On the Second K-group of an Elliptic Curve

by Klaus Rolshausen (München) and Norbert Schappacher (Strasbourg)

The abstract formalism used to define higher algebraic K-groups is more often than not the envy and the despair of the mathematician eager to work with them. In fact, on the one hand, finite generation results tend to be as clearly expected as they are impossible to prove, and, on the other, the general theory makes it usually quite hard to produce explicit non-zero elements in specific K-groups. Recent progress in the spirit of Zagier’s polylogarithm conjecture now has provided some (in part conjectural) remedies for the second one of these diametrically opposed problems. Research in this area is moving fast at the moment. We will concentrate here on the group K_{2}(E), for an elliptic curve E defined over \overline{Q}, where solid progress has been realized by A. Gončarov, A. Levin, and J. Wildeshaus over the last two years.

The aim of our note is to present a down-to-earth treatment of the case of the elliptic analogue of Zagier’s polylogarithm conjecture pertaining to K_{2}(E)—see section 4 below for the final result, which enables one to systematically produce elements of K_{2}(E) from certain divisors on E. We thus complete the first author’s thesis [Rolshausen 1996]. More precisely, outside of K-theory, we use only the classical theory of elliptic functions according to Eisenstein, Kronecker and Weierstrass, as well as algebraic number theory. And from K-theory, except for a few basic exact sequences and well-known properties of K-groups of fields, or of varieties over finite fields, we only need [Schappacher, Scholl 1991, §3]. However, the way we set things up is of course modelled on (earlier versions of) the papers [Gončarov, Levin, 1997] and [Wildeshaus 1997b]. Contrary to Gončarov and Levin, we will systematically neglect the torsion in the groups under consideration, tensoring with Q.

The original motivation of the first author’s thesis [Rolshausen 1996] was to check Beilinson’s conjecture numerically for certain examples of elliptic curves over Q—see section 5 below as well as the tables in [Rolshausen 1996].

Since our approach is deliberately down-to-earth, we do not discuss the recent preprints [Bloch 1997], [Gončarov 1997]. Nor do we go into the higher conjectures which concern the special values L(Sym^{n}E, n + 1), for n ≥ 2. By Beilinson’s general formalism, the corresponding absolute cohomology groups are the groups H_{M}^{n+1}(Sym^{n}E, Q(n + 1)) attached to the motive Sym^{n}E which can be defined as the sub-motive of h(E^{k}′) on which the permutations of S_{k+1} act via the sign character. Here, E^{k}′ is the sub-variety of E^{k+1} where the sum of the components is 0. Thus, H_{M}^{n+1}(Sym^{n}E, Q(n + 1)) can be described as a sub-vector space of K_{n+1}(E^{n}′) ⊗_{Z} Q. The peculiar problems of these higher cases and some conjectural solutions are discussed in the preprints [Gončarov 1995] and [Wildeshaus 1997c].

We thank S. Bloch, A. Gončarov and J. Wildeshaus for very helpful discussions. The second author thanks the Max-Planck-Institut für Mathematik in Bonn for a very productive stay in August 1997.

1. Recalling K_{2}(E). Let F be a number field over which the elliptic curve E is defined, and let E/𝔬_{F} be a regular minimal model of E over the ring of integers of F. The second K-groups of E and of E/F are connected by the localization exact sequence

(1.1)  ⋯ ⟶ ∐_{𝔭} K′_{2}(E_{𝔭}) ⟶ K_{2}(E) ⟶ K_{2}(E) ⟶^{∂} ∐_{𝔭} K′_{1}(E_{𝔭}) ⟶ ⋯,

the direct sums being taken over the finite places of F. As to the fibres E_{𝔭} of E at the finite primes, it is well-known that all the K′_{2}(E_{𝔭}) are torsion groups, and that K′_{1}(E_{𝔭}) is non-torsion if and only if K′_{1}(E_{𝔭})⊗_{Z} Q is a Q-line (and the 𝔭-component of ∂ is surjective), and that this happens if and only if E has split multiplicative reduction at 𝔭—see for instance [Schappacher, Scholl 1991, §3] or [Rolshausen 1996, I.3].

1.2. Beilinson’s Conjecture, I. Denote by r_{F} the number of infinite places of F. Then one expects that

dim_{Q}(K_{2}(E/𝔬_{F}) ⊗_{Z} Q) =^{?} ord_{s=0} L(E, s) = r_{F} .

Or, equivalently, that

dim_{Q}(K_{2}(E/F) ⊗_{Z} Q) =^{?} r + #{𝔭 | E has split multiplicative reduction mod 𝔭}.

The conjecture is fleshed out by considering a regulator map. To define it in the case at hand use the following short exact sequence, which is derived from the localization sequence of E with respect to its generic fibre,

(1.3)  0 ⟶ K_{2}(E/F) ⊗_{Z} Q ⟶ K_{2}(F (E)) ⊗_{Z} Q ⟶^{T} ∐_{P ∈E}(k(P)^{∗} ⊗_{Z} Q),

where P varies over the \overline{F} -valued points of E, and k(P) denotes the field of definition of P. The zero on the left follows from the fact that the second K-group of a number field is torsion. By Matsumoto’s theorem (which the second author learned by presenting it in Martin Kneser’s Oberseminar, some twenty years ago),

K_{2}(F (E)) = \frac{⋀^{2} F (E)^{∗}}{⟨f ∧ 1 − f | f ∈ F (E), f ≠ 0, 1⟩_{Z}} ∋ \{f, g\}.

In terms of these generators, the P -component of the tame symbol T in (1.3) is explicitly given by the formula

(1.4)  T_{P} (\{f, g\}) = (−1)^{v_{P} (f)v_{P} (g)} \frac{f^{v_{P} (g)}}{g^{v_{P} (f)}} (P).

We can now define the regulator. In the spirit of this note, we simply present it as a mapping

reg_{ω} : K_{2}(E/F) ⊗_{Z} Q ⟶ R^{r_{F}} = ⨁_{v|∞} F_{v}

which depends on the choice of a holomorphic differential form ω on E defined over F. It was S. Bloch who first defined it in [Bloch 1977], on the generators {f, g} of K_{2}(F (E)). Write the divisors of the functions as (f) = ∑ a_{x}[x], (g) = ∑ b_{y}[y]. Fix an embedding τ : F ↪ C, and call v the infinite place of F given by τ. Via τ, we have over C, (E, ω) ≅ (C/Γ, dz) with a unique lattice Γ = ω_{1}Z ⊕ ω_{2}Z ⊂ C. If x ∈ E(C), denote by z(x) the corresponding class in C/Γ. The v-component of the regulator is defined to be

the real part of the integral reg_{ω,v}(\{f, g\}) = Re(\frac{1}{πi} ∫_{C/L} log |f| \overline{(\frac{dg}{g} )} ∧ dz). According to [Bloch 1977], this can be transformed into the following expression—see also [Rohrlich 1987]:

(1.5)  reg_{ω,v}({f, g}) = −Re(A(Γ)^{2} ∑_{x,y} a_{x}b_{y}K_{1}(0, z(x) − z(y), 2, Γ)).

