CLASSICAL AND ELLIPTIC POLYLOGARITHMS

AND SPECIAL VALUES OF L-SERIES

Don Zagier and Herbert Gangl^{1}

The Dirichlet class number formula expresses the residue at s = 1 of the Dedekind zeta function ζ_{F} (s) of an arbitrary algebraic number field F as the product of a simple factor (involving the class number of the field) with the determinant of a matrix whose entries are logarithms of units in the field. On the other hand, if F is a totally real number field of degree n, then a famous theorem by Klingen and Siegel says that the value ζ_{F} (m) for every positive even integer m is a rational multiple of π^{mn}. In [52] and [53], a conjectural generalization of these two results was formulated according to which the special value ζ_{F} (m) for arbitrary number fields F and positive integers m can be expressed in terms of special values of a transcendental function depending only on m, namely the m^{th} classical polylogarithm function. These instances are expected to form part of a much more general picture in which a special value of an L-series of “motivic origin” is expressed in terms of some transcendental function. In this survey we collect some pieces fitting into and illustrating this picture.

The paper is organized as follows.

In Part I we review the polylogarithm conjecture and survey some of the known results. The case m = 2, which is related to volumes of hyperbolic manifolds and has a more geometric flavor, is discussed in detail in §1. In the next two sections we formulate the conjecture for general m and describe some of the numerical and theoretical evidence which supports it, while in §4 we discuss a refinement involving a “lifting” from ℝ to ℂ/(2πi)^{m}ℚ of the m^{th} polylogarithm function. The natural setting for all of this is algebraic K-theory and the conjectures about polylogarithms lead to a purely algebraic (conjectural) description of the higher K-groups of fields.

In Part II we simultaneously generalize and specialize by considering partial zeta functions ζ_{F,A}(s) (i.e. the analogue for m ≥ 1 of the famous Stark conjectures about values of L-series at s = 1) but restricting our attention to the case where F is an imaginary quadratic field. This case, which for m = 1 is the Kronecker limit formula, can be treated much more concretely than the general case, both experimentally and theoretically. We describe experimental computations for small m and results of Deninger and Levin for m = 2. An interesting aspect here is that the existence of the above-mentioned “lifting” of the polylogarithms suggests that the partial zeta-values D_{F}^{m−1/2}ζ_{F,A}(m) should also have a natural lifting to ℂ/π^{2m}ℚ. It turns out that this “lifted” partial zeta-value really can be defined in a natural way (using Eisenstein series) and that the natural conjecture “lifted zeta-value = lifted polylog value” is supported by the numerical data (cf. §6). In this way the unproved polylogarithm conjecture leads to a definition of a new invariant for ideal classes in imaginary quadratic number fields.

Finally, in Part III we describe the numerical and theoretical evidence for conjectures expressing special values of L-series associated to elliptic curves in terms of “elliptic polylogarithm functions” which are obtained by an averaging process from the classical polylogarithms. Again the natural way to understand the results is in terms of algebraic K-theory, this time of the elliptic curve. We also describe particularly interesting examples of the conjectural picture coming from the recent work of Deninger, Boyd and Villegas on Mahler measures.

1Text based on the lectures given by D. Zagier at Banff

Part I. Polylogarithms, zeta-values, and algebraic K-theory

1. Dilogarithms, hyperbolic manifolds, ζ_{F} (2) and K_{3}(F)

Imaginary quadratic fields and Humbert’s formula. We first consider the case of an imaginary quadratic field F = ℚ(\sqrt{−d}) of discriminant −d. The group SL_{2}(O_{F}) acts as a discrete group of isometries on the hyperbolic 3-space ℍ^{3}, and it is well-known that

ζ_{F} (2) = \frac{4π^{2}}{d\sqrt{d}} Vol(ℍ^{3}/SL_{2}(O_{F}))

(originally found by Humbert—modulo a gap which can be corrected [28]—in 1919, and nowadays proved with the help of Tamagawa numbers or residues of Eisenstein series).

On the other hand, any hyperbolic 3-dimensional manifold can be triangulated, and we can assume that the vertices of the tetrahedra are algebraic numbers. By a formula of Lobachevsky (as given in detail in Thurston/Milnor [46]), the volume of a general hyperbolic tetrahedron can be expressed as a combination of a fixed number (24, to be precise) of values of the Bloch-Wigner dilogarithm

D(z) = ℑ(Li_{2}(z) + log |z| log(1 − z)) ,

evaluated at arguments which are algebraic functions of the coordinates of the vertices; here Li_{2} is the dilogarithm function, which is given in the unit disk by

Li_{2}(z) = ∑_{n>0} \frac{z^{n}}{n^{2}} , |z| < 1,

and can be extended analytically to a multivalued function on ℂ − {0, 1}, while the modified function D(z) turns out to be one-valued.

Combining these two facts, we can express ζ_{F} (2) in terms of D(z) with algebraic arguments z. As an example, we give the case F = ℚ(\sqrt{−7}). Here various descriptions of the manifold ℍ^{3}/Γ for some torsionfree subgroup Γ ⊂ SL_{2}(O_{F}) (as a link complement, as a union of two hyperbolic prisms, etc.) are given in [46]. We can use them to compute the volume and then, combining with Humbert’s formula as given above, obtain e.g.

ζ_{ℚ(\sqrt{−7})}(2) = \frac{4π^{2}/3}{7^{3/2}}D(ξ_{7}) ,  where ξ_{7} = 2[\frac{1 + \sqrt{−7}}{2}] + [\frac{−1 + \sqrt{−7}}{4}] .  (1)

(Here and in the following we use the evident linear extension of a function on a set X to a function on ℤ[X].)

For other fields the formulas quickly become rather complicated, e.g.

ζ_{ℚ(\sqrt{−23})}(2) = \frac{4π^{2}/3}{23^{3/2}} D(ξ_{23}) ,  (2)

with

ξ_{23} = 21[\frac{1 + \sqrt{−23}}{2}] + 7 [ 2 + \sqrt{−23} ] + [\frac{3 + \sqrt{−23}}{2}] − 3[\frac{5 + \sqrt{−23}}{2}] + [ 3 + \sqrt{−23} ] .

General number fields. Now let F be a general number field of discriminant D_{F} and degree [F : ℚ] = n = r_{1} + 2r_{2}. If the number r_{2} of pairs of complex embeddings of F equals 1, then a similar argument applies: we can associate to F a certain discrete subgroup Γ ⊂ SL_{2}(ℂ) and a manifold (now even compact) M = ℍ^{3}/Γ, with the property that Vol(M) ∼_{ℚ^{×}} |D_{F} |^{1/2}π^{−2(n−1)}ζ_{F} (2), where ∼_{ℚ^{×}} means that the term on the right is a non-zero rational multiple of the one on the left. (Take Γ to be the group of units of an order in a quaternion algebra over F ramified at all real places.) We again deduce that |D_{F} |^{1/2}π^{−2(n−1)}ζ_{F} (2) is a ℚ-linear combination of values of D(x) with x ∈ \overline{ℚ}.

For r_{2} > 1, we can get in the same way a discrete subgroup Γ of SL_{2}(ℂ)^{r_{2}} such that the quotient M := (ℍ^{3})^{r_{2}}/Γ has volume a rational multiple of |D_{F} |^{1/2}π^{−2(r_{1}+r_{2})}ζ_{F} (2). We then use a lemma which asserts that such a quotient can be triangulated into a finite union of products Δ_{1} × ⋯ × Δ_{r_{2}} where each Δ_{i} is a tetrahedron in ℍ^{3}; the volume is therefore expressible as a finite sum of r_{2}-fold products of D’s with algebraic arguments. The details are given in [52].

The Bloch group. Any triangulation of a complete hyperbolic 3-manifold M provides, together with the above-mentioned formula for the volume of a hyperbolic 3-simplex, an expression for the volume of M in terms of dilogarithms,

Vol(M) = ∑_{i=1}^{N} D(z_{i})

for some z_{i} ∈ ℂ (we can assume the z_{i} to actually lie in \overline{ℚ}, considered as a subfield of ℂ).

An analysis of the triangulation yields that the z_{i} occurring in the formula are always subject to the algebraic condition

∑_{i} z_{i} ∧ (1 − z_{i}) = 0 ∈ ⋀^{2}ℂ^{×}.  (3)

This follows, for example, from the combinatorial description of triangulations in [37] or from the results of [18].

We reformulate this fact by stating that each pair (manifold, triangulation) produces an element ξ ∈ A(\overline{ℚ}) with Vol(M) = D(ξ) , where we define for any field F

A(F) = { ∑ n_{i}[z_{i}] ∈ ℤ[F] | ∑ n_{i} (z_{i} ∧ (1 − z_{i})) = 0 }

(here [1] and [0] are to be interpreted as belonging to A(F)) and the function D : ℂ → ℝ is extended to ℤ[ℂ] by linearity as above. What happens when we change the triangulation? We will get different z_{i}’s and a different expression for the volume but the two expressions will be obtained from one another by a finite number of applications of the five term relation

D(x) + D(y) + D(\frac{1 − x}{1 − xy} ) + D(1 − xy) + D(\frac{1 − y}{1 − xy} ) .  (4)

This relation, which was discovered many times in the 19^{th} century (in the more complicated version using Li_{2} rather than D), can be verified easily by differentiating it (w.r.t. x, say), since the derivative of D(z) is an elementary function, but there is a prettier geometric way. The value of a general hyperbolic tetrahedron, as already mentioned, is a sum of 24 D-values, but the volume of an ideal tetrahedron (one with all its vertices in ∂\overline{ℍ^{3}} ≅ ℙ^{1}(ℂ)) is equal to a single value of D, namely D(r(a, b, c, d)) where a, b, c, d ∈ ℙ^{1}(ℂ) are the vertices and r denotes the cross-ratio. Now if we take five points a_{1}, …, a_{5} in ℙ^{1}(ℂ), then the “signed volume”

∑_{i=1}^{5}(−1)^{i}Vol(Δ(a_{1}, …, â_{i}, …, a_{5})) = ∑_{i=1}^{5}(−1)^{i}D(r(a_{1}, …, â_{i}, …, a_{5}))

vanishes, and this is equivalent to the five term relation. This is a special configuration, but one can show that the effect on the volume formula of subdividing any hyperbolic simplex into smaller pieces is also a consequence of the five term relation.

These considerations motivate defining, for any field F, the Bloch group B(F) as the quotient

B(F) = \frac{A(F)}{C(F)} ,

where C(F) is the group generated by the 5-term relation, i.e.

C(F) = \{[x] + [y] + [\frac{1 − x}{1 − xy} ] + [1 − xy] + [\frac{1 − y}{1 − xy} ] | x, y ∈ F, xy ≠ 1\} .

(The reader should check that C(F) ⊂ A(F) .) By (2), it is obvious that D is a well-defined linear function from B(ℂ) to ℝ. We have shown that to M there is associated a canonical element ξ_{M} ∈ B(\overline{ℚ}), independent of the choice of triangulation, with Vol(M) = D(ξ_{M}).

Algebraic K-theory. The discussion shows that if F is a number field with r_{2} = 1, then ζ_{F} (2) equals |D_{F} |^{1/2}π^{2(n−1)}D(ξ) for some ξ ∈ B(\overline{ℚ}) ⊗ ℚ. However, much more is true: ξ actually can be chosen to lie in the subgroup B(F) ⊗ ℚ, and there is a corresponding result also for fields with r_{2} > 1. To see this, the geometric method with triangulations is inadequate and one must use algebraic K-theory instead. To any ring R, there are associated algebraic K-groups K_{0}(R), K_{1}(R) = R^{×}, K_{2}(R), K_{3}(R), … The definition of these groups (which we will not use explicitly and hence do not describe) is complicated and highly non-constructive. However, in the case when R = F is a number field, Borel [9] showed that the higher K-groups, modulo torsion, are free abelian groups of rank given by

dim_{ℚ} (K_{n}(F) ⊗ ℚ) = \{ \begin{matrix}0, & n ≥ 2 even, \\ n_{∓}, & n = 2m − 1 > 1, ∓ = (−1)^{m−1} , \end{matrix}

where n_{+} = r_{1}+r_{2} and n_{−} = r_{2} are the dimensions of the ±-eigenspaces of complex conjugation on F ⊗_{ℚ} ℝ ≅ ℝ^{r_{1}} × ℂ^{r_{2}}. Furthermore, for each m > 1 there is a map reg_{m} : K_{2m−1}(ℂ) → ℝ which is (−1)^{m−1}-invariant w.r.t. complex conjugation on ℂ, and the composite map

\xymatrix{reg_{m,F} : K_{2m−1}(F) \ar[r] & (K_{2m−1}(ℝ))^{r_{1}} × (K_{2m−1}(ℂ))^{r_{2}} & ^{reg^{m}} \ar[r] & ℝ^{n_{∓}}}

maps K_{2m−1}(F)/(torsion) isomorphically onto a cocompact lattice Reg_{m,F} ⊂ ℝ^{n_{∓}} whose covolume is a rational multiple of |D_{F} |^{1/2}ζ_{F} (m)/π^{mn_{±}}.

In the case n = 3, Bloch [5] gave a map between K_{3}(F) and B(F), and it was shown later by Suslin [44] that in fact the two groups are canonically isomorphic (at least up to torsion; in this article we will consistently ignore torsion). Moreover, under this isomorphism the Borel regulator map on K_{3} corresponds to the map D on B(F), giving (the ∼ denotes an isomorphism up to finite kernel and cokernel)

[figure: B(F); ∼; (D ∘ σ_{1}, …, D ∘ σ_{r_{2}}); K_{3}(F); _{reg_{2,F}}; ℝ^{r_{2}}]

where σ_{i} : F ↪ ℂ, i = 1, …, r_{2}, denote the different complex embeddings (up to conjugation). Together with Borel’s theorem, this yields in particular the Theorem. Let F be a number field with r_{1} real and r_{2} complex places. Then we have:

(1) The group B(F) is finitely generated of rank r_{2}.

(2) Let ξ_{1}, …, ξ_{r_{2}} be a ℚ-basis of B(F) ⊗ ℚ and σ_{1}, …, σ_{r_{2}} a set of complex embeddings (containing no pair of complex conjugate ones) of F into ℂ. Then

ζ_{F} (2) ∼_{ℚ^{×}} |D_{F} |^{1/2} · π^{2(r_{1}+r_{2})} · det (D(σ_{i}(ξ_{j}))_{1≤i,j≤r_{2}}) .

It is this statement which we wish to generalize to higher zeta-values.

2. Formulation of the polylogarithm conjecture

Higher polylogarithms. We now turn to the conjectures relating K_{2m−1}(F) and ζ_{F} (m) for higher values of m to the m^{th} polylogarithm function Li_{m}(z) = ∑_{n>0} z^{n}/n^{m}. The first step is to have at our disposal a single-valued version ℒ_{m} of this function generalizing the function D =: ℒ_{2}. Ramakrishnan [38] showed that such a one-valued version exists, and explicit formulas were given in [50], [54], [53]. We will use the function

ℒ_{m}(z) = ℜ_{m}( ∑_{k=0}^{m−1} \frac{2^{k}B_{k}}{k!} log^{k} |z| Li_{m−k}(z)) ,

where ℜ_{m} denotes ℑ or ℜ according as m is even or odd and B_{k} is the k^{th} Bernoulli number. The series defining Li_{r} converge exponentially for |z| < 1 and ℒ_{m}(z^{−1}) = (−1)^{m−1}ℒ_{m}(z), so we can easily compute ℒ_{m} numerically to high accuracy. (A yet better method is indicated in [13], Proposition 1.) Extensive experiments suggested a definition of higher Bloch groups B_{m}(F). The idea behind the definition was that, by analogy, B_{m}(F) should be written as a quotient A_{m}(F)/C_{m}(F) of a group A_{m}(F) of “allowable” elements (corresponding to the condition ∑ n_{i} (z_{i} ∧ (1 − z_{i})) = 0 in the case m = 2) by a group C_{m}(F) of “concrete relations” coming from functional equations of the m-logarithm ℒ_{m} (the relations in the case m = 2 being induced by the five term relation).

Since ℒ_{m} is (−1)^{m−1}-symmetric with respect to complex conjugation, the map ℒ_{m} defines for any number field F a map ℒ_{m,F} : B_{m}(F) → ℝ^{n_{∓}} and we expect the following diagram to commute:

[figure: B_{m}(F); ∼; _{ℒ_{m,F}}; (B_{m}(ℂ)^{Σ})_{+} = B_{m}(ℝ)^{r_{1}} × B_{m}(ℂ)^{r_{2}}; (ℒ_{m},...,ℒ_{m}); K_{2m−1}(F); _{reg_{m,F}}; ℝ^{n_{∓}}]

Here Σ = Σ_{F} denotes the set of embeddings F ↪ ℂ and ( )_{+} the +-eigenspace with respect to the joint action of complex conjugation on Σ_{F} and B_{m}(ℂ), and ∓ is (−1)^{m−1}. If this picture is correct, then, by Borel’s theorem, the evident analogues of the above-stated theorem hold: the vector space B_{m}(F) ⊗ ℚ has dimension n_{∓} over ℚ, and if ξ_{1}, …, ξ_{n_{∓}} is a basis of this vector space, then the determinant of the matrix ℒ_{m}(σ_{i}(ξ_{j}))_{i,j} is a non-zero rational multiple of |D_{F} |^{1/2}π^{−mn_{±}}ζ_{F} (m).

