NUMERICAL VERIFICATION OF BEILINSON’S CONJECTURE FOR K_{2} OF HYPERELLIPTIC CURVES.

TIM DOKCHITSER, ROB DE JEU, AND DON ZAGIER

Abstract. We construct families of hyperelliptic curves over ℚ of arbitrary genus g with (at least) g integral elements in K_{2}. We also verify the Beilinson conjectures about K_{2} numerically for several curves with g = 2, 3, 4 and 5.

1. Introduction

Let k be a number field, with r_{1} real embeddings and 2r_{2} complex embeddings into ℂ, so that [k : ℚ] = r_{1} + 2r_{2}. It is a well known classical theorem that, if O_{k} is the ring of algebraic integers in k, then O_{k}^{∗} is a finitely generated abelian group of rank r = r_{1} + r_{2} − 1. If u_{1}, …, u_{r} form a basis of O_{k}^{∗}/torsion, and σ_{1}, …, σ_{r+1} are the complex embeddings of k up to complex conjugation, then the regulator of O_{k}^{∗} is defined by

R = \frac{2^{r_{2}}}{[k : ℚ]} |det (\begin{matrix}1 & log |σ_{1}(u_{1})| & … & log |σ_{1}(u_{r})| \\ ⋮ & ⋮ & & ⋮ \\ 1 & log |σ_{r+1}(u_{1})| & … & log |σ_{r+1}(u_{r})| \end{matrix})| ,

and one has that Res_{s=1}ζ_{k}(s) = \frac{2^{r_{1}}(2π)^{r_{2}}R |Cl(O_{k})|}{w\sqrt{Δ_{k}}}, where Δ_{k} is the absolute

value of the discriminant of k and w the number of roots of unity in k. Because K_{0}(O_{k}) ≅ Cl(O_{k}) ⊕ ℤ and K_{1}(O_{k}) ≅ O_{k}^{∗}, so |Cl(O_{k})| = |K_{0}(O_{k})_{tor}| and w = |K_{1}(O_{k})_{tor}|, this can be interpreted as a statement about the K-theory of O_{k}, and it is from this point of view that it can be generalized to ζ_{k}(n) for n ≥ 2. Namely, in [12], Quillen proved that K_{n}(O_{k}) is a finitely generated abelian group for all n. Borel in [4] computed its rank, showing that this rank is zero for even n ≥ 2 and is equal to r_{±} for odd n = 2m − 1 > 1, where (−1)^{m} = ±1 and r_{−} = r_{1} + r_{2}, r_{+} = r_{2}. Moreover, for those odd n he showed (see [5]) that a suitably defined regulator of K_{2m−1}(O_{k}) is a non-zero rational multiple of ζ_{k}(m)/π^{mr_{∓}} \sqrt{Δ_{k}}.

Inspired by this, Bloch in [2] considered K_{2} of elliptic curves E defined over ℚ with complex multiplication, and showed that there is a relation between a regulator associated to certain elements in K_{2}(E) and the value of L(E, 2). Beilinson then proposed a very general conjecture about similar relations between certain regulators of K-groups of projective varieties over number fields and values of their L-functions at integers, see [13, §5].

Those conjectures were tested numerically for K_{2} of elliptic curves over ℚ by Bloch and Grayson in [3], which led to a modification of Beilinson’s original conjecture. Some more numerical work has been done in this direction, e.g., Young carried out similar calculations for elliptic curves over certain real quadratic number fields (as well as over ℚ) in his thesis [15], and Kimura worked out the case of the genus two curve y^{2} − y = x^{5} in [9].

The goal of this paper is to verify Beilinson’s conjecture numerically for K_{2} of a number of hyperelliptic curves of genus greater than 1 defined over ℚ. For such curves it is much more difficult to write down enough elements in K_{2} than in the case of genus 1.

In Sections 2 and 3 we will review the statement of Beilinson’s conjecture for the case of K_{2} of curves defined over ℚ. In the following four sections we show how to construct curves with interesting elements in K_{2}. In Section 8 we deal with a technical condition in Beilinson’s conjecture, the integrality condition, and in Section 9 we discuss how to compute the regulator numerically. The final section is devoted to examples.

The first two authors would like to thank the Collège de France for a visit to work on this paper, as well as the EC network Arithmetic Algebraic Geometry for travel support. The second author would also like to thank the Newton Institute for a productive stay during which this paper was worked on.

2. Curves and their L-functions (review)

Let C be a non-singular, projective, geometrically irreducible curve over ℚ of genus g. Its L-function is the Dirichlet series, convergent for Re(s) > \frac{3}{2}, defined by the Euler product

L(C, s) = ∏_{p prime} L_{p}(C, s) (Re(s) > \frac{3}{2}) ,

where the Euler factor L_{p}(C, s), for those primes p for which the equations defining C over ℚ can be reduced modulo p and still define a smooth curve of genus g over 𝔽_{p}, is defined by the formula

L_{p}(C, s) = exp ( ∑_{n=1}^{∞} (p^{n} + 1 − #C(𝔽_{p^{n}}))\frac{p^{−ns}}{n} )

and is known to be the reciprocal of a polynomial in p^{−s} of degree 2g with constant term 1. For the remaining (finitely many) primes p, L_{p}(C, s)^{−1} is also a polynomial in p^{−s} with constant term 1, but now of degree at most 2g. Finally, one can associate to the curve C/ℚ a positive integer N called its conductor which plays a role in the functional equation of its L-function, see below. We do not give precise definitions of the remaining Euler factors or of N since they are involved and we will be computing all of these quantities experimentally and not from their definitions.

Conjecture 2.1. (Hasse-Weil) The function

L^{∗}(C, s) = \frac{N^{s/2}}{(2π)^{gs}} Γ(s)^{g} L(C, s)

extends to an entire function of s and satisfies L^{∗}(C, s) = wL^{∗}(C, 2 − s), where w = +1 or −1.

Note that, if this conjecture holds, then L^{(0)}(C, 0) = ⋯ = L^{(g−1)}(C, 0) = 0 and

\frac{L^{(g)}(C, 0)}{g!} = lim_{s→0} \frac{L(C, s)}{s^{g}} = L^{∗}(C, 0) = wL^{∗}(C, 2) = \frac{wN}{(2π)^{2g}} L(C, 2) ≠ 0 ,

BEILINSON’S CONJECTURE FOR K_{2} OF HYPERELLIPTIC CURVES 3

because Γ(s) has a pole of order one at s = 0 with residue 1.

Remark 2.2. In this paper we need the value of L(C, s) only at s = 2, where the defining Euler product is absolutely convergent. However, this convergence is very slow and in practice we will calculate the value of L(C, 2) numerically later on by assuming that Conjecture 2.1 holds and using the algorithms described in [7] and [8]. These algorithms also compute N, w, and the Euler factors L_{p}(C, s) for “bad” primes, under the assumption that the functional equation of L^{∗}(C, s) holds for some choices of these quantities from a certain list, thus providing convincing numerical checks of the validity of the resulting functional equation for L^{∗}(C, s).

3. K-theory, regulators, and the Beilinson conjectures

In this section we will give definitions of the various K-groups occurring that are more elementary and explicit than the usual ones, but that are equivalent in our situation. We will indicate the relations to these standard definitions as we go along.

Let F be a field. Then K_{2}(F) can be defined^{1} as

F^{∗} ⊗_{ℤ} F^{∗}/⟨a ⊗ (1 − a), a ∈ F, a ≠ 0, 1⟩ ,

where ⟨⋯ ⟩ denotes the subgroup generated by the elements indicated. The class of a ⊗ b is denoted {a, b}, so that K_{2}(F) is an abelian group (written additively), with generators {a, b} for a and b in F^{∗}, and relations

{a_{1}a_{2}, b} = {a_{1}, b} + {a_{2}, b}
{a, b_{1}b_{2}} = {a, b_{1}} + {a, b_{2}}
{a, 1 − a} = 0 if a is in F, a ≠ 0, 1.

It is a nice exercise to show that those relations imply that {a, b} = −{b, a} and {c, −c} = 0 for a, b and c in F^{∗}.

Now consider a (non-singular, projective, geometrically irreducible) curve C defined over ℚ. Associated to C are K-groups K_{n}(C) whose definition is a little complicated, but for this paper we need only a certain quotient group^{2} K_{2}^{T} (C) of K_{2}(C) (“T” for “tame”), which can be described in a simpler way. Set

K_{2}^{T} (C) = ker(K_{2}(F) ⟶^{T} ⨁_{x∈C(\overline{ℚ})} \overline{ℚ}^{∗}) ,

where F = ℚ(C) is the function field of C and where the x-component of the map T is the tame symbol at x, defined by

(3.1)  T_{x} : \{a, b\} ↦ (−1)^{ord_{x}(a)ord_{x}(b)} \frac{a^{ord_{x}(b)}}{b^{ord_{x}(a)}} (x) .

Note that this definition makes sense since \frac{a^{ord_{x}(b)}}{b^{ord_{x}(a)}} has order zero at x and hence is defined and non-zero at x. It is clear that T_{x} is a map on F^{∗} ⊗_{ℤ} F^{∗}, and checking that it is trivial on the symbols {a, 1 − a} for a ≠ 0, 1 is a good exercise (which also explains why we want to have the power of −1 in the formula), so T_{x} defines a map on K_{2}(F).

1The actual definition of K_{2} (of arbitrary rings) is more complicated (see [11, §5]), and its equivalence for fields with the definition in terms of the “symbols” {a, b} is a famous theorem of Matsumoto, see Theorem 11.1 of loc. cit.

2usually denoted H^{0}(C, \underline{K}_{2})

We note that if α is an element of K_{2}(ℚ(C)), then

(3.2)  ∏_{x∈C(\overline{ℚ})} T_{x}(α) = 1 ,

a result known as the product formula, see [1, Theorem 8.2].

Beilinson, generalizing work by Bloch [2], defined regulators of the K-groups of C (see [13]). We will describe these in elementary terms for K_{2}^{T} (C).

We start with a map from F^{∗} × F^{∗} to the group of almost everywhere defined 1-forms on the Riemann surface X = C(ℂ) by putting

(3.3)  η(a, b) = log |a| d arg b − log |b| d arg a ,

where arg a is the argument of a. Note that this is well defined (the argument arg is defined up to multiples of 2π, but these map to zero under d) and is a smooth (indeed, real-analytic) 1-form on the complement of the set of zeros and poles of a and b. It is clear that η(a_{1}a_{2}, b) = η(a_{1}, b) + η(a_{2}, b) and η(a, b_{1}b_{2}) = η(a, b_{1}) + η(a, b_{2}), so that η induces a map (still denoted η) on F^{∗} ⊗_{ℤ} F^{∗}. Also, η(a, b) is closed, since dη(a, b) = Im(d log a ∧ d log b) = 0.

For any smooth closed 1-form ω defined on the complement of a finite set S ⊂ X, and any smooth oriented loop γ in X \ S, we have a pairing

(γ, ω) = \frac{1}{2π} ∫_{γ} ω

which depends only on the homology class of γ in X \ S. As γ moves across a point x in S, the value of (γ, ω) jumps by (C_{x}, ω), where C_{x} denotes a small circle around x. A simple calculation shows that (C_{x}, η(a, b)) = log |T_{x}({a, b})|. It follows that if α = ∑_{i} a_{i} ⊗ b_{i} is an element of F^{∗} ⊗_{ℤ} F^{∗} such that T_{x}(α) = 1 for all x in X, then ( · , η(α)) is a well defined map from H_{1}(X; ℤ) to ℝ.

Next, one has to check that this map vanishes if α = a ⊗ (1 − a) and hence gives us a pairing

⟨ · , · ⟩ : H_{1}(X; ℤ) × K_{2}^{T} (C) → ℝ

given by ⟨γ, α⟩ = (γ, η(α)). This follows from the fact that η(a, 1 − a) = d D(a), where D(z) is the Bloch-Wigner dilogarithm function.