Here we have used the notation from [Weil 1976] for the Kronecker-Eisenstein series: If we take the basis ω_{1}, ω_{2} of the lattice such that ω_{1} is real and y = Im(ω_{2}/ω_{1}) > 0, then A(Γ) = \frac{ω_{1}^{2}y}{π}. The Pontryagin duality between Γ and C/Γ is afforded by ⟨γ, x⟩ = exp(\frac{γ\overline{x}−x\overline{γ}}{A(Γ)} ). Then for z ∈ C/Γ, and an integer ν ≥ 0, one has the convergent series in the half-plane Re(s) > \frac{ν}{2} + 1 ,

(1.5.1)  K_{ν}(0, z, s, Γ) = ∑_{γ∈Γ}′⟨γ, z⟩\frac{\overline{γ}^{ν}}{(γ\overline{γ})^{s}} .

(The extra parameter which we set equal to zero here because we will not need it, is relevant for the analytic continuation of these series.)

The analogues of these regulator maps for the finite places are the components of the boundary map ∂ in (1.1). They were determined by Bloch and Grayson, and more generally in [Schappacher, Scholl 1991]. They are zero unless E has split multiplicative reduction at 𝔭. Suppose this is so, i.e., the fibre E_{𝔭} is a Néron N-gon for some N. Just as we chose a differential ω in order to obtain the regulators at the infinite places as real-valued functions, we now pick a basis of the Q-line K′_{1}(E_{𝔭}) ⊗_{Z} Q. Specifically, we number the components of E_{𝔭} and take, in the notation of [Schappacher, Scholl 1991], the basis element \frac{1}{3N} · Φ_{1}^{1} in order to identify K′_{1}(E_{𝔭}) ⊗_{Z} Q = Q. Then the component map ∂_{𝔭} on K_{2}(E) is obtained by linearity from the following rule on Steinberg symbols {f, g} which have the property that the closure of the support of the divisors (f) = ∑ a_{x}[x], (g) = ∑ b_{y}[y] are contained in the smooth part of E:

(1.6.1)  {f, g} ↦ − ∑_{x,y} a_{x}b_{y}B_{3,𝔭}(x − y),

where

(1.6.2)  B_{3,𝔭}(x) = B_{3}(⟨\frac{ν}{N} ⟩),

if and only if the section of E defined by x meets the Néron N-gon E_{𝔭} in its ν-th side. Here B_{3}(t) = t^{3} − \frac{3}{2}t^{2} + \frac{1}{2}t is the third Bernoulli polynomial. Cf. [Schappacher, Scholl 1991, 3.1 (p. 815)].

1.7 Beilinson’s Conjecture, II. (i) Let S be the set of places of F where E has split multiplicative bad reduction. The Q-vector space

(⨁_{v|∞} reg_{ω,v} ⊕ ⨁_{𝔭∈S} B_{3,𝔭})(K_{2}(E/F) ⊗_{Z} Q)

is a Q-structure of ⨁_{v|∞} R ⊕ ⨁_{𝔭∈S} R.

(ii) ⋀^{r_{F}} (reg_{ω}(K_{2}(E/𝔬_{F}) ⊗_{Z} Q)) = L^{(r_{F})}(E/F, 0) · Q ⊂ R = ⋀^{r_{F}} (R^{r_{F}}).

One may enhance the analogy between the infinite places and those in S by building a factor of log(N𝔭) into the definition of B_{3,𝔭}, and then relating the total regulator (including the S-components) to the derivative at s = 0 of the imprimitive L-function, with Eulerfactors at places in S removed. This was proposed, in slightly greater generality, as an “S-Beilinson conjecture” in [Ramakrishnan 1989]. But we find this formal procedure rather pointless in the case at hand.

It is not known at present for a single curve E, that K_{2}(E) ⊗_{Z} Q or K_{2}(E) ⊗_{Z} Q are finite dimensional. The strongest results towards (ii) known to date still are, on the one hand, Bloch’s treatment of the CM case in [Bloch 1977] which was refined in [Rohrlich 1987] and generalized in [Deninger 1989], and on the other, Beilinson’s theorem on modular curves, and thus on modular elliptic curves—see [Schappacher, Scholl 1988].

1.8 Remark on the injectivity of the regulator. It is sometimes heuristically useful—say, in the case of an elliptic curve E/Q—to assume that the Beilinson regulator reg_{∞} is injective on all of K_{2}(E) ⊗ Q—cf. [Bloch, Grayson 1986, p. 84, note (*)] for a case in point. But, as Bloch pointed out to us, this hypothesis inspires less confidence, from a theoretical point of view, than conjecture 1.7.(i), because it postulates the injectivity of a real map on a Q-vector space which is not always going to be one-dimensional. In fact, compare 1.7.(i) to the injectivity of the classical regulator for a number field, i.e., the injectivity of the map on the right in the commutataive triangle of theorem 3.4 below.

2. The basic diagram. The obvious idea to try and construct elements in K_{2}(E) is to use the exact sequence (1.3). Let us pass to the direct limit over all number fields. This gives the exact sequence

(2.1)  0 ⟶ K_{2}(E/\overline{Q}) ⊗_{Z} Q ⟶ K_{2}(\overline{Q}(E)) ⊗_{Z} Q ⟶^{T} \overline{Q}^{∗} ⊗_{Z} I_{E},

where the tame symbol now sends

\{f, g\} ↦ ∑_{x∈E(\overline{Q})}(−1)^{v_{x}(f)v_{x}(g)} (\frac{f^{v_{x}(g)}}{g^{v_{x}(f)}} |_{x} ⊗ x).

Here we have written (tensoring with Q a notation used by Gončarov and Levin) I_{E} for the augmentation ideal of the group algebra Q[E(\overline{Q})], i.e., for the divisors of degree 0 on E with Q-coefficients. Weil reciprocity guarantees that the tame symbol does indeed land in this subspace.

Factoring out I_{E} by principal divisors, i.e., by I_{E}^{2}, and inserting Matsumoto’s theorem for the middle group, we obtain the exact sequence

(2.2) 0 ⟶ K_{2}(E/\overline{Q})⊗_{Z}Q ⟶ \frac{⋀_{Z}^{2} \overline{Q}^{∗}}{(\overline{Q}^{∗} ∧ \overline{Q}(E)^{∗})⟨f ∧ (1 − f)⟩_{Z}} ⊗_{Z}Q ⟶^{T̃} \overline{Q}^{∗}⊗_{Z}E(\overline{Q})⊗_{Z}Q.