Definition of the higher Bloch groups. The only question is how to define the groups A_{m}(F) and C_{m}(F). We describe the conjectural answer which was found by numerical experimentation and presented in [53]. For m = 2 the definition of A_{m}(F) = A(F) can be written more compactly as A_{2}(F) = ker β_{2}, where

β_{2} : ℤ[F] → ⋀^{2}F^{×} , [z] ↦ \{ \begin{matrix}z ∧ (1 − z) , & z ≠ 1 , \\ 0 , & z = 0, 1 . \end{matrix}

The next simplest case occurs for m = 3 and totally real F. Here we set A_{3}(F) = ker β_{3} where β_{3} is the map

β_{3} : ℤ[F] → F^{×} ⊗ ⋀^{2}F^{×} , [z] ↦ \{ \begin{matrix}z ⊗ (z ∧ (1 − z)) , & z ≠ 1 , \\ 0 , & z = 0, 1 . \end{matrix}

For instance, if F = ℚ then, as the reader can easily check, the elements

ξ_{1} = [\frac{1}{2}] , ξ_{2} = 2[3] − [−3] , ξ_{3} = 6[\frac{2}{3}] + 3[\frac{3}{4}] − 3[\frac{1}{2}] − [\frac{8}{9}] − 2[\frac{1}{3}] + [ − \frac{1}{3}]

are in A_{3}(ℚ), and one finds ℒ_{3}(ξ_{1}) = \frac{7}{8}ζ(3) , ℒ_{3}(ξ_{2}) = \frac{13}{6}ζ(3) and ℒ_{3}(ξ_{3}) = \frac{67}{24}ζ(3) . Similarly, if F = ℚ(\sqrt{5}) , then the elements [1] and [\frac{1+\sqrt{5}}{2}] belong to the kernel of β_{3} and one finds

( \begin{matrix}ℒ_{3}(1) & ℒ_{3}(1) \\ ℒ_{3}(\frac{1+\sqrt{5}}{2}) & ℒ_{3}(\frac{1−\sqrt{5}}{2}) \end{matrix} ) = ( \begin{matrix}ζ(3) & ζ(3) \\ \frac{1}{10}ζ(3) + \frac{25}{48} \sqrt{5}L(3, χ_{5}) & \frac{1}{10}ζ(3) − \frac{25}{48}L(3, χ_{5}) \end{matrix} )

which has determinant −\frac{25}{24} \sqrt{5} ζ(3)L(3, χ_{5}) = −\frac{25}{24} \sqrt{5} ζ_{F} (3) . (Here L(s, χ_{5}) denotes the Lseries of the quadratic character χ_{5} associated to ℚ(\sqrt{5}) .) We can instead choose a basis like ξ_{+} = [\frac{1+\sqrt{5}}{2}]+[\frac{1−\sqrt{5}}{2}] and ξ_{−} = [2−\sqrt{5}]−[2+\sqrt{5}]+3([\frac{1+\sqrt{5}}{2}]−[\frac{1−\sqrt{5}}{2}]) , with ξ_{+} invariant and ξ_{−} anti-invariant under Gal(F/ℚ); then ℒ_{3}(ξ_{+}) is a rational multiple (=\frac{1}{5}) of ζ(3) and ℒ_{3}(ξ_{−}) a rational multiple (= \frac{75}{32}) of \sqrt{5}L(3, χ_{5}), and this choice of basis diagonalizes the matrix.

For general number fields the condition ξ ∈ ker(β_{3}) turns out (experimentally) to be no longer sufficient to make ξ land in a lattice with the desired properties, and in fact one needs a further condition, as was pointed out by Deligne. Observe first that if ξ = ∑ n_{i}[x_{i}] ∈ ker(β_{3}) , then for every homomorphism ϕ : F^{×} → ℤ the element ι_{ϕ}(ξ) := ∑ n_{i}ϕ(x_{i})[x_{i}] ∈ ℤ[F] belongs to A(F) = ker(β_{2}). The needed extra condition is that ι_{ϕ}(ξ) in fact belongs to the subgroup C(F) ⊂ A(F). This is not a restriction for F totally real since then A(F)/C(F) ≅ K_{3}(F) ≅ ℤ^{r_{2}} = {0}, but in the general case we obtain a group

A_{3}(F) := {ξ ∈ ℤ[F] | ι_{ϕ}(ξ) ∈ C(F) for all homomorphisms ϕ : F^{×} → ℤ}

which is a proper subgroup of ker(β_{3}).

We now make the analogous definition for all m, setting

A_{m}(F) := {ξ ∈ ℤ[F] | ι_{ϕ}(ξ) ∈ C_{m−1}(F) ∀ ϕ ∈ Hom(F^{×}, ℤ)} .  (5)

This inductive definition requires knowing the previous “C”-group. The subgroup C_{m}(F) is supposed to be the group generated by the functional equations of ℒ_{m}, just as C(F) was generated by the five term relation. However, though one can give a formal definition based on this idea (see “Functional equations” below), it is not effective since the functional equations of the higher polylogarithms are not known explicitly. We therefore give a practical (i.e. numerically computable) definition of C_{m}(F) for number fields F as

C_{m}(F) := {ξ ∈ A_{m}(F) | ℒ_{m}(σ(ξ)) = 0 ∀ σ ∈ Σ_{F} } .  (6)

We can then formulate the polylogarithm conjecture for arbitrary m:

Conjecture. Let F be a number field. Then the image of the map

ℒ_{m,F} : A_{m}(F) ⟼ ℝ^{n_{∓}}

is commensurable with the Borel regulator lattice Reg_{m,F} ⊂ ℝ^{n_{∓}}. In particular, (1) The group B_{m}(F) := A_{m}(F)/C_{m}(F) is finitely generated of rank n_{∓}.

(2) Let ξ_{j} (1 ≤ j ≤ n_{∓}) be a ℚ-basis of B_{m}(F)⊗ℚ and σ_{i} (1 ≤ i ≤ n_{∓}) run through Σ. Then

ζ_{F} (m) ∼_{ℚ^{×}} |D_{F} |^{1/2} · π^{mn_{±}} · det (ℒ_{m}(σ_{i}(ξ_{j}))_{1≤i,j≤n_{∓}}) .  (7)

We will discuss the evidence for this conjecture in §3. In the rest of this section we mention some supplementary aspects of its formulation.

Functional equations. The definition of C_{m}(F) which was given above leads to a formulation of the polylogarithm conjecture which can be tested numerically, but is rather artificial and works only for number fields. As already mentioned, C_{m}(F) should really be defined as “the group of all functional equations of the m-logarithm.” We explain here how to do this. The new (and presumably equivalent) definition works for all fields and fits more naturally into a cohomological framework.

It is easy to see by induction that A_{m}(F) defined above is always a subgroup of the kernel of the map

β_{m} = β_{m,F} : ℤ[F] → Sym^{m−2}F^{×} ⊗ ⋀^{2}F^{×} , [z] ↦ \{ \begin{matrix}z^{⊗(m−2)} ⊗ (z ∧ (1 − z)) , & z ≠ 1 , \\ 0 , & z = 0, 1 . \end{matrix}

The “homotopic” definition of C_{m}(F) is then

C_{m}^{∗}(F) = {ξ(1) − ξ(0) | ξ(t) ∈ ker(β_{m,F (t)})} .

One can show [53, Prop. 1, p. 411] that an element ξ(t) ∈ ℤ[F (t)] is in the kernel of β_{m,F (t)} if and only if the map t ↦ ℒ_{m}(ξ(t)) is constant, so that the elements of C_{m}^{∗}(F) really are the specializations to F of all functional equations of ℒ_{m}. Clearly C_{m}^{∗}(F) ⊆ C_{m}(F), but the equality is known only for m = 2.

It is highly non-trivial to find functional equations for ℒ_{m}. Kummer [30] gave special equations for m = 3, 4, 5. Goncharov [23] found a functional equation in 3 variables for ℒ_{3} which has a beautiful geometric interpretation in terms of configurations and is believed to generate C_{3}^{∗} (i.e., to play the same role for the trilogarithm as does the five term relation for the dilogarithm). Examples of functional equations for higher m (up to m = 7) were given in [20]. A typical example for m = 6 (cf. also [55]) is

∑_{(a,b)∈T} \frac{n_{a,b}}{μ_{a,b}} ∑_{{w∈W \atop z∈Z}} ℒ_{6}((\frac{w}{z} )^{a}(\frac{1 − w}{1 − z} )^{b}) = (independent of y) ,

where x and y are independent variables,

T = {(−2, 3), (−2, 1), (1, −2), (1, 1), (1, 0), (0, −1), (−1, 1)} ,

W = \{x(1 − x), −\frac{(1 − x)}{x^{2}}, −\frac{x}{(1 − x)^{2}} \} , Z = \{y, 1 − y, \frac{1}{1 − y} , \frac{y}{y − 1}, \frac{y − 1}{y}, \frac{1}{y} \} ,

and the coefficients n_{ab} and μ_{ab} are defined by

∑_{(a,b)∈T} n_{ab}(aX + bY)^{5} = 0 ∈ ℤ[X, Y] , μ_{ab} = log_{2}(1 + ab + a^{2} + b^{2}) .

For m > 7, however, only the “trivial” (distribution and inversion) relations are known.

Goncharov’s complexes. The definition of A_{m}(F) can be written more elegantly as

A_{m}(F) = ker (ℤ[F] ⟶^{ι} F^{×} ⊗ ℤ[F]/C_{m}(F)) ,

where ι : ℤ[F] → F^{×} ⊗ ℤ[F] maps [x] to x ⊗ [x] and [0] to 0. The group B_{m}(F) = A_{m}(F)/C_{m}(F) then becomes ker (G_{m}(F)⟶^{∂}F^{×} ⊗ G_{m−1}(F)), where G_{m}(F) = ℤ[F]/C_{m}(F) and ∂ is the map induced by ι. More generally, Goncharov [23] (using the notation R_{m}(F) for our C_{m}(F) and B_{m}(F) for our G_{m}(F)) has defined for any field F the complex

G_{m}(F) ⟶^{∂} F^{×}⊗G_{m−1}(F) ⟶^{∂} ⋀^{2}F^{×}⊗G_{m−2}(F) ⟶^{∂} … ⟶^{∂} ⋀^{m−2}F^{×}⊗G_{2}(F) ⟶^{∂} ⋀^{m}F^{×}

where ∂ maps x_{1} ∧ ⋯ ∧ x_{r} ⊗ [x] to x_{1} ∧ ⋯ ∧ x_{r} ∧ x ⊗ [x] (except in the last step, where x_{1} ∧ ⋯ ∧ x_{m−2} ⊗ [x] is mapped to x_{1} ∧ ⋯ ∧ x_{m−2} ∧ x ∧ (1 − x)). The cohomology of this complex at the first point is then B_{m}(F), and Goncharov sharpens the conjecture of [53] to say that the cohomology groups of the complex give the graded pieces of a certain natural filtration on the algebraic K-groups, the Adams filtration. Of course, this only makes sense with the algebraic definition of C_{m}(F) in terms of functional equations, because the “practical” definition (6) applies to number fields only. Since the Adams filtration in the case of a number field is trivial after the first step, we can test Goncharov’s conjecture in this case by testing the exactness of the above complex. The first non-trivial case is that

G_{4}(F) ⟶^{∂} F^{×} ⊗ G_{3}(F) ⟶^{∂} ⋀^{2}F^{×} ⊗ G_{2}(F)

should be exact. This has been checked experimentally for F = ℚ in [21], confirming the expected result.

Connection to hyperbolic volumes for m > 2. The connection to hyperbolic volumes in the case m = 2 does not simply extend to higher m. It is known by results of Schläfli that the formula for the volume of a hyperbolic 2m-simplex reduces to the formula for dimension 2m − 1, so the volume of a 4-dimensional simplex can be expressed in terms of D. For hyperbolic simplices of dimension 5 and 6 it has been established that the volume can be expressed as a sum of values of trilogarithmic expressions (Böhm [8] (implicitly), Müller [36] (in principle, but with missing terms) and Kellerhals [29], cf. also Goncharov [24]). The volume of the 7-simplex, though, is not expressible in terms of Li_{4} alone (Goncharov, private communication). However, Goncharov conjectures [24] that the volume of any hyperbolic manifold of finite volume in dimension 2m − 1 or 2m should be expressible in terms of ℒ_{m}.

3. Evidence for the conjecture

Numerical evidence. The conjecture in the form given in §2 can be tested experimentally. Starting from m = 2, we inductively want at each stage that (i) the images under ℒ_{m,F} of the combinations found span a lattice, i.e. a discrete subgroup of ℝ^{n_{∓}} of maximal rank, and (ii) the covolume is a simple multiple of ζ_{F} (m). The discreteness means that we can check numerically whether a given element of A_{m} belongs to C_{m} (its coordinates with respect to a ℤ-basis of the lattice are integers, so can be checked to vanish by a low accuracy calculation), and this is just what we need to compute the next group A_{m+1}(F) and check that it again satisfies (i); and (ii) then provides a further test of the conjecture if the value of the zeta function is known.

Note that for a given element ξ = ∑ n_{i}[x_{i}] ∈ ℤ[F^{×}] there are only finitely many conditions to be tested to determine whether ξ belongs to C_{m}(F), since in (6) one can restrict to ϕ running over a basis of Hom(G, ℤ), where G ⊂ F^{×} is the group generated by the x_{i}. More precisely, if we fix subgroups G ⊆ G′ ⊂ F^{×} of ranks r and r′, respectively, and denote by X the set of all x ∈ G with 1 − x ∈ G′, then the number of conditions for an element ξ ∈ ℤ[X] to lie in A_{m}(F) is

\binom{r + m − 2}{m − 1}r′ − \binom{r + m − 1}{m} + ∑_{k=1}^{m−2} \binom{r + k − 1}{k}n_{(−1)^{m−k−1}} ,  (8)

and the number of conditions for ξ to lie in C_{m}(F) is given by the same expression but with the sum starting at k = 0. (The first two terms give the rank of Sym^{m−1}(X) ⊗ X′/Sym^{m}(X) which is isomorphic to the target space of the map β_{m} in Sym^{m−2}(X) ⊗ ⋀^{2}(X′).) This expression is polynomial in r and linear in r′, and a theorem of Erdős–Stewart–Tijdeman ([19]; cf. also [53, p. 425ff] or [55, p. 390ff]) guarantees that the cardinality of X grows much faster than this (at least for F = ℚ, but experimentally for all number fields), so that in practice we can always find enough elements to generate the full lattice and test the conjecture.

This inductive process has been successfully carried out numerically in thousands of cases. Some examples will be described at the end of the section.

Compatibility with Galois descent. A consequence of the conjecture and the wellknown Galois descent for K-groups is the corresponding property for higher Bloch groups, i.e. B_{m}(F) = B_{m}(E)^{G} for a Galois extension E/F with Galois group G, and this property, which is highly non-trivial, can be checked in examples. For instance, consider F = ℚ(\sqrt{−7}) ⊂ E = ℚ(ζ_{7}) with G = ℤ/3ℤ. The element [ζ_{7}] trivially belongs to A_{m}(E), so its trace ξ_{cyc} = [ζ_{7}]+[ζ_{7}^{2}]+[ζ_{7}^{4}] ∈ B_{m}(E)^{G} should be equal to an element of B_{m}(F), which can be checked experimentally by computing its image in ℝ under the map ℒ_{m}. For m = 2 we can check explicitly that ξ_{cyc} is equal modulo the five term relation to \frac{4}{7}ξ_{7} with ξ_{7} as in (1). Other examples will be given in §5.

Compatibility with Lichtenbaum’s conjecture. Assuming both the conjecture above and Lichtenbaum’s conjecture (which interprets the quotient of ζ-value and regulator in terms of orders of K-groups) it is possible to not only verify inductively the above picture with computer experiments but it also allows to conjecturally predict orders of higher K-groups (cf. [11], [22], where the orders predicted agree with the few theoretical results known) and has moreover proved useful to detect a mistake in a proof concerning Lichtenbaum’s conjecture.

Cyclotomic fields. Let F = ℚ(ζ_{N}), N > 2. Here n_{+} = n_{−} = \frac{1}{2}ϕ(N) (Euler ϕ-function). Take the ξ_{j} to be [ζ_{N}^{l}] (1 ≤ l ≤ N/2, (l, N) = 1). Clearly ξ_{j} is in A_{m}(F) for all m (since ζ_{N}^{l} is torsion in F^{×}) and formula (7) is true in this case, as one sees by a standard calculation (cf. [53, p. 421ff]) using three facts:

i) The Dedekind zeta function ζ_{F} (s) is the product of L(s, χ) with χ running over all primitive Dirichlet characters of conductor dividing N,

ii) For a primitive character χ of conductor f, we have

L(m, χ) = ∑_{n=1}^{∞} \frac{χ(n)}{n^{m}} = \frac{1}{G(χ̄)} ∑_{n=1}^{∞} ∑_{r=1}^{f} \frac{χ̄(r) e^{2πirn/f}}{n^{m}} = \frac{1}{G(χ̄)} ∑_{r=1}^{f} χ̄(r) Li_{m}(e^{2πir/f}) ,

where G(χ̄) = ∑_{r=1}^{f} χ̄(r)e^{2πir/f} denotes the Gauss sum associated to χ̄.

iii) The Frobenius formula

det (t_{gh^{−1}})_{g,h∈G} = ∏_{χ∈Hom(G,ℂ^{×})} ∑_{g∈G} χ(g)t_{g} ,

valid for any finite abelian group G and complex numbers (t_{g})_{g∈G}.