Finally, we observe that, if c denotes complex conjugation on X, then c^{∗}(η(α)) = −η(α) for any α in K_{2}^{T} (C), since c^{∗}(log |a|) = log |a| and c^{∗}(d arg b) = −d arg b: a and b are in ℚ(C)^{∗}, so (c^{∗}b)(z) = b(\overline{z}) = \overline{b(z)}. This means that if γ is in H_{1}(X; ℤ)^{+}, the c-invariant part of H_{1}(X; ℤ), then ⟨γ, α⟩ = 0 for any α in K_{2}^{T} (C):

⟨γ, α⟩ = \frac{1}{2π} ∫_{γ} η(α) = \frac{1}{2π} ∫_{c∘γ} η(α) = \frac{1}{2π} ∫_{γ} c^{∗}(η(α)) = −\frac{1}{2π} ∫_{γ} η(α) = −⟨γ, α⟩ .

We therefore have to compute ⟨γ, α⟩ only for γ in H_{1}(X; ℤ)/H_{1}(X; ℤ)^{+}. In practice, we can just as well compute it for all γ in H_{1}(X; ℤ)^{−}, the anti-invariants in H_{1}(X; ℤ) under the action of c, giving us finally the regulator pairing

(3.4)  ⟨ · , · ⟩ : H_{1}(X; ℤ)^{−} × K_{2}^{T} (C)/torsion → ℝ .

It is easy to see that H_{1}(X; ℤ)^{−} has rank g. Beilinson originally conjectured^{3} that the rank of K_{2}^{T} (C)/torsion is also equal to g, that the pairing in (3.4) was nondegenerate, and that there was a relation between L(C, 2) and the absolute value of the determinant of the matrix of this pairing with respect to bases of H_{1}(X, ℤ)^{−} and K_{2}^{T} (C)/torsion. Unfortunately, this conjecture was wrong, as K_{2}^{T} (C)/torsion can have rank bigger than g already for g = 1, as was discovered by Bloch and Grayson in [3]. They found that one should consider a certain subgroup of K_{2}^{T} (C)/torsion defined by an additional condition, which we now proceed to describe.

The extra condition comes from the fact that one should not consider the curve over ℚ, but instead a model of it over ℤ. This is analogous to the situation in Section 1, where one has to consider K_{1}(O_{k}) ≅ O_{k}^{∗} instead of K_{1}(k) ≅ k^{∗} in order to get the correct regulator.

So let C be a regular proper model of C over ℤ, i.e., a regular, proper, irreducible two-dimensional scheme over ℤ such that the generic fiber C_{ℚ} is isomorphic to C, see, e.g., [10, Chapter 10]. For each prime p, let C_{p} be the fiber of C over 𝔽_{p}. For each irreducible component D of the curve C_{p}, let 𝔽_{p}(D) denote its field of rational functions over 𝔽_{p}. Then we define

(3.5)  K_{2}^{T} (C) = ker (K_{2}^{T} (C) → ⨁_{p, D⊆C_{p}} 𝔽_{p}(D)^{∗}) ,

where the map to 𝔽_{p}(D)^{∗} is given as follows. The order of vanishing along D gives rise to a discrete valuation on F, v_{D}. The component of the map in (3.5) corresponding to D is given by the tame symbol corresponding to D,

(3.6)  T_{D} : \{a, b\} ↦ (−1)^{v_{D}(a)v_{D}(b)} \frac{a^{v_{D}(b)}}{b^{v_{D}(a)}} (D) ,

in complete analogy with (3.1). Finally, we set

(3.7)  K_{2}(C; ℤ) = K_{2}^{T} (C)/torsion ,

a subgroup of K_{2}^{T} (C)/torsion. It is the same as what is sometimes denoted by K_{2}(C)_{ℤ}/torsion (cf. [14, § 3.3]), and is therefore independent of the choice of the regular proper model C of C (see [13, page 13]), justifying the notation.

Remark 3.8. We could have defined K_{2}^{T} (C) in a single step as

(3.9)  K_{2}^{T} (C) = ker (K_{2}(F) → ⨁_{D} 𝔽(D)^{∗}) ,

where D runs through all irreducible curves on C and 𝔽(D) stands for the residue field at D. Any such D is either “vertical”, in which case it is a component of some C_{p} and T_{D} is the map in (3.6), or else “horizontal”, in which case it corresponds to the Gal(\overline{ℚ}/ℚ)-orbit of some x in C(\overline{ℚ}) and T_{D} being trivial is equivalent to T_{y} being trivial for all y in that Gal(\overline{ℚ}/ℚ)-orbit.

3Strictly speaking, Beilinson conjectured K_{2}^{T} (C) ⊗_{ℤ} ℚ to have dimension g over ℚ. However, K_{2}^{T} (C)/K_{2}(ℚ) is expected to be finitely generated by virtue of a conjecture of Bass, and since K_{2}(ℚ) is a torsion group, this would imply the equivalence of Beilinson’s formulation and our formulation.

We can now restrict the pairing (3.4) to

(3.10)  ⟨ · , · ⟩ : H_{1}(X; ℤ)^{−} × K_{2}(C; ℤ) → ℝ ,

and formulate our description of Beilinson’s conjecture, as modified in accordance with [3], as follows:

Conjecture 3.11. Let C be a non-singular, projective, geometrically irreducible curve of genus g defined over ℚ, and let X = C(ℂ). Then:

(1) The group K_{2}(C; ℤ) is a free abelian group of rank g and the pairing (3.10) is non-degenerate;

(2) Let R denote the absolute value of the determinant of this pairing with respect to ℤ-bases of H_{1}(X; ℤ)^{−} and K_{2}(C; ℤ), and let L^{∗}(C, 0) be defined as in Section 2. Then L^{∗}(C, 0) = QR for some non-zero rational number Q.

Remark 3.12. The definition of L^{∗}(C, 0) requires the analytic continuation of L(C, s), but since the analytic continuation and the expected functional equation of L(C, s) would imply that L^{∗}(C, 0) is rationally proportional to π^{−2g}L(C, 2), Beilinson’s conjecture could be formulated without any assumptions about the analytic continuation of L(C, s).

Remark 3.13. In practice, the conjecture is rather untractable, as it seems impossible to compute K_{2}(C; ℤ) even after tensoring this group with ℚ. Indeed, we neither can guarantee finding enough elements to generate K_{2}(C; ℤ), nor can we necessarily determine the rank of a subgroup generated by finitely many elements, as we do not know any practical method for determining if a given combination of elements in F^{∗} ⊗_{ℤ} F^{∗} can be written as a sum of Steinberg symbols a ⊗ (1 − a). But we can try to find g elements in K_{2}(C; ℤ) and compute R as in the conjecture using those elements rather than a basis of K_{2}(C; ℤ). If R is non-zero numerically we can check the relation with L^{∗}(C, 0) as in Conjecture 3.11. Also, if we have more than g elements α_{j} in K_{2}(C; ℤ), the conjecture implies that the maps ⟨ · , α_{j}⟩ : H_{1}(X; ℤ)^{−} → ℝ should be linearly dependent over ℤ, and this too can be checked numerically. Both types of verification will be carried out in Section 10.

Remark 3.14. We have restricted ourselves to the statement of Beilinson’s conjecture for a curve over ℚ, but the conjecture can be formulated equally well for any number field k. Suppose that C/k is a (non-singular, projective, geometrically irreducible) curve of genus g. Let O_{k} be the ring of integers of k, and let C be a model of C/k over O_{k}. Then one defines K_{2}^{T} (C) as in (3.9), with the sum over irreducible curves D in C. Again, for the “horizontal” curves D, the T_{D} correspond to the T_{x} for x in C(\overline{ℚ}), up to conjugation under Gal(\overline{ℚ}/k). Once again, K_{2}^{T} (C)/torsion is independent of the choice of the model C, and is denoted by K_{2}(C; ℤ). One expects K_{2}(C; ℤ) ≅ ℤ^{g [k:ℚ]}. The main difference is that the Riemann surface involved will no longer be connected. Instead, let X be the Riemann surface obtained from all points in C over ℂ, using all embeddings of k into ℂ. This is a disjoint union of [k : ℚ] connected Riemann surfaces, each of genus g. Complex conjugation acts on this either by swapping conjugate pairs of complex embeddings of k, or by acting on the Riemann surface corresponding to a real embedding of k. Then H_{1}(X; ℤ)^{−} ≅ ℤ^{g [k:ℚ]}, and there is a pairing H_{1}(X; ℤ)^{−} × K_{2}(C; ℤ) → ℝ given by ⟨γ, α⟩ = \frac{1}{2π} ∫_{γ} η(α). Here η(α) is the 1-form on X which for α = {a, b} is given on the connected Riemann surface corresponding to σ : k → ℂ by log |a_{σ}|d arg b_{σ} − log |b_{σ}|d arg a_{σ}. (The subscripts indicate that we consider the functions on the Riemann surface obtained by applying σ to the coefficients involved in a and b.) One again expects ⟨ · , · ⟩ to be non-degenerate, defines R to be the absolute value of the determinant of the matrix of ⟨ · , · ⟩ with respect to ℤ-bases of H_{1}(X; ℤ)^{−} and K_{2}(C; ℤ), and conjectures that L^{∗}(C, 0) = QR for some non-zero rational number Q.

4. Constructing elements of K_{2} from torsion divisors

The first problem in testing Beilinson’s conjecture as formulated in 3.11 is that it is not at all clear how to construct elements of K_{2}^{T} (C) on a given curve C over ℚ, that is, how to produce rational functions f_{i}, g_{i} on C such that ∑_{i}{f_{i}, g_{i}} satisfies the tame symbol condition at every point of C.

To understand the condition, consider one symbol {f, g} in K_{2}(ℚ(C)). If div(f) and div(g) have disjoint support, then this symbol lies in K_{2}^{T} (C) if and only if f(P)^{ord_{P} (g)} = 1 for every zero or pole P of g and g(Q)^{ord_{Q}(f)} = 1 for every zero or pole Q of f. Essentially this says that f equals one (or a root of unity, if |ord_{P} (g)| > 1) whenever g has a zero or a pole and vice versa.

To try to satisfy these conditions, it is natural to look at functions which have only very few zeros and poles. The simplest case is given by functions f and g which have only one (multiple) zero and one (multiple) pole. If one of these points is common for f and g, then it turns out that simply renormalizing the functions is sufficient to satisfy the tame symbol conditions. For this we use the product formula (3.2). All of our examples are then based on the following construction.

Construction 4.1. Let C/ℚ be a curve. Assume P_{1}, P_{2}, P_{3} ∈ C(ℚ) are distinct points whose pairwise differences are torsion divisors. Thus there are rational functions f_{i} with

div(f_{i}) = m_{i}(P_{i+1}) − m_{i}(P_{i−1}), i in ℤ/3ℤ ,

where m_{i} is the order of (P_{i+1})−(P_{i−1}) in the divisor group Pic^{0}(C). We then define three elements of K_{2}(ℚ(C)) by

S_{i} = \{\frac{f_{i+1}}{f_{i+1}(P_{i+1})}, \frac{f_{i−1}}{f_{i−1}(P_{i−1})}\}, i in ℤ/3ℤ .

The functions f_{i} are unique up to constants, so the symbols S_{i} are uniquely defined by the points P_{i}. It turns out that they satisfy the tame symbol condition everywhere:

Lemma 4.2. The S_{i} are elements of K_{2}^{T} (C).

Proof. The components of S_{i} are normalized to make the tame symbol trivial at P_{i−1} and P_{i+1}. By the product formula (3.2) it is also trivial at P_{i}, this being the only other point in the support of the divisors of f_{i−1} and f_{i+1}. □

Next we show that the three elements S_{i} generate a subgroup of rank at most 1 of K_{2}^{T} (C)/torsion.

Proposition 4.3. We keep the notation of Construction 4.1.

(1) There is a unique element {P_{1}, P_{2}, P_{3}} of K_{2}^{T} (C)/torsion such that in this group

S_{i} = \frac{lcm(m_{1}, m_{2}, m_{3})}{m_{i}} \{P_{1}, P_{2}, P_{3}\}, i = 1, 2, 3 .

(2) The element {P_{1}, P_{2}, P_{3}} is unchanged under the even permutations and changes sign under the odd permutations of the points.

