Kronecker double series and the dilogarithm

Andrey Levin alevin@mpim-bonn.mpg.de

June 29, 2000

Abstract

In this article we give an explicit expression for the value of a certain Kronecker double series at any point of complex multiplication as a sum of dilogarithms whose arguments are values of some modular unit of higher level at the corresponding points. This result can be interpreted in the spirit of the Zagier conjecture. The special value of the Kronecker double series is equal to the value of the partial ζ-function of an ideal class for an order in an imaginary quadratic field. The values of the above mentioned modular unit belong to the ray class field corresponding to this order. So we get an explicit formula for the value of a partial ζ-function at s = 2 as a combination of dilogarithms of algebraic numbers.

1 Introduction.

We start by fixing notations. 1.1. Modular part.

1.1.1. First, we recall some standard facts about modular curves. We

set H = \{τ ∈ ℂ|ℑ(τ) > 0\}, M = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Mat_{2}(ℤ) \ \{0\} acts by

M(τ) = \frac{aτ+b}{cτ+d}. Denote by L = L_{τ} the lattice in ℂ generated by τ and 1, and denote by E = E_{τ} the corresponding elliptic curve ℂ/L_{τ}. The matrix M defines a map of lattices: M^{τ}: L_{M(τ)} → L_{τ}; w → (cτ + d)w and an isogeny M_{τ}: E_{τ} → E_{M(τ)}; ξ → (ad − bc)(cτ + d)^{−1}ξ. If τ is a fixed point of M, then M_{τ} is a map of the curve E_{τ} onto itself. In this case we will omit the subscript τ in our notation.

A point ξ of E_{τ} defines a character χ_{ξ} of the lattice L_{τ}: χ_{ξ}(w) = exp (\frac{2πi(w\overline{ξ}−\overline{w}ξ)}{τ−\overline{τ}}) . This pairing is Mat_{2}(ℤ)-invariant: χ_{M(ξ)}(w) = χ_{ξ}(M(w)).

1.1.2. Definition. The second Kronecker double series K_{2}(ξ; τ) is the C^{∞}-function on ℂ × H defined by the convergent series

K_{2}(ξ; τ) = (\frac{τ − \overline{τ}}{2πi} )^{2} ∑_{w∈L}′ \frac{χ_{ξ}(w)}{|w|^{4}}.  (1)

It is easy to check that this function is invariant with respect to the abovedefined action of SL_{2}(ℤ) on ℂ × H.

1.1.3. The Weierstrass ℘-function is defined by the convergent series

℘(ξ; τ) = \frac{1}{ξ^{2}} + ∑_{w∈L_{τ}}′ (\frac{1}{(w + ξ)^{2}} − \frac{1}{w^{2}} ) .

This function is elliptic. 1.2. Dilogarithms.

1.2.1. Definition. The Euler dilogarithm Li_{2}(z) is the multi-valued analytic function on ℙ^{1} \ {0, 1, ∞} defined as the analytic continuation of the series ∑_{j≥1} \frac{z^{j}}{j^{2}} (this series converges if |z| < 1).

1.2.2. Definition. The Bloch-Wigner dilogarithm D_{2}(z) is the (singlevalued) continuous function on ℙ^{1}, smooth on ℙ^{1} \ {0, 1, ∞}, defined by the formula D_{2}(z) = ℑ(Li_{2}(z)) + arg(1 − z) · log |z| for z ≠ 0, 1, ∞, D_{2}(0) = D_{2}(1) = D(∞) = 0.

We can extend the function D_{2} linearly to a function on the ℚ-vector space ℚ[ℂ].

1.2.3. Define a map δ from ℚ[ℂ \ {0, 1}] to ⋀^{2} ℂ^{∗} by the formula: [x] → x ∧ (1 − x). 1.3. Results.

1.3.1. Main Theorem. Let τ be a fixed point of M = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ≠ 0, 1.

Set m = det(M), n = det(M − 1). Then

−(m + 1)(n + 1)\frac{c(τ − \overline{τ})}{i}K_{2}(0; τ)

= 4mnD_{2} (\frac{ad − bc}{cτ + d}) + ∑_{\begin{matrix}α∈Ker(M)\0 \\ β∈Ker(M−1)\0 \end{matrix}} ∑_{k=1}^{m} ∑_{l=1}^{n} D_{2} (\frac{℘(kα + lβ) − ℘(α)}{℘(β) − ℘(α)});

and the argument of the dilogarithm belongs to the kernel of the map δ:

δ (4mn × [cτ + d] + ∑_{α,β,k,l} [\frac{℘(kα + lβ) − ℘(α)}{℘(β) − ℘(α)}]) = 0 ∈ ∧^{2}(ℂ^{∗} ⊗_{ℤ} ℚ).

1.3.2. Our proof is based on introducing some new function L_{1,1}, which we call the elliptic (1, 1)-logarithm, which will be defined in Section 2. The Main Theorem follows from Theorems A and B below.

1.3.3. Theorem A. Let τ, M, m , n be as in the Theorem 1.3.1. Then

\frac{c(τ − \overline{τ})}{4i} \frac{(m + 1)(n + 1)}{mn}K_{2}(0; τ) = −D_{2} (\frac{ad − bc}{cτ + d})+ ∑_{\begin{matrix}α∈KerM\0 \\ β∈Ker(M−1)\0 \end{matrix}} L_{1,1}(α, β, 0).

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1.3.4. Theorem B. Let α and β be two nontrivial different torsion points on an elliptic curve E_{τ}, say mα = nβ = 0 for some m, n ∈ ℕ. Then

∑_{k=1}^{m} ∑_{l=1}^{n} D_{2} (\frac{℘(kα + lβ) − ℘(α)}{℘(β) − ℘(α)}) = −2mn(L_{1,1}(α, β, 0) + L_{1,1}(−α, β, 0)).

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1.4. General remarks, structure of the article and acknowledgement.

1.4.1. The fact that the value of the Kronecker double series K_{2} at a CM-point can be expressed as a combination of dilogarithms is not new. It can be derived from a result of C. Deninger in the following way. Any CMpoint τ defines the ring R of endomorphisms of the corresponding lattice L_{τ}; this ring is an order in some imaginary quadratic extension F (= R ⊗_{ℤ} ℚ) of ℚ. Extend the field F by the value of the j-invariant at the point τ. Deninger constructed [D1] an element in the third algebraic K-group of the field F (j(τ)) such that the regulator maps this element to the value of the K_{2} at this CM-point. Due to Suslin and Bloch, we know that the regulator on K_{3} of a number field is given by the Bloch-Wigner dilogarithm. So we can conclude that the value of the K_{2} equals a combination of the values of the dilogarithm at numbers in F (j(τ)). The arguments of the dilogarithm in our formula belong to some extension of the field F (j(τ)), but the collection of them is Galois invariant.

Our proof is in some sense parallel to the Deninger construction.

1.4.2. We can also interpret the Main Theorem independently of algebraic K-theory. The value of K_{2}(0; τ) for a CM-point τ is (a rational multiple of) the value at s = 2 of the partial zeta function ζ_{F,A}(s) = ∑_{𝔞∈A} N(𝔞)^{−s} for some ideal class A. The fact that this value can be written as a combination of dilogarithms with arguments belonging to the associated class field over F is a special case of the refined version of Zagier’s conjecture according to which the values of all partial zeta functions at an arbitrary integer argument s = m can be expressed in terms of m-logarithms.

For the case at hand, the same result, of course, also follows from the theorems of Deninger (existence of elements in K_{2} with required values of regulator) and Bloch-Suslin (structure of K_{2} and description of regulator from K_{2}). Our proof, as well as giving an explicit formula, also has the advantage of avoiding algebraic K-theory.

It is possible in principle that this method could be applied for higher values of m where it is not known that the regulator may be expressed in terms of polylogarithms.

1.4.3. Theorem A reflects a general phenomenon. Another reflection of this phenomenon is the following fact. For any elliptic modular curve over ℚ the value of its L-function at s = 2 can be expressed as a combination of the values of a special function (the elliptic (1, 2)-logarithm) [G2], which is a “relative” of our elliptic (1, 1)-logarithm. On the other hand, for a CMcurve the value of the L-function is equal to a combination of the values of a certain Kronecker double series [D1]. Therefore this Kronecker double series must be equal to a combination of the values of the elliptic (1, 2)-logarithm.