To see this (cf. the proof of (3.7) in [Gončarov, Levin 1997]), note first that a Steinberg symbol of the form \{c, g\}, with a constant c ∈ \overline{Q}^{∗}, is mapped by T to c ⊗ div(g), and therefore lies in the kernel of T̃. But the kernel of the restriction of T to

(\overline{Q}^{∗} ∧ \overline{Q}(E)^{∗}) ⊗_{Z} Q ⟶ \overline{Q}^{∗} ⊗_{Z} I_{E}^{2}

is K_{2}(\overline{Q}) ⊗_{Z} Q = 0, again by Matsumoto’s theorem, and T thus restricted to a map (\overline{Q}^{∗} ∧ \overline{Q}(E)^{∗}) ⊗_{Z} Q = \overline{Q}^{∗} ⊗_{Z} I_{E}^{2} ⟶ \overline{Q}^{∗} ⊗_{Z} I_{E}^{2} is simply the identity.

Note that, if we work over a field of definition F of E which has finite degree over Q, then (2.2) can be simply cut down by taking invariants under G_{F}: = G(\overline{Q}/F), because the K-groups tensored with Q have Galois descent. Note also that all the maps reg_{ω,v} and B_{3,𝔭} vanish on symbols of the form {c, h}.

The exact sequence (2.2) immediately gives what used to be known as

2.3 Bloch’s Lemma. A Steinberg symbol {f, g} such that the divisors of f and g are supported on torsion points of E, can be corrected by elements of the form {c, h} to come from K_{2}(E) ⊗_{Z} Q.

For a long time this was the only method to produce explicit elements in K_{2}(E). If E has complex multiplication (and therefore in particular no primes of bad multiplicative reduction), such elements conjecturally suffice—according to (1.7)—to generate all of K_{2}(E)—see [Rohrlich 1987]. However, for elliptic curves with sufficiently big image of G_{F} in Aut (E_{tors}), the elements given by Bloch’s lemma will all lie in the kernel of ⨁_{v|∞} reg_{ω,v} ⊕⨁_{𝔭} B_{3,𝔭}, and therefore conjecturally be zero—see [Mestre, Schappacher 1991, 1.3.1]. Beilinson’s Eisenstein Symbol—see [Deninger 1989]—is essentially a generalization and refinement of Bloch’s lemma.

A rather short-lived and somewhat frustrating phase in the development of our current knowledge of K_{2}(E), roughly between 1989 and 1994, saw a number of ad hoc constructions of non-trivial elements in K_{2}(E) from certain points of infinite order on special elliptic curves without complex multiplication, working directly from the sequence (2.1) or (2.2) (unpublished constructions by Schappacher-Nekovář, R. de Jeu, J-F. Mestre; also R. Ross obtained an element on the Fermat elliptic curve with CM). We will briefly discuss the oldest such example in section 5 below..

We now write down the commutative diagram which will give us the most general way to obtain elements in K_{2}(E) from divisors on the elliptic curve. It is suggested by [Gončarov, Levin 1997], although our presentation of it blends in some notions from [Wildeshaus 1997b]. It begins with the line (2.2). The rest will be explained as we go on.

K_{2}(E/\overline{Q}) ⊗_{Z} Q ↪ \frac{⋀_{Z}^{2} \overline{Q}^{∗}}{(\overline{Q}^{∗} ∧ \overline{Q}(E)^{∗})⟨f ∧ (1 − f)⟩_{Z}} ⊗_{Z} Q ⟶^{T̃}  \overline{Q}^{∗} ⊗_{Z} E(\overline{Q}) ⊗_{Z} Q
↑  ↓ β  ≅ ↑ φ_{2} ⊗ id
|  \frac{I_{E}^{4}}{β⟨f ∧ (1 − f)⟩_{Q}}  ⟶ ker d_{2} ⊗_{Q} (E(\overline{Q}) ⊗_{Z} Q)
|  ↑ p  ||
|  I_{E}^{4}  ⟶ ker d_{2} ⊗_{Q} (E(\overline{Q}) ⊗_{Z} Q)
|  ∩  ∩
ker d_{3}^{#}  ↪  L_{3}^{#}  ⟶^{d_{3}^{#}} L_{2} ⊗_{Q} (E(\overline{Q}) ⊗_{Z} Q)

Commutative diagram (2.4)

The column on the right will be defined and constructed in the next section—see 3.6. Let us discuss the central column. We implicitly encountered the map β when we were discussing regulators. It is defined by

β(f ∧ g) = ∑_{x,y} a_{x}b_{y} [x − y] ∈ Q[E(\overline{Q})],

where the divisors of the functions are written (f) = ∑ a_{x}[x], (g) = ∑ b_{y}[y]. Since the principal divisors make up I_{E}^{2}, the image of β coincides with the fourth power I_{E}^{4} of the augmentation ideal.

The vertical map p is the natural projection, and the space at the bottom of the central column is defined, following [Wildeshaus 1997b], by

L_{3}^{#} = Q[E(\overline{Q}) − \{0\}].

The inclusion of I_{E}^{4} into it is defined by forgetting the contribution of 0 in the divisors. Since they are all of degree 0, this does not erase any information.

We now turn to the right column of the diagram. This will take a certain amount of preparation …

3. The multiplicative group of a number field. Choose again a number field F over which the elliptic curve E is defined. Put

L_{2}^{#} = Q[E(F) − {0}],

and write elements of this vector space in the form ∑_{i} a_{i}{x_{i}}_{2}^{#}. Define a Q-linear map by its effect on the basis:

\begin{aligned}d_{2}^{#} : & L_{2}^{#} & ⟶ & E(F) ⊗_{Z} E(F) ⊗_{Z} Q \\ & \{x\}_{2}^{#} & ↦ & x ⊗ x ⊗ 1. \end{aligned}

The kernel of d_{2}^{#} contains a few obvious types of elements: the divisors supported on torsion points of E(F), the relations {kx} − k^{2}{x} (for k ∈ Z and x ∈ E(F) not killed by k), and the ‘parallelogram equations’ 3.1.1 below. It is elementary to show that certain among these elements already suffice to generate all of kerd_{2}. See for instance [Gončarov, Levin 1997, 4.6].