Proof. (1) Uniqueness is obvious, so we only need to show existence. Replacing the functions f_{i} by f_{i}/f_{i}(P_{i}) if necessary, we can assume that f_{i}(P_{i}) = 1, so that S_{i} = {f_{i+1}, f_{i}}. Now let

(4.4)  L = lcm(m_{1}, m_{2}) = lcm(m_{1}, m_{3}) = lcm(m_{2}, m_{3}) ;

the three least common multiples are equal since every integer which kills both (P_{i})−(P_{i+1}) and (P_{i+1})−(P_{i+2}) in Pic^{0}(C) also kills (P_{i})−(P_{i+2}), which is their sum. It follows that the quotients

r_{i} = L/m_{i}, i = 1, 2, 3

are pairwise relatively prime. Then

div(f_{i}^{r_{i}}) = L(P_{i+1})−L(P_{i−1}),

so the function f_{1}^{r_{1}}f_{2}^{r_{2}}f_{3}^{r_{3}} has trivial divisor, and is therefore a constant:

f_{1}^{r_{1}}f_{2}^{r_{2}}f_{3}^{r_{3}} = c . (4.5)

Next, we have, in K_{2}^{T} (C)

r_{1}S_{2} = {f_{3}, f_{1}^{r_{1}}}, r_{2}S_{1} = {f_{2}^{r_{2}}, f_{3}} = −{f_{3}, f_{2}^{r_{2}}} ,

so

r_{1}S_{2} − r_{2}S_{1} = {f_{3}, f_{1}^{r_{1}}f_{2}^{r_{2}}} = {f_{3}, cf_{3}^{−r_{3}}} = {f_{3}, (−1)^{r_{3}}c} ,

because {f_{3}, −f_{3}} is trivial. Since the left-hand side is in K_{2}^{T} (C), so is {f_{3}, (−1)^{r_{3}}c}. This implies that c^{2m_{3}} = 1, so {f_{3}, (−1)^{r_{3}}c} is torsion. Therefore, modulo torsion, for i, j = 1, 2, 3,

r_{i}S_{j} =r_{j}S_{i} .

Now choose integers α_{1}, α_{2}, α_{3} such that ∑ α_{i}r_{i} =1 and set

T = α_{1}S_{1} + α_{2}S_{2} + α_{3}S_{3} .

Then

r_{1}T = α_{1}r_{1}S_{1} + α_{2}r_{1}S_{2} + α_{3}r_{1}S_{3} = α_{1}r_{1}S_{1} + α_{2}r_{2}S_{1} + α_{3}r_{3}S_{1} = S_{1},

and, similarly, r_{2}T = S_{2} and r_{3}T = S_{3}, all modulo torsion. In particular, the subgroup ⟨S_{1}, S_{2}, S_{3}⟩ of K_{2}^{T} (C)/torsion is generated by T alone.

(2) The first statement follows from the construction in part (1). For the second, let f_{i} be as above, chosen so that f_{i}(P_{i})=1.

Take P̃_{1} = P_{2}, P̃_{2} = P_{1}, P̃_{3} = P_{3} and the functions f̃_{1} = f_{2}^{−1}, f̃_{2} = f_{1}^{−1}, f̃_{3} = f_{3}^{−1}, so that α̃_{1} = α_{2}, α̃_{2} = α_{1} and α̃_{3} = α_{3}. The corresponding symbols are

S̃_{1} = {f̃_{2}, f̃_{3}} = {f_{1}^{−1}, f_{3}^{−1}} = −{f_{1}, f_{3}^{−1}} = {f_{1}, f_{3}} = −{f_{3}, f_{1}} = −S_{2},

and, similarly, S̃_{2} ={f_{3}^{−1}, f_{2}^{−1}}=−S_{1} and S̃_{3} ={f_{2}^{−1}, f_{1}^{−1}}=−S_{3}, so

T̃ = α̃_{1}S̃_{1} + α̃_{2}S̃_{2} + α̃_{3}S̃_{3} = −α_{2}S_{2} − α_{1}S_{1} − α_{3}S_{3} = −T .
□

Proposition 4.6. Let C/ℚ be a curve and P_{1}, P_{2}, P_{3}, P_{4} be four distinct points in C(ℚ) such that all (P_{i})−(P_{j}) are torsion divisors. Then the four elements

{P_{1}, …, P̂_{i}, …, P_{4}} (1 ≤ i ≤ 4)

are linearly dependent. More precisely, if m_{ij} is the order of (P_{i})−(P_{j}), then

(4.7)  ∑_{i=1}^{4}(−1)^{i} C_{i} {P_{1}, …, P̂_{i}, …, P_{4}} = 0

holds in K_{2}^{T} (C)/torsion with

(4.8)  C_{i} = gcd(m_{ij}, m_{ik}, m_{il})  ({i, j, k, l}={1, 2, 3, 4}) .

Proof. Choose f_{ij} with div(f_{ij}) = m_{ij}(P_{i})−m_{ij}(P_{j}). The discussion surrounding (4.5) shows that

(\frac{f_{ij}}{f_{ij}(P_{k})})^{\frac{M}{m_{ij}}} (\frac{f_{jk}}{f_{jk}(P_{i})})^{\frac{M}{m_{jk}}} (\frac{f_{ki}}{f_{ki}(P_{j})})^{\frac{M}{m_{ki}}} = 1

if {i, j, k} ⊂ {1, 2, 3, 4} and M is any integer divisible by 2 lcm(m_{ij}, m_{jk}, m_{ik})^{2}. Choose such positive M common for all triples {i, j, k} and let F_{ij} = f_{ij}^{M/m_{ij}}. Then

div(F_{ij}) = M(P_{i}) − M(P_{j}) ({i, j} ⊂ {1, 2, 3, 4})

and

\frac{F_{ij}}{F_{ij}(P_{k})} \frac{F_{jk}}{F_{jk}(P_{i})} \frac{F_{ki}}{F_{ki}(P_{j})} = 1 (\{i, j, k\} ⊂ \{1, 2, 3, 4\}) . (4.9)

Also define elements of K_{2}^{T} (C) by

S_{i,j,k} = \{\frac{F_{ki}}{F_{ki}(P_{j})}, \frac{F_{ij}}{F_{ij}(P_{k})}\} = \{\frac{F_{ij}}{F_{ij}(P_{k})}, \frac{F_{jk}}{F_{jk}(P_{i})}\} = \{\frac{F_{jk}}{F_{jk}(P_{i})}, \frac{F_{ki}}{F_{ki}(P_{j})}\} .

The asserted relation (4.7), but with C_{i} replaced by

(4.10)  C′_{i} = \frac{lcm(m_{lj}, m_{lk}, m_{jk})}{m_{lj}m_{lk}m_{jk}}  (\{i, j, k, l\} = \{1, 2, 3, 4\}) ,

then follows from the statement that, in K_{2}^{T} (C)/torsion,

S_{1,2,3} − S_{1,2,4} + S_{1,3,4} − S_{2,3,4} = 0 . (4.11)

To prove (4.11), let f =F_{14}/F_{14}(P_{2}), g =F_{24}/F_{24}(P_{3}) and h=F_{34}/F_{34}(P_{1}). Renormalizing if necessary, we may also assume that F_{12} = fg^{−1}, F_{23} = gh^{−1} and F_{31} = hf^{−1}. Finally, let

α = f(P_{3}), β = g(P_{1}), γ = h(P_{2}) .

Now we apply (4.9) for {i, j, k} = {1, 2, 4}, {1, 3, 4}, {2, 3, 4} and {1, 2, 3}. This gives, respectively, that

(fg^{−1})(P_{4}) = β^{−1}, (gh^{−1})(P_{4}) = γ^{−1}, (hf^{−1})(P_{4}) = α^{−1} and αβγ = 1 .

Finally, using all of these we expand the symbols S_{i,j,k} in terms of the 8 generators,

v = ({f, g}, {f, h}, {g, h}, {f, β}, {g, α}, {g, β}, {h, α}, {α, β}) .

We find

\begin{aligned}S_{1,2,3} = & \{fg^{−1}/α, gh^{−1}/β\} & = (1, −1, 1, −1, 1, 1, −1, 1) · v , \\ −S_{1,2,4} = & \{f, g/β\} & = (−1, 0, 0, 1, 0, 0, 0, 0) · v , \\ S_{1,3,4} = & \{f/α, h\} & = (0, 1, 0, 0, 0, 0, 1, 0) · v , \\ −S_{2,3,4} = & \{g, h/γ\} & = (0, 0, −1, 0, −1, −1, 0, 0) · v . \end{aligned}

Therefore the left-hand side of (4.11) reduces to one symbol {α, β}. This symbol can be also rewritten in terms of the original functions. One shows that

\{α, β\} = ±\{(\frac{f_{kl}(P_{i})}{f_{kl}(P_{j})})^{\frac{M}{m_{kl}}} , (\frac{f_{il}(P_{j})}{f_{il}(P_{k})})^{\frac{M}{m_{il}}} \} .

for all combinations of four distinct indices i, j, k and l with |i − k| = |j − l| = 2. In any case this element comes from K_{2} of a number field, which is torsion. Hence the asserted relation holds.

It remains to show that the numbers C_{i} and C′_{i} defined by (4.8) and (4.10) are proportional. Equivalently, if γ_{i} = v_{p}(C_{i}) and γ′_{i} = v_{p}(C′_{i}) denote the valuations of C_{i} and C′_{i} at some prime p, then we must show that γ_{i} − γ′_{i} is independent of i. The numbers γ_{i} and γ′_{i} are given by

(4.12) γ_{i} = min_{j≠i}{ν_{ij}},  γ′_{i} = max_{j,k≠i}{ν_{jk}} − ∑_{j,k≠i, j<k} ν_{jk}  (i = 1, 2, 3, 4) ,

where ν_{ij} denotes v_{p}(m_{ij}). If we use the six numbers ν_{ij} to label the edges of a tetrahedron T in the obvious way, then the relation (4.4) says that the two largest labels of the sides of any face of T are equal. From this we find easily that there are two possibilities: if three incident edges of T have the same label, then, possibly after renumbering, we have ν_{12} = ν_{13} = ν_{14} = a, ν_{23} = ν_{24} = b, ν_{34} = c for some integers a ≥ b ≥ c, while if this does not happen then, again up to renumbering, we have ν_{13} = ν_{14} = ν_{23} = ν_{24} = a, ν_{12} = b, ν_{34} = c with a > b, c. From (4.12) we then find that (γ_{1}, …, γ_{4}) and (γ′_{1}, …, γ′_{4}) are given by (a, b, c, c) and (−b − c, −a − c, −a − b, −a − b) in the first case and by (b, b, c, c) and (−a − c, −a − c, −a − b, −a − b) in the second case, so that in both cases γ_{i} − γ′_{i} = a + b + c for all i. This completes the proof of Proposition 4.5.

□

Corollary 4.13. Let C be a curve defined over ℚ and P_{1}, …, P_{n} ∈ C(ℚ) points such that all (P_{i}) − (P_{j}) are torsion. Then the subspace of K_{2}^{T} (C) ⊗ ℚ generated by all elements {P_{i}, P_{j}, P_{k}} is already generated by those of the form {P_{1}, P_{i}, P_{j}}.

The corollary implies that the space spanned by the n(n − 1)(n − 2) symbols {P_{i}, P_{j}, P_{k}}, which already by part (2) of Proposition 4.3 had dimension at most \binom{n}{3}, in fact has dimension at most \binom{n−1}{2}.

Remark 4.14. If C/ℚ is a curve, it is sometimes convenient to consider points P_{i} in C(\overline{ℚ}). Then we have to work in K_{2}^{T} (C\overline{ℚ}), which is defined as the kernel of the tame symbol (given by (3.1)),

K_{2}^{T} (C\overline{ℚ}) = ker(K_{2}(\overline{ℚ}(C)) → ⨁_{x∈C(\overline{ℚ})} \overline{ℚ}^{∗}) .

The product formula (3.2) still holds for elements in K_{2}(\overline{ℚ}(C)). All the results in this section remain true in this context. Moreover, the inclusion of ℚ(C) into \overline{ℚ}(C) induces a map from K_{2}(ℚ(C)) to K_{2}(\overline{ℚ}(C)). Because the tame symbol on both groups is given by the same formula, we can check if an element in K_{2}(ℚ(C)) lies in K_{2}^{T} (C) using the tame symbol of its image in K_{2}(\overline{ℚ}(C)).

The final result of this section, a strengthening of Corollary 4.13, says that we cannot construct any more elements of K_{2}^{T} (C\overline{ℚ})⊗ℚ using only functions whose zeros and poles differ by torsion divisors than those already given by Construction 4.1.

