (2πi)^{2}η^{4}(τ)(dξ)^{2}.

Now we will give the analytic description of the Deligne pairing [∗, ∗]. Let L_{a} be the line bundle corresponding to the divisor (a) − (0). Choose a representative α ∈ ℂ of a (α is defined up to ℤ + ℤτ). Let us define L_{α} on E as the quotient of the trivial line bundle on ℂ under the action of ℤ + ℤτ : 1 acts trivially and τ acts by multiplication by exp(2πiα). We identify L_{α+1} with L_{α} trivially and L_{α+τ} with L_{α} by multiplication by exp(2πiξ).

The fiber of the Poincare line bundle [L_{α}, L_{β}] on J × J over the point (α, β) is equal to L_{α}|_{β} ⊗ L_{α}^{−1}|_{0}. This line bundle is described as a quotient

\frac{ℂ × ℂ × ℂ}{(ℤ ⊕ ℤτ) ⊕ (ℤ ⊕ ℤτ)}

(α, β, λ) ⟼ (α + m + nτ, β + m′ + n′τ, λ · exp(2πi(nβ + n′α + nn′τ)))

Let us show that L_{α} = L_{a}. Consider a slight modification of the Jacobi θ-function (z = exp(2πiξ)):

θ(ξ) = θ(ξ; τ) = q^{1/12}z^{− \frac{1}{2}} ∏_{j≥0}(1 − q^{j}z) ∏_{j>0}(1 − q^{j}z^{−1})

Then \frac{θ(ξ − α)}{θ(ξ)θ(α)} is a section of L_{α} with the required divisor (a) − (0).

Our recipe for the calculation of the element

< (a) − (0), (a) − (0) >∈ (T_{0}^{∗}E)^{⊗2} ⊗_{k^{∗}} [a · a]

leads to the expression

\frac{dξ · θ′(0)}{θ^{2}(α)}(\frac{θ(−α)}{dξ · θ′(0)θ(α)})^{−1} = −(\frac{dξ · θ′(0)}{θ(α)})^{2}.

Notice that θ′(0) = 2πiη^{2}(τ) and the chosen analytic trivialization of (T_{0}^{∗}E)^{2} is (2πi)^{2}η(τ)^{4}(dξ)^{2}. So the final answer is −θ(α)^{−2}.

2. The Tate curves. Let K be a field complete with respect to a discrete valuation v. Let O = {a ∈ K|v(a) ≤ 0} be the ring of integers, I = {a ∈ K|v(a) < 0} the maximal ideal, k = O/I the residue field.

Let q ∈ I. According to Tate, the group K^{∗}/q^{ℤ} is isomorphic to the group of points of the elliptic curve E_{q} over K given by equation y^{2} + xy = x^{3} + a_{4}x + a_{6}, where

a_{4} = −5 ∑_{j≥1} \frac{j^{3}q^{j}}{1 − q^{j}}; a_{6} = −\frac{1}{12} ∑_{j≥1} \frac{(7j^{5} + 5j^{3})q^{j}}{1 − q^{j}}

The discriminant and j-invariant of this curve are: Δ = q ∏_{j≥1}(1−q^{j})^{24}, j = \frac{1}{q} + 744 + 196884q + · · ·. The map of K^{∗}/q^{ℤ} to the group of points of E_{q} is defined by the following expressions:

x(u) = ∑_{j∈ℤ} \frac{q^{j}u}{(1 − q^{j}u)^{2}} − 2 ∑_{j≥1} \frac{jq^{j}}{1 − q^{j}}; y(u) = ∑_{j∈ℤ} \frac{q^{2j}u^{2}}{(1 − q^{j}u)^{3}} + ∑_{j≥1} \frac{jq^{j}}{1 − q^{j}} .

The unity 1 of K^{∗} maps to neutral element 0 of the curve.

Define a function T (u) on K^{∗} by the formula:

T (u) = ∏_{j≥0}(1 − q^{j}u) ∏_{j>0}(1 − q^{j}u^{−1})

This function vanishes on {q^{ℤ}}. It is quasiperiodic: T (uq) = −u^{−1}T (u).

Let a ∈ O \ qO. Denote by the same symbol its image in E_{q}. Then a section s of the bundle O_{E_{q}}((a) − (0)) can be represented by a function f on K^{∗} such that f(uq) = af(u). Indeed, the periodic function f(u)T (u)(T (ua^{−1}))^{−1} has the required divisor.