Abelian fields. If F is abelian, then one obtains a partial result. By the Kronecker-Weber theorem, F is a subfield of some cyclotomic field E = ℚ(ζ_{N}). The Dedekind zeta function of F still splits as ∏_{χ} L(s, χ) where χ runs over a certain set of primitive characters (namely those which are trivial on Gal(E/F) ⊂ Gal(E/ℚ) = (ℤ/Nℤ)^{×}). The same calculation as for the cyclotomic case then gives us a lattice contained in B_{m}(E)^{Gal(E/F)} of the expected rank n_{∓}(F) and which maps under ℒ_{m,F} to a lattice in ℝ^{n_{∓}} of the desired covolume.

General theoretical results. Finally, there are two main theoretical results in support of the conjecture. First of all, Beilinson and Deligne [4] and, independently, de Jeu [17] have shown that we can map the higher Bloch groups (in a somewhat different version than the one presented here) to the corresponding higher K-groups, and that the Borel regulator map in K-theory really is given by the polylogarithms. In particular, this suffices to show that there is a ℚ-linear relation among the polylogarithms of any n_{∓} + 1 elements of the m^{th} Bloch group. What is not known in general is the surjectivity, hence the expressibility of ζ_{F} (m) as a determinant of polylogarithms. In what is certainly the most important progress towards the general conjecture, Goncharov [23] proved this surjectivity for the case m = 3. He also gave an explicit functional equation for the function ℒ_{3}(z) with 22 terms in 3 variables which is believed to be the universal functional equation describing the subgroup C_{m}(F) in this case.

Examples. We illustrate the numerical procedure described at the beginning of the section. Example 1. The statement of the conjecture is non-trivial even for F = ℚ, although

the mere expressibility of the zeta-value in terms of polylogarithms is uninteresting in this case because ζ(m) is a rational multiple of π^{m} for m even and equals ℒ_{m}(1) for m odd. We illustrate a non-trivial verification up to the level of the 7-logarithm. Choose G = ⟨−1, 2, 3⟩ and G′ = ⟨−1, 2, 3, 5, 7⟩. On the one hand, there are (at least) 29 rational numbers x_{1} = \frac{1}{3}, x_{2} = −\frac{1}{3}, x_{3} = −\frac{1}{9}, … x_{29} = −\frac{2}{243} satisfying x ∈ G and 1 − x ∈ G′. On the other hand, the number of conditions imposed on A_{7}(ℚ) by formula (8) with r = 2, r′ = 4, and m = 7 is 28, so we must get at least one element ξ = ∑ n_{i}[x_{i}] in A_{7}(ℚ). In fact, we find exactly one such combination (the coefficients, given explicitly on p. 386 of [55], are integers of up to 12 digits), and evaluating ℒ_{7}(ξ) we find to very high accuracy a (huge) integer multiple of \frac{1}{96}ζ(7).

Example 2. A yet more impressive example is given in [13], where elements of the lattice A_{m}(F) are computed up to m = 16 for a field F of degree 10, namely the field generated by Lehmer’s famous Salem number α (the algebraic number of conjecturally minimal height). Here the group G is taken to be the group of rank 1 generated by −1 and α (i.e., this example is a so-called “ladder”), while G′ is a group of small rank (generated by the units and a few small primes) which is chosen to contain many elements of the form 1 ± α^{i}. At each stage, the (many) elements in A_{m}(F) found give vectors in ℝ^{n_{∓}} (with n_{+} = 6, n_{−} = 4) lying, within an accuracy of several hundred digits, in a lattice of the right rank. If we had missed a condition in the “correct” definition of A_{m}(F), it is very likely that it would have shown up before the level m = 16. Very recently, Bailey and Broadhurst [1] extended this example one level higher, to the 17-logarithm.

Example 3. The third example, another ladder, is simpler than the two previous ones and will be used again later, so we give it in more detail. Let F = ℚ(θ), where θ^{3} − θ − 1 = 0, be the cubic field of discriminant −23. Here r_{1} = r_{2} = 1. We denote by σ_{j} (0 ≤ j ≤ 2) the embeddings

F ↪ ℂ sending θ to θ_{0} = 1.32471 … (the real root), θ_{1} = \frac{θ_{0}}{2} ( − 1 + \frac{\sqrt{−23}}{2θ_{0} + 3}), and θ_{2} = θ̄_{1},

respectively. Let G (=the group of units in F) be the group generated by −1 and θ. The six elements

x_{0} = 1, x_{1} = θ, x_{2} = −θ, x_{3} = θ^{3}, x_{4} = −θ^{4}, x_{5} = θ^{5}

(and of course their inverses) have the property that both x and 1 − x belong to G. (We could also include x_{6} = θ^{2}, but because of the “distribution relation” 2^{1−m}ℒ_{m}(θ^{2}) = ℒ_{m}(θ) + ℒ_{m}(−θ) it would not add anything new.) Since G has rank 1, each x_{i} ∧ (1 − x_{i}) (1 ≤ i ≤ 5) vanishes (up to torsion), giving 6 elements α_{i} = [x_{i}] (0 ≤ i ≤ 5) in A_{2}(F). The conjecture says that each D(σ_{1}(α_{i})) = D(σ_{1}(x_{i})) should be a rational multiple of π^{−4}23^{3/2}ζ_{F} (2), and in fact we find

D(σ_{1}(α_{i})) = λ_{i} · \frac{3 · 23^{3/2}}{8π^{4}} ζ_{F} (2) ,  (λ_{0}, …, λ_{5}) = (0, 1, −2, 2, 1, −1) .  (9)

(As already mentioned, polylogarithms can be computed rapidly using the formulas in [13]. We will indicate in §5 how to compute ζ_{F} (s) to high accuracy. It is easy to check directly that α_{i} equals λ_{i}α_{1} in the Bloch group.)

At the next stage, the condition for ∑_{i=0}^{5} n_{i}α_{i} ∈B_{3}(F) is that the element ∑ n_{i}ε_{i}α_{i} ∈ A_{2}(F) should map to 0 in B_{2}(F), i.e. ∑ n_{i}ε_{i}λ_{i} = 0, where ε_{0}, …, ε_{5} = 0, 1, 1, 3, 4, 5 denote the exponents in x_{i} = ±θ^{ε_{i}}. This gives five non-trivial elements β_{j} = [x_{j}] − λ_{j}ε_{j}[x_{1}] in A_{3}(F). We find

ℒ_{3}(σ_{0}(β_{j})) = μ_{j} · \frac{ζ(3)}{3} + ν_{j} · \frac{23^{5/2}}{144 π^{3}} \frac{ζ_{F} (3)}{ζ(3)} , ℒ_{3}(σ_{1}(β_{j})) = μ_{j} · \frac{ζ(3)}{3} − ν_{j} · \frac{23^{5/2}}{288 π^{3}} \frac{ζ_{F} (3)}{ζ(3)} ,

with

(μ_{j})_{j=0,2,...,5} = (3, 0, 3, 2, 1) , (ν_{j})_{j=0,2,...,5} = (0, 3, −15, −13, 13) .

Thus, in accordance with the conjecture, the elements which we have found in B_{3}(F) lie in a 2-dimensional lattice ℤ ξ_{1} + ℤ ξ_{2} with ξ_{1} = β_{0} = [1], ξ_{2} = β_{5} − 4β_{2} = [x_{5}] − 4[x_{2}] − 3[x_{1}] and

| \begin{matrix}ℒ_{3}(σ_{0}(ξ_{1})) & ℒ_{3}(σ_{0}(ξ_{2})) \\ ℒ_{3}(σ_{1}(ξ_{1})) & ℒ_{3}(σ_{1}(ξ_{2})) \end{matrix} | = | \begin{matrix}ζ(3) & \frac{ζ(3)}{3} + \frac{23^{5/2}}{144π^{3}} \frac{ζ_{F} (3)}{ζ(3)} \\ ζ(3) & \frac{ζ(3)}{3} − \frac{23^{5/2}}{288π^{3}} \frac{ζ_{F} (3)}{ζ(3)} \end{matrix} | ∼_{ℚ^{×}} π^{−3} \sqrt{23} ζ_{F} (3) .

We now have three linear combinations [x_{3}] + 5[x_{2}] + 4[x_{1}] − [x_{0}], 3[x_{4}] + 13[x_{2}] + 14[x_{1}] − 2[x_{0}] and [x_{5}] + [x_{4}] + [x_{1}] − [x_{0}] of the β_{j} belonging to ker(ℒ_{3,F}). In order to deduce elements in A_{4}(F) from them, we first eliminate [x_{0}] by taking appropriate linear combinations and then formally replace [x_{i}] by \frac{1}{ε_{i}} [x_{i}] (“pseudo-integration with respect to θ”). We are left with γ_{4} = 9[x_{4}] − 8[x_{3}] + 36[x_{2}] + 72[x_{1}] and γ_{5} = [x_{5}] − 4[x_{4}] + 3[x_{3}] − 46[x_{2}] − 57[x_{1}], which are mapped

by ℒ_{4} ∘ σ_{1} to \frac{3}{128} \frac{23^{7/2}}{π^{4}} \frac{ζ_{F} (4)}{ζ(4)} and 0, respectively. Finally, by “pseudo-integrating” γ_{5}, we obtain

the combination δ_{5} = \frac{1}{5}[x_{5}] − [x_{4}] + [x_{3}] − 46[x_{2}] − 57[x_{1}] ∈ A_{5}(F) ⊗ ℚ, and the elements [x_{0}] and

5(δ_{5} − [x_{0}]) in A_{5}(F) map under ℒ_{5,F} to ζ(5) · (1, 1) and \frac{1}{128} \frac{23^{9/2}}{π^{5}} \frac{ζ_{F} (5)}{ζ(5)} · (−2, 1), respectively.

4. The “enhanced” polylogarithm

The enhanced regulator lattice. The regulator map reg_{m} : K_{2m−1}(ℂ) → ℝ is known to have a natural lift to ℂ/ℚ(m), and one can show (cf. [3]) that also the polylogarithm map ℒ_{m} : B_{m}(ℂ) → ℝ enjoys this property (here we adopt the usual notation ℚ(m) = (2πi)^{m}ℚ). We will call this lifted function, whose explicit definition will be described in this section, the enhanced polylogarithm and denote it by ℒ̂_{m}. If F is a number field, then we can combine ℒ̂_{m} as before with the embeddings F ↪ ℂ to get a map ℒ̂_{m,F} from B_{m}(F) to (ℂ/ℚ(m))_{+}^{Σ}. Moreover,

the construction of ℒ̂_{m} is such that the indeterminacy on B_{m}(F) has bounded denominator, i.e. N · ℒ̂_{m,F} takes values in (ℂ/ℤ(m))_{+}^{Σ} for some integer N = N(m, F). Note that (ℂ/ℤ(m))_{+}^{Σ} is topologically the product of the Euclidean space ℝ^{n_{∓}} with a torus (ℝ/(2π)^{m}ℤ)^{n_{±}}. The inverse image under the projection ℝ^{n} = ℂ_{+}^{Σ} → (ℂ/ℤ(m))_{+}^{Σ} of the image of A_{m}(F) under N · ℒ̂_{m,F} is then a cocompact lattice R̂_{m,F} ⊂ ℝ^{n};

0 → ℤ(m)^{n_{±}} ⟶ R̂_{m,F} ⟶ N · ℒ̂_{m,F} (A_{m}(F)) ⟶ 0
∥  ⋂  ⋂
0 → ℤ(m)^{n_{±}} ⟶ ℝ^{n} ⟶ ℝ^{n_{∓}} × (ℝ/ℤ(m))^{n_{±}} ⟶ 0

The regulator lattice R_{m,F} discussed in §2 had rank n_{∓}. The new (“enhanced”) lattice R̂_{m,F} has rank n = n_{+} + n_{−} and is (up to commensurability, as usual) the semi-direct product of R_{m,F} by the lattice ℤ(m)^{n_{±}}. In particular, its covolume contains no new information since it is, up to a rational factor, just π^{mn_{±}} times the covolume of R_{m,F} : The fact that the lattice R̂_{m,F} is a semidirect product means that the torus ℝ^{n}/R̂_{m,F} is a torus bundle over ℝ^{n_{∓}}/R_{m,F} with fibre (ℝ/ℤ(m))^{n_{±}}, and the volumes multiply. Equivalently, the matrix representing R̂_{m,F} with respect to the standard basis of ℝ^{n} has the form

( \begin{matrix}(2π)^{m}I_{n_{±}} & ∗ \\ 0 & matrix of R_{m,F} \end{matrix} ) ,

so the determinant is just (2π)^{mn_{±}} times the previous determinant. The enhanced lattice is nevertheless of interest, for several reasons:

(1) It contains more information, as described by the off-diagonal block, whose elements are in general non-trivial (see example below).

(2) It gives a more elegant formulation of the main conjecture since the covolume of R̂_{m,F} is now a rational multiple of |D_{F} |^{1/2}ζ_{F} (m), without the extra factor π^{mn_{±}}.

(3) One may sometimes have an independent construction of a candidate for the regulator lattice, and if this candidate also has an enhanced version then one can study the stronger conjecture that the enhanced lattices agree. We will see an example of this in §7.

Summarizing the above, we have

Using the m^{th} enhanced polylogarithm on the Bloch group, we can associate to any number field F of degree n a regulator lattice R̂_{m,F} ⊂ ℝ^{n} for every m ≥ 2 whose covolume is given (up to a rational number) by the value |D_{F} |^{1/2} ζ_{F} (m) .

Lifting the dilogarithm. We describe explicitly how to construct the enhanced polylogarithm ℒ̂_{m} in the case m = 2. The first observation is that the ambiguity of the many-valued function

F (u) = Li_{2}(e^{u}) − u Li_{1}(e^{u}) , (u ∈ ℂ)

is contained in ℤ(2) = (2πi)^{2}ℤ. (Here Li_{2}(e^{u}) and Li_{1}(e^{u}) are to be defined by analytic continuation along the same path.) This is because the derivative of F is the meromorphic function u/(1 − e^{−u}), which has only simple poles (at u = 2πin, n ∈ ℤ) with residue (= 2πin) in 2πiℤ. We can think of F as a holomorphic function on ℂ − (2πiℤ − {0}) with values in ℂ/ℤ(2). It satisfies the functional equations

F (u) + F (−u) = \frac{π^{2}}{3} + \frac{u^{2}}{2}  (10)

(Proof: both sides have derivative u and equal \frac{π^{2}}{3} at 0) and

F (u + 2πi) − F (u) = 2πi log(1 − e^{u}) ∈ ℂ/(2πi)^{2}ℤ  (11)

(Proof: both sides have derivative \frac{2πi}{1 − e^{−u}} and vanish at −∞). The well-definedness of F modulo

ℤ(2) says that the combination Li_{2}(x) − log(x) Li_{1}(x) becomes well-defined modulo ℤ(2) as soon as we choose a branch of log(x) , i.e. a lift of x to the abelian cover {u ∈ ℂ | e^{u} ≠ 1} of ℂ − {0, 1}, and hence that Li_{2} itself is well-defined modulo ℤ(2) on the maximal abelian cover X = {(u, v) ∈ ℂ^{2} | e^{u} + e^{v} = 1} of ℂ − {0, 1}(= ℙ^{1}(ℂ) − {0, 1, ∞}). Instead of Li_{2} we use the function F̂ : X → ℂ/ℤ(2) defined by

F̂(u, v) = F (u) − \frac{1}{2}uv .

Its ambiguity with respect to changes of the logarithmic determinations u and v is given by

F̂(u + 2πir, v + 2πis) = F̂(u, v) + πi(rv − su) + 2π^{2}rs  (r, s ∈ ℤ) .  (12)

and its imaginary part is given by:

ℑ(F̂(u, v)) = D(e^{u}) + \frac{1}{2}ℑ(\overline{u}v) ,  (13)

as one easily verifies.

If we now consider an element ξ = ∑ n_{j}[x_{j}] of ker(β_{2}), form the group G = ⟨x_{j}, 1−x_{j} | j⟩ and choose an arbitrary lifting of each x_{j} to X, i.e. choose a logarithm u_{j} of x_{j} and a logarithm v_{j} of 1−x_{j} for each j, then the complex values u_{j} and v_{j} all belong to the group Ĝ = {t ∈ ℂ | e^{t} ∈ G}, a free ℤ-module of rank =rk(G) + 1. (Here, as earlier, G is the subgroup of ℂ^{×} generated by all the x_{j} and 1 − x_{j}.) The property ∑ n_{j}(x_{j} ∧ (1 − x_{j})) = 0 now becomes

∑ n_{j} (u_{j} ∧ v_{j}) = 2πi ∧ A ∈ ⋀^{2}(Ĝ) ,

with A ∈ Ĝ being well-defined modulo the subgroup 2πiℤ ⊂ Ĝ. We define the “enhanced dilogarithm” by

D̂(ξ) = ∑ n_{j}F̂(u_{j}, v_{j}) − πiA ∈ ℂ/2π^{2}ℤ .

Notice that we have taken the range of D̂ to be ℂ/2π^{2}ℤ rather than ℂ/4π^{2}ℤ, because A is defined only modulo 2πiℤ. It is easily verified that replacing u_{j} and v_{j} by u_{j} + 2πir_{j} and v_{j} + 2πis_{j} for some integers r_{j}, s_{j} changes A to A+2πi ∑ n_{j}(r_{j}v_{j} −s_{j}u_{j}). Formula (12) then implies that D̂(ξ) is independent of the chosen lifting (u_{j}, v_{j}) of x_{j}, while formula (13) gives ℑ(D̂(ξ)) = D(ξ), since ∑ n_{j} ℑ(\overline{u_{j}}v_{j}) = 0. The 5-term relation for D automatically remains valid for D̂. The function D̂ is then our desired lifting of D : B_{2}(ℂ) → ℝ to ℂ/2π^{2}ℤ.