Proposition 4.15. Let S ⊆ C(\overline{ℚ}) and P_{0} ∈ S be such that (P) − (P_{0}) is a torsion divisor for all P in S. Let V be the subspace of K_{2}^{T} (C\overline{ℚ}) ⊗ ℚ generated by all elements \{P, Q, P_{0}\} with P, Q ∈ S and W the subspace of K_{2}(\overline{ℚ}(C)) ⊗ ℚ generated by all symbols {f, g} with div(f) and div(g) supported in S. Then

W ∩ K_{2}^{T} (C\overline{ℚ}) ⊗ ℚ = V .

Proof. Let ξ = ∑_{i}\{f_{i}, g_{i}\} be an element of W ∩ K_{2}^{T} (C\overline{ℚ}) ⊗ ℚ and S′ ⊆ S be the set of all zeros and poles of the f_{i} and g_{i}. By assumption, there exists an integer N > 0 and functions f_{P} ∈ \overline{ℚ}(C)^{∗}, f_{P_{0}} = 1, such that div(f_{P}) = N(P) − N(P_{0}) for all P ∈ S′. Then each f_{i}^{N} or g_{i}^{N} is, up to a scalar factor, a multiplicative combination of the functions f_{P}, so N^{2}ξ is a linear combination of symbols of the form \{f_{P} , f_{Q}\}, \{f_{P} , c\} and \{c, c′\} with P, Q ∈ S′ and c, c′ ∈ \overline{ℚ}^{∗}. But \{c, c′\} = 0 in K_{2}^{T} (\overline{ℚ}(C)) ⊗ ℚ and \{f_{P} , f_{Q}\} is the sum of \{f_{P} , f_{Q}(P)\} + \{f_{P} (Q), f_{Q}\} and a multiple of {P, Q, P_{0}} ∈ V, so ξ can be written as ∑_{P ∈T} {f_{P} , c_{P} } + ξ′ with ξ′ ∈ V for some constants c_{P} ∈ \overline{ℚ}^{∗}. Since the tame symbol of \{f_{P} , c_{P} \} at x is c_{P}^{−N} for x = P and trivial for all other x ∈ C(\overline{ℚ}) \ \{P_{0}\}, the fact that some multiple of ξ (and hence also of ξ − ξ′) has trivial tame symbol everywhere implies that each c_{P} is a root of unity. Hence ξ ∈ V. □

5. Torsion divisors on hyperelliptic curves

Now suppose that C is a hyperelliptic curve of genus g ≥ 1 defined over ℚ. The hyperelliptic involution on C determines a double cover ϕ : C → ℙ^{1} ramified at 2g + 2 points, the fixed points of the involution. Assume that one of these points ∞_{C} is defined over ℚ. After a change of coordinates on ℙ^{1} we can assume that ϕ(∞_{C}) = ∞_{ℙ^{1}}. Then C gets a model

y^{2} = c_{2g+1}x^{2g+1} + c_{2g}x^{2g} + … + c_{1}x + c_{0}, c_{2g+1} ≠ 0, (5.1)

where the polynomial on the right has coefficients in ℚ and has no multiple roots. This equation can be seen either as defining a double cover of ℙ^{1} or as a curve (singular for g > 1) in ℙ^{2} whose normalization is C. The point ∞=∞_{C} is then the unique point at infinity of this normalization. The cover ϕ is given by the function x on C and its ramification points are ∞ and T_{i} =(α_{i}, 0), where α_{1}, …, α_{2g+1} are the roots of the right-hand side of (5.1) in \overline{ℚ}.

We will look for points P such that the divisor (P) − (∞) is m-torsion for some m. Such points will be called (m-)torsion points. Of course, if P, Q are torsion points, then (P) − (Q) is also a torsion divisor. If we succeed in constructing many torsion points P_{i}, then we get many elements {∞, P_{i}, P_{j}} ∈ K_{2}^{T} (C)/torsion by using Proposition 4.3. Here are some examples of curves with torsion points:

Example 5.2: 2-torsion. With notation as before, any difference (P) − (Q) for P, Q ∈ {∞, T_{1}, …, T_{2g+1}} is 2-torsion:

2(T_{i}) − 2(T_{j}) = div(\frac{x−α_{i}}{x−α_{j}} ), 2(T_{i}) − 2(∞) = div(x−α_{i}) .

So the T_{i} are 2-torsion points, though not necessarily defined over ℚ.

Conversely, suppose given an arbitrary curve C over ℚ and two distinct points P, Q ∈ C(ℚ) such that the divisor (P)−(Q) is 2-torsion. Say div(ϕ) = 2(P)−2(Q) with ϕ defined over ℚ. Then ϕ : C → ℙ^{1} is a double cover defined over ℚ. It follows that C is hyperelliptic and admits a model (5.1) with Q = ∞, P ∈ {T_{i}}.

Example 5.3: (2g+1)-torsion. Assume that the hyperelliptic equation (5.1) has the special form

y^{2} = c x^{2g+1} + (b_{g}x^{g} + … + b_{1}x + b_{0})^{2} . (5.4)

If necessary, we can scale x and y to make c = −1.

The substitution y ↦y + ∑ b_{i}x^{i} transforms this equation to

y^{2} + 2(b_{g}x^{g} + … + b_{1}x + b_{0})y = cx^{2g+1} . (5.5)

The point O = (0, 0) (corresponding to (0, b_{0}) on the curve (5.4)) lies on this curve and the function y has divisor (2g + 1)(O)−(2g + 1)(∞), so O is a (2g + 1)-torsion point.

In fact, any curve C as in (5.1) with a non-trivial rational (2g + 1)-torsion point O is isomorphic to a curve of the form (5.4) and hence (5.5). Indeed, given such a curve C, consider the local system

L = H^{0}(C, O_{C}((2g + 1)∞)).

By Riemann-Roch, the dimension of this as ℚ-vector space is 1−g +2g +1 = g +2. It is easy to see that {1, x, …, x^{g}, y} is a basis. So for a point O = (x_{O}, y_{O}) ∈ C(ℚ) the divisor (2g + 1)(O) − (2g + 1)(∞) is principal if and only if there is an element of L,

h(x, y) = y − b̃_{g}x^{g} − … − b̃_{1}x − b̃_{0} ,

which vanishes to order exactly 2g + 1 at O. (Note that necessarily it cannot be expressed in terms of x alone because the order of such a function is at most 2g.) Then the curve defined by h(x, y) = 0 can have only the point (x_{O}, y_{O}) in common with C, and after substituting y = b̃_{g}x^{g} + … + b̃_{1}x + b̃_{0} in (5.1), we see that the equation of the curve is of the form

y^{2} = c(x − x_{O})^{2g+1} + (b̃_{g}x^{g} + … + b̃_{1}x + b̃_{0})^{2} .

After a translation we can assume x_{O} = 0 and the equation of the curve becomes of the form (5.4) with O = (0, b_{0}).

Example 5.6: (2g+2)-torsion. Similar to the previous example, a curve

y^{2} = −c^{2}x^{2g+2} + (cx^{g+1} + b_{g}x^{g} + … + b_{1}x + b_{0})^{2} ,

with b_{g}, c ≠ 0 is of the form (5.1) and it has a model

y^{2} + 2(cx^{g+1} + b_{g}x^{g} + … + b_{1}x + b_{0})y = −c^{2}x^{2g+2} .

In this model div(y) = (2g +2)(O)−(2g +2)(∞), so the curve has a (2g +2)-torsion point O =(0, 0). Again x and y can be scaled to make −c^{2} = −1.

As in Example 5.3, one can use the linear system

L = H^{0}(C, O_{C}((2g + 2)∞))

to show that any curve (5.1) with a rational (2g + 2)-torsion point is of this form.

6. Elements of K_{2} for hyperelliptic curves

In this section we study elements of K_{2}^{T} (C) constructed from torsion points on a non-singular hyperelliptic curve C/ℚ of genus g. We show that using only 2torsion points is not sufficient to construct non-torsion elements. So we will use curves which also have rational torsion points of order 2g + 1 or 2g + 2.

As discussed in Examples 5.3 and 5.6, such a curve C/ℚ can always be given by an equation of the form

y^{2} + f(x) y + x^{d} = 0 (6.1)

in one of the following two cases:

\begin{matrix}(a) & d = 2g + 1, & f(x) = b_{g}x^{g} + … + b_{1}x + b_{0} ; \\ (b) & d = 2g + 2, & f(x) = 2x^{g+1} + b_{g}x^{g} + … + b_{1}x + b_{0} and b_{g} ≠ 0 . \end{matrix} (6.2)

This curve is isomorphic to y^{2} = t(x), where t(x) is the 2-torsion polynomial,

t(x) = −x^{d} + f(x)^{2}/4 .

Since we want C to be non-singular, we assume that t(x) has no multiple roots, and, in particular, that b_{0} ≠ 0.

On the curve (6.1) we have a rational point O = (0, 0) and the divisor d(O)−d(∞) is principal, namely equal to div(y). There is also a “reflected” d-torsion point

O′ = (0, −f(0)),

the image of O under the hyperelliptic involution (x, y) ↦ (x, −y − f(x)).

Moreover, every rational root α of t(x) gives a rational 2-torsion point T_{α} = (α, −f(α)/2). All the pairwise differences of the points O, O′, ∞ and the T_{α} are torsion divisors. So the results of Section 4 apply and we get elements of K_{2}^{T} (C)/torsion, such as

{T_{α}, O, ∞}, {O, O′, ∞}, {T_{α}, T_{β}, T_{γ}}, ….

If we replace K_{2}^{T} (C) with K_{2}^{T} (C\overline{ℚ}) then, by Remark 4.14, we can also use points T_{α} for any root α ∈ \overline{ℚ} of the 2-torsion polynomial t(x), not necessarily rational.

Proposition 6.3. Assume g ≥ 2, and let V be the ℚ-subspace of K_{2}^{T} (C\overline{ℚ}) ⊗ ℚ generated by the elements of the form {P_{i}, P_{j}, P_{k}} where P_{i}, P_{j}, P_{k} run through the points ∞, O, O′ and the T_{α} for α ∈ \overline{ℚ} a root of t(x). Then V is already generated by the elements of the form {∞, O, T_{α}}. More precisely, in addition to the tetrahedron relation in Proposition 4.6, if we let ε = gcd(2, d) (so ε = 1 or 2 and d = 2g + ε), then we have the relations

{∞, T_{α}, T_{β}} = 0 ,
{∞, O, T_{α}} + {∞, O′, T_{α}} = 0

and

ε{∞, O, O′} = ∑_{α}{∞, O, T_{α}} .

Furthermore, if f(0)^{2} = 1, then the element

𝕄 = {−f(0)y, −x}

is in K_{2}^{T} (C), and its class in K_{2}^{T} (C\overline{ℚ})/torsion satisfies

d 𝕄 = ε{∞, O, O′} .

Proof. It follows from the tetrahedron relation in Proposition 4.6 that all elements \{P_{i}, P_{j}, P_{k}\} of K_{2}^{T} (C\overline{ℚ}) ⊗ ℚ are combinations of those of the form \{∞, P_{i}, P_{j}\}. So it suffices to look at symbols where one of the points is ∞.

First, the elements {∞, T_{α}, T_{β}} are trivial, since

\{∞, T_{α}, T_{β}\} = \{\frac{x − β}{α − β} , \frac{β − α}{x − α}\} = \{\frac{x − α}{β − α}, \frac{x − β}{α − β} \} = \{\frac{x − α}{β − α}, 1 − \frac{x − α}{β − α}\} = 0 .

This relation together with the tetrahedron relation shows that also all symbols of the form \{T_{α}, T_{β}, T_{γ}\} are trivial in K_{2}^{T} (C\overline{ℚ})/torsion. This proves that, in our construction, 2-torsion points are not sufficient to construct non-torsion elements of K_{2}^{T} (C) or K_{2}^{T} (C\overline{ℚ}) on any hyperelliptic curve of the form y^{2} = −x^{2g+1} + · · ·.

In order to unravel the definitions in Construction 4.1 and Proposition 4.3, we determine the order d′ of (O) − (O′) = 2(O) − 2(∞), and the order d′′_{α} of (O) − (T_{α}) and (T_{α}) − (O′) = (O) − (T_{α}) in Pic^{0}(C).