Like in the analytic case, the total space of the Poincaré line bundle T_{(a,b)} is isomorphic to the quotient K^{∗} × K^{∗} × K^{∗} modulo the action of the group ℤ ⊕ ℤ generated by the following transformations:

(a, b, λ) → (qa, b, bλ); (a, b, λ) → (a, qb, aλ).

The corresponding group structure on the collection T_{(a,·)} is defined by the law: (b_{1}, λ_{1}) × (b_{2}, λ_{2}) = (b_{1}b_{2}, λ_{1}λ_{2}).

Let us calculate the expression < (a) − (0), (a) − (0) >. The divisor of the section T (u)^{−1}(T (ua^{−1})) equals (a) − (0). The regularized value of this expression at the divisor (a) − (0) is equal to:

\frac{−\frac{d u}{u}(∏_{j>0}(1 − q^{j}))^{2}}{∏_{j≥0}(1 − q^{j}a) ∏_{j>0}(1 − q^{j}a^{−1})} × (\frac{∏_{j≥0}(1 − q^{j}a^{−1}) ∏_{j>0}(1 − q^{j}a)}{−\frac{d u}{u}(∏_{j>0}(1 − q^{j}))^{2}} )^{−1} =

= − (\frac{\frac{d u}{u}(∏_{j>0}(1 − q^{j}))^{2}}{a^{− \frac{1}{2}} ∏_{j≥0}(1 − q^{j}a) ∏_{j>0}(1 − q^{j}a^{−1})})^{2} .

In this calculation notice that (1 − u) · u = −(u − 1) modulo (u − 1)^{2}, so the factor 1 − u leads to −\frac{du}{u}.

The trivialization of (T_{0}^{∗}E)^{⊗12} is defined by the section

(\frac{d u}{u})^{12}Δ = ((\frac{d u}{u})^{2} q^{\frac{1}{6}} (∏_{j>0}(1 − q^{j}))^{4})^{6}

Hence the needed expression is:

− (\frac{\frac{d u}{u}(∏_{j>0}(1 − q^{j}))^{2}}{a^{− \frac{1}{2}} ∏_{j≥0}(1 − q^{j}a) ∏_{j>0}(1 − q^{j}a^{−1})})^{2} ((\frac{d u}{u})^{2} q^{\frac{1}{6}} (∏_{j>0}(1 − q^{j}))^{4})^{−1}

= −(q^{\frac{1}{12}} a^{− \frac{1}{2}} T (a))^{−2}.

3. Calculation of the canonical height. a) Archimedean case. Let us calculate the archimedean height. The torsor T_{(a,0)} is trivialized. So the group T_{(a,·)} is isomorphic to the quotient of ℂ × ℂ^{∗} with coordinates (β, λ) by the action of the group ℤ + ℤτ:

1: (β, λ) → (β + 1, λ), τ: (β, λ) → (β + τ, λ × exp(2πiα))

A homomorphism |(·, ·)|_{α} : ℂ × ℂ^{∗} → ℝ^{+} which is invariant under the action of ℤ + ℤτ and coincides with the norm | · | on ℂ^{∗} if β = 0 is given by

|(β, λ)|_{α} = exp(−πi\frac{(α − ᾱ)(β − β̄)}{τ − τ̄})|λ|

It defines a homomorphism T_{(α,·)} → ℝ^{+}. Therefore the value of our height at α equals to

log |(α, −(\frac{1}{θ(α)})^{2})|_{α} = 2[−(log |θ(α)| + \frac{πi}{2} \frac{(α − ᾱ)^{2}}{τ − τ̄})]

It coincides with 2 times the canonical Néron height from [Sil].

b) The nonarchimedean case. Let q ∈ I^{n} \ I^{n+1}. The Tate curve over Spec(O) has a singular fiber over Spec(k). The singular fiber C = ∪C_{j} of the minimal Neron model is an n-gon. The set I^{j}\I^{j+1} represents the points of the curve E over Spec(K) whose restrictions to the singular fiber belong to the j-th component C_{j} of the n-gon.

Consider the map

h̃_{v} : K^{∗} × K^{∗} × K^{∗} → ℤ; h̃_{v}(a, b, λ) = v(q)v(λ) − v(a)v(b).

This map is invariant with respect to the action of ℤ⊕ℤ; therefore it detemines a map h_{v} from T_{(·,·)} to ℤ. The map h_{v} is a group homomorphism on T_{(a,·)}, its image is discrete, hence its kernel is a maximal compact subgroup.