If we look at an element ξ = ∑ n_{j}[x_{j}] with all x_{j} real, then D̂(ξ) is real by (13) and because D(x) = 0 for x real. Modulo π^{2}/2 it is given by

D̂(ξ) = ∑_{j} n_{j} R(x_{j}) (mod π^{2}/2) ,

where R : ℝ → ℝ is the Rogers dilogarithm function, defined by the formulas

R(x) = \{ \begin{matrix}Li_{2} + \frac{1}{2} log(x) log(1 − x) & if 0 < x < 1, \\ π^{2}/3 − R(1/x) & if x > 1, \\ −π^{2}/6 + R(1/(1 − x)) & if x < 0, \end{matrix}

together with the values R(0) = 0, R(1) = π^{2}/6 dictated by continuity. We have R(−∞) = −π^{2}/6, R(+∞) = π^{2}/3, so that R gives a continuous function from ℙ^{1}(ℝ) = ℝ∪\{∞\} to ℝ/\frac{1}{2}π^{2}ℤ. Thus, in the case of real arguments there is a version of the real part of the dilogarithm which is defined modulo π^{2}/2 even for single arguments x and we gain only a factor of 4 by considering elements of A_{2}(ℝ), whereas in the complex case there is no lifting of individual dilogarithms at all and it is only on the subgroup A_{2}(ℂ) of ℤ[ℂ] that we get a lifting of the real part of the dilogarithm to the quotient of ℝ by a lattice of rank 1.

Example. We return to Example 3 of §3. There we gave six elements α_{i} ∈ A_{2}(F), each of which is mapped to 0 under D̂∘σ_{0} and to a rational multiple of π^{−4}\sqrt{23} ζ_{F} (2) under D̂∘σ_{1}. Let us consider the image of one of these elements, say α_{1} = [θ], under the enhanced dilogarithm. (The other α_{j} give nothing new since α_{j} = λ_{j}α_{1} in the Bloch group.) Let t_{0} = log θ_{0} = 0.281199 …, t_{1} = log θ_{1} = −0.140599 … + 2.437734 . . . · i. We have 1 − θ = −θ^{−4}, so for the real embedding we can take u = t_{0}, v = −4t_{0} + πi, obtaining

D̂(σ_{0}(α_{1})) = F̂(t_{0}, −4t_{0} + πi) + \frac{πit_{0}}{2} = F (t_{0}) + 2t_{0}^{2} = R(θ_{0}) = 2.10466 …,

and similarly for the complex embedding

D̂(σ_{1}(α_{1})) = F̂(t_{1}, −4t_{1} + πi) + \frac{πit_{1}}{2} = −11.7444041862 … + 0.471353681388 . . . · i .

Note that the original regulator lattice R_{2,F} ⊂ ℝ is the rank 1 group generated by the real value D(θ_{1}) = ℑ(D̂(θ_{1})), whereas the new lattice R̂_{2,F} ⊂ ℝ × ℂ is generated by the three vectors (D̂(θ_{0}), D̂(θ_{1})), (π^{2}, 0) and (0, π^{2}), and (presumably) does not split over ℚ into the product of sublattices of ℝ and ℂ, since the ratio of D̂(θ_{0}) and π^{2} is (presumably) not a rational number. (We do, however, have the relation D̂(θ_{0}) + D̂(θ_{1}) + D̂(θ_{2}) = −13 π^{2}/6.)

Remark. We saw in §1 above that by triangulating a hyperbolic 3-manifold M one gets a well-defined element ξ_{M} ∈ B(ℂ) whose image under D is the value of M. The enhanced dilogarithm D̂ then gives us a finer invariant D̂(ξ_{M}) whose imaginary part is the volume of M and whose real part is a well-defined number in ℂ/2π^{2}ℤ. According to a suggestion of Thurston, it should be (up to a normalizing factor) the Chern-Simons invariant of M. Cf. [47], [37], [51].

Lifting the polylogarithm. We begin by giving a different formula for the enhanced dilogarithm which is less precise than the above (it is defined modulo π^{2}ℚ only instead of 2π^{2}ℤ) but is simpler and easier to generalize. For any complex number U congruent to log x modulo 2πiℚ, we set

F(x, U) = Li_{2}(x) − ULi_{1}(x) = F (u) − (U − u)Li_{1}(x) ∈ ℂ/ℚ(2) ,

where u is any choice of logarithm of x. This is well-defined for a given choice of u because U −u ∈ 2πiℚ and the ambiguity of Li_{1}(x) lies in 2πiℤ, and it is independent of the choice of u by (11). If now ξ = ∑ n_{j}[x_{j}] is any element of ker(β_{2}), and if U_{j} and V_{j} are any complex numbers satisfying

U_{j} ≡ log x_{j} (mod ℚ(1)) , V_{j} ≡ log(1 − x_{j}) (mod ℚ(1))

(rather than actual choices of log x_{j} and log(1 − x_{j}) like our previous u_{j}, v_{j}), and if we define A ∈ ℂ^{×}/ℚ(1) by ∑ n_{j}(U_{j} ∧ V_{j}) = 2πi ∧ A as before, then one checks immediately that the expression

ℒ̂_{2}(ξ) := ∑_{j} n_{j}(F(x_{j}, U_{j}) − \frac{1}{2}U_{j}V_{j}) − πiA ∈ ℂ/ℚ(2)

is independent of the choices of U_{j} and V_{j} and is the reduction modulo ℚ(2) of the previously defined invariant D̂(ξ) ∈ ℂ/\frac{1}{2}ℤ(2). The advantage of the new construction is that there is a way of choosing U_{j} and V_{j} for which the number A automatically vanishes, leading to a simpler definition. Let L be any homomorphism ℂ^{×} → ℂ satisfying

L(x) ≡ log(x) (mod ℚ(1)) ∀ x ∈ ℂ^{×} ,

i.e. L is any lift

[figure: ℂ; ℂ^{×}; L; ℂ^{×}/torsion; exp; ∼; ℂ/ℚ(1)]

(Defining such a lift on all of ℂ^{×} would require the axiom of choice, but we will only ever be using its restriction to a finitely generated subgroup G ⊂ ℂ^{×}—specifically, the group generated by the x_{j} and 1−x_{j} for all x_{j} occurring in some elements of the Bloch group of a number field—and then the lift can be defined simply by splitting G as the sum of a free abelian and a finite group and choosing arbitrary logarithms of the generators of the free part.) Then with the choices U_{j} = L(x_{j}), V_{j} = L(1 − x_{j}), we find immediately that the condition ∑ n_{j}(x_{j} ∧ (1 − x_{j})) = 0 ∈ ⋀^{2}(ℂ^{×}) ⊗ ℚ implies A = 0, so we can define ℒ̂_{2}(ξ) simply as

ℒ̂_{2}(ξ) := ∑_{j} n_{j}(F(x_{j}, L(x_{j})) − \frac{1}{2}L(x_{j})L(1 − x_{j})) ∈ ℂ/ℚ(2) .

We can now generalize this to higher m. The function F (u) is replaced by

F_{m}(u) = ∑_{n=0}^{m−1} \frac{(−u)^{n}}{n!} Li_{m−n}(e^{u}) ,

whose derivative \frac{(−1)^{m}}{(m − 1)!} \frac{u^{m−1}}{1 − e^{−u}} has residues in \frac{1}{(m − 1)!}ℤ(m − 1) and which therefore is a

well-defined holomorphic map from ℂ to ℂ/\frac{1}{(m − 1)!}ℤ(m − 1). Its change under u ↦ u + 2πi is

given by

F_{m}(u + 2πi) = ∑_{n=0}^{m−1} \frac{(−2πi)^{n}}{n!} F_{m−n}(u) ,

and using this we can check that the expression

F_{m}(x, U) = ∑_{n=0}^{m−1} \frac{(log x − U)^{n}}{n!} F_{m−n}(log x) ∈ ℂ/ℚ(m) (U ∈ ℂ, U ≡ log x (mod 2πiℚ)) ,

is independent of the choice of log x. In fact we can define F_{m}(x, U) directly, without choosing log x at all.

Proposition 1. Let x ∈ ℂ − {0, 1} and U ∈ ℂ with U ≡ log(x)(mod ℚ(1)). Then the quantity

F_{m}(x, U) = ∑_{n=0}^{m−1} \frac{(−U)^{n}}{n!} Li_{m−n}(x) ,

where all the polylogarithms Li_{m−n}(x) are defined by analytic continuation along the same path, is well-defined. The effect of changing U is given by the formula

F_{m}(x, U + λ) = ∑_{n=0}^{m−1} \frac{(−λ)^{n}}{n!} F_{m−n}(x, U)  (λ ∈ ℚ(1)) .  (14)

Proof: The polylogarithm can be written as

Li_{m}(x) = \frac{1}{(m − 1)!} ∫_{0}^{∞} \frac{t^{m−1}dt}{x^{−1}e^{t} − 1}

(Proof: for |x| < 1 expand \frac{1}{x^{−1}e^{t} − 1} in a geometric series and integrate term by term), so

F_{m}(x, U) = \frac{1}{(m − 1)!} ∫_{0}^{∞} \frac{(t − U)^{m−1}dt}{x^{−1}e^{t} − 1} .

The many-valuedness of this comes only from the choice of path of integration from 0 to ∞, and since the residues {(t−U)^{m−1} | e^{t} = x} of the integrand are all in ℚ(m−1), the ambiguity which this produces is contained in ℚ(m), as asserted (and in fact with a controlled denominator: at most N^{m−1}(m − 1)! if xe^{−U} is an N^{th} root of unity). The second statement of the proposition is an immediate consequence of the integral representation. □

We now define the enhanced polylogarithm for ξ = ∑ n_{j}[x_{j}] ∈ A_{m}(ℂ) by

ℒ̂_{m}(ξ) = ∑_{j} n_{j}(F_{m}(x_{j}, L(x_{j})) − \frac{(−1)^{m}}{m!} L(x_{j})^{m−1}L(1 − x_{j})) ,

where L as before is any lift to ℂ of the logarithm map ℂ^{×} → ℂ/ℚ(1).

Proposition 2. Let ξ ∈ A_{m}(ℂ), m ≥ 2. Then:

(i) The value of ℒ_{m}(ξ) is independent of the choice of L .

(ii) ℜ_{m}(ℒ̂_{m}(ξ)) = ℒ_{m}(ξ) .

(iii) ℒ_{m}(ξ) = 0 if ξ ∈ C_{m}(ℂ) .

Proof: We prove only (i), the ideas for (ii) and (iii) being similar. We must show that the number ℒ̂_{m}^{∗}(ξ) obtained by replacing L by L^{∗} = L + λ, where λ is a homomorphism from ℂ^{×} to ℚ(1), is equal to ℒ̂_{m}(ξ) (modulo ℚ(m)) for ξ ∈ A_{m}(ℂ). We have

L^{∗}(x)^{m−1}L^{∗}(1 − x) − L(x)^{m−1}L(1 − x)

≡ ∑_{n=0}^{m−2} \binom{m − 1}{n}L(x)^{n}λ(x)^{m−n−1}L(1 − x) + ∑_{n=1}^{m−1} \binom{m − 1}{n}L(x)^{n}λ(x)^{m−n−1}λ(1 − x)

= ∑_{n=1}^{m−1} \binom{m − 1}{n}L(x)^{n−1}λ(x)^{m−n−1}(L(x)λ(1 − x) − L(1 − x)λ(x))

+ ∑_{n=1}^{m−1} \binom{m}{n}L(x)^{n−1}λ(x)^{m−n}L(1 − x) ,

where ≡ denotes congruence modulo ℚ(m) and in the second line we have used the identity \binom{m−1}{n−1} + \binom{m−1}{n} = \binom{m}{n}. From this and (14) it follows that

ℒ̂_{m}^{∗}(ξ) − ℒ̂_{m}(ξ) = ∑_{n=1}^{m−2} \frac{(−1)^{n}}{n!} ℒ̂_{m−n}( ∑_{j} n_{j}λ(x_{j})^{n}[x_{j}])

− \frac{(−1)^{m}}{m!} ∑_{n=1}^{m−1} \binom{m − 1}{n} ∑_{j} n_{j} L(x_{j})^{n−1}λ(x_{j})^{m−n−1}(L(x_{j})λ(1 − x_{j}) − L(1 − x_{j})λ(x_{j})) .

The first sum vanishes by inductive use of (iii) because ∑ n_{j}λ(x_{j})^{n} [x_{j}] ∈ C_{m−n}(ℂ), and the second because ∑ n_{j}[x_{j}] ∈ ker β_{m}. □

The content of the proposition, and of this whole section, can be summarized by the following diagram:

[figure: C_{m}(ℂ) ⊗ ℚ; 0; A_{m}(ℂ) ⊗ ℚ; _{ℒ̂_{m}}; ℚ[ℂ]; _{ℒ_{m}}; ℂ/ℚ(m); ℜ_{m}; ℝ]

Example. Again, we take up Example 3 of §3 and compute the enhanced trilogarithm for the elements β_{j} of B_{3}(F) (j = 0, 2, …, 5). We need only consider σ_{1} since the imaginary part of ℒ̂_{3}(ξ) changes sign under ξ ↦ \overline{ξ}. Choosing L(θ_{1}) = log(θ_{1}) = −0.1405 … + 2.4377 … i, we obtain

ℒ̂_{3}(σ_{1}(β_{j})) = μ_{j} \frac{ζ(3)}{3} + ν_{j}( − \frac{23^{5/2}}{288π^{2}} \frac{ζ_{F} (3)}{ζ(3)} + f · i) + ω_{j} · \frac{(2πi)^{3}}{24}

for a certain real number f = 15.13531974616481725314880543 …, where the integers μ_{j} and ν_{j} are as in Example 3 of §3 and (ω_{j})_{j=0,2,... ,5} = (0, 0, 3, 19, 23) . If we choose instead L^{∗}(θ_{1}) = L(θ_{1}) − iπ, then we find the same formula with (ω_{j})_{j} = \frac{1}{2}(0, 12, −15, −14, 11). The number

f can be considered as an “imaginary part” of −\frac{23^{5/2}}{288π^{2}} \frac{ζ_{F} (3)}{ζ(3)} , defined only modulo π^{3}ℚ (or

more precisely, \frac{π^{3}}{3} ℤ). We will give an interpretation of this somewhat cryptic statement in the following chapter.

Part II. A higher Kronecker limit formula for imaginary quadratic fields

5. A conjectural higher Kronecker limit formula

Let E/F be an extension of number fields, which we may assume to be Galois. Then the zeta function of E can be written as the product of so-called Artin L-series L_{F} (s, ρ) associated to an irreducible representation ρ of the Galois group G. In particular, the residue res_{s=1}ζ_{E}(s) is a product of res_{s=1}ζ_{F} (s) and special values L_{F} (1, ρ). On the other hand, the residue is the covolume of the group of units O_{E}^{×} on which G acts, which implies that O_{E}^{×} (at least after tensoring with ℂ or \overline{ℚ}) decomposes as a direct sum of irreducible representations of G and consequently that the regulator decomposes into a corresponding product of factors. Comparing these two facts, one is led to the famous

Conjecture (Stark). The value L_{F} (1, ρ) is expressible as a simple multiple (a power of π times an algebraic number) of the determinant of a matrix whose entries are logarithms of units in E.

This conjecture which was formulated in far greater precision by Stark [43] is totally open except in very special cases, but one can nevertheless generalize it to higher special values. The number ζ_{E}(m) is both a product of special values of L_{F} (m, ρ) and, by Borel’s theorem, the covolume of the regulator lattice Reg_{m,E} on which G acts. Therefore it is natural to guess that each special value L_{F} (m, ρ) is the volume of one of the lattices into which the G-action splits Reg_{m,E}. This conjecture, which was made by Gross in 1974 [27], combined with the conjectural description of the Borel regulator map in terms of polylogarithms, gives the

Strengthened polylogarithm conjecture. The value at s = m of any Artin L-function L_{F} (s, ρ) is a simple multiple of the determinant of a matrix whose entries are polylogarithms evaluated at elements in the higher Bloch group B_{m}(E).

Again, one can easily make this a little more precise, and could probably make a very detailed analysis along the lines of the one given in [45] for m = 1 if one wished, but this seems pointless as long as even the case m = 1 remains unapproachable.

A particularly interesting special case arises if we concentrate on abelian extensions of an imaginary quadratic field K, where we can formulate a much more precise statement which in the case m = 1 reduces to a famous classical theorem. For simplicity we restrict to the Hilbert class field H of K. We first observe that the zeta function ζ_{H}(s) of H can be expressed as the determinant of a matrix whose entries are the partial zeta functions of the quadratic field. This follows easily from the well-known decomposition of ζ_{H} into L-series associated to ideal class characters χ:

ζ_{H}(s) = ∏_{χ:Cℓ_{K}→ℂ^{×}} L_{K}(s, χ) = ∏_{χ} ( ∑_{A∈Cℓ_{K}} χ(A) ζ_{K,A}(s))

= det ((ζ_{K,AB^{−1}}(s))_{A,B∈Cℓ_{K}} ) .