In fact, we have

d′ = d/ε  and  d′′_{α} = 2d/ε .

Namely, if d = 2g + 1 then d′ = d and if d = 2g + 2 then d′ = d/2. And if d = 2g + 1, then (O) − (T_{α}) = [(O) − (∞)] − [(T_{α}) − (∞)] must have order d′′_{α} = 2d, but if d = 2g + 2 the order can be d or d/2. However, it can be d/2 = g + 1 only if (g+1)[(O)−(∞)] = (T_{α})−(∞) = (∞)−(T_{α}), so that (g+1)(O)+(T_{α})−(g+2)(∞) is the divisor of some non-zero h in H^{0}(C, O_{C}((g + 2)∞)). Since g ≥ 2, g + 2 ≤ 2g, so h lies in H^{0}(C, O_{C}(2g∞)), which has basis {1, x, …, x^{g}} (cf. Example 5.3). But that implies that div(h) is invariant under the hyperelliptic involution whereas (g + 1)(O) + (T_{α}) − (g + 2)(∞) is not because O ≠ O′. So d′′_{α} = d/2 is impossible.

From this and the definition in Proposition 4.3 we see that

(6.4)  \{∞, O, T_{α}\} = \{ \frac{y}{−f(α)/2}, \frac{x − α}{−α} \} ,
\{∞, O′, T_{α}\} = \{ \frac{−f(x) − y}{−f(α)/2} , \frac{x − α}{−α} \}

and

ε\{∞, O, O′\} = \{ \frac{y}{−f(0)}, \frac{y + f(x)}{f(0)} \} .

The second relation in the proposition now follows from

\{\frac{y}{−f(α)/2}, \frac{x − α}{−α}\} + \{\frac{−f(x) − y}{−f(α)/2} , \frac{x − α}{−α}\} = \{\frac{y(f(x) + y)}{−f(α)^{2}/4}, 1 − \frac{x}{α}\}

= \{\frac{−x^{d}}{−α^{d}} , 1 − \frac{x}{α}\} = d\{\frac{x}{α}, 1 − \frac{x}{α}\} = 0 .

In order to prove the third relation, we notice that

\begin{matrix}2\{ \frac{y}{−f(α)/2} , \frac{x − α}{−α} \} = \{ \frac{y^{2}}{f(α)^{2}/4} , 1 − \frac{x}{α} \} = \{ \frac{−y^{2}}{−α^{d}} , 1 − \frac{x}{α} \} \\ = \{ \frac{y^{2}}{α^{d}} , 1 − \frac{x}{α} \} − d\{ \frac{x}{α} , 1 − \frac{x}{α} \} = \{ \frac{y^{2}}{x^{d}} , 1 − \frac{x}{α} \} , \end{matrix} (6.5)

so that taking the sum over all roots α ∈ \overline{ℚ} of t(x) gives

(6.6)  \begin{matrix}∑_{t(α)=0} \{ \frac{y^{2}}{x^{d}} , 1 − \frac{x}{α} \} = \{ \frac{y^{2}}{x^{d}} , ∏_{t(α)=0}(1 − \frac{x}{α}) \} \\ = \{ \frac{y^{2}}{x^{d}} , \frac{t(x)}{t(0)} \} = \{ \frac{y^{2}}{x^{d}} , \frac{(y + f(x)/2)^{2}}{f(0)^{2}/4} \} , \end{matrix}

where we used that t(x) = (y + f(x)/2)^{2}. Also

(6.7)  \begin{aligned}2\{ \frac{y}{−f(0)}, \frac{y + f(x)}{f(0)} \} & = 2\{ \frac{y}{−f(0)}, \frac{y (y+f(x))}{f(0)^{2}} \} \\ & = \{ \frac{y^{2}}{f(0)^{2}} , \frac{−x^{d}}{f(0)^{2}} \} = \{ \frac{y^{2}}{x^{d}} , \frac{−x^{d}}{f(0)^{2}} \} , \end{aligned}

so that 2ε{∞, O, O′} − 2 ∑_{α}{∞, O, T_{α}} equals

\{ \frac{y^{2}}{x^{d}} , \frac{−x^{d}}{f(0)^{2}} \} − \{ \frac{y^{2}}{x^{d}} , \frac{(y + f(x)/2)^{2}}{f(0)^{2}/4} \} = \{ \frac{y^{2}}{x^{d}} , \frac{−x^{d}}{(2y + f(x))^{2}} \} . (6.8)

That this last element is zero in K_{2}^{T} (C) can be seen by applying the identity

\{ \frac{a}{1 − a}, \frac{a(a − 1)}{(1 − 2a)^{2}} \} = \{ (\frac{1 − a}{a})^{2} , 1 − (\frac{1 − a}{a})^{2} \} − \{ \frac{a}{1 − a}, −\frac{a}{1 − a} \}

− 2{ a, −a } + 2{ 1 − a, a } = 0 ,

valid in K_{2} of any field if a, 1 − a and 1 − 2a are non-zero, to a = −y/f(x), so that 1 − a = −x^{d}/f(x)y and 1 − 2a = (2y + f(x)/f(x)). This comples the proof of the third relation.

Finally, if f(0)^{2} = 1, then all the tame symbols T_{P} of 𝕄 = {−f(0)y, −x} are trivial for P different from O = (0, 0), O′ = (0, −f(0)) and ∞, and for those three points we find that

\begin{aligned}T_{(0,0)}(𝕄) & = & (−1)^{d} \frac{−f(0)y}{(−x)^{d}} |_{(0,0)}= 1 , \\ T_{(0,−f(0))}(𝕄) & = & (−1)^{0} \frac{−f(0)y}{(−x)^{0}} |_{(0,−f(0))}= f(0)^{2} = 1 , \\ T_{∞}(𝕄) & = & 1 . \end{aligned}

The first equation uses that y(y + f(x)) = −x^{d}, so that the function −y/x^{d} equals 1/(y + f(x)) and therefore assumes the value 1/f(0) at (0, 0). The third equation follows from the first two and the product formula. Then (6.7) shows that

2d𝕄 = 2d\{−f(0)y, −x\} = \{y^{2}, −x^{d}\} = 2\{ \frac{y}{−f(0)}, \frac{y+f(x)}{f(0)} \} = 2ε\{∞, O, O′\} ,

which finishes the proof. □

Remark 6.9. When g = 1, the statements of Proposition 6.3 still hold, but when d = 2g + 2 = 4, the identity ε{∞, O, O′} = ∑_{α}{∞, O, T_{α}} is replaced by

2{∞, O, O′} = {∞, O, T_{α}} + {∞, O, T_{α′}} + 2{∞, O, T_{β}} ,

where T_{α}, T_{α′} and T_{β} are the points of order 2, and 2(O) − 2(∞) = (T_{β}) − (∞) in Pic^{0}(C\overline{ℚ}). The proof is the same, using that 2\{∞, O, O′\} = \{\frac{y}{−f(0)}, \frac{y+f(x)}{f(0)}\}, \{∞, O, T_{γ}\} = \{\frac{y}{−f(γ)/2}, \frac{x−γ}{−γ} \} for γ = α or α′ but 2\{∞, O, T_{β}\} = \{\frac{y}{−f(β)/2}, \frac{x−β}{−β} \}, and similarly for O′ instead of O in the last two identities. In (6.8) one takes 4{∞, O, O′} − 2{∞, O, T_{α}} − 2{∞, O, T_{α′}} − 4{∞, O, T_{β}} as the left-hand side, whereas the right-hand side vanishes as before.

Now we return from K_{2}^{T} (C\overline{ℚ})/torsion to K_{2}^{T} (C)/torsion, the group that we are interested in. Here we have elements {∞, O, T_{α}}, where α is a rational root of t(x) but, in fact, we have more. As made explicit in (6.4), {∞, O, T_{α}} is the class of an element of K_{2}^{T} (C), namely \{\frac{y}{−f(α)/2}, \frac{x−α}{−α}\}. (If g = 1 and d = 4 this statement has to be slightly modified according to Remark 6.9.) If m(x) is a rational factor of the 2-torsion polynomial t(x), then the element ∑_{m(α)=0}\{\frac{y}{−f(α)/2}, \frac{x−α}{−α}\} (where the sum is taken over the roots of m(x) in \overline{ℚ}) in K_{2}^{T} (C\overline{ℚ}), can be shown to come from K_{2}^{T} (C). But if we multiply it by 2 and use (6.5), then, by a calculation similar to (6.6), we get explicitly that

2 ∑_{m(α)=0} \{\frac{y}{−f(α)/2}, 1− \frac{x}{α}\} = \{\frac{y^{2}}{x^{d}} , \frac{m(x)}{m(0)}\} . (6.10)

This computation shows that we get an element of K_{2}^{T} (C) for every rational irreducible factor m(x) of the 2-torsion polynomial t(x), not necessarily linear. Thus we can hope to get up to k linearly independent elements of K_{2}^{T} (C)/torsion where k is the number of irreducible rational factors of the 2-torsion polynomial of the curve in (6.1). Let us now summarize the construction:

Construction 6.11. Let C/ℚ be given by (6.1), (6.2). Let m_{1}, …, m_{k} be the irreducible factors in ℚ[x] (up to multiplication by ℚ^{∗}) of the 2-torsion polynomial t(x) = c x^{d} + f(x)^{2}/4. To each of them we associate an element of K_{2}^{T} (C)/torsion,

M_{j} = \{\frac{y^{2}}{x^{d}} , \frac{m_{j}(x)}{m_{j}(0)}\}, 1 ≤ j ≤ k . (6.12)

Extending by linearity gives a map

\begin{matrix}ℤ^{k} & ⟶ & K_{2}^{T} (C)/torsion \\ (n_{1}, …, n_{k}) & ⟼ & ∑_{i=1}^{k} n_{i}M_{i} = \{\frac{y^{2}}{x^{d}} , ∏_{i=1}^{k} \frac{m_{i}(x)^{n_{i}}}{m_{i}(0)^{n_{i}}} \} . \end{matrix} (6.13)

Remark 6.14. With notation as in Construction 6.11 and Proposition 6.3, if f(0) = ±1, then the class of 𝕄 = {−f(0)y, −x} in K_{2}^{T} (C)/torsion satisfies

2d𝕄 = ∑_{j=1}^{k} M_{j} .

This follows from (6.10) and the two identities in Proposition 6.3 involving ε{∞, O, O′} and (6.4) (or the corresponding statements in Remark 6.9 if g = 1 and d = 4).

Remark 6.15. In case d = 2g + 2 there is a universal relation between the M_{j} in our construction. First note that the 2-torsion polynomial factors as

t(x) = −x^{2g+2} + \frac{f(x)^{2}}{4} = (\frac{f(x)}{2} − x^{g+1})(\frac{f(x)}{2} + x^{g+1}) .

Let H = −y/x^{g+1}, so that H + 1/H = f(x)/x^{g+1}. Then, in K_{2}^{T} (C),

\{\frac{y^{2}}{x^{d}} , \frac{f(x) − 2x^{g+1}}{f(x) + 2x^{g+1}} \} =\{H^{2}, \frac{H − 2 + 1/H}{H + 2 + 1/H} \} = 2\{H, \frac{(1 − H)^{2}}{(1 + H)^{2}} \}

= 4{H, 1 − H} − 4{−H, 1 − (−H)} = 0 .

So after numbering and normalizing the m_{i} in such a way that

\frac{f(x)}{2} − x^{g+1} = m_{1} ⋯ m_{l}, \frac{f(x)}{2} + x^{g+1} = m_{l+1} ⋯ m_{k} ,

we find that in K_{2}^{T} (C)/torsion,

M_{1} + … + M_{l} − M_{l+1} − … − M_{k} = 0 .

In other words, the map (6.13) has a kernel of rank at least one. The numerical results of Section 10 suggest that this is the only universal relation among the M_{j}, and that, in particular, no such phenomenon occurs for the case d=2g + 1.