3.1 Lemma. kerd_{2}^{#} is generated by all divisors of the following form, where x, y ∈ E(F).

(3.1.1)  {x + y}_{2}^{#} + {x − y}_{2}^{#} − 2{x}_{2}^{#} − 2{y}_{2}^{#}  for x ≠ ±y
(3.1.2)  {2x}_{2}^{#} − 4{x}_{2}^{#}  for 2x ≠ O
(3.1.3)  {2x}_{2}^{#} − 2{x}_{2}^{#} − 2{−x}_{2}^{#}  for 2x ≠ O
(3.1.4)  −4{x}_{2}^{#}  for 2x = O

3.2 Remark. Note that all divisors in kerd_{2}^{#} are such that the global Néron-Tate height ĥ on E over F, extended linearly from points to divisors, vanishes on them. — If for a basis x_{1}, …, x_{ρ} of the Mordell-Weil group E(F) modulo torsion the heights ĥ(x_{i}) are Q-linearly independent real numbers, then kerd_{2}^{#} = {∑ a_{i}{x_{i}}_{2}^{#} | ∑ a_{i}ĥ(x_{i}) = 0}.

3.3 Notation. For any place v of F, denote by h_{v} the local Néron height on E/F_{v}, extended Q-linearly to divisors. For a finite place v, log_{v} : F^{∗} ⟶ Z is the additive valuation onto Z, and if v is the infinite place given by the embedding τ : F ↪ C, then log_{v} : F^{∗} ⟶ R is the function log | |_{τ}. In either case, we extend by Q-linearity to F^{∗}⊗_{Z}Q.

The local Néron height at the infinite places is related to Kronecker double series: If the infinite place v of F is induced by the embedding τ : F ↪ C, and if the point x ∈ E(F)↪^{τ}E(C) = C/Γ corresponds to the complex number z, modulo the lattice Γ, then we have (notations being as in (1.5.1) above)

(3.3.1)  h_{v}(x) = \frac{A(Γ)}{2}K_{0}(0, z, 1, Γ).

This follows for instance from a comparison of [Zagier 1990, Thm. 1.3] with [Silverman 1994, chap. VI].

3.4 Theorem. [Elliptic Analogue of Zagier’s Polylogarithm Conjecture for K_{1}(F).] There is a unique Q-linear map φ_{2}^{#} rendering the following diagram commutative:

φ_{2}^{#} : ker d_{2}^{#}  ⟶  F^{∗} ⊗_{Z} Q
⨁vh_{v} ↘  ↙ _{⨁v} log_{v}

⨁_{v inf.}R ⊕ ⨁_{v fin.}Q

3.4.1 Remark. G. Faltings has pointed out to us that this proposition (along with its algebro-geometric prrof via the Poincaré bundle) generalizes verbatim to abelian varieties E defined over F. The only additional complication is that the local heights depend in general on the choice of a section of the ample line bundle. Proof: There are now several proofs of (variants of) this proposition—see [Gončarov, Levin 1997] and [Wildeshaus 1997b, Thm. 2] for a proof using the Poincaré bundle and Deligne’s symbol, and [Wildeshaus 1997d] for a motivic approach. We sketch here the proof which seems to us to be the most elementary. It was first envisaged in [Rolshausen 1996].

The uniqueness of φ_{2}^{#} follows from Dirichlet’s Unit Theorem. Also, one can easily define φ_{2}^{#} on generators of kerd_{2}^{#}, say those listed in 3.1, in such a way that the triangle commutes, by invoking standard formulae from the theory of local heights—see for instance [Silverman 1994, chap. VI, in particular ex. 6.3]: Fix a Weierstrass model for E/F of the very classical shape Y^{2} = 4X^{3} − g_{2}X − g_{3} (g_{2}, g_{3} ∈ F), denote by X, Y the coordinate functions with respect to this model, by Δ its discriminant. Then we want to prescribe the following values in F^{∗} ⊗ Q.

(3.4.2) \begin{aligned}φ_{2}^{#}(\{x + y\}_{2}^{#} + \{x − y\}_{2}^{#} − 2\{x\}_{2}^{#} − 2\{y\}_{2}^{#}) & = (X(x) − X(y))^{−6}Δ ⊗ \frac{1}{6} \\ φ_{2}^{#}(\{2x\}_{2}^{#} − 4\{x\}_{2}^{#}) & = (−Y (x))^{−4}Δ ⊗ \frac{1}{4} \\ φ_{2}^{#}(\{2x\}_{2}^{#} − 2\{x\}_{2}^{#} − 2\{−x\}_{2}^{#}) & = (−Y (x))^{−4}Δ ⊗ \frac{1}{4} \\ φ_{2}^{#}(−4\{x\}_{2}^{#}) & = (−3X(x)^{2} + \frac{1}{4}g_{2})^{−3}Δ ⊗ \frac{1}{3} \end{aligned}

So we have to show that there exists a well-defined map on kerd_{2}^{#} which takes all those values. For this we may work over C, fixing any embedding of F. The idea is to exponentiate an analytic formula for the local archimedean height. But since these formulae involve some theta function, one has to be careful about writing down an expression which is independent of the representatives in C chosen for the points of C/Γ, where Γ is a lattice corresponding to E. Specifically, given ξ = ∑_{j} a_{j}{x_{j}}_{2}^{#} ∈ kerd_{2}^{#}, choose Ξ = ∑_{k} A_{k}[z_{k}] ∈ Q[C − Γ] such that Ξ ↦ ξ under z_{k} ↦ (z_{k} mod Γ), and such that Ξ lies in the kernel of the Q-linear map Q[C − Γ] ⟶ C ⊗_{Q} C defined by z ↦ z ⊗ z. (This last condition was introduced in [Wildeshaus 1997b, p. 374–375], emending the treatment given in [Rolshausen 1996, V.4.4].) For instance, if ξ is a generator of type (3.1.1) and if x = z mod Γ, y = z′ mod Γ, then Ξ = [z + z′] + [z − z′] − 2[z] − 2[z′] will do. Similarly, one sees that such representatives Ξ exist for all divisors ξ ∈ kerd_{2}^{#}; generators of type (3.1.4) can be recuperated over C (where all 2-torsion points are rational) from (3.1.1), for various distinct 2-torsion points x and y.