(Here Cℓ_{K} denotes the ideal class group, L_{K}(s, χ) = ∑_{𝔞} χ(𝔞)N(𝔞)^{−s} the associated L-series and ζ_{K,A}(s) = ∑_{𝔞∈A} N(𝔞)^{−s} the partial zeta function associated to the ideal class A. The final identity is a special case of the theorem of Frobenius about group determinants and can be proved easily since conjugation by the matrix (χ(A))_{χ,A} diagonalizes the matrix of partial zeta functions.) But by the conjecture stated in §2 we expect ζ_{H}(m) ( m = 2, 3, 4, …) to be a multiple of the determinant of an h × h-matrix, where h = #Cℓ_{K} = [H : K] = n_{±}(H) , whose entries are polylogarithms evaluated at arguments in H. (If H were an arbitrary abelian extension of K instead of the maximal unramified one, then Cℓ_{K} would be replaced by a ray class group, but the analysis would be otherwise identical.) The case s = 1 where the values of the partial zeta functions can be written in terms of logarithms of units in F is a consequence of Kronecker’s limit formula and the theory of complex multiplication, and the construction of units in the Hilbert class field from special values of modular forms constitutes one of the high points of 19^{th} century number theory.

It is therefore natural to guess that in fact the individual entries agree and that each partial zeta-value ζ_{K,A}(m) is a linear combination of polylogarithm values, and in fact, numerical examples suggested its statement. We consider K as embedded into ℂ. Then there is a distinguished embedding of H into ℂ as K(j(z_{1})) where j : ℍ/SL(2, ℤ) → ℂ is the classical modular invariant and z_{1} = \frac{D+\sqrt{D}}{2} ∈ ℍ, the upper half plane. The other embeddings of H into ℂ are obtained from this one by conjugating it with the elements of Gal(H/K) and with complex conjugation. Moreover, by class field theory Gal(H/K) is isomorphic to Cℓ_{K} via the Artin symbol. A fairly precise formulation of the conjecture indicated above is then:

Conjecture. Let K be an imaginary quadratic number field and m > 1. Then there is an element ξ = ξ_{m,K} ∈ B_{m}(H) ⊗ ℚ, where H is the Hilbert class field of K, such that

ζ_{K,A}(m) = 2 \frac{(−1)^{[ \frac{m−1}{2} ]} (2π)^{m}}{(m − 1)!} |D_{K}|^{\frac{1}{2} −m} ℒ_{m}(σ_{A}(ξ))  (15)

for every ideal class A of K, where σ_{A} ∈ Gal(H/K) is the Artin symbol associated to A.

For m = 1 the statement must be modified because each ζ_{F} (m) has a simple pole, but ζ_{H}(s)/ζ_{K}(s) = ∏_{χ≠1} L_{m}(s, χ) is finite at s = 1 and can be expressed as a determinant of values (ζ_{K,A}(s) − ζ_{K,B}(s))|_{s=1} where A and B denote ideal classes in K, and the Kronecker limit formula expresses each of these differences as the logarithm of a certain element in H defined by complex multiplication theory.

Remarks. 1. ξ_{m,K} is unique if it exists (assuming the general polylogarithm conjecture), since by specifying ℒ_{m}(σ(ξ)) for all σ ∈ Gal(H/K) we have specified the image of ξ under the full map ℒ_{m,H} : A_{m}(H) → ℝ^{h}, and the polylogarithm conjecture says that this map is injective (up to torsion). It also follows that c(ξ_{m,K}) = (−1)^{m−1}ξ_{m,K} , where c ∈ Gal(H/ℚ) denotes complex conjugation.

2. The partial zeta function ζ_{K,A}(s) in the case of an imaginary quadratic field K can also be written as an Epstein zeta function: the norms of the integral ideals in A are just the values attained by some quadratic form Q(x, y) = ax^{2} +bxy +cy^{2} (a, b ∈ ℤ) with discriminant b^{2} −4ac = D_{K}, so

ζ_{K,A}(s) = ζ_{Q}(s) = \frac{1}{2} \begin{matrix}∑ & ′ \\ x,y \end{matrix} \frac{1}{Q(x, y)^{s}} .

(Here the prime means that (x, y) = (0, 0) is to be omitted and the factor \frac{1}{2} takes care of the automorphism (x, y) ↦ (−x, −y). If K = ℚ(i) or K = ℚ(\sqrt{−3}), it must be replaced by \frac{1}{4} or \frac{1}{6}, but we will ignore these cases since the conjecture is only interesting if h > 1.) Thus the conjecture says in particular:

Let Q be any positive definite binary quadratic form with rational coefficients and let m ≥ 2. Then |disc(Q)|^{−1/2}π^{−m}ζ_{Q}(m) is a ℚ-linear combination of values of ℒ_{m}(x) with x algebraic.

3. The partial zeta function can also be written as a non-holomorphic Eisenstein series: if Q(m, n) = am^{2} + bmn + cn^{2} is a form in the class associated to A, and we set z_{Q} = \frac{b+\sqrt{D_{K}}}{2a} ∈ ℍ, then Q(m, n) = |D_{K}/4|^{1/2} |mz_{Q} + n|^{2}/ℑ(z_{Q}), so

ζ_{Q}(s) = |D_{K}/4|^{−s/2}E(z_{Q}, s) ,  (16)

where

E(z, s) = \frac{1}{2}\begin{matrix}∑ & ′ \\ m,n∈ℤ \end{matrix} \frac{ℑ(z)^{s}}{|mz + n|^{2s}} (z ∈ ℍ, ℜ(s) > 1),

is the non-holomorphic Eisenstein series of weight 0. This is useful because the Fourier expansion of E(z, s) is explicitly known and converges with exponential rapidity (at least like ∑ e^{−nπ\sqrt{3}}), so that we can compute special values of partial zeta functions very easily.

Examples. 1. Let K = ℚ(\sqrt{−23}). The class number is 3 and the Hilbert class field is H = K(θ) = K · F, where F = ℚ(θ) with θ^{3} − θ − 1 = 0 is the field considered in Example 3 of §3. The class group Cℓ_{K} consists of the classes A_{0} = {(λ)} (the principal class), A_{1} = {(λ)℘} and A_{−1} = \{(λ) ¯℘\} where ℘ = (2, \frac{1+\sqrt{−23}}{2}) is one of the prime ideals dividing 2. The corresponding zeta functions ζ_{j}(s) = ζ_{K,A_{j}}(s) (j ∈ ℤ/3ℤ) are given by

ζ_{0}(s) = \frac{1}{2} \begin{matrix}∑ & ′ \\ m,n \end{matrix} \frac{1}{(m^{2} + mn + 6n^{2})^{s}} , ζ_{±1}(s) = \frac{1}{2} \begin{matrix}∑ & ′ \\ m,n \end{matrix} \frac{1}{(2m^{2} + mn + 3n^{2})^{s}} .

These partial zeta functions are related to the Dedekind zeta functions of K and F by

ζ_{0}(s) + 2ζ_{1}(s) = ζ_{K}(s) , ζ_{0}(s) − ζ_{1}(s) = \frac{ζ_{F} (s)}{ζ(s)} ,

so by formulas (2) and (9) with i = 1, their values at s = 2 are expressible in terms of dilogarithms as

ζ_{0}(2) = \frac{4π^{2}}{9 · 23^{3/2}} (D(ξ_{23}) + 24D(θ_{1})) ,  ζ_{1}(2) = \frac{4π^{2}}{9 · 23^{3/2}} (D(ξ_{23}) − 12D(θ_{1})) , (17)

where we have labelled the conjugates of θ as θ_{j} (j ∈ ℤ/3ℤ) as in §3. We can check these formulas numerically since ζ_{0}(2) and ζ_{1}(2) can be computed rapidly using (16) and (24) below:

ζ_{0}(2) = \frac{π^{4}}{90} + \frac{8π}{23\sqrt{23}}(\frac{1}{2}ζ(3) + ∑_{n=1}^{∞}(−1)^{n}σ_{−3}(n) (1 + πn\sqrt{23}) e^{−πn\sqrt{23}} )
= 1.2192662262804613691002600581331744360615022454119 …,
ζ_{1}(2) = \frac{π^{4}}{360} + \frac{16π}{23\sqrt{23}}(\frac{1}{2}ζ(3) + ∑_{n=1}^{∞}(−1)^{n}σ_{−3}(2n) (1 + πn\sqrt{23}) e^{−πn\sqrt{23}} )
= 0.5444665316326198108031691850861304122679869701278 …,

where σ_{−3}(n) = ∑_{d|n} d^{−3}. Equation (17) verifies the conjecture for this field and m = 2 with

ξ_{2,K} = \frac{1}{18}([ξ_{23}] + 12[σ(θ)] − 12[σ^{2}(θ)]) ,  (18)

where σ is a generator of the (cyclic) Galois group Gal(H/K). Of course this above representation as a sum of dilogarithms is not unique. We give a different expression which is more homogeneous with respect to the action of the Galois group. The unit group of H is generated by θ_{0}, θ_{1} and θ_{2} with the relation θ_{0}θ_{1}θ_{2} = 1. Then the elements

ξ_{H,−23}^{(j)} = 8[−θ_{j}^{4}θ_{j+1}^{2}] − 5[−θ_{j}^{4}θ_{j+1}^{3}] + 21[−θ_{j}^{2}θ_{j+1}] (j ∈ ℤ/3ℤ) ,

satisfy

D(ξ_{H,−23}^{(j)}) = −\frac{3 · 23^{3/2}}{2π^{2}} ζ_{j}(2)

and are cyclically permuted by Gal(H/K), so that the element ξ_{2,K} := −\frac{1}{12}ξ_{H,−23} has the property required by the conjecture.

2. For other K = ℚ(\sqrt{−d}) with class number 3 we again find that the sum ζ_{0}(2) + 2ζ_{1}(2) = ζ_{K}(2) has a complicated expression as a combination of dilogarithms of numbers in K, but that the difference ζ_{0}(2) − ζ_{1}(2) = ζ_{F} (2)/ζ(2) equals 4π^{2}d^{−3/2}D(σ_{1}(ξ_{F})) for some very simple element ξ_{F} of the Bloch group of the totally real subfield F of the Hilbert class field of K, e.g.:

\begin{matrix}d & F = ℚ(θ) & ξ_{F} \\ 23 & θ^{3} − θ − 1 = 0 & [θ] \\ 31 & θ^{3} − θ^{2} − 1 = 0 & [θ] \\ 59 & θ^{3} − 2θ^{2} − 1 = 0 & 4[θ] − 2[−θ^{2}] \\ 83 & θ^{3} − 2θ^{2} − 2θ − 1 = 0 & 6[θ] − 2[θ^{2}] \\ 107 & θ^{3} − 4θ^{2} + 2θ − 1 = 0 & 8[θ] − [θ^{2}] \end{matrix}

In each of these cases, the conjecture is verified for m = 2 with ξ_{2,K} = \frac{1}{3}ξ_{K} + \frac{1}{3}(σ_{1}(ξ_{F})−σ_{2}(ξ_{F})), where ξ_{K} ∈ B_{2}(K) is an element with D(ξ_{K}) = 4π^{2}d^{−3/2}ζ_{K}(2). There are also examples like ξ_{H,−23} above, e.g. the combination ξ_{H,−31}^{(j)} = 19[−θ_{j−1}^{2}θ_{j}]−3[−θ_{j−1}^{5}θ_{j}^{3}]+8[θ_{j−1}^{3}]−8[θ_{j}^{3}] for d = 31 with

D(ξ_{H,−31}^{(j)}) = −\frac{3 · 31^{3/2}}{2π^{2}} ζ_{j}(2)  (j = 0, 1, 2) ,  θ_{j}^{3} − θ_{j}^{2} − 1 = 0 .

3. We can proceed similarly for higher polylogarithms. For instance, for d = 23 and the trilogarithm we have

ζ_{0}(3) + 2ζ_{1}(3) = ζ_{K}(3) = L(3, (\frac{·}{23})) ζ(3) = \frac{96π^{3}}{23^{5/2}} ℒ_{3}(1)

(which is much simpler than the corresponding formula (2) for m = 2, the reason being that for odd m the group B_{m}(K) coincides up to torsion with B_{m}(ℚ)) and

ζ_{0}(3) − ζ_{1}(3) = ζ_{F} (3)/ζ(3) = \frac{72π^{3}}{23^{5/2}} (ℒ_{3}(σ_{0}(β_{2})) − ℒ_{3}(σ_{1}(β_{2})))

by Example 3 of §3, and taking linear combinations, we obtain each partial zeta function ζ_{j}(3) as a combination of trilogarithms. A different element which exhibits the Galois equivariance property stated in the conjecture above is ξ = [θ] − [θ^{2}] + [θ^{5}], for which we have

\frac{96π^{3}}{23^{5/2}} ℒ_{3}(σ_{j}(ξ)) = ζ_{j}(3) (j ∈ ℤ/3ℤ) ,

which is again much simpler than the corresponding formula for the dilogarithm.

4. We now consider examples with class number h > 3, for which the partial zeta functions ζ_{A}(s) can not in general be expressed in terms of Dedekind zeta functions of subfields of H. For simplicity we restrict to d prime. Then h is odd and ζ_{H}(s)/ζ_{K}(s) is the square of the quotient ζ_{F} (s)/ζ(s) which is a product of \frac{h−1}{2} L-series associated to characters on Cℓ(F):

ζ_{H}(s) = ζ_{K}(s)L(s)^{2} ,  ζ_{F} (s) = ζ(s)L(s) ,  L(s) := ∏_{χ∈X} L_{K}(s, χ) ,

where X denotes a set of representatives for the non-trivial characters of Cℓ_{F} modulo complex conjugation. Again the expression for ζ_{K}(m), especially for m even, is typically more complicated than the expressions for L(m), so we concentrate on the latter.

For simplicity we assume that Cℓ_{K} = ⟨A⟩ is cyclic. Then, denoting by χ a generator of the character group of Cℓ_{F} and by ζ_{j}(s) the partial zeta function ζ_{A^{j}}(s), we have ζ_{H}(s) = ζ_{K}(s)L(s)^{2} , ζ_{F} (s) = ζ(s)L(s) with

L(s) := ∏_{j=1}^{(h−1)/2} L_{K}(s, χ^{j}) = ± det((ζ_{i−j}(s) − ζ_{i+j}(s))_{i,j=1,...,(h−1)/2}) ,

for instance, for h = 5 we have (denoting by ρ a primitive 5^{th} root of unity)

L(s) = ∏_{j=1}^{2}(ζ_{0}(s) + (ρ^{j} + ρ^{−j})ζ_{1}(s) + (ρ^{2j} + ρ^{3j})ζ_{2}(s)) = | \begin{matrix}ζ_{0}(s) − ζ_{2}(s) & ζ_{1}(s) − ζ_{2}(s) \\ ζ_{1}(s) − ζ_{2}(s) & ζ_{0}(s) − ζ_{1}(s) \end{matrix} | .

A consequence of the conjecture is that there exist elements ξ_{j} = −ξ_{−j} ∈ B_{2}(F) ⊗ ℚ such that

D(σ^{i}(ξ_{j})) = \frac{d^{3/2}}{16π^{2}} (ζ_{i−j}(2) − ζ_{i+j}(2)) for all i and j, where F is considered as embedded as a

subfield of ℂ via the real embedding and σ is the generator of Gal(H/K) corresponding to A. (Take ξ_{j} = σ^{−j}(ξ)−σ^{j}(ξ) with ξ as in the conjecture; by the first remark following the conjecture, this can also be written as (1 + c)σ^{j}(ξ) with c the generator of Gal(H/F), and hence belongs to B_{2}(F).) The conjecture is easier to check in this form than in the original one, because (as we already saw in the h = 3 examples) the value of ζ_{K}(2) = ∑_{j} ζ_{j}(2) is typically much harder to express in terms of dilogarithms than the differences ζ_{i}(2) − ζ_{j}(2).

As a numerical example (one of many we have computed), consider the field K = ℚ(\sqrt{−47}) with class number 5. Its partial zeta functions are ζ_{j}(s) = ζ_{K,A^{j}}(s) = ζ_{Q_{j}}(s), j = −2, .., 2, with A = [2, \frac{−1+\sqrt{−47}}{2}], A^{2} = [3, \frac{1+\sqrt{−47}}{2}], and

Q_{0}(x, y) = x^{2} + xy + 12y^{2} , Q_{±1}(x, y) = 2x^{2} ± xy + 6y^{2} , Q_{±2}(x, y) = 3x^{2} ∓ xy + 4y^{2} .

The Hilbert class field of K is H = K(α), where α^{5} +2α^{4} +2α^{3} +α^{2} −1 = 0. Taking for α the real root 0.5764 … gives the real embedding of the subfield F = ℚ(α). The other embeddings of F into ℂ send α to σ(α) = −1.105 …+0.5954 … i, σ^{2}(α) = −0.1828 …+1.032 … i, σ^{3}(α) = \overline{σ^{2}(α)} and σ^{4}(α) = \overline{σ(α)}, where σ is a generator of Gal(H/K). The group of units of F is generated (up to torsion) by α and β := α + 1, and the dilogarithms of the conjugates of the elements

ξ_{1} = −[α^{−1}β] − [α^{3}β^{2}] , ξ_{2} = [α^{4}β] + [α^{3}β^{3}]

in B_{2}(F) are related to the values at s = 2 of the partial zeta functions by

( \begin{matrix}D(σ(ξ_{1})) & D(σ^{2}(ξ_{1})) \\ D(σ(ξ_{2})) & D(σ^{2}(ξ_{2})) \end{matrix} ) = ( \begin{matrix}1.7702513872 … & 0.2095840007 … \\ 0.2095840007 … & 1.5606673865 … \end{matrix} )

= \frac{47^{5/2}}{16π^{2}} ( \begin{matrix}ζ_{0}(2) − ζ_{2}(2) & ζ_{1}(2) − ζ_{2}(2) \\ ζ_{1}(2) − ζ_{2}(2) & ζ_{0}(2) − ζ_{1}(2) \end{matrix} ) .