Remark 6.16. The map K_{2}(ℚ(C)) → K_{2}(\overline{ℚ}(C)) has torsion kernel, which allows us to identify K_{2}(ℚ(C)) ⊗ ℚ with the subspace of K_{2}(\overline{ℚ}(C)) ⊗ ℚ where Gal(\overline{ℚ}/ℚ) acts trivially. If we use this identification, and we let W ⊆ K_{2}(\overline{ℚ}(C)) ⊗ ℚ be as in Proposition 4.15 (with S = {∞, O, O′, T_{α_{1}} …, T_{α_{2g+1}}}, where α_{1}, …, α_{2g+1} are the roots of t(x) in \overline{ℚ}, and ∞ playing the role of P_{0}), we have that

W ∩ K_{2}^{T} (C) ⊗ ℚ = ⟨M_{1}, …, M_{k}⟩_{ℚ} ,

where the M_{j} are as in Construction 6.11, and ⟨ ⋯ ⟩_{ℚ} indicates the ℚ-vector space they generate. Therefore the M_{j}’s give us everything in K_{2}^{T} (C) ⊗ ℚ that we can get from combinations of symbols \{f, g\} where f and g are in \overline{ℚ}(C)^{∗} and div(f) and div(g) are supported in {∞, O, O′, T_{α_{1}} …, T_{α_{2g+1}}}.

Namely, Proposition 4.15 shows that V = K_{2}^{T} (C\overline{ℚ}) ⊗ ℚ ⋂ W is spanned by the {∞, P, Q} where P and Q are in {O, O′, T_{α_{1}}, …, T_{α_{2g+1}}}, and by Proposition 6.3 and its proof those elements can be expressed in the \{∞, O, T_{α_{i}}\} or the \{\frac{y^{2}}{x^{d}} , \frac{x−α_{i}}{−α_{i}} \}. Therefore, if k = ℚ(α_{1}, …, α_{2g+1}), then the action of Gal(\overline{ℚ}/ℚ) on V factors through Gal(k/ℚ), and V^{Gal(\overline{ℚ}/ℚ)} is generated by the elements

∑_{σ∈Gal(k/ℚ)} σ (\{\frac{y^{2}}{x^{d}} , \frac{x − α_{i}}{−α_{i}} \}) = ∑_{σ∈Gal(k/ℚ)}\{\frac{y^{2}}{x^{d}} , \frac{x − σ(α_{i})}{−σ(α_{i})}\} = δ_{i}\{\frac{y^{2}}{x^{d}} , \frac{n_{i}(x)}{n_{i}(0)}\} ,

where n_{i}(x) is the minimal polynomial of α_{i} over ℚ and δ_{i} · deg(n_{i}(x)) = [k : ℚ]. Because n_{i}(x) is an irreducible rational factor of t(x) this equals δ_{i}M_{j} for some j.

7. Constructing good polynomials

Our goal is to use Construction 6.11 to produce explicit families of hyperelliptic curves C with as many elements of K_{2}^{T} (C) as possible. This comes down to constructing polynomials of the right form, which we will refer to as good polynomials. This is addressed in this section.

Problem 7.1. Construct (families of) polynomials of the form

t(x) = −x^{d} + f(x)^{2}/4

which have many rational factors and no multiple roots. Here d ≥ 5, and we want that t(x) has degree d − 1 for d even, and degree d for d odd.

Remark 7.2. To get potentially interesting examples for the Beilinson conjecture, we need at least g = genus(C) linearly independent elements of K_{2}^{T} (C)/torsion. Thus we want t(x) to have at least g rational factors if d = 2g + 1 is odd and at least g + 1 of them if d = 2g + 2 is even (cf. Construction 6.11 and the universal relation in Remark 6.15). Such a t(x) is what we will call a good polynomial.

We have seen in Examples 5.3 and 5.6 that, for our purposes, we might just as well consider, for c ≠ 0,

t(x) = cx^{2g+1} + f(x)^{2} where f(x) = b_{g}x^{g} + … + b_{1}x + b_{0}

when d = 2g + 1, and

t(x) = −c^{2}x^{2g+2} + f(x)^{2} where f(x) = cx^{g+1} + b_{g}x^{g} + … + b_{1}x + b_{0} with b_{g} ≠ 0

when d = 2g + 2. It will be more convenient to use those non-normalized versions because we can sometimes let c play a role in the construction of such t(x).

For the remainder of this section we keep the notations c, d, t(x) and f(x) as above. For d = 5 and d = 6 we will explain how to describe all such t(x) that factor completely over the rationals. For larger degrees we will give sporadic examples that factor completely or nearly completely, and also infinite families of good polynomials (with d = 2g + 2) for all genera g.

One way to produce good polynomials is to start with a general polynomial t(x) and force it to have given rational roots. We illustrate this with one example:

Example 7.3: d = 5. Take

t(x) = x^{5} + (b_{2}x^{2} + b_{1}x + b_{0})^{2} .

and force the quintic to have two rational roots, so that we get (at least) 3 rational factors.

If α is a root of t(x), then −α^{5} is a square, so α = −k^{2} for some k ∈ ℚ. We want t(x) to have two distinct rational roots −m^{2} and −n^{2}, so

−m^{10} + (b_{2}m^{4} − b_{1}m^{2} + b_{0})^{2} = 0 and − n^{10} + (b_{2}n^{4} − b_{1}n^{2} + b_{0})^{2} = 0 .

Taking roots yields

m^{5} = b_{2}m^{4} − b_{1}m^{2} + b_{0} and n^{5} = b_{2}n^{4} − b_{1}n^{2} + b_{0} .

(We can take the positive signs by replacing m by −m or n by −n if necessary.) Solve this linear system for b_{0} and b_{1} and set b_{2} = (k + m^{2} + mn + n^{2})/(m + n). Finally, multiply by (m + n)^{2} to obtain

(m+n)^{2}t(x) = (m+n)^{2}x^{5} +((k+m^{2}+mn+n^{2})x^{2} +(km^{2}+m^{2}n^{2}+kn^{2})x+km^{2}n^{2})^{2} .

This is a family on 3 parameters m, n, k ∈ ℚ although it gives only a 2-dimensional family of curves y^{2} = t(x): this equation is homogeneous of multi-degree (1, 1, 2, 2, 5) in (m, n, k, x, y), so letting

m ↦ λm, n ↦ λn, k ↦ λ^{2}k ,

gives a polynomial which corresponds to an isomorphic curve. Thus one can, for instance, assume that m, n, k are integers or instead that, say, m = 1. Note that kmn ≠ 0 (for otherwise the quintic has a double root α = 0) and that m ≠ ±n.

Remark 7.4. We could also have forced t(x) to have a third rational root α = −l^{2}. This condition gives a third linear equation for b_{0}, b_{1} and b_{2}. Then the system can be solved uniquely, producing a 3-parameter family of polynomials with 4 rational factors which we do not write down here. We have been unable to find additional examples in this family for which we could verify Beilinson’s conjecture, either because the coefficients were too large or because we did not obtain enough elements in K_{2}(C; ℤ) of the corresponding curve C. Besides, there is a somewhat neater construction, which gives more rational roots:

Example 7.5: d = 5. In degree 5 it is in fact possible to describe the polynomials of the desired form which factor completely over the rationals as follows. Recall from Example 7.3 that every rational root of t(x) is of the form −r^{2} for some rational r, so if t(x) factors completely then we must have

t(x) = x^{5} + (b_{2}x^{2} + b_{1}x + b_{0})^{2} = ∏_{i=1}^{5}(x + r_{i}^{2}) .

Assume that the |r_{i}| are pairwise distinct and non-zero (otherwise t(x) has a double root). Then the two formulas for t(x) give two factorizations of −t(−x^{2}), so

(x^{5} + b_{2}x^{4} − b_{1}x^{2} + b_{0})(x^{5} − b_{2}x^{4} + b_{1}x^{2} − b_{0}) = ∏_{i=1}^{5}(x − r_{i}) ∏_{i=1}^{5}(x + r_{i}) .

Call the two quintics on the left q(x) and −q(−x) respectively. Clearly in every pair ±r_{i} one of the numbers is a root of q(x) and the other a root of q(−x). By changing the signs of some of the r_{i} if necessary, we can assume that all of them are roots of q(x). As the quintic q(x) has no x^{3} and x terms, Newton’s formulae imply that the numbers s_{i} =r_{i}^{−1} satisfy the two equations

∑_{i=1}^{5} s_{i} = 0 and ∑_{i=1}^{5} s_{i}^{3} = 0 . (7.6)

Conversely, any five-tuple {s_{i}} of non-zero rationals satisfying (7.6) and such that s_{i} ≠ ±s_{j} for i ≠ j gives rise to t(x) = ∏(x + s_{i}^{−2}) of the desired form. Note that the tuple {−s_{i}} gives the same polynomial.

Now in order to find rational solutions to (7.6), note that these equations define a cubic surface in ℙ^{4}. So given two rational solutions to (7.6), the line joining them intersects the surface in a third point which is also rational. Thus, starting with some obvious solutions such as (a, −a, b, −b, 0) and its permutations or some other experimentally found small solutions like

(1, 5, −7, −8, 9), (2, 4, −7, −9, 10), (1, 14, −17, −18, 20), … (7.7)

one can construct as many other solutions as one wants by using this “chord construction” and by scaling and permuting the coordinates.

Example 7.8: d even. Finally, here is a method to produce examples for even d = 2m. First, if

t(x) = −c^{2}x^{2m} + f(x)^{2}/4 = (cx^{m} + f(x)/2)(f(x)/2 − cx^{m}) (7.9) with f(x) = 2cx^{m}+b_{g}x^{g} +⋯+b_{0} and cb_{g} ≠ 0, then we could take the first factor to be of the form 2c(x − a_{i}) … (x − a_{m}), which gives us at least m + 1 rational factors of t(x). In order to get more rational factors, it is easier to put f̃(x) = x^{m}f(1/x), so that, with g(x) = f̃(x)/2 + c and k = 2c,

x^{2m}t(1/x) = f̃(x)^{2}/4 − c^{2} = g(x)(g(x) − k) , (7.10)

and the problem simply becomes to find polynomials g(x) and non-zero constants k such that g(x) and g(x) − k have no multiple roots and a lot of rational factors. Strictly speaking we should also ensure that (7.10) has no constant term and a non-zero linear term, so that (7.9) is of degree 2m − 1, a condition which has to be satisfied to get the correct shape of t(x). But this can be always achieved by a translation if there is a rational root.

Scaling g(x) and k does not change our problem, so we can assume g(x) has leading coefficient 1. If we choose g(x) = (x − b_{1}) … (x − b_{m}) we get a family of good polynomials with at least m + 1 rational factors, having b_{1}, …, b_{m} and k as parameters. Now the question is how to improve this by forcing a polynomial of the form

g(x) − k = (x − b_{1}) ⋯ (x − b_{m}) − k

to have more rational factors. If we choose k = ∏(−b_{i}), then g(x) − k has a factor x, so we get a family with at least m+2 rational factors, namely x, x−b_{1}, …, x−b_{m} and the remaining factor of degree m − 1.

For d = 6 (m = 3) this remaining factor is quadratic. So with a suitable rational parametrization one can construct a “universal” family of polynomials which factor completely over the rationals. One example of such a family is given by

\begin{aligned}g(x) & = & (x − 1)(x − r − s)(x − rs), \\ k & = & rs(r − 1)(s − 1), \\ g(x) − k & = & (x − r)(x − s)(x − rs − 1) . \end{aligned}

For larger degrees, examples can be found as follows. Without loss of generality we may assume that b_{1}, …, b_{m} are integers. Choose bounds B, C and search through all integers −B ≤ b_{1} < ⋯ < b_{m} ≤ B. Let g(x) = (x − b_{1}) … (x − b_{m}) and compute g(−C), …, g(C − 1), g(C). If a number k ≠ 0 occurs more than once in this list of values, then g(x) − k has several integral roots in the range from −C to C, each of which yields one linear factor. This gives a simple method to look for polynomials (7.10) with even more rational roots.