1.4.4. The paper is organized as follows. In the second section we define the elliptic (1, 1)-logarithm for an arbitrary elliptic curve and study its elementary properties. In the third section we realize the Kronecker series as an integral over the square of the elliptic curve, and reduce this integral to an integral over the elliptic curve itself if this curve has complex multiplication. In the fourth section we compare the elliptic (1, 1)-logarithm and the dilogarithm. In the fifth section we check the vanishing of the map δ for the arguments of dilogarithms on the right hand side of the Main Theorem, thus completing its proof. In the last section we prove a more general formula, relating values of K_{2} at torsion points to the dilogarithm.

1.4.5. Acknowledgment. I wish to thank Sasha Goncharov, who explained me his ideas about Chow polylogarithms. I am grateful to Don Zagier and Herbert Gangl for very stimulating discussions and computer experiments at a crucial moment for this work during my stay at the Max-Planck-Institut für Mathematik, Bonn in 1997. I wish to thank the MPI for hospitality.

2 Elliptic (1, 1)-logarithm.

In this section we define and study properties of the elliptic (1, 1)-logarithm. Some motivation for considering this function is Goncharov’s integral representation of the Bloch-Wigner dilogarithm. This representation uses the special case of a very general differential operation which we call A_{∗}. 2.1. The operations A_{n}.

Let ϕ_{1},⋯,ϕ_{n} be smooth functions on a complex variety X.

2.1.1. Definition.

A_{n}(ϕ_{1}, ⋯, ϕ_{n}) = Alt_{n} (∑_{j=0}^{n−1}(−1)^{j}ϕ_{1}∂ ϕ_{2} ∧ ⋯ ∧ ∂ ϕ_{n−j} ∧ \overline{∂} ϕ_{n−j+1} ∧ ⋯ ∧ \overline{∂} ϕ_{n}) ,

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where Alt_{n} denotes the alternation with respect to the action of the symmetric group S_{n} with a factor (n!)^{−1} :

Alt_{n}(F (x_{1}, ⋯, x_{n}) = \frac{1}{n!} ∑_{σ∈S_{n}} sign(σ)F (x_{σ(1)}, ⋯, x_{σ(n)}).

2.1.2. Remark. If ϕ_{j} = log |f_{j}|^{2} for analytic functions f_{j}, then A_{n} is (up to a factor 2 for odd n and 2i for even n) the so-called Beilinson-Deligne product of the f_{j}’s.

2.1.3. Remark. The (p, q)-component of A_{n} equals (−1)^{q} \frac{p!q!}{(p+q)!} times the (p, q)-component of Alt_{n}(ϕ_{1}d ϕ_{2} ∧ ⋯ ∧ d ϕ_{n}).

An important property of the operations A_{n} is the following:

2.1.4. Proposition. For n > 1

\begin{aligned}d A_{n}(ϕ_{1}, ⋯, ϕ_{n}) & = & ∂ ϕ_{1} ∧ ∂ ϕ_{2} ∧ ⋯ ∧ ∂ ϕ_{n} + (−1)^{n−1}\overline{∂} ϕ_{1} ∧ \overline{∂} ϕ_{2} ∧ ⋯ ∧ \overline{∂} ϕ_{n} \\ & & + ∑_{j=1}^{n}(−1)^{j}\overline{∂}∂ ϕ_{j} ∧ A_{n−1}(ϕ_{1}, ⋯, ϕ̂_{j}, ⋯, ϕ_{n}). \end{aligned}

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The proof is a straightforward computation. 2.2. Goncharov’s integral representation of the Bloch-Wigner dilogarithm.

2.2.1. Lemma. [G1] The value of the Bloch-Wigner dilogarithm at a point a ≠ 0, 1, ∞ is equal to the following convergent integral:

\frac{1}{4π} ∫_{ℙ^{1}} A_{3}(log |z|^{2}, log |1 − z|^{2}, log |a − z|^{2}).

Proof. As we integrate over a curve, only the (1, 1)-component gives a contribution. So we can replace A_{3} by

−\frac{1}{6} (log |z|^{2}d log |1 − z|^{2} ∧ d log |a − z|^{2} − log |1 − z|^{2}d log |z|^{2} ∧ d log |a − z|^{2}

+ log |a − z|^{2}d log |z|^{2} ∧ d log |1 − z|^{2}) .

First, we will prove that the integral converges. The integrand is smooth outside the points 0, 1, a and ∞. Let (r, ϕ) be a polar coordinate system near one of the first three points. Then any term of the integrand has an asymptotic either log |r|rd r ∧ d ϕ, or r^{−1}rd r ∧ d ϕ and is integrable. As A_{3} is trilinear and totally antisymmetric, we can replace the collection of arguments (log |z|^{2}, log |1−z|^{2}, log |a−z|^{2}) by the collection (log |z|^{2}, log |1−z|^{2}− log |z|^{2}, log |a−z|^{2}−log |z|^{2}) = (log |z|^{2}, log |1−z^{−1}|^{2}, log |az^{−1}−1|^{2}); for this collection the convergence at ∞ can be checked by the same consideration as above.

By Stokes’s formula we reduce the integral to

−\frac{1}{16π} ∫ (log |z|^{2}d log |1 − z|^{2} ∧ d log |a − z|^{2} − log |1 − z|^{2}d log |z|^{2} ∧ d log |a − z|^{2}) .

Evidently, the (1, 1)-component of d ϕ_{1} ∧ d ϕ_{2} is the negative of the (1, 1)- component of (∂ − \overline{∂})ϕ_{1} ∧ (∂ − \overline{∂})ϕ_{2}. So the integral is equal to

\frac{1}{16π} ∫ (log |z|^{2}(∂ − \overline{∂}) log |1 − z|^{2} − log |1 − z|^{2}(∂ − \overline{∂}) log |z|^{2})∧(∂−\overline{∂}) log |a−z|^{2}.

An easy calculation shows that

d D_{2}(z) = −i (log |z|(∂ − \overline{∂}) log |1 − z| − log |1 − z|(∂ − \overline{∂}) log |z|) .

So, the integral is equal to

\frac{i}{4π} ∫ d D_{2}(z) ∧ (∂ − \overline{∂}) log |a − z|^{2} = \frac{1}{4πi} ∫ D_{2}(z)d (∂ − \overline{∂}) log |a − z|^{2}.

As d (∂ −\overline{∂}) log |a−z|^{2} = 2\overline{∂}∂ log |a−z|^{2} = 4πi(δ_{a}(z)−δ_{∞}(z)) and D_{2}(∞) = 0, the proof is finished.

2.2.2. Remark. As follows from the proof, for any compact curve X and any two meromorphic functions f and g on it we have the integral representation:

∑_{p∈X} ord_{p}(g)D_{2}(f(p)) = \frac{1}{4π} ∫_{X} A_{3}(log |f|^{2}, log |1 − f|^{2}, log |g|^{2}). (6)

2.2.3. Remark. For any curve C

∫_{C} A_{3}(ϕ_{1}, ϕ_{2}, ϕ_{3}) = ∫_{C} \frac{1}{2}ϕ_{1}d ϕ_{2} ∧ d ϕ_{3},

so this integral is equal to zero if ϕ_{j} = const for some j.

2.2.4. Lemma. For any three different points a, b, c ∈ ℂ, we have

∫_{ℙ^{1}} A_{3}(log |z − a|^{2}, log |z − b|^{2}, log |z − c|^{2}) = 4πD_{2} (\frac{c − a}{b − a}) . (7)

Proof. Denote the ratio \frac{c−a}{b−a} by r. Make the change of variable x = \frac{z−a}{b−a}. Then log |z −a|^{2} = log |x|^{2} +log |b−a|^{2}, log |z −b|^{2} = log |x−1|^{2} +log |b−a|^{2} and log |z−c|^{2} = log |x−r|^{2}+log |b−a|^{2}. According to the previous Remark, terms including the constant log |b − a|^{2} vanish. So, we reduce the integral to the integral from the previous Lemma.

2.2.5. Remark. The integrand in (7) formally vanishes if any of a, b and c are the same by antisymmetry (we say formally because we are not really allowed to multiply the 1-forms if their singularities coincide). If only two of a, b, c are equal, then the r.h.s. of (7) is also zero, since D_{2}(0) = D_{2}(1) = D_{2}(∞) = 0. Along the triple diagonal the expression in (7) has a discontinuity. However, it is continuous on the blow-up of ℂ^{3} along the triple diagonal. 2.3. The elliptic (1, 1)-logarithm.