Restricting to such representatives Ξ = ∑_{k} A_{k}[z_{k}] of ξ, the expression

(3.4.3)  φ_{2}^{#}(ξ) = ∑_{k}[(e^{−6z_{k}η(z_{k})}σ(z_{k})^{12} · Δ) ⊗ (−\frac{A_{k}}{12} )] ∈ C^{∗} ⊗_{Z} Q,

where η, σ are the classical Weierstrass functions relative to the lattice Γ, is well-defined. Indeed, if Ξ, Ξ′ are two such representatives of ξ, then Ξ−Ξ′ is of the form ∑_{l} c_{l}([z_{l}]−[z_{l} +

γ_{l}]), for suitable c_{l} ∈ Q, z_{l} ∈ C, and γ_{l} ∈ Γ. Write the complex numbers z_{l} and γ_{l} in terms of a fixed basis of the lattice: z_{l} = A_{l}ω_{1} + B_{l}ω_{2} (A_{l}, B_{l} ∈ R), γ_{l} = C_{l}ω_{1} + D_{l}ω_{2} (C_{l}, D_{l} ∈ Z). The transformation formula for our theta function—see for instance [Silverman 1994, VI.3.1.(b)]—combined with Legendre’s period relation then shows that formula (3.4.3) evaluated on Ξ − Ξ′ yields

exp(∑_{l} c_{l}{A_{l}D_{l} − C_{l}B_{l}} · πi) ∈ C^{∗} ⊗_{Z} Q.

So it suffices to show that ∑_{l} c_{l}{A_{l}D_{l}−C_{l}B_{l}} ∈ Q. On the other hand, ∑_{l} c_{l}([z_{l}]−[z_{l}+γ_{l}]) goes to 0 in C⊗_{Q} C, since this is true by hypothesis for Ξ and Ξ′. This yields in particular the relations

∑_{l,A_{l}∉Q} c_{l}D_{l}(A_{l}ω_{1} ⊗ ω_{2}) = 0 = ∑_{l,B_{l}∉Q} c_{l}C_{l}(ω_{1} ⊗ B_{l}ω_{2}),

and thus ∑_{l,A_{l}∉Q} c_{l}D_{l}A_{l} = ∑_{l,B_{l}∉Q} c_{l}C_{l}B_{l} = 0, which is enough to prove what we want.

To see that (3.4.3) does reproduce in C all the values prescribed algebraically on individual generators in (3.4.2), use for the type (3.1.1) the divisor Ξ = [z + z′] + [z − z′] − 2[z] − 2[z′] and the classical addition theorem for the Weierstrass σ-function:

\frac{σ(z + z′)σ(z − z′)}{σ(z)^{2}σ(z′)^{2}} = ℘(z) − ℘(z′),

and similarly, deriving this addition formula, for the other types of generators.

Note that “Siegel’s function”, used by Wildeshaus in Proposition 2 in loc. cit., does agree with (3.4.3) for our divisors Ξ, as can be seen by going through the proof of [Silverman 1994, Thm. VI.3.4].

3.5 Distribution relations. The map φ_{2}^{#} satisfies the following compatibility with isogenies of elliptic curves—see [Wildeshaus 1997b] and cf. [Wildeshaus 1997a, I-5], where the term ‘norm compatibility’ is used: Let ψ : E ⟶ E′ be an isogeny defined over F and let ξ = ∑_{i} a_{i}{x_{i}}_{2}^{#} ∈ L_{2}^{#}. Then ψ(ξ) = ∑_{i} a_{i}{ψ(x_{i})}_{2}^{#} belongs to kerd_{2}^{#} on E′ if and only if one has d_{2}^{#}(∑_{i} a_{i} ∑_{t∈kerψ}{x_{i} +t}_{2}^{#}) = 0 on E. Furthermore, if these equivalent conditions hold, then one has for the maps φ_{2}^{#} on E, resp. on E′, the following identity in F^{∗} ⊗_{Z} Q:

(3.5.1)  φ_{2}^{#}(∑_{i} a_{i} ∑_{t∈kerψ}{x_{i} + t}_{2}^{#}) = φ_{2}^{#}(∑_{i} a_{i}{ψ(x_{i})}_{2}^{#}).

The direct proof of this distribution relation from (3.4.3) is somewhat nontrivial—see [Jarvis 1996], cf. the appendix to [Coates 1991]. Note that, at least if ψ is multiplication by d on E, the distribution relation follows from the corresponding property of the local heights—see [Silverman 1994, ex. 6.4], for the infinite places see also [Mestre, Schappacher 1991, 2.4.2].

3.6 Back to diagram 2.4. We can now complete our discussion of diagram 2.4 above. First, divide by the kernel of φ_{2}^{#}:

(3.6.1)

d_{2} : L_{2} = L_{2}^{#}/ker φ_{2}^{#}  ⟶  E(F) ⊗_{Z} E(F) ⊗_{Z} Q,
φ_{2} : ker d_{2} = \frac{ker d_{2}^{#}}{ker φ_{2}^{#}}  ⟶  F^{∗} ⊗_{Z} Q.

Elements of L_{2} will be written ∑_{j} a_{j}{x_{j}}_{2}. Then, following the usual inductive procedure of Zagier’s polylogarithm conjecture, Wildeshaus defines the mapping analogous to d_{2}^{#}, one level up, by

(3.6.2)  \begin{aligned}d_{3}^{#} : & L_{3}^{#} = Q[E(F) − \{0\}] & ⟶ & L_{2} ⊗_{Q} (E(F) ⊗_{Z} Q) \\ & \{x\}_{3}^{#} & ↦ & \{x\}_{2} ⊗ x ⊗ 1. \end{aligned}

This is the definition over any fixed field of definition F of the curve E. In diagram 2.4, notations refer to the case F = \overline{Q}. 3.6.3. As in the diagram, consider I_{E}^{4} as included in L_{3}^{#} by forgetting the 0-component. Then

d_{3}^{#}(I_{E}^{4}) ⊂ ker(d_{2} ⊗ id_{E(\overline{Q})⊗_{Z}Q}) = kerd_{2} ⊗_{Q} (E(\overline{Q}) ⊗_{Z} Q).

Indeed, to see that d_{2}^{#} ⊗ id(d_{3}^{#}(∑_{x≠y} a_{x}b_{y}{x − y})) = 0, if ∑_{x} a_{x}{x} and ∑_{y} b_{y}{y} are principal divisors, is just an exercise in regrouping terms, and adding the missing contributions for x = y (which average out), once the tensors have been expanded bilinearly. That d_{3}^{#} factors through p will be a byproduct of the commutativity 3.6.5 below.

Thus all solid arrows in diagram 2.4 are defined.

3.6.4 Surjectivity of φ_{2} over \overline{Q}. For every α ∈ F^{∗}, there exists an extension field

F ⊆ F′ ⊆ \overline{Q} of degree [F′ : F] ≤ 4 and an element ξ ∈ [(ker d_{2}^{#})_{/F′}]^{G(\overline{Q}/F)} such that

φ_{2}^{#}(ξ) = α ⊗ 1.

This follows from the first formula in (3.4.2). As a consequence, φ_{2} is an isomorphism for F = \overline{Q}.