Here are a few examples with m = 4 and m = 5 where both g(x) and g(x) − k factor completely, sorted according to k. The columns in the two tables below contain k, the roots of g(x) and the roots of g(x) − k respectively.

\begin{aligned}& m = 4 \\ k & g(x) = 0 & g(x) = k \\ 2520 & 9, 7, −4, −10 & 11, 2, 0, −11 \\ 7560 & 13, 12, −5, −14 & 16, 7, −2, −15 \\ 10080 & 14, 12, −4, −15 & 17, 6, 0, −16 \\ 10080 & 17, 14, −9, −17 & 19, 11, −7, −18 \\ 12600 & 17, 12, −6, −17 & 19, 8, −3, −18 \\ 13860 & 15, 12, −4, −17 & 18, 5, 1, −18 \\ 15840 & 15, 14, −3, −17 & 19, 5, 3, −18 \\ 18720 & 19, 12, −5, −19 & 21, 7, −1, −20 \\ 25200 & 19, 14, −3, −20 & 22, 5, 4, −21 \\ 27720 & 21, 12, −5, −22 & 23, 6, 0, −23 \end{aligned}  \begin{aligned}& m = 5 \\ k & g(x) = 0 & g(x) = k \\ 50400 & 12, 11, −1, −6, −14 & 14, 6, 4, −9, −13 \\ 50400 & 14, 10, 1, −9, −15 & 15, 6, 5, −11, −14 \\ 110880 & 15, 13, −3, −4, −16 & 17, 8, 4, −9, −15 \\ 110880 & 16, 10, −3, −7, −16 & 17, 5, 4, −12, −14 \\ 272160 & 17, 16, −1, −7, −20 & 20, 8, 7, −11, −19 \\ 327600 & 20, 11, −5, −9, −21 & 21, 5, 4, −15, −19 \\ 393120 & 23, 17, −6, −15, −24 & 24, 15, −3, −19, −22 \\ 554400 & 21, 19, 2, −11, −24 & 24, 11, 9, −14, −23 \\ 554400 & 22, 17, −1, −9, −24 & 24, 11, 6, −13, −23 \\ 1058400 & 23, 22, −1, −8, −27 & 27, 13, 8, −13, −26 \end{aligned}

One should note here that such a search produces many more examples than just the ones above. The tables actually start

\begin{aligned}& m = 4 \\ k & g(x) = 0 & g(x) = k \\ 180 & 5, 4, −3, −4 & 6, 2, −1, −5 \\ 360 & 6, 4, −3, −5 & 7, 1, 0, −6 \\ 504 & 7, 5, −4, −6 & 8, 3, −2, −7 \\ 720 & 7, 4, −4, −7 & 8, 1, −1, −8 \\ … & … & … \end{aligned}  \begin{aligned}& m = 5 \\ k & g(x) = 0 & g(x) = k \\ 5040 & 8, 7, −1, −5, −9 & 9, 5, 1, −7, −8 \\ 10080 & 9, 7, −2, −4, −10 & 10, 4, 2, −7, −9 \\ 50400 & 12, 11, −1, −6, −14 & 14, 6, 4, −9, −13 \\ 50400 & 14, 10, 1, −9, −15 & 15, 6, 5, −11, −14 \\ … & … & … \end{aligned}

However, most of the examples are induced in the sense that modulo a linear substitution, p(x) = g(x)(g(x) − k) is of the form q(x^{2}) for some q(x) of degree m. For instance, for the first entry for m=4 (k =180) we have

p(x) = (x − 5)(x − 4)(x + 3)(x + 4) · (x − 6)(x − 2)(x + 1)(x + 5)

and, after rearranging the factors, we find that

\begin{matrix}p(−x+ \frac{1}{2}) = & (x− \frac{11}{2})(x− \frac{9}{2})(x− \frac{7}{2})(x− \frac{3}{2})(x+ \frac{3}{2})(x+ \frac{7}{2})(x+ \frac{9}{2})(x+ \frac{11}{2}) \\ = & x^{8} − 65x^{6} + \frac{10979}{8}x^{4} − \frac{164385}{16}x^{2} + \frac{4322241}{256} . \end{matrix}

The curve y^{2} = t(x) is therefore a cover of an elliptic curve y^{2} = x^{4} −65x^{3} +· · ·. In the same way, the first two lines for m=5 simply come from genus 2 curves, which we have already seen in the cubic surface construction: they are given by the first two 5-tuples in (7.7).

Finally, for m = 6 (g =5, d=12) one can easily find several non-trivial examples where g(x)(g(x)−k) factors completely except for one quadratic factor. It is harder to find non-trivial examples which factor completely, but they do exist. In the notation of (7.10) the smallest one is

\begin{aligned}& (x^{6}+2x^{5}−787x^{4}−188x^{3}+150012x^{2}−149040x−3326400)^{2} − 3326400^{2} \\ = & (x−22)(x−20)(x−18)(x−12)(x−10)(x−1)x(x+7)(x+15)(x+18)(x+23)(x+24) . \end{aligned}

8. Integrality of the elements

Construction 6.11 shows how to construct hyperelliptic curves C/ℚ with elements in K_{2}^{T} (C)/torsion. However, we are not yet in a position to compute regulators and test Beilinson’s conjecture 3.11, because we still need our elements to be integral, that is, to lie in K_{2}(C; ℤ) ⊆ K_{2}^{T} (C)/torsion. (See (3.7).)

Let C/ℤ be a regular proper model of C/ℚ. Recall that an element α of K_{2}(ℚ(C)) lies in K_{2}^{T} (C) if the tame symbol T_{P} (α) is trivial for all P ∈ C(\overline{ℚ}). Recall also that α lies in K_{2}^{T} (C) if, in addition, the tame symbol is trivial for each irreducible curve D in C, see (3.5) or (3.9). This means that in addition to being trivial for “horizontal” curves (which come from points P ∈ C(\overline{ℚ})), the tame symbol of α must be trivial for all irreducible components of the fibers of C → Spec ℤ. This gives additional conditions on α for all primes p of ℤ. (One can show though that, to a given α, one can always add an element in K_{2}(ℚ) such that the sum satisfies this condition for all primes p for which the fiber C_{p} is smooth over 𝔽_{p}.)

So, in general, given a curve C/ℚ and α ∈ K_{2}^{T} (C) the way to verify that α gives rise to an element of K_{2}(C; ℤ) is to find a regular model C/ℤ and then check that the tame symbol for the vertical components of C is trivial. In practice, finding such a model means starting with any equation of C with integer coefficients, which defines an arithmetic surface, and performing blow-ups until we obtain a regular surface. This, however, is a complicated process, so we will try to deduce integrality from the behaviour of α on the original (possibly singular) arithmetic surface.

This can be done in fair generality for the families of examples that we are interested in. We repeat our notation for the sake of convenience in reference, so we consider a curve of genus g as in Construction 6.11, defined by

y^{2} + f(x)y + x^{d} = 0 (8.1)

in one of the two cases

\begin{matrix}(a) & d = 2g + 1, & f(x) = b_{g}x^{g} + … + b_{1}x + b_{0} ; \\ (b) & d = 2g + 2, & f(x) = 2x^{g+1} + b_{g}x^{g} + … + b_{1}x + b_{0} and b_{g} ≠ 0 , \end{matrix} (8.2)

where t(x) = −x^{d} + f(x)^{2}/4 has no multiple roots, so that in particular b_{0} ≠ 0. Assume further that f(x) has integer coefficients, so (8.1) defines an arithmetic surface over ℤ.

Theorem 8.3. Let C/ℚ be defined by (8.2). Assume that b_{0}, …, b_{g} are integers with no common factor. Let m(x) ∈ ℤ[x] be a factor of the 2-torsion polynomial t(x) = −x^{d} +f(x)^{2}/4, irreducible over ℤ, and let M be the associated element of K_{2}^{T} (C)/torsion,

M = \{ \frac{y^{2}}{x^{d}} , \frac{m(x)}{m(0)} \} .

We have (1) if m(0) = ±1 then M ∈ K_{2}(C; ℤ);

(2) if m(0) ≠ ±1 then nM ∉ K_{2}(C; ℤ) for any n ≠ 0.

If, in addition, b_{0} = ±1, then the class of 𝕄 = {−b_{0}y, −x} is in K_{2}(C; ℤ) if d = 2g + 1, and the class of 2𝕄 = {y^{2}, x} is in K_{2}(C; ℤ) if d = 2g + 2.

Proof. Denote

M̃ = \{f_{1}, f_{2}\}, f_{1} = \frac{y^{2}}{x^{d}} , f_{2} = \frac{m(x)}{m(0)} .

M̃ is an element of K_{2}^{T} (C) (see (6.10)), and M its class in K_{2}^{T} (C)/torsion. We shall show that, for a specific regular proper model C of C, M̃ ∈ K_{2}^{T} (C) if m(0) = ±1, so that M ∈ K_{2}(C; ℤ). But, for the same model, we shall show that if m(0) ≠ ±1, then T_{D}(M̃) is not torsion for some irreducible D of a fiber C_{p}. Therefore no nonzero multiple of M can lie in K_{2}(C; ℤ) ⊆ K_{2}^{T} (C)/torsion, because, up to torsion, K_{2}^{T} (C) is a subgroup of K_{2}^{T} (C) that does not depend on the choice of C.

Let p be a prime and F be the fiber above p of the arithmetic surface defined by (8.1). We start by showing that the function y^{2}/x^{d} is non-constant along every irreducible component D of F. From the shape of the equation it follows that either such a D equals F or it is given by an equation of the form y = h(x). In either case, it is a curve which dominates the x-axis. If y^{2}/x^{d} is constant along D, then y^{2} = kx^{d} on D for some constant k. Clearly k ≠ 0. Then f(x)y = −(k + 1)x^{d} and squaring this equation yields

f(x)^{2}kx^{d} = (k + 1)^{2}x^{2d} on D.

Since D dominates the x-axis, this is only possible if this is an identity of polynomials, which implies that b_{g}x^{g} + ⋯ + x_{0} = 0 mod p, contradicting the assumption on the b_{j}.

Now let C/ℤ be the regular model of C/ℚ obtained from the arithmetic surface defined by (8.1) by repeatedly blowing up the singularities. If D ⊂ C is an irreducible curve which maps onto D, then the function f_{1} = \frac{y^{2}}{x^{d}} is non-constant along D as well.

We are now ready to prove the theorem. First assume that m(0) ≠ ±1 and take a prime p | m(0). Let D and D be as above. Then v_{D}(f_{1}) = 0 and v_{D}(f_{2}) < 0. So the tame symbol T_{D}(M̃) is (up to a sign) a non-zero power of f_{1} and is therefore non-constant and hence non-torsion.

For the other implication assume that m(0) = ±1. Then for every prime p and D as above, v_{D}(f_{1}) = 0 and v_{D}(f_{2}) = 0 and, as a consequence, T_{D}(M̃)=1.

However, the argument above does not prove that v_{D}(f_{1}) = v_{D}(f_{2}) = 0 for every component D of the fiber of C → Spec ℤ above p. There are in general curves D on C which map to singular points P of the model (8.1), rather than onto irreducible components D. In this case if, say, f_{1} happens to have a zero or a pole passing just through this point P, then it can happen that v_{D}(f_{1}) ≠ 0.

We claim that for every singular point P on the surface defined by (8.1) which lies on the fiber above p, either f_{1} or f_{2} is regular and equal to 1 at P. Then T_{D}(M̃) = 1 for every irreducible curve D of C mapping to P by (3.6).

First, consider the point at infinity in the fiber above p. The arithmetic surface defined by (8.1) has a chart at infinity which can be obtained by letting x = 1/x̃ and y = ỹ/x̃^{g+1}. So for d = 2g + 1, the equation at infinity is

ỹ^{2} + (b_{g}x̃ + … + b_{1}x̃^{g} + b_{0}x̃^{g+1}) ỹ + x̃ = 0

and the point at infinity (x̃ = ỹ = 0 mod p) is non-singular. If d = 2g + 2, then the equation is

ỹ^{2} + (2 + b_{g}x̃ + … + b_{1}x̃^{g} + b_{0}x̃^{g+1}) ỹ + 1 = 0

and the function f_{1} = ỹ^{2} is regular and equal to 1 at (x̃ = 0, ỹ = −1 mod p), the point at infinity.

For finite x, if P_{0} = (x_{0}, y_{0}) mod p is a singularity of (8.1), then f(x_{0}) = −2y_{0}. Then either x_{0} ≠ 0, in which case f_{1} is regular and equal to 1 at P from the equation of the curve, or x_{0} = 0, in which case f_{2} is regular and equal to 1 at P because p does not divide m(0). This completes the proof of (1) and (2).