We will define a real-valued function L_{1,1}(α, β, γ; τ) on the third power of an elliptic curve (more precisely, on the fibered third power of the universal elliptic curve over the modular curve), invariant with respect to the diagonal action of the elliptic curve by shifts and antisymmetric with respect to permutations of the variables. This function is called the elliptic (1, 1)-logarithm.

2.3.1. A natural generalization of the function log |x − a|^{2} for an elliptic curve E_{τ} is the Kronecker double series

L_{1}(ξ; τ) := log |θ̃(ξ, τ)|^{2} + \frac{1}{2} \frac{2πi}{τ − \overline{τ}} (ξ − \overline{ξ})^{2} = −\frac{τ − \overline{τ}}{2πi} ∑_{w∈L}′_{e} \frac{χ_{ξ}(w)}{|w|^{2}}

where ∑_{e} denotes Eisenstein summation; see [W],

θ̃(ξ, τ) = \frac{θ(ξ, τ)}{η(τ)} = q^{1/12}(z^{\frac{1}{2}} − z^{− \frac{1}{2}} ) ∏_{j=1}^{∞}(1 − q^{j}z)(1 − q^{j}z^{−1}).

q = exp(2πiτ) (more precisely, q^{\frac{1}{12}} = exp(\frac{1}{6}πiτ)), and z = exp(2πiξ) (more precisely, z^{± \frac{1}{2}} = exp(±πiξ)). The notation L_{1} for this function is not standard. It is meant to emphasize that this function is the elliptic 1-logarithm.

2.3.2. The function L_{1} is the Green function for the operator \overline{∂}∂ on an elliptic curve

\overline{∂}∂(L_{1}(ξ; τ)) = 2πiδ_{0} + \frac{2πi}{τ − \overline{τ}} d ξ ∧ d \overline{ξ}.

Here δ_{0} denotes the delta function at zero.

2.3.3. The function L_{1} satisfies the distribution relation

∑_{α:M_{τ} (α)=β} L_{1}(α; τ) = L_{1}(β; M(τ)).

2.3.4. We have the “theta decomposition” of any elliptic function f (ord_{β}(f) = 0)

log |f(ξ)|^{2} − log |f(β)|^{2} = ∑ ord_{α}(f) (L_{1}(ξ − α) − L_{1}(β − α)) . (8)

So the natural elliptic generalization of the function D_{2} (\frac{c−a}{b−a}) is the following function

2.3.5. Definition. The elliptic (1, 1)-logarithm L_{1,1}(α, β, γ; τ) is the following convergent integral:

\frac{1}{4π} ∫_{E_{τ}} A_{3} (L_{1}(ξ − α), L_{1}(ξ − β), L_{1}(ξ − γ)) .

The convergence of the integral can be checked by the same consideration as for D_{2}(a).

2.3.6. Remark. As above, only the (1, 1)-component of the integrand gives a nontrivial contribution, so

L_{1,1}(α, β, γ; τ) = −\frac{1}{8π} ∫_{E_{τ}} L_{1}(ξ − α)d L_{1}(ξ − β) ∧ d L_{1}(ξ − γ).

We can give another definition of this function based on the Fourier expansion.

2.3.7. Lemma. The function L_{1,1}(α, β, γ; τ) considered as a distribution is equal to the following series

\frac{(τ − \overline{τ})^{2}}{16π^{2}i} × ∑_{\begin{matrix}w_{1}+w_{2}+w_{3}=0 \\ w_{i}≠0 \end{matrix}} \frac{χ_{α}(w_{1})χ_{β}(w_{2})χ_{γ}(w_{3})(w_{2}\overline{w}_{3} − \overline{w}_{2}w_{3})}{|w_{1}|^{2}|w_{2}|^{2}|w_{3}|^{2}}.

A series of this kind was introduced by Deninger [D2]. In contrast to his case, however, our series is not absolutely convergent.

Proof. According to the previous Remark we can compute the following integral

−\frac{1}{8π} ∫ (−\frac{τ − \overline{τ}}{2πi} ∑_{w_{1}≠0} \frac{χ_{ξ−α}(w_{1})}{|w_{1}|^{2}} ) ( ∑_{w_{2}≠0} \frac{χ_{ξ−β}(w_{2})(\overline{w}_{2}d ξ − w_{2}d \overline{ξ})}{|w_{2}|^{2}} )

∧ ( ∑_{w_{3}≠0} \frac{χ_{ξ−γ}(w_{3})(\overline{w}_{3}d ξ − w_{3}d \overline{ξ})}{|w_{3}|^{2}} )

= −\frac{1}{8π} \frac{τ − \overline{τ}}{2πi} ∫ ∑_{w_{i}≠0} \frac{χ_{ξ−α}(w_{1})χ_{ξ−β}(w_{2})χ_{ξ−γ}(w_{3})(w_{2}\overline{w}_{3} − \overline{w}_{2}w_{3})}{|w_{1}|^{2}|w_{2}|^{2}|w_{3}|^{2}}d ξ ∧ d \overline{ξ}.

The integrals of the terms with w_{1} + w_{2} + w_{3} ≠ 0 vanish, as integrals of nontrivial harmonics over a torus, so we get

−\frac{1}{8π} \frac{τ − \overline{τ}}{2πi} ∑_{\begin{matrix}w_{1}+w_{2}+w_{3}=0 \\ w_{i}≠0 \end{matrix}} \frac{χ_{α}(w_{1})χ_{β}(w_{2})χ_{γ}(w_{3})(w_{2}\overline{w}_{3} − \overline{w}_{2}w_{3})}{|w_{1}|^{2}|w_{2}|^{2}|w_{3}|^{2}} ∫ d ξ ∧ d \overline{ξ}

= −\frac{1}{8π} \frac{τ − \overline{τ}}{2πi} ∑_{\begin{matrix}w_{1}+w_{2}+w_{3}=0 \\ w_{i}≠0 \end{matrix}} \frac{χ_{α}(w_{1})χ_{β}(w_{2})χ_{γ}(w_{3})(w_{2}\overline{w}_{3} − \overline{w}_{2}w_{3})}{|w_{1}|^{2}|w_{2}|^{2}|w_{3}|^{2}}×(−(τ − \overline{τ})).

2.3.8. Remark. This Fourier series is antisymmetric with respect to the permutations of α, β and γ, since for w_{1} + w_{2} + w_{3} = 0

w_{2}\overline{w}_{3} − \overline{w}_{2}w_{3} = \frac{1}{3} | \begin{matrix}1 & 1 & 1 \\ w_{1} & w_{2} & w_{3} \\ \overline{w}_{1} & \overline{w}_{2} & \overline{w}_{3} \end{matrix} | .

2.3.9. The function L_{1,1} is smooth outside the diagonals; this follows from the general formula for differentiation of an integral with respect to a parameter. On the diagonals the function L_{1,1} is equal to zero, as A_{3} is antisymmetric.

2.3.10. Lemma. The function L_{1,1} is continuous on the complement of the triple diagonal α = β = γ.

Proof. We will prove that for γ ≠ 0

lim_{t→0} L_{1,1}(tα, tβ, γ; τ) = 0.

Let us choose some rather small ϵ and represent the integral as the sum of the integral over the disk of radius ϵ around 0 and the integral over the complement of this disk inside the elliptic curve. The second integral tends to zero, as the integral of any term of A_{3} converges and A_{3} is antisymmetric. Inside the disk, L_{1}(ξ − tα) equals log |ξ − tα|^{2} + ϕ(ξ, tα), where ϕ(ξ, tα), is a smooth function, the same is true for L_{1}(ξ −tβ). Substitute this decomposition into A_{3}; we get several types of summands: 1) all arguments of A_{3} are smooth, 2) one argument is singular and 3) two arguments are singular. In the first two cases the integral tends to zero by the same reason as above. In order to estimate the last summand, perform the change of variable z = t^{−1}ξ

∫_{|ξ|<ϵ} A_{3}(log |ξ − tα|^{2}, log |ξ − tβ|^{2}, L_{1}(ξ − γ))

= ∫_{|ξ|<t^{−1}ϵ} A_{3}(log |z − α|^{2}, log |z − tβ|^{2}, L_{1}(tz − γ) − L_{1}(−γ))

+ ∫_{|ξ|<t^{−1}ϵ} A_{3}(log |z − α|^{2}, log |z − tβ|^{2}, L_{1}(−γ)),

the second integral tends to zero as L_{1}(−γ) is a constant.