6. Known results for m = 2

For m = 2 the conjecture described in §5 can be formulated in terms of K-theory, since we know that the Bloch group B_{2}(H) is isomorphic to K_{3}(H) and that the dilogarithm regulator map coincides with the Borel regulator. An abstract proof in this case has been given by Deninger [15], [16], who showed the existence of an element of K_{3} whose regulator is given by the special values at s = 2 of the partial zeta functions of K.

The problem of constructing explicit elements ξ ∈ B_{2}(\overline{ℚ}) (though not necessarily in B(H)) with the required image under D was solved very recently by A. Levin. To each ideal 𝔞 in the ideal class A of K there is associated an elliptic curve E = ℂ/𝔞 defined over H with complex multiplication by O_{K}. For any three points a, b, c ∈ E, Levin considers the quantity

γ_{a,b,c} = \frac{℘_{a} − ℘_{c}}{℘_{a} − ℘_{b}} .

Here ℘_{a} = ℘_{E}(a) is the value of the Weierstrass ℘-function of E at a. If a, b and c are torsion points, then γ_{a,b,c} belongs to a certain abelian extension H′ ⊇ H of K (a ray class field).

Proposition (Levin). Let 𝔞 be an ideal class in the imaginary quadratic field K and λ ∈ O_{K}, μ = 1 − λ. Then the element

ξ_{A,λ} = 4N(λ)N(μ) [λ] + ∑_{{α∈λ^{−1}𝔞/𝔞−\{0\} \atop β∈μ^{−1}𝔞/𝔞−\{0\}}}  ∑_{{l(mod N(λ)) \atop m(mod N(μ))}} [γ_{α,β,mα+nβ}]

of ℤ[H′] belongs to B(H′).

The key point in proving this is that γ_{a,b,c} and 1 − γ_{a,b,c} have a factorization

γ_{a,b,c} = \frac{σ_{a−c}σ_{a+c}σ_{b}^{2}}{σ_{a−b}σ_{a+b}σ_{c}^{2}} , 1 − γ_{a,b,c} = \frac{σ_{b−c}σ_{b+c}σ_{a}^{2}}{σ_{b−a}σ_{b+a}σ_{c}^{2}}

in terms of the Weierstrass σ-function. Using this, one can write the image of each term in the sum under β_{2} and show that the sum maps to 0. Moreover, from the theory of complex multiplication one sees fairly easily that ξ_{A,λ} depends only on the ideal class A of 𝔞 (as the notation suggests), is invariant under Gal(H′/H), and transforms under Gal(H/K) by σ_{A}(ξ_{B,λ}) = ξ_{AB,λ}. The general polylogarithm conjecture now suggests that the image of ξ_{A,λ} under D is a multiple of ζ_{K,A}(2). Numerical experiments suggested what the correct multiple should be, and the resulting conjectural formula was then proved in [33].

Theorem (Levin). With the above notation,

D(ξ_{A,λ}) = (N(λ) + 1)(N(μ) + 1) · ζ_{K,A}(2) .

The proof depends on relating the value of the dilogarithms to the values of a certain “elliptic (1,1)-logarithm” related to the elliptic polylogarithms which will be discussed in Part III. Nothing in the construction of the element ξ_{λ} or the proof of the theorem depends in a really essential way on the fact that m = 2, and it can be hoped that Levin’s construction will be generalized to give explicit elements of B_{m}(H′)^{Gal(H′/H)} ⊗ ℚ whose values under the m^{th} polylogarithm regulator map are as given in (15).

7. Enhanced partial zeta values and a new invariant for imaginary quadratic fields

A particular amusing consequence of the conjecture becomes apparent if we apply the “lifting” of ℒ_{m} as described in §3. Again, let K be imaginary quadratic with Hilbert class field H. By the conjecture in §5 (or, for m = 2, the theorems of Deninger and Levin just quoted), we have for each m a canonical element ξ_{m,K} ∈ B_{m}(H) satisfying (15). But from the discussion in §4 we know that on the m^{th} Bloch group there is a natural lifting of the function ℒ_{m} to an enhanced function ℒ̂_{m} with values in ℂ/ℚ(m). It is then natural to ask the Questions: 1. Is there a naturally defined invariant I_{m}(A) belonging to ℂ/ℚ(m) (or even ℂ/α_{m}ℤ(m) for some α_{m} ∈ ℚ independent of K) such that

ℜ_{m}(I_{m}(A)) = ±\frac{|D_{K}|^{m− \frac{1}{2}} (m − 1)!}{2(2π)^{m}}ζ_{K,A}(m)  ?  (19)

2. If so, does the “enhanced higher Kronecker limit formula” ℒ̂_{m}(σ_{A}(ξ_{m,K})) = I_{m}(A) hold? In this paragraph we show that the answers to 1. and 2. are “Yes” and “Experimentally, yes.”

Construction of the invariant I_{m}(A). We had already observed that ζ_{K,A}(s) for imaginary quadratic K and an ideal class A in K is expressible in terms of the non-holomorphic Eisenstein series E(z, s) of weight 0 by (16), where the argument z is the root in the upper halfplane of az^{2} +bz +c = 0 for some positive definite binary quadratic form Q(p, q) = ap^{2} +bpq +cq^{2} corresponding to an ideal in the ideal class A. The above question can therefore be rephrased:

Can the value of π^{−m}|D|^{(m−1)/2}E(z, m) ∈ ℝ (m = 2, 3, 4, …) be lifted in a natural way to ℂ/ℚ(m) when z ∈ ℍ/Γ is a CM-point of discriminant D?

To do this, we look at the holomorphic Eisenstein series

G_{2m}(z) = \frac{1}{2}ζ(1 − 2m) + ∑_{n=1}^{∞} σ_{2m−1}(n)e^{2πinz}

of weight 2m on SL_{2}(ℤ). (Here σ_{ν}(n) := ∑_{d|n} d^{ν}.) Its Eichler integral is any function

G̃_{2m}(z) = \frac{1}{2}ζ(1 − 2m) · \frac{(2πiz)^{2m−1}}{(2m − 1)!} + ∑_{j=0}^{2m−2} a_{j}z^{j} + ∑_{n=1}^{∞} σ_{1−2m}(n)e^{2πinz}  (20)

(a_{0}, …, a_{2m−2} for the moment arbitrary). From (\frac{1}{2πi} \frac{d}{dz} )^{2m−1}G̃_{2m} = G_{2m} and the modularity of G_{2m} it follows, as is well-known, that

(cz + d)^{2m−2}G̃_{2m}(\frac{az + b}{cz + d}) = G̃_{2m}(z) + P_{m,γ}(z) ∀ γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ

for some polynomial P_{m,γ} of degree ≤ 2m − 2. The important point is that if we choose

a_{0} = \frac{1}{2}ζ(2m − 1), a_{1} = ⋯ = a_{n−2} = 0

(this is not the only possible choice; see below), then the coefficients of the polynomials P_{m,γ}(z) lie in ℚ(2m − 1) for all γ ∈ Γ.

We now apply to G̃_{2m} the differential operator

𝔻_{m} = ∂_{−2} ∘ ∂_{−4} ∘ ⋯ ∘ ∂_{2−2m} = ∑_{ℓ=0}^{m−1} \frac{(m + ℓ − 1)!}{ℓ! (m − ℓ − 1)!} (\frac{1}{4πy} )^{ℓ} (\frac{1}{2πi} \frac{d}{dz} )^{m−1−ℓ} ,

where ∂_{k} = \frac{1}{2πi} \frac{∂}{∂z} − \frac{k}{4πy}, which has the property of mapping modular forms of weight k to modular forms of weight k + 2. The result is a new function

E_{m}(z) = 𝔻_{m}(G̃_{2m}(z))  (21)

which satisfies

E_{m}(γz) = E_{m}(z) + P_{m,γ}^{∗}(\frac{1}{2iy} , \frac{x}{iy} , \frac{|z|^{2}}{2iy} ) for all γ ∈ Γ ,

where P_{m,γ}^{∗} is a homogeneous polynomial of degree m − 1 with coefficients in ℚ(m). This has two consequences:

(i) The real part of E_{m}(z) is Γ-invariant. Using the explicit Fourier expansions, one shows that

ℜ(E_{m}(z)) = \frac{(m − 1)!}{2π^{m}} E(z, m)  ( z ∈ ℍ) .  (22)

(ii) If z satisfies a quadratic equation ax^{2} + bx + c = 0 with a, b, c ∈ ℤ and discriminant D = b^{2} − 4ac < 0, then the image of D^{(m−1)/2}E_{m}(z) in ℂ/ℚ(m) is unchanged under z ↦ γz,

γ ∈ Γ, because D^{(m−1)/2} P_{m,γ}^{∗}(\frac{1}{2iy} , \frac{x}{iy} , \frac{|z|^{2}}{2iy} ) = P_{m,γ}^{∗}(a, b, c) ∈ ℚ(m).

It follows from (i) and (ii) that the invariant

I_{m}(A) = D^{(m−1)/2} E_{m}(z_{A}) ,  (23)

where z_{A} is the image in ℍ/Γ of z_{Q} for any quadratic form Q in the class of A, is well defined modulo ℚ(m) (and even modulo α_{m}ℤ(m) for some α_{m} ∈ ℚ independent of D) and satisfies (19). Observe that by changing the choices of a_{j} in (20) by elements of ℚ(2m−1) we change the cocycle γ ↦ P_{m,γ} by a coboundary with coefficients in ℚ(m − 1); this does not affect the property that the indeterminacy of I_{m}(A) belongs to ℚ(m), but can change the specific lattice ℤ(m)α_{m}.

Explicit formulas for I_{m}(A). For m = 2 we have the Fourier expansions

E(z, 2) = \frac{π^{4}y^{2}}{90} + \frac{πζ(3)}{2y} + \frac{π}{y} ∑_{n=1}^{∞} σ_{−3}(n) (1 + 2πny) e^{−2πny} cos(2πnx) ,  (24)

G_{4}(z) = \frac{1}{240} + ∑_{n=1}^{∞} σ_{3}(n) e^{2πinz} ,  G̃_{4}(z) = \frac{(2πiz)^{3}}{1440} + \frac{ζ(3)}{2} + ∑_{n=1}^{∞} σ_{−3}(n) e^{2πinz} ,

E_{2}(z) = \frac{(2πiz)^{2}(iz + 3y)}{1440y} + \frac{ζ(3)}{4πy} + ∑_{n=1}^{∞} σ_{−3}(n) (n + \frac{1}{2πy} ) e^{2πinz} .  (25)

The function E_{2}(z) has real part \frac{1}{2π^{2}} E(z, 2) and satisfies the modular transformation formulas

E_{2}(z + 1) = E_{2}(z) + π^{2} \frac{|z|^{2} + x + 1/3}{120iy} ,  E_{2}(−1/z) = E_{2}(z) − \frac{π^{2}x}{72iy}  (26)

(the proof of (26) will be indicated below). It follows that the invariant I_{2}(Q) := \sqrt{D} E_{2}(z_{Q}) satisfies (19) and

I_{2}([a, b + 2a, c + b + a]) = I_{2}([a, b, c]) + \frac{π^{2}}{60} (c − \frac{b}{2} + \frac{a}{3}) , I_{2}([c, −b, a]) = I_{2}([a, b, c]) − \frac{π^{2}}{72} b ,

and hence that

I_{2}(A) = I_{2}([Q]) ∈ ℂ/\frac{π^{2}}{360}ℤ is well-defined .

We can improve upon the number 360 in the denominator by choosing a slightly different normalization:

E_{2}^{•}(z) = E_{2}(z) + \frac{iπ^{2}}{240} \frac{|z|^{2} + 5x/3 + 1}{y}

(corresponding to replacing G̃_{4} by G̃_{4}+ \frac{iπ^{3}}{120}(z^{2}+ \frac{5}{3}z+1) ), which still has real part (2π^{2})^{−1}E(z, 2) but now transforms under SL_{2}(ℤ) by

E_{2}^{•}(z + 1) = E_{2}^{•}(z) + \frac{iπ^{2}}{120} \frac{|z|^{2} − 1}{y} , E_{2}^{•}(−1/z) = E_{2}^{•}(z) ,

so that the invariant

I_{2}^{•}(Q) := \sqrt{D} E_{2}^{•}(\frac{b + \sqrt{D}}{2a} ) = I_{2}(Q) − \frac{π^{2}}{24} (\frac{a + c}{5} − \frac{b}{6})

satisfies

I_{2}^{•}([a, b + 2a, c + b + a]) = I_{2}^{•}([a, b, c]) + \frac{π^{2}(a − c)}{60} , I_{2}^{•}([c, −b, a]) = I_{2}^{•}([a, b, c])

and hence gives a class invariant I_{2}^{•}(A) = I_{2}^{•}([Q]) which is well-defined modulo \frac{π^{2}}{60} ℤ.

For m > 2 the formulas are more complicated. We give only the Fourier expansion of the function E_{m}(z) :

E_{m}(z) = ∑_{ℓ=0}^{m−1} \frac{(m + ℓ − 1)!}{ℓ!(m − ℓ − 1)!} (4πy)^{−ℓ} [\frac{ζ(1 − 2m)}{2} \frac{(2πiz)^{ℓ+m}}{(ℓ + m)!} + ∑_{n=0}^{∞} n^{m−1−ℓ} σ_{1−2m}(n) q^{n}] ,

where σ_{s}(0) := \frac{1}{2} ζ(−s). The modular transformation properties of this can be given explicitly, but for m > 2 we do not know how to add a coboundary to get a function E_{m}^{•}(z) with an optimal denominator in its transformation law with respect to 27.

We explain briefly how to prove the transformation formulas (26). (The calculation for E_{m} goes the same way.) These formulas follow by applying 𝔻_{2} = ∂_{−2} to the transformation properties

G̃_{4}(z + 1) = G̃_{4}(z) − \frac{π^{3}i(3z^{2} + 3z + 1)}{180} ,  z^{2}G̃_{4}(−1/z) = G̃_{4}(z) + \frac{π^{3}iz}{36} .  (27)

The first of these equations is evident since G̃_{4} is the sum of a periodic function and a multiple of z^{3}. For the second, we observe that the difference z^{2}G̃_{4}(−1/z) − G̃_{4}(z) is equal to a polynomial of degree ≤ 2, Az^{2} + Bz + C. Interchanging the role of z and −1/z, one deduces immediately C = −A. To compute the values, the easiest way is to notice that the function

g(y) = ∑_{n=1}^{∞} σ_{−3}(n)e^{−2πny} = G̃_{4}(iy) − \frac{ζ(3)}{2} + \frac{π^{3}}{180} y^{3}

satisfies the transformation law

g(y) = −y^{2}g(1/y) + A_{−1}y^{−1} + A_{0} + A_{1}y + A_{2}y^{2} + A_{3}y^{3}

with A_{−1} = A_{3} = \frac{π^{3}}{180} , A_{0} = A_{2} = A− \frac{1}{2}ζ(3) , A_{1} = −iB. The fact that g is exponentially small

at ∞ then implies that the Mellin transform g̃(s) = ∫_{0}^{∞} g(y)y^{s−1}dy converges for all ℜ(s) ≫ 0 and has a meromorphic continuation with simple poles of residue A_{ν} at s = −ν (ν = −1, …, 3) and no other poles. But g̃(s) is easily computed to be (2π)^{−s}Γ(s)ζ(s)ζ(s + 3), and by comparing the residues we find A = C = 0, B = iπ^{3}/36.

The enhanced conjecture. Now that we have defined the invariant I_{m}(A), we can formally state the conjecture which was suggested at the beginning of the section.

Conjecture. Let K be an imaginary quadratic field with Hilbert class field H, m ≥ 2. Then the numbers I_{m}(A) ∈ ℂ/ℚ(m) are equal to the enhanced polylogarithms ℒ̂_{m}(σ_{A}(ξ)) for some element ξ ∈ B_{m}(H) ⊗ ℚ.

More precisely, we should have N_{m}ξ ∈ B_{m}(H) and ℒ̂_{m}(ξ) ≡ I_{m}(A) mod N_{m}^{−1}ℤ(m) for some positive integer N_{m} depending on m but not on K.

Examples. 1. We consider our usual example K = ℚ(\sqrt{−23}) and, for m = 2, the element ξ = ξ_{2,K} ∈ B_{2}(K) constructed in Example 1 of §5. The real part of I_{2}(A_{0}) is a rational multiple of π^{2} and hence not interesting (this follows either by substituting z_{0} = \frac{1+\sqrt{−23}}{2} into the Fourier expansion of E_{2}(z) given above or else by noting that I_{2}(A^{−1}) = −\overline{I_{2}(A)} modulo ℚ(2) for any ideal class A). For A_{1} we find by (23) and (25)

I_{2}(A_{1}) ≡ α + \frac{23^{3/2}}{8π^{2}} ζ_{1}(2) · i (mod ℚ(2)) ,

with

α = \frac{7π^{2}}{576} − \frac{2}{π} ∑_{n=0}^{∞}(−1)^{n}σ_{−3}(2n + 1) (1 + π(n + \frac{1}{2})\sqrt{23}) e^{−π(n+ \frac{1}{2} )\sqrt{23}}
= 0.11703683929364456681742552529827002050121888426989 ….