3.6.5 The commutativity of diagram 2.4. For all elliptic functions f, g with divisors (f) = ∑_{x} a_{x}[x], (g) = ∑_{y} b_{y}[y], putting ξ = ∑_{x≠y} a_{x}b_{y}{x − y}_{3}^{#}, we have

(φ_{2} ⊗ id) ∘ d_{3}^{#} (ξ) = T̃(f ∧ g).

Proof. Any expression f ∧g can be written as a linear combination of f_{i}∧g_{i} such that, for all i, no inclusion holds between the supports supp(f_{i}), supp(g_{i}) (multiply f, g by functions introducing new, disjoint zeroes or poles). We are therefore reduced to checking 3.6.5 under this additional assumption on the supports of f and g. Assuming this, we may choose divisors D(f) = ∑_{k} A_{k}[z_{k}] ∈ Q[C] and D(g) = ∑_{l} B_{l}[z′_{l}] ∈ Q[C] such that:

(i) D(f), resp. D(g), projects to the divisor of f, resp. g, on C/Γ, (ii) ∑_{k} A_{k} = ∑_{k} A_{k}z_{k} = ∑_{l} B_{l} = ∑_{l} B_{l}z′_{l} = 0 in C, (iii) for all k, l, if z_{k} − z′_{l} ∈ Γ, then z_{k} = z′_{l}.

By (ii), Ξ = ∑_{k,l} A_{k}B_{l}[z_{k} − z′_{l}], and therefore also Ξ̃ = ∑_{k,l;z_{k}≠z′_{l}} A_{k}B_{l}[z_{k} − z′_{l}], goes to zero in C ⊗_{Q} C. Then, in view of (i), (iii), φ_{2}(∑_{x≠y} a_{x}b_{y}{x − y}_{2}) is given by formula (3.4.3) evaluated on Ξ̃. And more importantly: refining in a straightforward manner the arguments of the proof of 3.4, for the well-definedness and the validity of formula (3.4.3), one sees that also the image of d_{3}^{#}(ξ) = ∑_{x≠y} a_{x}b_{y}({x − y}_{2} ⊗ (x − y)) under φ_{2} ⊗ id is

computed by the same theta function, using Ξ̃.

Furthermore, defining

\begin{matrix}F (z) = & ∏_{k} (e^{− \frac{1}{2} (z−z_{k})η(z−z_{k})}σ(z − z_{k}) · Δ^{\frac{1}{12}} )^{A_{k}} , \\ G(z) = & ∏_{l} (e^{− \frac{1}{2} (z−z′_{l})η(z−z′_{l})}σ(z − z′_{l}) · Δ^{\frac{1}{12}} )^{B_{l}} , \end{matrix}

a classical computation shows (using (i) and (ii)) that F is in fact a meromorphic function on C/Γ, with the same divisor as f, and likewise for G and g. It will therefore suffice to check 3.6.5 for F ∧ G instead of f ∧ g. Expanding bilinearly

d_{3}^{#}(ξ) = (∑_{x≠y} a_{x}b_{y} \{x − y\}_{2} ⊗ (x − y)) ∈ L_{2} ⊗_{Q} (E(\overline{Q}) ⊗ Q)

and taking first an x in the support of f which is distinct from all the y’s, we find in (φ_{2} ⊗ id)(d_{3}^{#}(ξ)) the contribution

G(x) ⊗ x ⊗ − ∑_{k} A_{k},

with the sum extended over all k’s such that x = z_{k} (mod Γ). This is precisely the contribution at x of T̃(F ∧ G). Similarly, 3.6.5 works out at all points y in the support of g which do not figure among the zeroes or poles of f.

Finally, take x ∈ supp(f) ∩ supp(g), say, x = z_{1} (mod Γ) = z′_{1} (mod Γ). Then, by (iii), (φ_{2} ⊗ id)(d_{3}^{#}(ξ)) contains the term

\{(\frac{F (z)}{σ(z_{1} − z)^{A_{1}}} )^{B_{1}}(\frac{σ(z − z′_{1})^{B_{1}}}{G(z)} )^{A_{1}}\}|_{z=z_{1}=z′_{1}} ⊗ x ⊗ 1,

which again is precisely the contribution at x of T̃(F ∧ G)—even with the correct sign (−1)^{A_{1}B_{1}} in front (which we could have slobbered working, as we do, in C^{∗} ⊗ Q), because the Weierstrass sigma function is an odd function of z.

This concludes the proof of the commutativity of diagram 2.4, as far as the solid arrows in it are concerned. Next, we turn to the dotted arrow …

4. The elliptic polylogarithm conjecture for K_{2}(E). As before, E denotes an elliptic curve defined over the number field F. Recall the maps ∂_{𝔭} from (1.1) (the components of ∂), reg_{v} from (1.5) (for a fixed choice of differential ω, which is used to determine the lattices Γ_{v} attached to the elliptic curve with respect to the different embeddings of F into C), and B_{3,𝔭}(x) from 1.6. Write K_{2,1}^{(v)} for the Q-linear extension to divisors of the function which, in the notation of 1.5, takes a point z mod Γ_{v} on the elliptic curve to the complex number A(Γ_{v})^{2}K_{1}(0, z, 2, Γ_{v}). Recall also the notation introduced in (3.6.1–2).

4.1 Theorem. [Elliptic Analogue of Zagier’s Polylogarithm Conjecture for K_{2}(E).] There exists a Q-linear map φ_{3}^{#} : kerd_{3}^{#} ⟶ K_{2}(E) ⊗_{Z} Q making diagram 2.4 and the following triangle commute, and satisfying the distribution relation 4.2 below.

kerd_{3}^{#}  ⟶^{φ_{3}^{#}}  K_{2}(E) ⊗_{Z} Q
H_{A} = ⨁_{v inf.}Re(K_{2,1}^{(v)}) ⊕ ⨁_{𝔭 fin.}B_{3,𝔭} ↘  ↙ ⨁_{v inf.}reg_{v} ⊕ ⨁_{𝔭 fin.}∂_{𝔭} = reg_{A}

⨁_{v inf.}R ⊕ ⨁_{v fin.}Q

4.2 Distribution relation. Let ψ : E ⟶ E′ be an isogeny defined over F and let ξ = ∑_{i} a_{i}{x_{i}}_{3}^{#} ∈ L_{3}^{#}. Then ψ(ξ) = ∑_{i} a_{i}{ψ(x_{i})}_{3}^{#} ∈ kerd_{3}^{#} on E′ if and only if one has d_{3}^{#}(∑_{i} a_{i} ∑_{T ∈kerψ}{P_{i} + T }_{3}^{#}) = 0 on E. (This follows from 3.5.) Furthermore, if these equivalent conditions hold, then one has for the maps φ_{3}^{#} on E, resp. on E′, that

ψ_{∗} ∘ φ_{3}^{#}(∑_{i} a_{i} ∑_{t∈kerψ}{x_{i} + t}_{3}^{#}) = φ_{3}^{#}(∑_{i} a_{i}{ψ(x_{i})}_{3}^{#}),

where ψ_{∗} : K_{2}(E) ⊗ Q ⟶ K_{2}(E′) ⊗ Q is the Gysin map of ψ. For instance, if ψ is multiplication by d on E, then ψ_{∗} also multiplies the elements of K_{2}(E) by d.