For rather small t the first integral can be represented as a sum of the integral over the disk of radius \sqrt{t^{−1}ϵ} and the integral over the annulus \sqrt{t^{−1}ϵ} < |z| < t^{−1}ϵ. In the second integral replace log |z − β|^{2} by log |z − β|^{2} − log |z − α|^{2}. The integral over the disk is small since L_{1}(tz − γ) and also its derivatives are small; the integral over the annulus is small as | log |z − β|^{2} − log |z − α|^{2}| and also its derivatives are small.

2.3.11. Lemma. Let α, β and γ be points on an elliptic curve such that not all three of them coincide. Then the limit of L_{1,1}(tα, tβ, tγ; τ) at t → 0 is equal to D_{2} (\frac{γ−α}{β−α}).

Proof. Let us fix some rather small ϵ and represent the integral as the sum of the integral over the disk of radius ϵ around 0 and the integral over the complement of this disk inside the elliptic curve. The second integral tends to zero, as A_{3} is antisymmetric. Inside the disk, L_{1}(ξ − tα) equals log |ξ − tα|^{2} + ϕ(ξ, tα), where ϕ(ξ, tα) is a smooth function, and the same is true for L_{1}(ξ − tβ) and for L_{1}(ξ − tγ). As in the proof of the previous Lemma, only the summand A_{3}(log |ξ − tα|^{2}, log |ξ − tβ|^{2}, log |ξ − tγ|^{2}) gives a nontrivial contribution in the limit. Hence

lim_{t→0} L_{1,1}(tα, tβ, tγ; τ) = \frac{1}{4π} lim_{t→0} ∫_{|ξ|<ϵ} A_{3}(log |ξ−tα|^{2}, log |ξ−tβ|^{2}, log |ξ−tγ|^{2})

= \frac{1}{4π} lim_{t→0} ∫_{|ξ|<t^{−1}ϵ} A_{3}(log |t^{−1}ξ − α|^{2}, log |t^{−1}ξ − β|^{2}, log |t^{−1}ξ − γ|^{2})

= \frac{1}{4π} ∫_{ℂ} A_{3}(log |z − α|^{2}, log |z − β|^{2}, log |z − γ|^{2}) = D_{2} (\frac{γ − α}{β − α}) .

2.3.12. Remark. Evidently, L_{1,1}(α, β, γ; τ) is invariant with respect to “diagonal” shift of the arguments and to changing the sign of the arguments, so we get that the elliptic (1, 1)-logarithm is a function on the moduli space \overline{M}_{1,3} of curves of genus 1 with three marked points.

3 From the Kronecker series to the elliptic (1, 1)- logarithm.

The reduction of the Kronecker series K_{2} to the elliptic (1, 1)-logarithm for an elliptic curve with complex multiplication splits up into two steps.

First, we represent K_{2}(ξ; τ) as an integral over the square of the elliptic curve. This representation is valid for any elliptic curve and for any point ξ on it.

Second, we reduce the integral over the square of the elliptic curve to an integral over the elliptic curve itself. This is possible for a curve with complex multiplication and uses the existence of an extra projection of the square of the curve onto itself . 3.1. An integral representation of the Kronecker series.

We start from the simplest example which illustrates the main idea of this representation.

3.1.1. Proposition.

K_{2}(α; τ) = \frac{τ − \overline{τ}}{2πi} ∫_{E_{τ} ×E_{τ}} L_{1}(α−η_{1}−η_{2})\frac{∂ L_{1}(η_{1})}{∂ η_{1}} \frac{∂L_{1}(η_{2})}{∂ \overline{η}_{2}} \frac{d η_{1} ∧ d \overline{η}_{1}}{τ − \overline{τ}} ∧\frac{d η_{2} ∧ d \overline{η}_{2}}{τ − \overline{τ}} .

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Proof. Let us use the Fourier expansions of L_{1} and its derivatives

\frac{τ − \overline{τ}}{2πi} ∫ (−\frac{τ − \overline{τ}}{2πi} ∑_{w_{1}∈L}′_{e} \frac{χ_{α−η_{1}−η_{2}}(w_{1})}{|w_{1}|^{2}} ) ( ∑_{w_{2}∈L}′_{e} \frac{χ_{η_{1}}(w_{2})}{w_{2}} ) (− ∑_{w_{3}∈L}′_{e} \frac{χ_{η_{2}}(w_{3})}{\overline{w}_{3}} )

×\frac{d η_{1} ∧ d \overline{η_{1}}}{τ − \overline{τ}} ∧ \frac{d η_{2} ∧ d \overline{η_{2}}}{τ − \overline{τ}} .

The integral of the term with w_{1} ≠ w_{2}, w_{1} ≠ w_{3} vanishes, as an integral of a nontrivial harmonic over a torus, so we get the integral

\frac{τ − \overline{τ}}{2πi} ∫ (−\frac{τ − \overline{τ}}{2πi} ∑_{w_{1}∈L}′_{e} \frac{χ_{α}(w_{1})}{|w_{1}|^{2}} ) (\frac{1}{−w_{1}} ) (−\frac{1}{−\overline{w}_{1}} ) \frac{d η_{1} ∧ d \overline{η_{1}}}{τ − \overline{τ}} ∧\frac{d η_{2} ∧ d \overline{η_{2}}}{τ − \overline{τ}}

= (\frac{τ − \overline{τ}}{2πi} )^{2} ∑_{w_{1}∈L} \frac{χ_{α}(w_{1})}{|w_{1}|^{4}} (−1)^{2} = K_{2}(ξ; τ).

At the end of this subsection we will discuss a more symmetric representation of K_{2} for any τ and ξ; this result will not be used further.

Now we formulate the main result of this subsection.

3.1.2. Lemma. Let τ be a fixed point of M = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ≠ 0, 1;

m = detM, n = det(M − 1). Denote by η_{3} the expression η_{1} + M_{τ} (η_{2}) − α and by η_{4} the expression η_{1} + η_{2} − β. Then

\begin{aligned}∫_{E_{τ}^{2}} A_{3} (L_{1}(η_{1}), L_{1}(η_{2}), L_{1}(η_{3})) × (\frac{d η_{4} ∧ d \overline{η}_{4}}{τ − \overline{τ}} ) & = & \frac{πic(τ − \overline{τ})}{m}K_{2}(−α; τ); \\ ∫_{E_{τ}^{2}} A_{3} (L_{1}(η_{1}), L_{1}(η_{2}), L_{1}(η_{4})) × (\frac{d η_{3} ∧ d \overline{η}_{3}}{τ − \overline{τ}} ) & = & −πic(τ − \overline{τ})K_{2}(−β; τ); \\ ∫_{E_{τ}^{2}} A_{3} (L_{1}(η_{1}), L_{1}(η_{3}), L_{1}(η_{4})) × (\frac{d η_{2} ∧ d \overline{η}_{2}}{τ − \overline{τ}} ) & = & \frac{πic(τ − \overline{τ})}{mn}K_{2}(M_{τ} (β) − α; τ); \\ ∫_{E_{τ}^{2}} A_{3} (L_{1}(η_{2}), L_{1}(η_{3}), L_{1}(η_{4})) × (\frac{d η_{1} ∧ d \overline{η}_{1}}{τ − \overline{τ}} ) & = & −\frac{πic(τ − \overline{τ})}{n}K_{2}(β − α; τ). \end{aligned}

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Proof. We will prove the third equation; the other equations can be proved by the same consideration.

The last factor of the integrand has type (1, 1) and is closed, so we can replace the expression A_{3} (L_{1}(η_{1}), L_{1}(η_{3}), L_{1}(η_{4})) by \frac{1}{2}L_{1}(η_{3})d L_{1}(η_{1}) ∧ d L_{1}(η_{4}).