The numbers ℜ(D̂(θ_{0})) = 2.10466 … and ℜ(D̂(θ_{1})) = −11.7444 … appearing in Example 3 of

§4 are equal (to high accuracy) to α + \frac{29}{144}π^{2} and −\frac{1}{2}α − \frac{341}{288}π^{2} , respectively, and this verifies

the above conjecture with ξ as given in equation (18).

2. Similarly, for m = 3, we find that the invariants I_{3}(A_{j}) are given by

I_{3}(A_{0}) ≡ −\frac{23^{2}}{4π^{3}} ζ_{0}(3) − \frac{N_{0}}{7560}π^{3} · i (mod ℚ(3)) ,
I_{3}(A_{1}) ≡ −\frac{23^{2}}{4π^{3}} ζ_{1}(3) + ( − 4 · f + \frac{N_{1}}{3780}π^{3}) · i (mod ℚ(3)) ,

where f = 15.1353 … is the number given in the example at the end of §4, N_{0} = 1021 and N_{1} = 7319 to high accuracy. The proportionality of ζ_{0}(3) − ζ_{1}(3) with ℒ_{3}(β_{1}) verified in §4 thus lifts (experimentally) to the enhanced values, as predicted.

Part III. Elliptic curves and elliptic polylogarithms

8. The elliptic dilogarithm conjecture

The elliptic dilogarithm and L(E, 2). The elliptic dilogarithm D^{E} associated to the elliptic curve E/ℂ, as introduced by Bloch [6], is an “averaged” version of the Bloch-Wigner dilogarithm D: more precisely one represents E(ℂ) as the Tate curve ℂ^{×}/q^{ℤ} and averages D over the action of ℤ on ℂ^{×}, i.e.

D^{E}(P) = ∑_{l=−∞}^{∞} D(q^{l}x) ,

where P ∈ E(ℂ) is the image of x ∈ ℂ^{×}.

Since the series is invariant under x ↦ qx and converges absolutely, this is independent of the choice of x. The series converges rapidly, so that D^{E}(P) is easily calculable. Another representation of D^{E}(P) is given by writing E = ℂ/ℤτ + ℤ, τ ∈ ℍ (so q = e^{2πiτ}) and P = z (mod ℤτ + ℤ) (and so x = e^{2πiz}); then D^{E} is given by the Kronecker-Eisenstein series

D^{E}(P) = \frac{ℑ(τ)^{2}}{π} ℜ(\begin{matrix}∑ & ′ \\ m,n∈ℤ \end{matrix} \frac{exp(2πi(nξ − mη))}{(mτ + n)^{2}(mτ̄ + n)})  (28)

where z = ξτ + η, as one shows by computing the Fourier expansion of the right-hand side with respect to τ (cf. [6] or [54]).

The L-series of E at the point s = 1 is well-known to be linked with the height pairing on the rational points of E. On the other hand, this pairing is given in terms of logarithms of theta functions which can be seen as the elliptic analogue of the logarithm. It was Bloch’s idea that L(E/ℚ, 2) might be linked to the elliptic dilogarithm, in the following way

L(E/ℚ, 2) =^{?} π × rational linear combination of D^{E}(P_{i}) , P_{i} ∈ E(\overline{ℚ}) .

This was proved by him for CM elliptic curves and by Beilinson [2] for modular curves (see below). However, these authors considered only torsion points on E(\overline{ℚ}). This would be the analogue in the number field case of considering values of polylogarithms only at roots of unity, which would suffice to describe ζ_{F} (m) for all abelian F, but would not lead to the discovery of the higher Bloch groups B_{m}(F) and the corresponding conjectural description of K_{2m−1}(F).

This consideration suggests that one should look for conditions describing linear combinations of points on E such that the elliptic dilogarithm evaluated on them gives a rational multiple of the L-value. Numerical experiments carried out some 10 years ago with H. Cohen suggested that one had

∑_{i} n_{i}D^{E}(P_{i}) ∈ ℤ · R_{E} ,  R_{E} = \frac{N}{8π} L(E, 2)  (29)

(N =conductor of E), whenever the divisor ∑ n_{i}(P_{i}) satisfies the following conditions:

(i) ∑ n_{i}P_{i} = 0 on E
(ii) ∑ n_{i}(P_{i})^{3} = 0 in Sym^{3}(E)
(iii) ∑ n_{i}λ_{p}(P_{i})P_{i} = 0 on E for any prime p .

Here λ_{p}(P) is defined as the power of p dividing c, where c^{2} and c^{3} are the denominators of x(P) and y(P) in a minimal model of the curve E. Note that R_{E} = \frac{1}{2}π |L′(E, 0)| if E/ℚ is modular, by the functional equation of L(E, s).

Example. The modular curve X_{0}^{∗}(37) has the minimal model E = E_{37} : y^{2} − y = x^{3} − x. The Mordell-Weil group E(ℚ) is infinite cyclic with generator P = (0, 0). We have five multiples of P which are integral

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

(as well, of course, as their negatives) and two further multiples which involve only the single prime “2” in their denominators:

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

For each k, the combination

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

satisfies the conditions (i) and (ii) above. For k = 3, 4 and 6 the combinations

ξ_{3} = (3P) − 4(2P) + 5(P) , ξ_{4} = (4P) − 10(2P) + 16(P) , ξ_{6} = (6P) − 35(2P) + 64(P)

also satisfy (iii), since all the points involved are integral. The combinations ξ_{5} and ξ_{10} do not separately satisfy (iii) for p = 2, but the combination

ξ_{10}^{∗} = ξ_{10} − 4ξ_{5} = (10P) − 4(5P) − 85(2P) + 180(P)

does, since λ_{2}(5P) = 1, λ_{2}(10P) = 2. Computing numerically to high precision, we find:

D^{E}(ξ_{3}) = 8R_{E} , D^{E}(ξ_{4}) = 26R_{E} , D^{E}(ξ_{6}) = 90R_{E} , D^{E}(ξ_{10}^{∗}) = 248R_{E} ,  (30)

with R_{E} = \frac{37}{8π} L(E, 2) = 0.561748914579...

In the general case, conditions (i)–(iii) are not quite correct, as was pointed out by Schappacher–Rolshausen [40] and others. On the one hand, one needs an extra condition at certain primes of bad reduction (namely for p of split multiplicative reduction such that the Néron model is an N-gon with N > 1). On the other hand, (i) is not always necessary. For example (due to K. Rolshausen in his thesis), if E is the curve y^{2} + y = x^{3} − 325x + 6156 of conductor 4025, then the combination 16(P) − 10(2P) + (4P) for P = (20, 87) satisfies (i)–(iii) above but fails to give a rational multiple of the regulator, while 11(Q) + 2(2Q) − (3Q) for Q = (45, 287) does not satisfy (i) but nevertheless maps under D^{E} to \frac{4025}{8π}L(E, 2).

The right way is to replace λ_{p}(P) in (iii) by h_{p}(P), the local height at p in P. Here h_{p} denotes the local height associated to the valuation p. In this form, it is independent of the model of E used. For P of good reduction, this is just λ_{p}(P) log p, so there is no change, and for certain p of bad reduction (like p = 37 in the example above) the height condition at p reduces to (i) if all the P_{i} are p-integral, which is why (i) often occurred. The corrected conditions are then

(a) ∑ n_{i}(P_{i})^{3} = 0 in Sym^{3}(E) ,
(b) ∑ n_{i}h_{p}(P_{i})P_{i} = 0 on E for any prime p ,
(c) some integrality condition at primes p of split multiplicative reduction.

(31)

We then expect that (29) holds for any ξ = ∑ n_{i}(P_{i}) ∈ ℤ[E(ℚ)] satisfying (a)–(c).

More generally, we expect the same thing to happen for divisors ξ = ∑ n_{i}(P_{i}) ∈ ℤ[E(\overline{ℚ})] which are defined over ℚ (i.e. are Gal(\overline{ℚ}/ℚ)-invariant) but whose individual points P_{i} are defined only over \overline{ℚ}. (It is important to consider such more general divisors because there may not be enough points in E(ℚ) to produce linear combinations satisfying (a)–(c). For instance, E(ℚ) might be {0}.) We therefore define

A_{2}(E/ℚ) = \{ξ ∈ ∑ n_{i}(P_{i}) ∈ ℤ[E(\overline{ℚ})]^{Gal(\overline{ℚ}/ℚ)} | ξ satisfies (a)–(c)\}

where the words “prime p” must be replaced by “archimedean or non-archimedean place v” in (b) and by “non-archimedean place v” in (c). (Condition (a) implies that the sum over all v of the expression in (b) vanishes, so that we can always omit one place; this is why in formulating (b) for ℚ we could restrict attention to finite primes.) Then we expect:

Conjecture. The image of the map D^{E} : A_{2}(E/ℚ) → ℝ is a lattice commensurable with ℤ · R_{E}.

Functional equations and the elliptic Bloch group. As for the case of the usual dilogarithm, we want to find a subgroup C_{2}(E/ℚ) of A_{2}(E/ℚ) generated by some universal relations of D^{E} and to conjecture that the “elliptic Bloch group” B_{2}(E/ℚ) = A_{2}(E/ℚ)/C_{2}(E/ℚ) maps isomorphically onto the regulator lattice.

Two special relations are evident: from the inversion relation D(x^{−1}) = −D(x) of the usual dilogarithm we obtain D^{E}(P) = −D^{E}(−P), and from the duplication relation D(x^{2}) = 2D(x) + 2D(−x) we easily find that D^{E}(2P) = 2 ∑_{T ∈E[2]} D^{E}(P + T). (Combine x, −x, \sqrt{q}x and −\sqrt{q}x in the definition of D^{E}.) More interesting is Bloch’s relation [6], whose proof will be recalled in §9, that D^{E} also vanishes on the elements η_{f} = (f) ∗ (1 − f)^{−} for all f ∈ ℚ(E). Here ξ ∗ η^{−} is defined for any two divisors ξ = ∑ n_{i}(P_{i}), η = ∑ m_{j}(Q_{j}) as ∑ n_{i}m_{j}(P_{i} − Q_{j}). It is expected that these form a full set of relations, so that if we set

C_{2}(E/ℚ) = ⟨ η_{f}, (P) + (−P) , (2P) − 2 ∑_{T ∈E[2]}(P + T)⟩ ,

then the quotient

B_{2}(E/ℚ) := A_{2}(E/ℚ)/C_{2}(E/ℚ)

should map isomorphically under D^{E} to a lattice commensurable with ℤ · R^{E}.

As in the number field case, one can also write the group B_{2}(E/ℚ) as the kernel of the map (P) ↦ (P) ⊗ P from G_{3}(E) to G_{2}(E) ⊗ E, where G_{3}(E) = ℤ[E]/C_{2} and G_{2} is the largest quotient of ℤ[E] on which all the local heights vanish, and this is part of a larger complex G_{3}(E) → G_{2}(E) ⊗ E → E ⊗ ⋀^{2}E → ⋀^{3}E. For details see [26].

Example: For the curve E_{37} : y^{2} −y = x^{3} −x investigated above, we find that the 4 functions

f_{1} = y , f_{2} = x + y , f_{3} = \frac{1 + y − 2x}{4} , f_{4} = \frac{x + 2y − 1}{2x}

have (f_{i}) and (1 − f_{i}) with support in E(ℚ), the expressions η_{i} = η_{f_{i}} being equal (up to the “trivial” relation (x) + (−x) = 0) to

η_{1} = −(6P) + (4P) + 8(3P) − 7(2P) − 8(P) ,
η_{2} = (6P) − 5(4P) + 5(3P) − 5(2P) + 9(P) ,
η_{3} = −η_{4} = (10P) − 3(6P) − 4(5P) − (4P) + 6(3P) + 6(2P) + 2(P) ,

respectively, as the reader can check easily by computing the divisors of f_{i} and 1−f_{i} in each case. Hence each identity in (30) implies the remaining three ones. (That η_{3} = −η_{4} is less surprising than it looks, since any η_{f} would have to be a vector ∑ n_{i}(iP) ∈ ℤη_{1}+ℤη_{2}+ℤη_{3} with ∑ |n_{i}| ≤ 27 and there are very few such vectors.) In other examples, one can find many more such f. For instance, for the curve y^{2} + xy + y = x^{3} − x^{2} − 180x + 1047 of conductor 350 with E(ℚ) = ℤ · P, P = [−1, 35], one finds 379 essentially different functions f (where “essentially different” means up to the group generated by f ↦ 1/f and f ↦ 1 − f) for which η_{f} ⊂ ℤ[E(ℚ)]; for each of these the element η_{f} is (up to the relation (−nP) = −(nP)) a linear combination of the 19 points (P), (2P), …, (16P), (18P), (21P), (22P), but in fact the η_{f} lie in an 11-dimensional sublattice of this 19-dimensional lattice and are all very short vectors in this sublattice, so that we obtain the same relations among the numbers D^{E}(nP) in many different ways.

Remark. If we let q → 0, the Tate curve ℂ^{×}/q^{ℤ} degenerates to 𝔾_{m}(ℂ) = ℂ^{×}. If f is any rational function on ℙ^{1} satisfying f(0) = f(∞), then the degeneration of the above condition (f)∗(1−f)^{−} = 0 still makes sense, with (P)+(Q) meaning simply (P ·Q), and the corresponding relation

∑_{a,b∈ℂ^{×}} ord_{a}(f) ord_{b}(1 − f) [\frac{a}{b}] = 0

remains true in the Bloch group B_{2}(ℂ). For the special function

f(t) = \frac{(t − a)(t − a′)(t − bb′)}{(t − b)(t − b′)(t − aa′)} , (a′ = 1 − a, b′ = 1 − b) ,

this reduces modulo the trivial relations [x] + [\frac{1}{x}] = 0 and [x] + [1 − x] = 0 to the five term relation, which is thus seen to be in a certain sense a degeneration of its elliptic analogue.

9. The elliptic dilogarithm and the regulator map for K_{2}(E)

In this section we explain some of the known theory behind the computations and conjecture discussed in §8.

The regulator map for K_{2} of a curve. For any curve X/ℂ, the Bloch-Beilinson regulator pairing between K_{2}(ℂ(X)) and the space of holomorphic 1-forms on X is defined on symbols {F, G}, F, G ∈ ℂ(X)^{∗}, by

⟨r_{2}^{X}(\{F, G\}), ω⟩ = \frac{1}{2π} ∫_{X(ℂ)} log |F | d log \overline{G} ∧ ω .  (32)

To see that this is defined on K-theory we must check that the integral vanishes for F = 1 − G. This holds because log |F | d log \overline{(1 − F)} is the sum of a holomorphic differential and of the exact differential dΦ(F (z)), where Φ(z) = log |z| log |1 − z| − iD(z). If X is defined over ℚ, then the group K_{2}(X/ℚ) is defined as a subgroup of K_{2}(ℚ(X)) generated by expressions α = ∑{f_{i}, g_{i}} (f_{i}, g_{i} ∈ ℚ(X)) satisfying certain local conditions (vanishing of the tame symbols) at the zeros and poles of the f_{i} and g_{i}, and there is a still smaller subgroup K_{2}(X/ℤ) ⊂ K_{2}(X/ℚ) defined by certain conditions on the symbols at primes of bad reduction of X. The Bloch-Beilinson conjecture in the case of a curve X/ℚ predicts in this case that K_{2}(X/ℤ) has rank g, where g is the genus of X, and that the determinant of the g × g-matrix of its pairings with the lattice of holomorphic 1-forms on X/ℤ (Néron differentials) is rationally proportional to an explicit multiple of L(X/ℚ, 2) or of L^{(g)}(X/ℚ, 0). Here L(X/ℚ, s) is the Hasse-Weil zeta function, which conjecturally satisfies a functional equation under s → 2 − s with Γ-factor A^{s}Γ(s)^{g}.

The case of elliptic curves. In particular, if E is an elliptic curve, K_{2}(E/ℤ) should have rank 1 and the pairing of any element in this group with dz should be a simple multiple of L(E/ℚ, 2). But in this case, writing E(ℂ) as ℂ/ℤτ + ℤ, we have Bloch’s formula

⟨r_{2}^{E}\{F, G\}, ω⟩ = \frac{1}{π} D^{E}((F) ∗ (G)^{−})  (33)

where the pairing ξ ∗η^{−} of divisors ξ = ∑ n_{i}(P_{i}), η = ∑ m_{j}(Q_{j}) is defined as ∑_{i,j} n_{i}m_{j}(P_{i} −Q_{j}). (To prove this, write F and G as products of Jacobi theta series, replace the theta series by their product expansion, and integrate them term by term. See e.g. [26], pp. 416-417.) One therefore expects L(E/ℚ, 2) to be expressible in terms of special values of D^{E}.

In the original formulation of Bloch and Beilinson, it was thought to be K_{2}(E/ℚ) which should have rank 1 and map under r_{2}^{E} to a multiple of L(E/ℚ, 2). Computer experiments of Bloch and Grayson [7] suggested that one has to consider the K-group of the Néron model E_{ℤ} of E instead. Schappacher and Scholl [41] gave a precise description of the necessary local conditions. On the other hand, by studying the structure of the divisors ∑(f_{i}) ∗ (g_{i})^{−} with ∑{f_{i}, g_{i}} ∈ K_{2}(E/ℚ), Goncharov and Levin showed

Theorem [26, Theorem 1.2]. For any elliptic curve E/ℚ the image of r_{2}^{E} on K_{2}(E/ℚ) is—up to tensoring with ℚ—the same as the image of D^{E} on ℚ-rational divisors satisfying the conditions (a) and (b).