4.3 Construction of φ_{3}^{#}. Recall from (3.6.2) that kerd_{3}^{#} is a subspace of divisors supported on F -rational points of E. Denote temporarily by D the G(\overline{Q}/F)-invariants of the space (kerd_{3}^{#})\overline{Q}, taken relative to the ground field \overline{Q}. Then, for any given ξ ∈ kerd_{3}^{#}, we can find a Q-linear combination D ∈ D of divisors of the form {x}_{3}^{#} − d ∑_{d·y=x}{y}_{3}^{#} (say, with 1 ≤ d ≤ 7), such that ξ −D is a G(\overline{Q}/F)-invariant element of the space I_{E}^{4} considered in diagram 2.4. This follows immediately from the fact that I_{E}^{4} is the space of divisors on E which go to zero under the three Q-linear maps Q[E(\overline{Q}) − \{0\}] ⟶ E(\overline{Q})^{⊗i} ⊗_{Z} Q given by x ↦ x^{⊗i}, for i = 1, 2, 3. The fact that D goes to zero under d_{3}^{#} follows from 3.5.

Now, map ξ − D up in diagram 2.4 to p(ξ − D) ∈ \frac{I_{E}^{4}}{β⟨f∧(1−f)⟩_{Q}} . This element is independent of our choice of D, because any linear combination of divisors of the form {x}_{3}^{#} − d ∑_{d·y=x}{y}_{3}^{#} that lies in I_{E}^{4} already belongs to ⟨f ∧ (1 − f)⟩_{Q}—see lemma 3.20 in [Gončarov, Levin 1997].

Since both ξ and D belong to the kernel of d_{3}^{#}, the commutativity of 2.4 shows that any element of β^{−1}(p(ξ − D)) has trivial tame symbol, and therefore lies in K_{2}(E/\overline{Q}) ⊗_{Z} Q. Since diagram 2.4 is clearly Galois-equivariant, and our elements are invariant under G(\overline{Q}/F), they lie in (K_{2}(E/\overline{Q}) ⊗_{Z} Q)^{G(\overline{Q}/F)} = K_{2}(E/F) ⊗_{Z} Q.

Thus, choosing any Q-linear section of β defines some mapping φ_{3}^{#} which makes the diagram 2.4 commute, and satisfies 4.2 by what was said in the preceding construction. Any such map also renders the triangle in 4.1 commutative, as we have already remarked in (1.5) and 1.6 above. This completes the proof of the theorem. But to be sure, it does not answer all our questions. Specifically, it does not say anything about the

4.4 Uniqueness of φ_{3}^{#}. If one believes Beilinson’s conjecture 1.7.(i), then the commutativity of the triangle in theorem 4.1 determines φ_{3}^{#} uniquely. This conjecture seems inaccessible at the moment. However, refining the analysis of a commutative diagram like 2.4 above, Gončarov and Levin have recently succeeded to prove (even over an arbitrary algebraically closed field) that one has the following sequence, which is exact modulo 2-torsion [Gončarov, Levin 1997, theorem 1.5]:

Tor(\overline{Q}^{∗}, E(\overline{Q})) ↪ \frac{H^{0}(E, K_{2})}{K_{2}(\overline{Q})} ⟶ \frac{I_{E,Z}^{4}}{β⟨f ∧ (1 − f)⟩_{Z}} ⟶ \overline{Q}^{∗}⊗_{Z}E(\overline{Q}) ⟶ ker(H^{1}(E, K_{2}) → \overline{Q}^{∗}) ⟶ 0

Here I_{E,Z} is the augmentation of the integral group ring Z[E(\overline{Q})], and K_{2} is the K_{2}-sheaf. Tensoring all groups with Q turns the second term simply into K_{2}(\overline{Q}) ⊗_{Z} Q, and kills the first term of the sequence, thus showing that no choice of splitting of β was needed for our construction … 5. Examples in a family. Now that we have at our disposal theorem 4.1, which we view in the first place as a systematic way to produce elements of K_{2}(E) ⊗ Q from certain divisors, we want to illustrate the result in a one-parameter family of elliptic curves which was written down by Jan Nekovář in 1990, following an idea of the second author. The first numerical experiments in this family were performed in the Spring of 1990 by D. Grayson and the second author. This was then later taken up by the first author in his thesis.

Similar numerical computations, for individual elliptic curves, were performed in 1991 by Don Zagier and Henri Cohen, with an intuition guided closely by the formalism of Zagier’s polylogarithm conjecture. They saw in particular that local heights impose conditions on the divisors that may create elements in K_{2}—see section 3 above. Beilinson, Gončarov, and Levin learned about this. Beilinson and Levin supplied the general motivic theory of elliptic polylogarithms, i.e., Kronecker double series, in [Beilinson, Levin 1994], which was complemented by explicit constructions in [Levin 1994a,b]. And in 1995, Gončarov and Levin, with the initial version of [Gončarov, Levin 1997], gave their first treatment of the elliptic polylogarithm conjecture in the case of K_{2}.

In contrast to this experimental prehistory of the work of Gončarov and Levin, Wildeshaus came up with the first version of [Wildeshaus 1997b] early in 1995 because he was seeking a motivically inspired elliptic analogue of Zagier’s polylogarithm conjecture, generalizing [Beilinson, Deligne 1994] directly. There the rôle of the local heights in the preliminary case of K_{1}(F) was only recognized by R. de Jeu in subsequent discussions.

5.1 An elementary construction. Let E be an elliptic curves over Q, and assume that P ∈ E(Q) is a rational point of infinite order. Let f ∈ Q(E)^{∗} be a function with divisor (f) = 2(P) + (−2P) − 3(0)—take for instance the equation of the tangent to E at P. Assume that we can find a function g ∈ Q(E)^{∗} such that, for some M ≥ 2,

(i) the divisor (g) is supported in E_{M}; (ii) g(P) = −g(−2P) ≠ 0. For every point Q in the support of (g), let F_{Q} denote the field generated over Q by the coordinates of Q, and let F be the composite of these fields F_{Q}. Then (i) insures that there is a function χ_{Q} ∈ F_{Q}(E)^{∗} such that (χ_{Q}) = M(Q) − M(O).