So we can calculate the integral

\frac{1}{2} ∫ (−\frac{τ − \overline{τ}}{2πi} ∑_{w_{3}∈L}′_{e} \frac{χ_{η_{1}+M_{τ} (η_{2})−α}(w_{3})}{|w_{3}|^{2}} )

× (( ∑_{w_{1}∈L}′_{e} \frac{χ_{η_{1}}(w_{1})d η_{1}}{w_{1}} ) ∧ (− ∑_{w_{4}∈L}′_{e} \frac{χ_{η_{1}+η_{2}−β}(w_{4})(d \overline{η}_{1} + d \overline{η}_{2})}{\overline{w}_{4}} )

+ ( ∑_{w_{1}∈L}′_{e} \frac{χ_{η_{1}}(w_{1})d \overline{η}_{1}}{\overline{w}_{1}} ) ∧ (− ∑_{w_{4}∈L}′_{e} \frac{χ_{η_{1}+η_{2}−β}(w_{4})(d η_{1} + d η_{2})}{w_{4}} ))∧\frac{d η_{2} ∧ d \overline{η_{2}}}{τ − \overline{τ}} .

As χ_{M_{τ}} (η_{2})(w) = χ_{η_{2}}(M(w)), only the terms with w_{3} + w_{1} + w_{4} = 0 and M(w_{3}) + w_{4} = 0 give nontrivial contributions in the integral. So we get

\frac{1}{2} \frac{(τ − \overline{τ})^{2}}{2πi} ∑_{w_{3}∈L}′ \frac{χ_{−α}(w_{3})}{|w_{3}|^{2}} × (\frac{1}{(M − 1)(w_{3})} × \frac{χ_{−β}(−M(w_{3}))}{−\overline{M(w_{3})}}

− \frac{1}{\overline{(M − 1)(w_{3})}} × \frac{χ_{−β}(−M(w_{3}))}{−M(w_{3})}) × ∫ \frac{d η_{1} ∧ d \overline{η_{1}}}{τ − \overline{τ}} ∧ \frac{d η_{2} ∧ d \overline{η_{2}}}{τ − \overline{τ}}

= −\frac{1}{2} \frac{(τ − \overline{τ})^{2}}{2πi} ∑_{w_{3}∈L}′ \frac{χ_{M(β)−α}(w_{3})}{|w_{3}|^{4}}×(\frac{1}{(cτ + d − 1)(c\overline{τ} + d)} − \frac{1}{(c\overline{τ} + d − 1)(cτ + d)})

= \frac{1}{2} \frac{(τ − \overline{τ})^{2}}{2πi} \frac{c(τ − \overline{τ})}{nm} ∑_{w_{3}∈L}′ \frac{χ_{M(β)−α}(w_{3})}{|w_{3}|^{4}} = \frac{πic(τ − \overline{τ})}{mn}K_{2}(M_{τ} (β) − α; τ).

We used above the following simple formula: for any auto-isogeny |cτ + d|^{2} = ad − bc = detM.

3.1.3. Remark. For general τ any auto-isogeny of E_{τ} is multiplication by some integer; and the r.h.s. vanishes. Hence the result is interesting only for elliptic curves with complex multiplication.

3.1.4. The function \frac{2πi}{τ−\overline{τ}} K_{2} can be treated as a component of the vector valued function (the elliptic trilogarithm):

L_{3}(ξ; τ) = \frac{τ − \overline{τ}}{2πi} ( \begin{matrix}∑_{w∈L} & ′ & \frac{χ_{ξ}(w)}{w^{3}\overline{w}} \\ ∑_{w∈L} & ′ & \frac{χ_{ξ}(w)}{w^{2}\overline{w}^{2}} \\ ∑_{w∈L} & ′ & \frac{χ_{ξ}(w)}{w\overline{w}^{3}} \end{matrix} ) ,

which takes values in the symmetric square S^{2}(H) of the homology group H of the elliptic curve E_{τ} with complex coefficients. This space is isomorphic to a direct summand of the second cohomology group of the square of the elliptic curve. A “natural” basis of this space is

f_{1} = \frac{d \overline{η}_{1} ∧ d \overline{η}_{2}}{(τ − \overline{τ})^{2}} ; f_{2} = \frac{d η_{1} ∧ d \overline{η}_{2} − d η_{2} ∧ d \overline{η}_{1}}{(τ − \overline{τ})^{2}} ; f_{3} = \frac{d η_{1} ∧ d η_{2}}{(τ − \overline{τ})^{2}} .

3.1.5. Proposition.

L_{3}(ξ; τ) = ∫_{E_{τ}^{2}} A_{3} (L_{1}(η_{1}), L_{1}(η_{2}), L_{1}(η_{1} + η_{2} − ξ)) ∧ ( \begin{matrix}f_{1} \\ f_{2} \\ f_{3} \end{matrix} ) .  (11)

The proof is a straightforward calculation. 3.2. Reduction to the elliptic (1, 1)-logarithm.

3.2.1. Let us consider the current Φ = \frac{1}{2π^{2}} A_{4} (L_{1}(η_{1}), L_{1}(η_{2}), L_{1}(η_{3}), L_{1}(η_{4})) on E_{τ}^{2}, where, as above, η_{3} = η_{1} + M_{τ} (η_{2}) − α and η_{4} = η_{1} + η_{2} − β for some

auto-isogeny M = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ≠ 0, 1. The wedge product of currents is not

defined, but for general α and β the divisors of the singularities of L_{1}’s are in general position and the wedge product is well defined; the formula for the differential of the wedge product is also true.

3.2.2. The differential d Φ of Φ is equal to

\frac{1}{2π^{2}} ×(−\overline{∂}∂L_{1}(η_{1}) ∧ A_{3} (L_{1}(η_{2}), L_{1}(η_{3}), L_{1}(η_{4})) + \overline{∂}∂L_{1}(η_{2}) ∧ A_{3} (L_{1}(η_{1}), L_{1}(η_{3}), L_{1}(η_{4}))

−\overline{∂}∂L_{1}(η_{3}) ∧ A_{3} (L_{1}(η_{1}), L_{1}(η_{2}), L_{1}(η_{4})) + \overline{∂}∂L_{1}(η_{4}) ∧ A_{3} (L_{1}(η_{1}), L_{1}(η_{2}), L_{1}(η_{3}))) ,

as the (4, 0)-part and the (0, 4)-part vanish on a surface. We split the \overline{∂}∂L_{1}’s into a δ-part and a smooth part. Integrals with δ-functions are integrals over the elliptic curve and are equal to sums of values of the elliptic (1, 1)- logarithm. Integrals of smooth parts of ∂\overline{∂}L_{1}’s are calculated in the previous Lemma.

3.2.3. Let us call by B_{1}, ⋯, B_{4} the δ-parts of the four components of d Φ and by B′_{1}, ⋯, B′_{4} the smooth ones. Let us first calculate the integrals with δ-functions.

∫ B_{1} = −\frac{2πi}{2π^{2}} ∫_{E_{τ}} A_{3} (L_{1}(η_{2}), L_{1}(η_{3}), L_{1}(η_{4}))|_{η_{1}=0}

= \frac{1}{πi} ∫_{E_{τ}} A_{3} (L_{1}(η_{2}), L_{1}(M_{τ} η_{2} − α), L_{1}(η_{2} − β))

= \frac{1}{πi} ∫_{E_{τ}} ∑_{α′:M_{τ} (α′)=α} A_{3} (L_{1}(η_{2}), L_{1}(η_{2} − α′), L_{1}(η_{2} − β)) = −4i ∑_{α′:M_{τ} (α′)=α} L_{1,1}(0, α′, β).

The same consideration shows that

B_{2} = 4iL_{1,1}(0, α, β), B_{3} = −4i ∑_{\begin{matrix}α′:M_{τ} (α′)=α \\ γ′:(M_{τ} −1)(γ′)=α−β \end{matrix}} L_{1,1}(α′, 0, γ′),

B_{4} = 4i ∑_{γ′:(M_{τ} −1)(γ′)=α−β} L_{1,1}(0, β, γ′).

3.2.4. As the smooth part of \overline{∂}∂L_{1}(ξ; τ) equals \frac{2πi}{τ − \overline{τ}} d ξ ∧ d \overline{ξ}, the inte-

grals of B′_{j} were already calculated in (10).