Now the extra condition by Schappacher and Scholl corresponds to the extra condition (c) above, so that D^{E}(A_{2}(E/ℚ))⊗ℚ coincides with r_{2}^{E}(K_{2}(E/ℤ))⊗ℚ and hence is conjecturally of rank 1.

The case of modular curves. Finally, we have to see why the generator of this regulator lattice is related to L(E/ℚ, 2). If E has complex multiplication, this is easy, because the Hasse-Weil zeta function of E is the Hecke L-series of a Grössencharacter for an imaginary quadratic field and its value at s = 2 can be written more directly as a finite linear combination of Kronecker-Eisenstein series (28) with P a torsion point ([6], [39]). Much less obvious is Beilinson’s theorem that L(E/ℚ, 2) can be written as a combination of values of the function D^{E} evaluated at torsion points whenever E is modular. The idea of the proof is as follows. Let X = ℍ/Γ be a modular curve and choose F and G to be modular units (i.e. modular functions X → ℙ^{1} with divisors concentrated at the cusps) and ω = f(τ)dτ for some Hecke eigenform f ∈ S_{2}(Γ). Then the logarithmic derivatives of F and G are Eisenstein series of weight 2 on X and the integral on the right of (32) can be computed by Rankin’s method as a special value of the convolution of the L-series of f with the L-series of a weight 2 Eisenstein series. For a suitable choice of the Eisenstein series, it equals L(f ⊗ χ, 1)L(f, 2) where χ is a Dirichlet character. If f corresponds to the modular elliptic curve E (i.e., f(τ)dτ is the pull-back of dz with respect to a modular parametrization X → E), then L(f ⊗ χ, 1) is an algebraic multiple (non-zero for suitable χ) of a period of E and L(f, 2) = L(E/ℚ, 2). On the other hand, the regulator pairing between {F, G} and ω = f(τ)dτ is the same as the pairing between the push-forward of {F, G} and dz, so by (33) it follows that π^{−1}L(E, 2) is expressed as a ℚ-linear combination of values of D^{E} (with arguments which are torsion points of E, since the divisors of F and G are torsion points in the Jacobian of X_{0}(N) by the Manin-Drinfeld theorem). Summarizing, we have:

Theorem (Beilinson). If E/ℚ is a modular elliptic curve, then there exists an element ξ of ℚ[E_{tors}] with D^{E}(ξ) = R_{E}.

Connection with Mahler measure. We end this section by describing a beautiful relationship, brought to light by the work of Deninger [14], Boyd [10] and Villegas [48], between the regulator map r_{2}^{E} and the “Mahler measure” of certain polynomials.

If P (x_{1}, …, x_{n}) is a polynomial in n variables, then the (logarithmic) Mahler measure of P is defined by

m(P) = ∫_{0}^{1} ⋯ ∫_{0}^{1} log |P (e^{2πiθ_{1}}, …, e^{2πiθ_{n}})| dθ_{1} ⋯ dθ_{n} .

For n = 1 and P (x) = (x − α_{1}) ⋯ (x − α_{n}), m(P) is equal by Jensen’s formula to the measure ∑_{j} log ( max(|α_{i}|, 1)) introduced by Lehmer [31] in 1928, which for irreducible P (x) ∈ ℤ[x] is simply the logarithm of the height of the algebraic integer α_{1}. The definition with an integral is due to Mahler [34].

The original purpose of the Mahler measure was for results in transcendence theory and one was only interested in the approximate size of m(P), but a number of calculations suggested that the actual value of m(P) might in many cases be an interesting number, and in particular was sometimes connected with special values of L-functions. In particular, C. Smyth [42] discovered that the value of m(P) for P (x, y) = x + y + 1 and P (x, y, z) = x + y + z + 1 are simple multiples of the Dirichlet L-series value L(2, χ_{3}) and the Riemann zeta-value ζ(3), respectively; and T. Chinburg [12] observed that even in certain n = 1 examples the value of m(P) is related to the value of an Artin L-function at s = 1.

In 1990, Deninger [14] discovered that the Mahler measures of arbitrary polynomials with algebraic coefficients are periods of mixed motives, and more specifically, that Mahler measures of certain polynomials in two variables could be expressed in terms of the regulator map for elliptic curves. This phenomenon was then studied experimentally in great detail by Boyd [10], who discovered a number of families of examples of this type.

For k ∈ ℤ, k > 0, k ≠ 4, let P_{k}(x, y) be the polynomial

P_{k}(x, y) = (x + y)(xy + 1) − kxy .

Let E_{k} be the elliptic curve over ℚ defined by P_{k}(x, y) = 0. Then one finds numerically to high precision

|L′(E_{k}, 0)| = B_{k} m(P_{k}) ,

with B_{k} ∈ ℚ. (Recall that |L′(E, 0)| = 2π^{−1}R_{E} = (2π)^{−2}N_{E} L(E, 2). We can compute L′(E_{k}, 0) using standard number theory software packages and m(P_{k}) = ∫_{0}^{1} arccosh(\frac{1}{2}k − cos πx) dx by numerical integration.) A small table of the experimentally obtained values of B_{k} is as follows:

\begin{matrix}k & 1 & 2 & 3 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 & ⋯ & 198 & 199 & 200 \\ B_{k} & 1 & 1 & \frac{1}{2} & \frac{1}{6} & 2 & 2 & \frac{1}{4} & 2 & 8 & 8 & \frac{1}{2} & 4 & 8 & 24 & \frac{1}{11} & 24 & 16 & 40 & 2 & ⋯ & 12416 & 23296 & 64 \end{matrix}

The explanation of Boyd’s examples was worked out by Villegas [48]. (All three papers— Boyd’s, Deninger’s, and Villegas’s—are extremely interesting and are highly recommended to the reader.) In a few words, it is as follows. The symbol {x, y} of the two generators x and y of ℚ(E) satisfies the necessary conditions on the tame symbol and at primes of bad reduction to define an element of K_{2}(E_{ℤ}), and then a calculation based on Jensen’s formula and integration by parts reveals that the regulator ⟨r_{2}^{E}({f, g}), ω⟩, where ω as usual is the Néron differential of E_{ℤ}, is equal to the Mahler measure m(P_{k}). (The details are given in [48].) The conjecture described above then implies that m(P_{k}) is a rational multiple of π^{−1}L(E_{k}/ℚ, 2). We remark that in this particular example the support of (x) ∗ (y)^{−}, and hence the set of arguments at which D^{E} is evaluated to obtain the regulator, is contained in E_{tors}, but that in other families introduced in [10] and analyzed in [48], such as P_{k}(x, y) = y^{2} − x^{3} − kxy − 1, this is not the case.

In the last section of [48], Villegas shows how to interpret m(P_{k}) in modular terms: if we think of k = k(τ) (where the elliptic curve defined by P_{k} = 0 is isomorphic to ℂ/(ℤτ + ℤ)) as a modular function, then m(P_{k}) is the real part of a holomorphic function in ℍ whose derivative is a modular form of weight 3.

10. The elliptic trilogarithm and L(Sym^{2}(E), 3)

Mestre and Schappacher [35] (for torsion points), Goncharov [25] and Wildeshaus [49] (in the general case) formulated conjectures expressing certain special values of the L-series of symmetric powers of an elliptic curve over ℚ in terms of higher elliptic polylogarithms. In this section we describe the conjecture briefly for the next case (i.e., the elliptic trilogarithm and Sym^{2}(E)) and provide a numerical example. A detailed exposition is given in [25].

The trilogarithmic picture. In this case, the general Beilinson conjecture says that L(Sym^{2}(E), 3) should be related to a 2-dimensional lattice, since the functional equation of L(Sym^{2}(E), s) shows that it has a double zero at s = 0. Similarly, the regulator is given by two functions on divisors of E: the analogue of D^{E}, defined by

ℒ_{3,1}^{E}(x) = ∑_{n∈ℤ} ℒ_{3}(q^{n}x) ,

and a second one (the analogue of Bloch’s J_{q}(x) averaging the function J(x) = log |x| log |1 − x|)

ℒ_{3,2}^{E}(x) = ∑_{n=0}^{∞} J_{3}(q^{n}x) + ∑_{n=1}^{∞} J_{3}(q^{n}x^{−1}) + \frac{log^{2} |x| log^{2} |q/x|}{4 log |q|} ,

where J_{3}(x) = log^{2} |x| log |1 − x| . This function satisfies ℒ_{3,2}^{E}(x) = ℒ_{3,2}^{E}(qx) because

J_{3}(x) − J_{3}(x^{−1}) = − log^{3} |x| = \frac{log^{2} |qx| log^{2} |x| − log^{2} |x| log^{2} |q/x|}{4 log |q|} .

The functions ℒ_{3,1}^{E} and ℒ_{3,2}^{E} also have expressions, analogous to formula (28) for D^{E}, as linear com-

binations of the Kronecker-Eisenstein series ∑_{ω} \frac{χ(ω)}{ω^{2}ω̄^{2}} and ∑_{ω} \frac{χ(ω)}{ωω̄^{3}}, where ω runs over the lattice

L defining E = ℂ/L and χ : L → S^{1} is the character corresponding to P. There are of course also higher elliptic polylogarithms, relevant for the conjectural formula for L(Sym^{m−1}(E), m), which can be expressed either as averages over q^{ℤ} of higher classical polylogarithmic functions or else as linear combinations of higher Kronecker-Eisenstein series ∑ χ(ω)/ω^{a}ω̄^{b}. These functions, for which explicit formulas can be found in [54], are the elliptic analogues of the one-valued polylogarithm functions ℒ_{m} : ℂ → ℝ. There are also elliptic analogues of the original many-valued polylogarithm function Li_{m}, studied in detail in the beautiful paper [32] of A. Levin.

We would like to characterize the conditions on ξ = ∑ n_{i}(P_{i}) that are needed in order for the image of ξ under ℒ⃗_{3}^{E} = (ℒ_{3}^{E}, J_{3}^{E}) to belong to the expected 2-dimensional lattice. Let us suppose for convenience that we are looking at a curve E like E_{37} for which the conditions (i)–(iii) of §7 are the correct ones for D_{2}^{E}. Then we find (both experimentally and based on the conjectural theory) that the conditions (i) and (ii) get replaced by

(i′) ∑ n_{i} = 0 ∈ ℤ , ∑ n_{i}(P_{i})^{2} = 0 in Sym^{2}(E) , ∑ n_{i}(P_{i})^{4} = 0 in Sym^{4}(E) ,

and the condition (iii) by

(ii′) ∑ n_{i}λ_{p}(P_{i}) (P_{i})^{2} = 0 in Sym^{2}(E) for every prime p .

but there is a further condition, analogous to the one needed in the usual polylogarithm case. Namely, (i′) and (ii′) imply that ι_{ϕ}(ξ) = ∑ n_{i}ϕ(P_{i})P_{i} belongs to A_{2}(E/ℚ) for every homomorphism ϕ : E(ℚ) → ℤ; the further condition is then, that this element ι_{ϕ}(ξ) maps to 0 in the Bloch group B_{2} = A_{2}/C_{2}. All of the conditions together can be formulated more succinctly as saying that ι_{ϕ}(ξ) ∈ C_{2}(E/ℚ) for all homomorphisms ϕ : E(\overline{ℚ}) → ℚ.

Example. We again take E_{37}. We had found 4 elements ξ_{3}, ξ_{4}, ξ_{6} and ξ_{10}^{∗} ∈ A_{2} and by (30) three relations 4ξ_{4} − 13ξ_{3} , 4ξ_{6} − 45ξ_{3} , ξ_{10}^{∗} − 31ξ_{3} in C_{2}. “Pseudo-integrating” (i.e. replacing ∑ n_{i}(iP) by ∑_{i≠0} \frac{n_{i}}{i} (iP) , cf. Example 3 in §3) and then adding a “constant of integration” n_{0}(O) to make the new divisor of degree 0 gives us three new elements

η_{4} = 3(4P) − 13(3P) + 18(2P) − 3(P) − 5(O) ,
η_{6} = 2(6P) − 45(3P) + 60(2P) + 93(P) − 110(O) ,
η_{10} = 3(10P) − 24(5P) − 310(3P) + 585(2P) + 750(P) − 1004(O) ,

which satisfy all the above conditions and are linearly independent. If the theory is correct, then the images of these three divisors under the map ℒ⃗_{3}^{E} should lie in a 2-dimensional lattice with a covolume related to L(Sym^{2}(E), 3). Evaluating ℒ_{3,1}^{E} and ℒ_{3,2}^{E} numerically, we get the values

ℒ⃗_{3}^{E}(η_{4}) = (−31.93647324246920545 …, 51.36032576407323769 …) ,
ℒ⃗_{3}^{E}(η_{6}) = (−270.92487957661031873 …, 29.85229569612024863 …) ,
ℒ⃗_{3}^{E}(η_{10}) = (−2374.45096970455445769 …, 165.95000973693576237 …) ,

and indeed we find a linear combination η_{10} − 9η_{6} + 2η_{4} which maps under ℒ⃗_{3}^{E} to (0, 0) to many decimal places. The determinant formed of the values of ℒ⃗_{3}^{E} for η_{4} and η_{6} is

Reg_{3}(E) = | \begin{matrix}ℒ_{3,1}^{E}(η_{4}) & ℒ_{3,2}^{E}(η_{4}) \\ ℒ_{3,1}^{E}(η_{6}) & ℒ_{3,2}^{E}(η_{6}) \end{matrix} | = 12961.41302992157 …

On the other hand, we can compute

L(Sym^{2}(E), 3) = 1.526262007533073 …

(see below) and we find, within the precision of the calculation, the relation

Reg_{3}(E) = \frac{37^{3}}{4} ℑ(τ)^{2} L(Sym^{2}(E), 3) .

Computing the L-value. We end by explaining how to compute L(Sym^{2}(E), 3). Let f = ∑ a_{n}q^{n} ∈ S_{2}(Γ_{0}(N)) be the modular form corresponding to E. Define the Dirichlet series L_{2}(f, s) by

L_{2}(f, s) := ζ(s − 1) L(Sym^{2}(E), s) = ∑_{n=1}^{∞} \frac{b_{n}}{n^{s}} ,

where the coefficients b_{n} can be obtained by

L_{2}(f, s) = (1 + N^{1−s}) ζ(2s − 2) ∑_{n=1}^{∞} \frac{a_{n}^{2}}{n^{s}} .

(This formula is for the N = 37 case; for other f the Euler factors for the primes of bad reduction might change.) The function L_{2}(f, s) can be shown by Rankin’s method to have the functional equation L_{2}^{∗}(f, s) = L_{2}^{∗}(f, 3 − s), where L_{2}^{∗}(f, s) = (2π)^{−2s}N^{s}Γ(s)Γ(s − 1)L_{2}(f, s). Hence, using the integral representation of Γ(s)Γ(s−1) as the Mellin transform of the K-Bessel function K_{1}(t) and splitting up the resulting integral into two pieces in the usual way, we find

L_{2}(f, 3) = C (A − \frac{A^{2}}{2} ) + \frac{1}{16} ∑_{n=1}^{∞} \frac{b_{n}}{n^{3}} G_{1}(4π\sqrt{\frac{nA}{N}}) + \frac{2^{8}π^{6}}{N^{3}} ∑_{n=1}^{∞} b_{n} G_{2}(4π\sqrt{\frac{n}{NA}}) ,

for any A > 0, where C = 2π^{2}N^{−1}L(Sym^{2}(E), 2) = 1.3296021859908 … and G_{1} and G_{2} are the functions

G_{1}(x) = ∫_{x}^{∞} t^{4}K_{1}(t)dt , G_{2}(x) = ∫_{x}^{∞} t^{−2}K_{1}(t)dt ,

both of which are bounded by a power of x times e^{−x} for x large. The numerical value of C can be obtained either from this formula (by evaluating it for two different values of A or by taking its derivative, e.g. at A = 1) or, by Rankin’s method, as a simple multiple (= (2π)^{3}/N^{2}) of the covolume of the lattice corresponding to E. The function G_{1}(x) can be expressed in terms of Bessel functions by

G_{1}(x) = (x^{4} + 8x^{2})K_{0}(x) + (4x^{3} + 16x)K_{1}(x)

and hence can be computed numerically using standard software packages. For G_{2}(x) we can use the rapidly convergent power series expansion

G_{2}(x) =\frac{1}{2x^{2}} − \frac{1}{4} log^{2}(x) − \frac{1}{2}(γ − log 2 − \frac{1}{2}) log x + C_{0}

− \frac{1}{4} ∑_{n=1}^{∞} \frac{(x/2)^{2n}}{n (n + 1)! n!}( log \frac{x}{2} + γ − ∑_{k=1}^{n} \frac{1}{k} − \frac{1}{2n} − \frac{1}{2(n + 1)})

(where γ is Euler’s constant and C_{0} = −0.3629591663513 …) for reasonably small x and the asymptotic expansion

G_{2}(x) ∼ \sqrt{\frac{π}{2x}}e^{−x}(\frac{1}{x^{2}} − \frac{17}{8x^{3}} + \frac{937}{128x^{4}} − ⋯ + \frac{r_{n}}{x^{n+2}} + …)

(where r_{−1} = 0, r_{n} + (n + \frac{3}{2})r_{n−1} = (−\frac{1}{2})^{n}n!\binom{1/2}{n}\binom{−3/2}{n} for n ≥ 0), for large x.

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