Let v(a) = j. The value of this map on < (a) − (0), (a) − (0) > equals to:

v(q)v(θ(a)^{−2}) − v(a)v(a) = n · (−2) · (\frac{1}{12}n − \frac{1}{2}j + v(1 − a)) − j^{2}

= −\frac{1}{6}n^{2} + jn − j^{2} − 2nv(1 − a) = −n^{2}B_{2}(\frac{j}{n}) − 2nv(1 − a)

where B_{2}(x) = x^{2} − x + 1/6 is the second Bernoulli polynomial. (Compare with the integrality condition given by the third Bernoulli polynomial).

4. Functoriality of the groups B_{2}(E)_{ℚ} under the isogenies. Let n be an integer prime to the characteristic of k. Let λ : E_{1} → E_{2} be an isogeny of order n between the elliptic curves E_{1} and E_{2}.

i) Pull back λ_{2}^{∗} : B_{2}(E_{2}) → B_{2}(E_{1})[\frac{1}{n}]. Let λ̂ be the dual isogeny J_{E_{2}} → J_{E_{1}}. The pull back of the basic extension 0 → k^{∗} → B_{2}(E_{1}) → S^{2}J_{E_{1}} → 0 (considered modulo n-torsion) under the homomorphism \frac{1}{n} λ̂ · λ̂ : S^{2}J_{E_{2}} → S^{2}J_{E_{1}} provides the map λ_{2}^{∗}:

0 → k^{∗} → B_{2}(E_{2}) → S^{2}J_{E_{2}} → 0
||  ↓ λ_{2}^{∗}  ↓ \frac{1}{n} λ̂ · λ̂
0 → k^{∗} → B_{2}(E_{1})[\frac{1}{n}] → S^{2}J_{E_{1}}[\frac{1}{n}] → 0

A more direct description of λ_{2}^{∗} can be spelled as follows. We have a natural morphism of k^{∗}- torsors:

λ^{∗} : [L, M]^{⊗n} → [λ^{∗}L, λ^{∗}M]; < s_{1}, s_{2} >^{⊗n} → < λ^{−1}s_{1}, λ^{−1}s_{2} >

To check that it is a map of torsors notice that

(< f · s_{1}, s_{2} >)^{⊗n} = f(s_{2})^{n}(< s_{1}, s_{2} >)^{⊗n}

and f(s_{2})^{n} = (λ^{−1}f)(λ^{−1}s_{2}). It is easy to see that these maps provide the pull-back map:

λ_{2}^{∗} : B_{2}(E_{2}) → B_{2}(E_{1})[\frac{1}{n}]; < s_{1}, s_{2} >→< λ^{−1}s_{1}, λ^{−1}s_{2} >^{⊗ \frac{1}{n}} .

Theorem 4.1 n · (λ_{2}^{∗}{λ(a)}_{2} − ∑_{γ∈Kerλ}{a + γ}_{2}) = 0

Proof. Any isogeny can be presented as the composition of cyclic isogenies. So we can assume that λ is cyclic. The expression

f_{n}(a) := n · (λ_{2}^{∗}{λ(a)}_{2} − ∑_{γ∈Kerλ}{a + γ}_{2})

is a nonvanishing function on the noncompact curve E_{1} \ Kerλ.

Let p_{n} : E → X_{0}(n) be the universal family of elliptic curves over the modular curve X_{0}(n), and Λ : E → E the universal cyclic n-isogeny. The curves E_{1} and E_{2} are the fibers of E over the points (E_{1}, Kerλ) and (E_{2}, E_{2}[n]/Kerλ) of X_{0}(n), and λ is the restriction of Λ. The construction above defines an algebraic function F on the universal curve E_{1}.

Consider the punctured formal neighborhood of the cusp point in which the universal isogeny is totally ramified. The j-invariant of the restriction of the universal curves to this neighborhood has a pole at the cusp; hence, this restriction can be described by the Tate curves E_{q} → E_{q^{n}} ([Sil]).

Let us prove that F_{n} equals 1 for the Tate curves. We are dealing with the isogeny K^{∗}/(q^{n})^{ℤ} → K^{∗}/q^{ℤ}. For the Tate curve we expressed in s. 4.2 the pairing < (a) − (0), (a) − (0) > in terms of the θ-function

θ_{q}(a) := q^{\frac{1}{12}} a^{− \frac{1}{2}} T (a) = q^{1/12}a^{− \frac{1}{2}} ∏_{j≥0}(1 − q^{j}a) ∏_{j>0}(1 − q^{j}a^{−1}).