3.2.5. As the integral of the differential of a current over a compact variety equals zero, we get ∑ B_{i} = − ∑ B′_{i}, so we have proved the following result:

3.2.6. Proposition. Let τ be a fixed point of M ≠ 0, 1;m = detM, n = det(M − 1) . Let α ≠ 0 and let β ≠ 0, α, M_{τ} (α). Then

\frac{c(τ − \overline{τ})}{4i} (\frac{1}{m}K_{2}(−α; τ) + K_{2}(−β; τ) + \frac{1}{mn}K_{2}(M_{τ} (β) − α; τ) + \frac{1}{n}K_{2}(β − α; τ))

is equal to

− ∑_{α′∈M^{−1}(α)}L_{1,1}(0, α′, β) + L_{1,1}(0, α, β) − ∑_{\begin{matrix}α′∈M^{−1}(α) \\ γ′∈(M−1)^{−1}(α−β) \end{matrix}}L_{1,1}(α′, 0, γ′)

+ ∑_{γ′∈(M−1)^{−1}(α−β)}L_{1,1}(0, β, γ′).

As the function K_{2} is continuous, we can “degenerate” this formula:

3.2.7. Theorem. Let τ be a fixed point of M ≠ 0, 1; m and n as above. Then

\frac{c(τ − \overline{τ})}{4i} (\frac{m + n + 1}{mn}K_{2}(−α; τ) + K_{2}(0; τ)) = − ∑_{\begin{matrix}α′∈M^{−1}(α) \\ γ′∈(M−1)^{−1}(α) \end{matrix}}L_{1,1}(α′, 0, γ′), α ≠ 0;

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\frac{c(τ − \overline{τ})}{4i} \frac{(m + 1)(n + 1)}{mn}K_{2}(0; τ) = −D_{2} (\frac{ad − bc}{cτ + d})− ∑_{\begin{matrix}α′∈Ker(M)\0 \\ γ′∈Ker(M−1)\0 \end{matrix}}L_{1,1}(α′, 0, γ′).

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Proof. The first formula is the result of substituting β = 0; and the second one is the limit of the first as α → 0. This completes the proof of Theorem A.

4 From the elliptic (1, 1)-logarithm to the dilogarithm.

In this section we relate the elliptic (1, 1)-logarithm to the dilogarithm. Specifically, we express the combination L_{1,1}(0, α, β; τ) + L_{1,1}(0, α, −β; τ) as a sum of dilogarithms for any torsion points α and β on any elliptic curve E_{τ}. The proof uses the representation of an elliptic curve as a covering of degree 2 of the projective line. 4.1. The dilogarithm as a combination of elliptic (1, 1)-logarithms.

We start by expressing the dilogarithm as a combination of elliptic (1, 1)- logarithms.

4.1.1. The Weierstrass ℘-function maps the elliptic curve onto ℙ^{1} and this covering has two branches. Let the points ±α on the elliptic curve be the preimages of a point a on ℙ^{1}: ℘(±α) = a; similarly, ℘(±β) = b, ℘(±γ) = c. Then by 2.2.4.,

D_{2} (\frac{c − a}{b − a}) = \frac{1}{4π} ∫_{ℙ^{1}} A_{3}(log |z − a|^{2}, log |z − b|^{2}, log |z − c|^{2})

= \frac{1}{2} \frac{1}{4π} ∫_{E_{τ}} A_{3}(log |℘(ξ) − ℘(α)|^{2}, log |℘(ξ) − ℘(β)|^{2}, log |℘(ξ) − ℘(γ)|^{2}).

The extra factor of 1/2 reflects the number of branches.

4.1.2. The “theta decomposition” implies

log |℘(ξ) − ℘(α)|^{2} = L_{1}(ξ + α) + L_{1}(ξ − α) − 2L_{1}(ξ) − 2L_{1}(α).

Let us substitute this expression into A_{3} and integrate. After a straightforward computation we get

4.1.3. Lemma.

D_{2} (\frac{℘(γ) − ℘(α)}{℘(β) − ℘(α)}) = L_{1,1}(α, β, γ) + L_{1,1}(−α, β, γ) + L_{1,1}(α, −β, γ)

+L_{1,1}(α, β, −γ) − 2 (L_{1,1}(0, β, γ) + L_{1,1}(0, −β, γ) + L_{1,1}(α, 0, γ)

+L_{1,1}(−α, 0, γ) + L_{1,1}(α, β, 0) + L_{1,1}(−α, β, 0)) .

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4.2. The elliptic (1, 1)-logarithms as a combination of dilogarithms. Now we combine the expressions from the previous Lemma in order to cancel almost all terms on the r.h.s.

4.2.1. Theorem. For any two nontrivial torsion points α ≠ ±β on an elliptic curve E_{τ}, say mα = nβ = 0 for some m, n ∈ ℕ, we have

∑_{k=1}^{m} ∑_{l=1}^{n} D_{2} (\frac{℘(kα + lβ) − ℘(α)}{℘(β) − ℘(α)}) = −2mn(L_{1,1}(α, β, 0) + L_{1,1}(−α, β, 0)).

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Sketch of proof. We must check that all except the two last terms on the r.h.s. of (14) cancel after summation. We show that the first one cancels:

L_{1,1}(α, β, kα + lβ) = L_{1,1}(α − (α + β), β − (α + β), kα + lβ − (α + β))

= L_{1,1}(−β, −α, (k − 1)α + (l − 1)β) = L_{1,1}(β, α, (1 − k)α + (1 − l)β)

= −L_{1,1}(α, β, (1 − k)α + (1 − l)β);

hence the first summands with arguments (k, l) and (1 − k, 1 − l) differ only by the sign and after averaging the first summands cancel. The arguments for the other terms are similar.

This completes the proof of Theorem B and hence of the formula for K_{2}(0, τ) in the Main Theorem.

4.2.2. Remark. We have proved that the combination L_{1,1}(α, β, 0) + L_{1,1}(−α, β, 0) for torsion points α and β is equal to a sum of dilogarithms. It is even true that the single term L_{1,1}(α, β, 0) is equal to a combination of dilogarithms; but we do not need this expression for the Main Theorem and it is rather complicated. We will derive it in Section 6.

5 Vanishing of the map δ.

In this section we prove that the argument of the dilogarithm in the Main Theorem belongs to the kernel of the map δ. 5.1. Values of the θ-function at torsion points.

5.1.1. The theta function θ̃(ξ, τ) = \frac{θ(ξ,τ)}{η(τ)} = q^{1/12}(z^{\frac{1}{2}} − z^{− \frac{1}{2}} ) ∏_{j=1}^{∞}(1 − q^{j}z)(1 − q^{j}z^{−1}) is not elliptic, it is only quasiperiodic

θ̃(ξ + 1, τ) = −θ̃(ξ, τ); θ̃(ξ + τ, τ) = −z^{−1}q^{−1/2} × θ̃(ξ, τ);

but for any torsion point ξ = rτ + s; r, s ∈ \frac{1}{N} ℤ of order N, we can redefine the value of the theta function at this point by the formula:

θ̃[ξ](τ) = z^{1/2+r}q^{−r^{2}/2−r}θ̃(ξ, τ).

The shift of the argument by a point of the lattice multiplies this value by some root of unity. Let us denote by ≡ the equality modulo multiplication by a root of unity.

5.1.2. Remark. If τ is imaginary quadratic over ℚ, then the numbers θ̃[ξ](τ) are algebraic.

5.1.3. For any two torsion points α and β the theta decomposition holds

℘(α; τ) − ℘(β; τ) ≡ \frac{θ̃′(0, τ)^{2} × θ̃[α − β](τ) × θ̃[α + β](τ)}{θ̃[α](τ)^{2} × θ̃[β](τ)^{2}}.

5.1.4. Let α be a torsion point on the elliptic curve E_{M(τ)}. Then

∏_{β∈M^{−1}α} θ̃[β](τ) ≡ θ̃[α](M(τ)), if α ≠ 0;

θ̃′(0, τ) × ∏_{β∈KerM\0} θ̃[β](τ) ≡ \frac{ad − bc}{cτ + d} × θ̃′(0, M(τ)).