We find for the tame symbol T of (2.1):

(5.1.1)  T (M{f, g}) = T (M{f, g(−2P)} + ∑_{Q}{f(Q)^{v_{Q}(g)}, χ_{Q}}).

Due to Galois descent in K_{2}(F (E)) ⊗ Q, the Galois invariant sum on the right, like every other term in the identity, defines an element of K_{2}(Q(E)) ⊗ Q. Therefore, correcting {f, g} by symbols with a constant component, it comes from an element of K_{2}(E) ⊗ Q.

5.2 Nekovář’s family. Take the following special case of 5.1. For a ∈ Q with a ≠ −\frac{1}{3}, let E_{a} : (3a + 1) y^{2} = x^{3} − (3a^{2} + 1) x + (2a^{3} + a + 1), and P_{a} = (a + 1, 1). P_{a} is a point of infinite order except for a = \frac{1}{3}, in which case E_{a} = X_{1}(11) and P_{a} is of order 5. Henceforth we assume that a ≠ ±\frac{1}{3}. Then −2P_{a} = (a − 1, −1), and we consider the symbol \{f_{a}, g_{a}\}, where f_{a} = y − x + a, and g_{a} = g = y, so that M = 2. By 5.1, {f_{a}, g_{a}} defines a element of K_{2}(E_{a}) ⊗ Q. The divisors of our functions are :

(f_{a}) = 2(P_{a}) + (−2P_{a}) − 3(0), (g) = −3(0) + ∑_{i=1}^{3}(Q_{i}),

where Q_{i} = (e_{i}, 0) are the 2-torsion points on E_{a}.

Let Z + τ_{a}Z, Im(τ_{a}) > 0, be the lattice corresponding to (E_{a}, \frac{ω_{a}}{Ω_{a}}), where ω_{a} is a non zero rational differential on E_{a} with real period Ω_{a} = | ∫_{E_{a}(R)^{0}} ω_{a}|. Furthermore, let z_{a} ∈ C correspond to the point P_{a} under the analytic parametrization C/Z + τ_{a}Z → E_{a}. And let t_{i} (i = 1, 2, 3) correspond to the 2-torsion points Q_{i}. Then the regulator reg_{ω}({f_{a}, g}) of any element of K_{2}(E_{a}) ⊗ Q mapping to {f_{a}, g} ∈ K_{2}(Q(E_{a})) ⊗ Q equals, in the notation of 4.1:

(5.2.1) \frac{Im(τ_{a})^{2}}{π^{2}} (−3 (2K_{2,1}(z_{a}) + K_{2,1}(−2z)) + ∑_{i=1}^{3} 2 K_{2,1}(z_{a} − t_{i}) + K_{2,1}(−2z_{a} − t_{i})).

By standard properties of the Kronecker series, this transforms into

(5.2.2)  \frac{Im(τ_{a})^{2}}{π^{2}} (−8 K_{2,1}(z_{a}) + 5K_{2,1}(2z_{a})) − \frac{1}{2}K_{2,1}(4z_{a})).

In order to test Beilinson’s conjectures 1.7, the first author computed the expression

(5.2.3)  \frac{π^{2} reg_{ω}(\{f_{a}, g\})}{N_{a} L(E_{a}, 2)},

where N_{a} is the conductor of the curve E_{a}, for small values of a using the software package PARI. In fact, in doing so one passes to a minimal model of the curve E_{a}, and the point on it corresponding to P_{a}. Beilinson’s conjectures (1.7)—together with the functional equation for the L-function L(E_{a}, s) (which is conjectural in that one cannot prove yet that all elliptic curves in Nekovář’s family are modular)—predict that the quantity (5.2.3) is a rational number whenever ∂_{p}({f_{a}, g}) = 0 at all primes p where E_{a} acquires split multiplicative reduction—see 1.1. This integrality obstruction is computed explicitly, using [Schappacher, Scholl 1991, §3], in [Rolshausen 1996, IV.3]. It turns out that

(5.2.4)  ∀p ∂_{p}({f_{a}, g}) = 0  ⇔  12a ∈ Z.

5.3 Relation to φ_{3}^{#}. In terms of the map φ_{3}^{#} : Q[E(Q)−{0}] ⟶ K_{2}(E)⊗Q constructed in section 4 above, the element −2{f_{a}, g} of K_{2}(E_{a}) ⊗ Q is given—at least modulo the kernel of the regulator—by the following divisor, suggested by (5.2.2):

ξ_{a} = 16{P_{a}}_{3}^{#} − 10{2P_{a}}_{3}^{#} + {4P_{a}}_{3}^{#}.

It is easy to see that ξ_{a} ∈ kerd_{3}^{#}. The tables in [Rolshausen 1996, chap. VI] show about one hundred curves E_{a}, for all a ∈ \frac{1}{12}Z with −16 ≤ a ≤ +16, where φ_{3}^{#}(ξ_{a}) is integral. In each of these cases, the quantity (5.2.2) is apparently a nonzero rational number.

For values of a where φ_{3}^{#}(ξ_{a}) is not integral, (5.2.2) seemed irrational (as far as any number does in the computer). But in these cases, it is more interesting to see whether theorem 4.1 allows us to produce other elements in K_{2}(E_{a}) which are integral, and compare their regulator to L(E_{a}, 2). The search for such elements is greatly facilitated by the fact that we could define φ_{3}^{#} on all of kerd_{3}^{#}, not just on divisors in I_{E}^{4}. Let us illustrate this with just one example—for more material, see [Rolshausen 1996, VI.2].

5.4 The example a = \frac{1}{7}. The minimal equation of E = E_{1/7} over Z is y^{2} + y = x^{3} − 325x + 6156. The conductor is N = N_{1/7} = 4025. Two points are easy to find on the minimal model: the standard point P = P_{1/7} = (20, 87), and another one, Q = (45, 287).

By (5.2.4), φ_{3}^{#}(16{P }_{3}^{#}−10{2P }_{3}^{#}+{4P }_{3}^{#}) does not lie in the integral part of K_{2}(E)⊗Q. But an easy search gives the element

ξ′ = −11{Q}_{3}^{#} − 2{2Q}_{3}^{#} + {3Q}_{3}^{#} ∈ kerd_{3}^{#}

which does pass the integrality test via (1.6.2) at p = 7, which is the only prime where split multiplicative reduction occurs for E. Note that ξ′ adds up to −12Q in E, so this element of kerd_{3}^{#} does not itself belong to I_{E}^{4}. A search for a corresponding element in I_{E}^{4} would have taken longer. Numerically one finds that

\frac{π^{2} reg_{ω}(φ_{3}^{#}(ξ′))}{N_{a} L(E_{a}, 2)} = \frac{1}{96}.

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