Remark. θ_{q}(a) is defined only up to a choice of sign, so only (θ_{q}(a))^{2} makes sense.

So we need to prove the following proposition.

Proposition 4.2

(\frac{∏_{0≤k<n} θ_{q}(t · q^{k})}{(∏_{0≤k<n} θ_{q^{n}}(t · q^{k}))^{n}} )^{2} = 1  (37)

It shows that the restriction of the function F_{n} to the preimage of neighborhood of the cusp point equals 1. Hence this function is equal to 1 on all universal curve X_{0}(N); the function f_{n} is the restriction of F_{n} to the fiber E; therefore f_{n} = 1.

Proof of the proposition. The θ-function has the following property:

θ_{q}(t · q^{l}) = (−1)^{k}a^{−k}q^{−k^{2}/2}θ_{q}(t)

Therefore ∏_{0≤k<n} θ_{q}(t · q^{k}) equals to

(−1)^{s_{1}(n)}t^{−s_{1}(n)}q^{−s_{2}(n)/2}q^{n/12}t^{−n/2} ∏_{j≥0}(1 − q^{j}t)^{n} ∏_{j>0}(1 − q^{j}t^{−1})^{n} (38)

Using the definition and notations

s_{1}(n) := 1+...+(n−1) = \frac{n(n − 1)}{2}; s_{2}(n) := 1^{2}+...+(n−1)^{2} = \frac{(n − 1)n(2n − 1)}{6}

we have

θ_{q^{n}}(t · q^{k}) = q^{n/12}t^{−1/2}q^{−k/2} ∏_{j≥0}(1 − q^{nj} · tq^{k}) ∏_{j>0}(1 − q^{nj} · t^{−1}q^{−k})

On the other hand ∏_{0≤k<n} θ_{q^{n}}(t · q^{k}) is equal to

q^{\frac{n^{2}}{12} − \frac{s_{1}(n)}{2}} t^{\frac{−n}{2}} · ∏_{0≤k<n} ∏_{j≥0}(1 − q^{nj+k}t) ∏_{0≤k<n} ∏_{j>0}(1 − q^{nj−k}t^{−1}) =

q^{\frac{n^{2}}{12} − \frac{s_{1}(n)}{2}} t^{\frac{−n}{2}} ∏_{j′≥0}(1 − q^{j′}t) ∏_{j′>0}(1 − q^{j′}t^{−1})  (39)

Comparing (38) and (39) we see that it remains to check that

(q^{\frac{n^{2}}{12} − \frac{s_{1}(n)}{2}} t^{−n/2})^{n} = q^{−s_{2}(n)/2+ \frac{n}{12}} · t^{− \frac{n}{2} −s_{1}(n)}

The statement of the proposition follows.

In particular when λ[m] is the isogeny of multiplication by m the theorem gives

Corollary 4.3 Suppose k̄ = k. Then for any a ∈ E(k) one has the ”distribution relation”

m({a}_{2} − ∑_{mb=a}{b}_{2}) = 0  (40)

Remark. In this case one can define [m]^{∗} : B_{2}(E_{2}) → B_{2}(E_{1})[\frac{1}{m}] using the map \frac{1}{m}[m] ∘ \frac{1}{m}[m] in the diagram defining λ_{2}. Thus we have the factor m instead of m^{2} in the formula (40).

ii) The transfer map λ_{2∗} : B_{2}(E_{1})[\frac{1}{n}] → B_{2}(E_{2})[\frac{1}{n}]. Notice that the group B_{2}(E)_{ℚ} does not satisfy the descent property. Namely, if k ⊂ K is a finite Galois extension then

B_{2}(E/k)_{ℚ} ↪ B_{2}(E/K)Gal(K/k)_{ℚ}

but this inclusion is not an isomorphism because the group S^{2}J(k)_{ℚ} does not have the descent property.

Suppose k = k̄. The transfer map λ_{2∗} should satisfy the projection formula

λ_{2}^{∗} ∘ λ_{2∗} = n · Id  (41)

and should fit into the following diagram, considered modulo n-torsion:

0 → k^{∗} → B_{2}(E_{1}) → S^{2}J_{E_{1}} → 0
↓ m_{n}  ↓ λ_{2∗}  ↓ λ · λ
0 → k^{∗} → B_{2}(E_{2}) → S^{2}J_{E_{2}} → 0

Here m_{n} : x → x^{n}. This is necessary in order to have (41) on the subgroup k^{∗} ⊂ B_{2}(E).