5.2. Computations.

5.2.1. First, we calculate the value of the map δ on \frac{℘(γ)−℘(α)}{℘(β)−℘(α)}

δ (\frac{℘(γ) − ℘(α)}{℘(β) − ℘(α)}) = (\frac{℘(γ) − ℘(α)}{℘(β) − ℘(α)}) ∧ (\frac{℘(γ) − ℘(β)}{℘(α) − ℘(β)})

≡ (\frac{θ̃[α − γ] × θ̃[α + γ] × θ̃[β]^{2}}{θ̃[α − β] × θ̃[α + β] × θ̃[γ]^{2}} ) ∧ (\frac{θ̃[β − γ] × θ̃[β + γ] × θ̃[α]^{2}}{θ̃[α − β] × θ̃[α + β] × θ̃[γ]^{2}} ) .

So the answer is the same as the result of the following procedure: 1. Define the map ν by the formula

{α, β, γ} → − ({α − β} ∧ {β − γ} + {β − γ} ∧ {γ − α} + {γ − α} ∧ {α − β}) .

2. Apply the map ν to the arguments of the function L_{1,1} on the r.h.s. of (14).

3. Apply the map ∧^{2}θ̃: ξ_{1} ∧ ξ_{2} → θ̃[ξ_{1}] ∧ θ̃[ξ_{2}] ∈ ⋀^{2} ℂ^{∗} to the result of the previous step.

5.2.2. The map ∧^{2}θ̃ ∘ ν satisfies the same properties as the map L_{1,1} with respect to shifts or permutations of arguments. So after summation over γ = jα + kβ we get

δ (∑_{k=1}^{m} ∑_{l=1}^{n} \{\frac{℘(kα + lβ) − ℘(α)}{℘(β) − ℘(α)}\}) ≡ −2mn ∧^{2} θ̃∘ ν(\{α, β, 0\} + \{−α, β, 0\})

≡ 2mn(β) (θ̃[α − β] ∧ θ̃[β] + θ̃[β] ∧ θ̃[α] + θ̃[α] ∧ θ̃[α − β]

+θ̃[α + β] ∧ θ̃[β] + θ̃[β] ∧ θ̃[α] + θ̃[α] ∧ θ̃[α + β]) .

5.2.3. Finally, we perform the last summation:

δ ( ∑_{\begin{matrix}α∈KerM\0 \\ β∈Ker(M−1)\0 \end{matrix}} ∑_{k=1}^{m} ∑_{l=1}^{n} \{\frac{℘(kα + lβ) − ℘(α)}{℘(β) − ℘(α)}\} )

≡ 4mn ∑_{\begin{matrix}α∈KerM\0 \\ β∈Ker(M−1)\0 \end{matrix}} (θ̃[α − β] ∧ θ̃[β] + θ̃[β] ∧ θ̃[α] + θ̃[α] ∧ θ̃[α − β]) .

now sum the first terms over α’s, the second terms over α’s and β’s and the third terms over β’s. We get

4mn ( ∑_{β∈Ker(M−1)\0} (θ̃[Mβ] ∧ θ̃[β] − θ̃[β] ∧ θ̃[β])

+ (\frac{(a − 1)(d − 1) − bc}{cτ + d − 1}) ∧ (\frac{ad − bc}{cτ + d})

+ ∑_{α∈KerM\0} (θ̃[α] ∧ θ̃[(M − 1)α] − θ̃[α] ∧ θ̃[α]))

= 4mn × (\frac{(a − 1)(d − 1) − bc}{cτ + d − 1}) ∧ (\frac{ad − bc}{cτ + d}) .

This expression is the negative of δ of the first term 4mn \{\frac{ad−bc}{cτ+d}\} in the formula (14).

6 Values of the Kronecker double series at torsion points.

In this section we express the value of the Kronecker double series K_{2}(α; τ) at a CM-point τ and a torsion point α on the elliptic curve E_{τ} as a sum of dilogarithms. We have proved (12) that this value of the Kronecker double series can be reduced to a combination of L_{1,1}(α′, 0, γ′), ( α′ ∈ M^{−1}(α), γ′ ∈ (M − 1)^{−1}(α)) and K_{2}(0, τ). So we will prove that L_{1,1}(α′, γ′, 0) equals to a sum of dilogarithms. 6.1. Reduction to “standard” functions.

6.1.1. Let α be a torsion point of exact order N(α) = 2^{ρ(α)} × N_{0}(α), N_{0}(α) is odd. Denote by f_{α} a function with divisor N(α)×(α−0). Evidently, log |f_{α}(η)|^{2} = const + NL_{1}(η − α) − NL_{1}(η). This yields the formula

6.1.2. Lemma. For any three different torsion points α, β and γ

\begin{matrix}\frac{1}{4π} ∫_{E_{τ}} A_{3}(f_{α}, f_{β}, f_{γ}) = N(α)N(β)N(γ) \\ × (L_{1,1}(α, β, γ) − L_{1,1}(0, β, γ) − L_{1,1}(α, 0, γ) − L_{1,1}(α, β, 0)) . \end{matrix}  (16)

6.1.3. Lemma. For any nontrivial non-equal torsion points α and β on the elliptic curve E_{τ}

∑_{k=1}^{N(α)} ∑_{l=1}^{N(β)} \frac{1}{N(α)^{2}N(β)^{2}N(kα + lβ)} ∫_{E_{τ}} A_{3}(f_{α}, f_{β}, f_{kα+lβ}) = −L_{1,1}(α, β, 0) .

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The proof is parallel to the proof of formula (15).

We will reduce the integrals in the l.h.s. of the previous equation to “dilogarithmic” integrals.

First, we have a decomposition of f_{α}(η) into a product of “standard” functions g_{ξ} = ℘(η − ξ) − ℘(ξ). Let λ = λ(α) be a natural number such that 2^{λ} ≡ 1mod(N_{0}) and denote by α_{0} the point 2^{ρ}α.

6.1.4. Lemma.

\begin{matrix}f_{α}(η)^{− \frac{2^{λ}−1}{N_{0}}} \\ = const ∏_{i=0}^{ρ−1} (℘(η − 2^{i}α) − ℘(2^{i}α))^{2^{ρ−i−1}(2^{λ}−1)} ∏_{j=0}^{λ−1} (℘(η − 2^{j}α_{0}) − ℘(2^{j}α_{0}))^{2^{λ−j−1}} \end{matrix}

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Proof. Compare divisors.

6.1.5. Remark. If F is a function on an elliptic curve and ξ is a torsion point we can define the operation of averaging with a factor 2 by the formula

Av_{2}^{ξ}(F)(ξ) = ∑_{j=0}^{∞} 2^{−j}F (2^{j}ξ) = ∑_{j=0}^{ρ(ξ)−1} 2^{−j}F (2^{j}ξ)+ ∑_{j=ρ(ξ)}^{ρ(ξ)+λ(ξ)−1} \frac{2^{−j}}{1 − 2^{−λ(ξ)}} F (2^{j}ξ).

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The last equality is nothing more than the formula for the sum of a geometric progression. So we can formally rewrite the statement of the previous lemma as

log(f_{α}(η)) = −\frac{N(α)}{2}Av_{2}^{α}(log(℘(η − α) − ℘(α))) ,

the superscript α denotes the variable with respect to which we average. 6.2. From standard functions to dilogarithms.

6.2.1. Lemma. Let k ⊂ K be an extension of fields of degree 2. Denote by σ the involution of K over k. Then for any elements f,g in K \k

f × \frac{g^{σ} − g}{fg^{σ} − gf^{σ}} + g × \frac{f^{σ} − f}{gf^{σ} − fg^{σ}} = 1;

and \frac{g^{σ} − g}{fg^{σ} − gf^{σ}} ∈ k.

The proof is evident.

6.2.2. We apply this lemma to the extension ℂ(ℙ^{1}) ⊂ ℂ(E) and “standard” functions. The involution σ is given by changing sign of the argument of a function.

6.2.3. Lemma. For the “standard” functions g_{α}(η) = ℘(η−α)−℘(α) and g_{β}(η) = ℘(η − β) − ℘(β),

\frac{g_{β}^{σ}(η) − g_{β}(η)}{g_{α}(η)g_{β}^{σ}(η) − g_{β}(η)g_{α}^{σ}(η)} = K(α, β) × \frac{(℘(η) − ℘(α))^{2}}{℘(η) − C(α, β)},

where

C(α, β) = \frac{℘(α)℘′(β) (℘(α + β) − ℘(α)) + ℘(β)℘′(α) (℘(α + β) − ℘(β))}{℘′(β) (℘(α + β) − ℘(α)) + ℘′(α) (℘(α + β) − ℘(β))},

K(α, β) = \frac{−℘′(β)}{(℘(α) − ℘(β)) × (℘′(β) (℘(α + β) − ℘(α)) + ℘′(α) (℘(α + β) − ℘(β)))}

Proof. Let us consider the numerator and the denominator on the l.h.s. separately.

The numerator g_{β}^{σ} − g_{β} is odd, so it vanishes at semi-periods. On the other hand it has double poles at η = ±β. Hence,

g_{β}^{σ} − g_{β} = c × \frac{℘′(η)}{(℘(η) − ℘(β))^{2}} .

The constant c can be calculated by considering the leading term at η = β and it is equal to −℘′(β).

The denominator g_{α}g_{β}^{σ} − g_{β}g_{α}^{σ} is also odd. It has a zero of order ≥ 2 at the point 0, and hence (by oddness) ≥ 3. On the other hand it has double poles at η = ±α, ±β. Hence,

g_{α}g_{β}^{σ} − g_{β}g_{α}^{σ} = c′ × \frac{℘′(η) × (℘(η) − C(α, β))}{(℘(η) − ℘(α))^{2} × (℘(η) − ℘(β))^{2}}

for some constants c′ and C(α, β). These constants can be calculated by computation of the leading terms at the points α and β;

c′ = − (℘(α) − ℘(β))×(℘′(β) (℘(α + β) − ℘(α)) − ℘′(α) (℘(α + β) − ℘(β))) .

6.2.4. Let us denote the function \frac{g_{β}^{σ}−g_{β}}{g_{α}g_{β}^{σ}−g_{β}g_{α}^{σ}} by F_{αβ}. So the “nonevident” zeroes ±ξ_{αβ} of the function F_{αβ} are defined by the condition ℘(±ξ_{αβ}) = C(α, β) (and ℘′(±ξ_{αβ}) = ±\sqrt{4C(α, β)^{3} − c_{4}(τ)C(α, β) − c_{6}(τ)}, c_{4} and c_{6} where are the coefficients of the Weierstrass equation).

6.2.5. Lemma. For any three points α, β and γ on an elliptic curve

g_{α} ∧ g_{β} ∧ g_{γ} + g_{α}^{σ} ∧ g_{β}^{σ} ∧ g_{γ}^{σ} = (g_{α}F_{αβ}) ∧ (g_{β}F_{βα}) ∧ g_{γ} + (g_{α}^{σ}F_{αβ}) ∧ (g_{β}^{σ}F_{βα}) ∧ g_{γ}^{σ}

− ((g_{α}F_{αγ}) ∧ F_{βα} ∧ (g_{γ}F_{γα}) + (g_{α}^{σ}F_{αγ}) ∧ F_{βα} ∧ (g_{γ}^{σ}F_{γα})
−F_{αβ} ∧ (g_{β}F_{βγ}) ∧ (g_{γ}F_{γβ}) + F_{αβ} ∧ (g_{β}^{σ}F_{βγ}) ∧ (g_{γ}^{σ}F_{γβ}))

−F_{αβ}∧F_{βα}∧(g_{γ}g_{γ}^{σ})+((g_{α}g_{α}^{σ}) ∧ F_{βα} ∧ F_{γα} + F_{αγ} ∧ F_{βα} ∧ (g_{γ}g_{γ}^{σ}) + 2 × (F_{αγ} ∧ F_{βα} ∧ F_{γα})

+F_{αβ} ∧ (g_{β}g_{β}^{σ}) ∧ F_{γβ} + F_{αβ} ∧ F_{βγ} ∧ (g_{γ}g_{γ}^{σ}) + 2 × (F_{αβ} ∧ (F_{βγ}) ∧ F_{γβ})) .

The proof is a straightforward computation.

6.2.6. The first six terms on the r.h.s. of the previous equation have type ϕ ∧ (1 − ϕ) ∧ ψ, so the integral of A_{3} of any such term equals the sum of dilogarithms by (6). The last seven terms contain only σ-invariant functions, so the corresponding integrals can be reduced to integrals over ℂℙ^{1} and are equal to combinations of dilogarithms also. On the other hand, the integral of A_{3} of the second term on the l.h.s is equal to the corresponding integral of the first one for evident geometrical reasons. 6.3. Results.

If we combine all previous results, we get the following.

6.3.1. Theorem. For any two different nontrivial torsion points α and β on an elliptic curve

L_{1,1}(α, β, 0) = D_{2}(Φ(α, β));  (20)

where

Φ(α, β) = \frac{1}{ord(α)ord(β)} ∑_{γ∈<α,β>\0} Av_{2}^{α}Av_{2}^{β}Av_{2}^{γ}Φ_{1}(α, β, γ);

where < α, β > denotes the subgroup generated by α and β, Av_{2}^{ξ}(F)(ξ) for a torsion point ξ is defined by (19), and

Φ_{1}(α, β, γ) = {G_{αβ}(γ)}_{2}

+Alt_{α,β}(2{G_{αγ}(β)}_{2} + 2{G_{αγ}(−β)}_{2} − {G_{αγ}(ξ_{βα})}_{2} − {G_{αγ}(−ξ_{βα})}_{2})

+8 \{\frac{℘(γ) − ℘(α)}{℘(β) − ℘(α)}\}_{2}+4Cyc_{α,β,γ} (\{\frac{℘(2γ) − ℘(α)}{℘(β) − ℘(α)}\}_{2})−4Cyc_{α,β,γ} (\{\frac{C(α, γ) − ℘(α)}{℘(β) − ℘(α)}\}_{2})

−2Alt_{α,β,γ} (\{\frac{℘(2γ) − ℘(α)}{C(β, γ) − ℘(α)}\}_{2}) + Alt_{α,β} (2 \{\frac{C(α, γ) − ℘(α)}{C(α, β) − ℘(α)}\}_{2}

+2 \{\frac{℘(γ) − C(α, γ)}{C(α, β) − C(α, γ)}\}_{2} + \{\frac{℘(2γ) − C(α, γ)}{C(α, β) − C(α, γ)}\}_{2} + \{\frac{C(α, γ) − ℘(2α)}{C(α, β) − ℘(2α)}\}_{2});

where

G_{α,β}(η) = (℘(η − α) − ℘(α) × K(α, β)\frac{(℘(η) − ℘(α))^{2}}{℘(η) − C(α, β)}.

Here C(α, β) and K(α, β) are as in Lemma 6.2.3., the points ±ξ_{αβ} are solutions of the equation ℘(±ξ_{αβ}) = C(α, β); Alt_{α,β}, Alt_{α,β,γ}, Cyc_{α,β,γ} denote the (anti)symmetrization with respect to S_{2}, S_{3}, A_{3} (with a factor 1).

6.3.2. Theorem. Let τ be a fixed point of M = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ≠ 0, 1;

m = detM, n = det(M − 1). Let α be a torsion point on the curve E_{τ}. Then

\frac{c(τ − \overline{τ})}{i} (\frac{m + n + 1}{mn}K_{2}(α; τ) + K_{2}(0; τ)) = D_{2} ( ∑_{\begin{matrix}α′∈M^{−1}(α) \\ γ′∈(M−1)^{−1}(α) \end{matrix}}Φ(α′, γ′) ) ,

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where Φ is defined in the previous Theorem. The argument of the dilogarithm belongs to the kernel of the map δ : {x}_{2} → x ∧ (1 − x).

We have proved all statements of this theorem except the last one. It can be proved by the same consideration as in Section 5.

References

[D1] Deninger C. Higher regulators and Hecke L-series of imaginary quadratic fields. I, II Invent. Math. 96(1989) no. 1, 1–69. Ann. of Math.(2) 132 (1990), no. 1, 131–158.

[D2] Deninger C. Higher operations in Deligne cohomology. Invent. Math. 120(1995) no. 2, 289–315.

[G1] Goncharov A.B. Chow polylogarithms and regulators. Math.Res.Let. 2(1995), no. 1, 95-112.

[G2] Goncharov A.B. Deninger’s conjecture of L-functions of elliptic curves at s = 3. Algebraic geometry, 4. J.Math.Sci. 81(1996), no 3, 2631-2656.

[W] Andre Weil, Elliptic functions according to Eisenstein and Kronecker, Springer–Verlag, Berlin–New York 1976.