Let us define the transfer map as follows:

λ_{2∗}{a}_{2} := {λ(a)}_{2} − (λ_{2}^{∗}{λa}_{2} − n{a}_{2})  (42)

Remark. Projection of λ_{2}^{∗}\{λa\}_{2} − n\{a\}_{2} to S^{2}J equals \frac{1}{n} λ̂ ∘ λ(a · a) − na · a = 0, so λ_{2}^{∗}{λa}_{2} − n{a}_{2} ∈ k^{∗}.

Let us show that formula (42) provides a transfer homomorphism. Suppose that ∑{a_{i}}_{2} + c = 0 in the group B_{2}(E_{1}), where c ∈ k^{∗} (we write the group B_{2}(E) additively). We have to prove that

∑_{i}{λ(a_{i})}_{2} − (∑_{i} λ_{2}^{∗}{λ(a_{i})}_{2} − ∑_{i} n{a_{i}}_{2}) + nc = 0

By the assumption ∑_{i}{a_{i}}_{2} = c^{−1} ∈ k^{∗}. As we have shown before, the expression in brackets always belongs to the subgroup k^{∗} ⊂ B_{2}(E_{2}). Therefore ∑_{i} λ_{2}^{∗}{λ(a_{i})}_{2} ∈ k^{∗}. Notice that λ_{2}^{∗} is injective modulo n-torsion and is the identity on the subgroup k^{∗} ⊂ B_{2}(E). Therefore modulo n-torsion ∑_{i}{λ(a_{i})}_{2} ∈ k^{∗} and ∑_{i}{λ(a_{i})}_{2} = ∑_{i} λ_{2}^{∗}{λ(a_{i})}_{2}.

Using proposition (4.1) one can easyly see that

λ_{2∗}\{a\}_{2} := \{λ(a)\}_{2} − \frac{1}{n} ∑_{γ∈Kerλ}(n\{a + γ\}_{2} − n\{a\}_{2})  (43)

5. A presentation of the group B_{2}(E) by generators and relations. Let P(a) (resp P′(a)) be the x-coordinate (resp. y-coordinate) of the point a ∈ E in Tate’s normal form of an elliptic curve over an arbitrary field k:

y^{2} + a_{1}xy + a_{3}y = x^{3} + a_{2}x^{2} + a_{4}x + a_{6}

Proposition 4.4 . a) If a ≠ b then

< (a + b) − (0), (a + b) − (0) > ⊗ < (a − b) − (0), (a − b) − (0) >

⊗ < (a)−(0), (a)−(0) >^{−2} ⊗ < (b)−(0), (b)−(0) >^{−2}= (Δ^{−1/6}(P(a)−P(b)))^{−2}

b). If a = b but 2a ≠ 0 then the left hand side is equal to (Δ^{−1/4}P′(a))^{−2}. If 2a = 0 then we get (Δ^{−1/3}P′′(a))^{−2}

Proof. We will prove part a). Part b) is similar. Let L_{a} be the line bundle corresponding to the divisor (a) − (0). Evidently

[L_{a+b}, L_{a+b}] ⊗ [L_{a−b}, L_{a−b}] ⊗ [L_{a}, L_{a}]^{−2} ⊗ [L_{b}, L_{b}]^{−2} = k^{∗}

so

< (a + b) − (0), (a + b) − (0) > ⊗ < (a − b) − (0), (a − b) − (0) >

⊗ < (a) − (0), (a) − (0) >^{−2} ⊗ < (b) − (0), (b) − (0) >^{−2}∈ k^{∗} ⊗ (T_{0}^{∗}E)^{⊗−4}

We want to calculate this element. One has

< (a + b) − (0), (a + b) − (0) > ⊗ < (a − b) − (0), (a − b) − (0) >
⊗ < (a) − (0), (a) − (0) >^{−2} ⊗ < (b) − (0), (b) − (0) >^{−2}=
< (a + b) − (a), (a + b) − (a) > ⊗ < (a + b) − (a), (a) − (0) >^{2}
⊗ < (a − b) − (a), (a − b) − (a) > ⊗ < (a − b) − (a), (a) − (0) >^{2}
⊗ < (b) − (0), (b) − (0) >^{−2}

We have

< (a + b) − (b), (a + b) − (b) > ⊗ < (a) − (0), (a) − (0) >^{−1}= 1 ∈ k^{∗} (44)

Indeed, the left hand side is a regular function in b on the elliptic curve and so it is a constant; its value at b = 0 is 1.

Therefore the first, third and last terms of the expression above that we need to compute cancel thanks to (44) and we get:

< (a + b) + (a − b) − 2(a), (a) − (0) >^{2}

Notice that (a+b)+(a−b)−2(a) is the divisor of the function Δ^{−1/6}(P(ξ− a)−P(b)). Its value at the point 0 is Δ^{−1/6}(P(a)−P(b)) and its generalized value at the point a is the trivialization we have chosen.

Corollary 4.5 Assume k = k̄. Then the homomorphism (7) is surjective.

Let us denote by R_{2}(E) the kernel of the homomorphism (7). Then

B_{2}(E) := \frac{ℤ[E(k)\0]}{R_{2}(E)}

Let R̃ be the subgroup of [E(k)\0] generated by the elements

{a, b} := {a + b} + {a − b} − 2{a} − 2{b}

Notice that {a, a} = {2a} − 4{a} and {a, a} − {a, −a} = 2({a} − {−a}).

Lemma 4.6 For any abelian group A the elements {a, b} and {a} − {−a} generate modulo 2-torsion the kernel of the surjective homomorphism ℤ[A] ⟶ S^{2}A {a} ⟼ a · a.

We will not use this fact later, so a (simple) proof is omitted.

Consider the homomorphism R̃ ⟶ k^{∗} defined by the formulas

{a, b} ⟼ Δ^{−1/3}(P(a)−P(b))^{2}, a ≠ b; {a, a} ⟼ Δ^{−1/2}(P′(a))^{2}, 2a ≠ 0

and {a, a} ⟼ (Δ^{−1/3}P′′(a))^{2} if 2a = 0. Thanks to corollary (4.5) this homomorphism is well defined. By definition the subgroup R_{2}(E) is its kernel.

Remark. This is a homomorphism to the multiplicative group k^{∗} of the field k defined via the additive structure of k.

6. A remark on the differential in the complex B^{∗}(E, 3). The restriction of δ_{3} to the subgroup B_{3}^{∗}(E) can be defined directly, without referring to the group B_{2}(E) and the homomorphism h. A more complicated formula is the price we pay.

Set for general a_{i} ∈ E(k)

δ_{3}(({a_{1}} − {0}) ∗ ({a_{2}} − {0}) ∗ ({a_{3}} − {0}) ∗ ({a_{4}} − {0}))= (45)

\frac{[P(a_{1} + a_{2}) − P(a_{3} − a_{4})][P(a_{1} + a_{3}) − P(a_{4})][P(a_{1} + a_{4}) − P(a_{3})]}{[P(a_{1}) − P(a_{3} − a_{4})][P(a_{1} + a_{2} + a_{3}) − P(a_{4})][P(a_{1} + a_{2} + a_{4}) − P(a_{3})]}⊗−1/2·a_{1}+...

where ... means three other terms obtained by cyclic permutation of indices. Here P(a) := x(a) is the x-coordinate of a point a.

The expression for (45) is symmetric in a_{1}, ..., a_{4}, which is not obvious from the formula. Over ℂ one can rewrite the right hand side of (45) in a more symmetric way using the θ-function:

\frac{θ(a_{1} + a_{2} + a_{3} + a_{4})θ(a_{1} + a_{2})θ(a_{1} + a_{3})θ(a_{1} + a_{4})}{θ(a_{1} + a_{2} + a_{3})θ(a_{1} + a_{2} + a_{4})θ(a_{1} + a_{3} + a_{4})θ(a_{1})} ⊗ a_{1} + ...

Morally the differential δ_{3} is given by the “formula” \{a\} ⟼ −\frac{1}{2}θ(a) ⊗ a which, unfortunately, makes no sence if we don’t use the group B_{2}(E). The relation with (45) is given by the classical formula

P(a) − P(b) = \frac{θ(a + b)θ(a − b)}{θ^{2}(a)θ^{2}(b)}, a ≠ ±b

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A.G.: Max-Planck-Institute fur mathematik, Bonn, 53115, Germany; sasha@mpim-bonn.mpg.de

Dept. of Mathematics, Brown Univercity, Providence, RI 02912 USA. A.L.: International Institute for Nonlinear Studies at Landau Institute

Vorobjovskoe sh. 2 Moscow 117940, Russia; andrl@landau.ac.ru