Now assume that b_{0} = ±1. That 𝕄 is in K_{2}^{T} (C) is part of Proposition 6.3, so, as above, we only have to consider T_{D}(𝕄) for D irreducible in a fiber of C_{p}. We see that the functions x and y have no poles or zeroes along any curve D mapping onto an irreducible component D of a fiber F of the arithmetic surface defined by (8.1), so, again, there can only be a problem at a singular point of that arithmetic surface. For a finite singularity (x_{0}, y_{0}) (modulo p), y_{0} ≠ 0 because y_{0} = 0 would lead to x_{0} = 0 and ±1 = b_{0} = f(x_{0}) = −2y_{0} = 0. If x_{0} = 0 then y_{0} ≠ x_{0} so that y_{0} = −f(x_{0}) = 2y_{0}, which would imply that y_{0} = 0. Therefore x_{0} ≠ 0 as well, and both −x and −f(0)y = ∓y have no poles or zeroes passing through (x_{0}, y_{0}). For d = 2g+1 there are no singularities at infinity, so 𝕄 defines a class in K_{2}(C; ℤ), but for d = 2g + 2 we, instead, consider the class of 2𝕄 = {y^{2}, x} = {y^{2}/x^{2g+2}, x} = {ỹ^{2}, 1/x̃}, and at the point at infinity, (x̃, ỹ) = (0, −1) modulo p, ỹ^{2} is regular and equal to 1. □

Remark 8.4. The proof that m(0) = ±1 implies the integrality of the element M = {y^{2}/x^{d}, m(x)/m(0)} does not use the assumption that the coefficients b_{j} have no common factor.

For the other implication, however, the assumption on the b_{j} cannot be dropped. In fact, if M ∈ K_{2}(C; ℤ), make the substitution x ↦ p^{−2}x, y ↦ p^{−d}y. It transforms C to an isomorphic curve

y^{2} + f̃(x)y + x^{d} = 0 (8.5)

and has the effect of multiplying m(0) and the b_{j} by positive powers of p. Of course, m̃(x) may be divisible by p, which could allow all powers of p to be cancelled, but by repeating this process if necessary, we can get an integral symbol on a curve as in (8.5) for which m̃(0) ≠ ±1.

9. Computing the Beilinson regulator

Let C be a hyperelliptic curve of genus g over ℂ, so the map ϕ : C → ℙ_{ℂ}^{1} has n = 2g + 2 points of ramification, P_{1}, …, P_{n}, and let X = C(ℂ) be the associated Riemann surface. It is not difficult to check that a basis of H_{1}(X; ℤ) consists of liftings to X of simple loops in ℙ_{ℂ}^{1} around exactly P_{i} and P_{i+1}, for i = 1, …, 2g. If a model of C is defined by an equation y^{2} = f(x) with f(x) in ℂ[x] of degree 2g + 1 without multiple roots, the point in C above the point at infinity in this model will be among the ramification points, and the other ones will be the roots of f(x) in ℂ ⊂ ℙ_{ℂ}^{1}. We therefore work with simple loops around exactly two roots of f(x) in ℂ ⊂ ℙ_{ℂ}^{1}, avoiding the roots of f(x), and lift them to loops on X.

In our case f(x) belongs to ℚ[x] and hence to ℝ[x], and we want to keep track of the action of complex conjugation on X. We will discuss the case when the leading coefficient of f(x) is positive, which we can always achieve by replacing x with −x if necessary. The real points in X, C(ℝ), are ∞_{C} together with the points above those x in ℝ ⊂ ℂ where f(x) ≥ 0. In Figure 9.1, the latter is indicated by the thick part of ℝ in ℂ, the thick dots being the roots of f(x).

If f(x) has only real roots P_{1} < P_{2} < ⋯ < P_{2g+1}, then let γ_{i} be a lift of a loop around P_{2i} and P_{2i+1} (i = 1, …, g) (this is illustrated on the left in Figure 9.1), and let δ_{i} be a lift of a loop around P_{2i−1} and P_{2i} (i = 1, …, g). As all of the γ_{i} have a fixed point under the action of complex conjugation, which lies above the intersection with the thick part of the real line, it is easy to check that the γ_{i} lie in H_{1}(X; ℤ)^{−} because complex conjugation reverses the orientation. Also, by shrinking the δ_{i} as much as possible, so they lift to a subset of C(ℝ), it is easy to see that the δ_{i} lie in H_{1}(X; ℤ)^{+}. So in this case H_{1}(X; ℤ) decomposes as a direct sum H_{1}(X; ℤ)^{−} ⊕ H_{1}(X; ℤ)^{+}, the γ_{i} form a basis of H_{1}(X; ℤ)^{−}, while the δ_{i} form a basis of H_{1}(X; ℤ)^{+}.

Figure 9.1

If f(x) does not have only real roots, let P_{1}, …, P_{2m+1} be the real roots of f(x), and let Q_{1}, \overline{Q_{1}}, …, Q_{g−m}, \overline{Q_{g−m}} be its nonreal roots, as pairs of complex conjugated numbers with Im(Q_{j}) > 0. Then it is easy to check that the lifts γ_{1}, …, γ_{m} of simple loops around P_{2} and P_{3}, P_{4} and P_{5}, …, P_{2m} and P_{2m+1}, and the lifts γ_{m+1}, …, γ_{g} of simple loops around Q_{1} and \overline{Q_{1}}, Q_{2} and \overline{Q_{2}}, etc., intersecting the thick part of the real axis as illustrated on the right in Figure 9.1, are in H_{1}(X; ℤ)^{−}. Also, if δ_{1}, …, δ_{g} are lifts to X of simple loops around P_{1} and P_{2}, …, P_{2m−1} and P_{2m}, as well as around P_{2m+1} and Q_{1}, Q_{1} and Q_{2}, Q_{2} and Q_{3}, etc., then the δ_{j} complement the γ_{j} to a basis of H_{1}(X; ℤ). Note that in Figure 9.2, δ′ ∘ σ = δ′ as in the case when f(x) has only real roots, but we do not get a splitting of H_{1}(X; ℤ) into + and −-parts. Namely, if δ is the lift of a loop around P_{2m+1} and Q_{1}, and γ is the lift of a loop around Q_{1} and \overline{Q_{1}} passing through the thick part of the real line to the right of P_{2m+1}, then δ − δ ∘ σ = γ, with γ in H_{1}(X; ℤ)^{−}, provided δ and γ are chosen in such a way that they are in the same branch of C(ℝ) above the thick part of the real line through which both loops in ℙ_{ℂ}^{1} pass. (An illustration of this is given on the right in the left part of Figure 9.2.) This holds as looping twice around a single point lifts to the trivial element in H_{1}(X; ℤ). If we could write δ = δ^{+} +δ^{−} with δ^{±}

δ′  δ  γ Figure 9.2

in H_{1}(X; ℤ)^{±}, then this would lead to 2δ^{−} = δ − δ ∘ σ = γ, which is not possible as γ is part of a basis of H_{1}(X; ℤ). In summary we find in this case that {γ_{1}, …, γ_{g}} is a basis of H_{1}(X; ℤ)^{−}, and that {δ_{1}, …, δ_{g}} is a basis of H_{1}(X; ℤ)/H_{1}(X; ℤ)^{−}.

For practical purposes, we choose all our loops in ℂ as concatenations of line segments, which makes computing and parametrizing their lifts to X using the analytic continuation of square roots particularly easy. Apart from the two types already described (lifts of a simple loop around two consecutive real roots of f(x), or of a simple loop around two conjugate non-real roots of f(x) intersecting the thick part of the real line, both illustrated in Figure 9.1), we also use a third type in order to avoid getting close to roots of f(x) unnecessarily, thus speeding up the calculations in numerical integration. If, for example, Q_{1} = x_{1} + iy_{1} and Q_{2} = x_{2} + iy_{2} are two non-real roots of f(x) which are close together, and 0 < y_{1} ≤ y_{2}, then, keeping a lift of a loop around the pair Q_{1}, \overline{Q_{1}} meeting the thick part of the real line, we can replace a lift of a loop around the pair Q_{2}, \overline{Q_{2}} with a loop around both pairs Q_{1}, \overline{Q_{1}}, Q_{2}, \overline{Q_{2}}, still meeting the thick part of the real line, as indicated on the right in Figure 9.2. This is just the sum of (compatible) lifts of a loop around Q_{1}, \overline{Q_{1}} and a loop around Q_{2}, \overline{Q_{2}}, both meeting the thick part of the real line in a common point. A similar method is used if more than two non-real roots of f(x) are close together.

Finally, we choose our loops also such that they do not pass through the image of the other torsion point under the map ϕ : X → ℙ_{ℂ}^{1}. This way, we can immediately compute ⟨γ, \{f, g\}⟩ = \frac{1}{2π} ∫_{γ} η(f, g), with η(f, g) as in (3.3), and integrate numerically over the lifts of the paths using the analytic continuation of y = \sqrt{f(x)} along the paths. The method works quite well in practice and we can easily obtain 12 or 15 decimals of precision for the regulator.

10. Examples

In this section we test Beilinson’s conjecture 3.11 as explained in Remark 3.13. We do this for hyperelliptic curves over ℚ, of genus 2, 3, 4 and 5. Using the results of Section 7, we construct curves of the form (8.1) and (8.2) whose 2-torsion polynomials have many rational factors. As Construction 6.11 shows, every such rational factor gives an element of K_{2}^{T} (C)/torsion. Theorem 8.3 shows which of these are in K_{2}(C; ℤ).

Our main source of examples are the constructions described in Examples 7.3 and 7.8. Unfortunately, the more sophisticated constructions of Examples 7.4 and 7.5 tend to produce either not enough symbols satisfying the integrality condition or else curves of very high conductor, for which we cannot compute the L-value.

Example 10.1: genus 2. Taking m = 1 and n = −2 in Example 7.3 yields a one-parameter family of curves

y^{2} = x^{5} + ((k + 3)x^{2} + (5k + 4)x + 4k)^{2} .

Further, if k is an integer which is divisible by 4, then the curve can be transformed to one of the form in Theorem 8.3. Thus write k = −4b with b ∈ ℤ. The curve is then isomorphic to

C_{b} : y^{2} + ((4b − 3)x^{2} − (5b − 1)x + b)y + x^{5} = 0 .

For non-zero b this is a non-singular genus 2 curve of discriminant

Δ(C_{b}) = 3^{2}b^{7}(8b + 3)^{2}(9b^{3} − 23b^{2} + 4b − 1) .

Its 2-torsion polynomial has 3 rational factors, namely (up to a constant)

\begin{aligned}m_{1} & = & x − 1, \\ m_{2} & = & 4x − 1, \\ m_{3} & = & x^{3} − (4b^{2}−6b+1)x^{2} + (5b^{2}−b)x − b^{2} . \end{aligned}

Recall from Construction 6.11 that each m_{i} gives an element M_{i} ∈ K_{2}^{T} (C)/torsion. They span a lattice in K_{2}^{T} (C)/torsion and intersecting it with K_{2}(C; ℤ) gives a sublattice of integral elements,

Λ_{M} = ⟨M_{1}, M_{2}, M_{3}⟩ ∩ K_{2}(C; ℤ).

Using Theorem 8.3 we can determine Λ_{M} exactly by inspecting the constant terms of the m_{i}. It follows that Λ_{M} = ⟨M_{1}, M_{2}, M_{3}⟩ for b = ±1 and Λ_{M} = ⟨M_{1}, M_{2}⟩ otherwise.

We can now test Beilinson’s conjecture for this family. We constructed at least two elements (M_{1} and M_{2}) of Λ_{M} and we might generally expect them to be linearly independent. If this is the case, Λ_{M} should be a subgroup of finite index in K_{2}(C; ℤ), which is supposed to have rank 2 by the first part of Beilinson’s conjecture 3.11. The second part would then imply that R(Λ_{M}), the regulator computed using a basis of Λ_{M}, is a non-zero rational multiple of L^{∗}(C_{b}, 0), the leading coefficient of L(C_{b}, s) at s=0. In summary, we expect

(a) rk_{ℤ} Λ_{M} = 2;

(b) R(Λ_{M}) = QL^{∗}(C_{b}, 0) for some Q ∈ ℚ^{∗}.

Finally, note that for b = ±1 we have Λ_{M} = ⟨M_{1}, M_{2}, M_{3}⟩, which is supposed to be of rank at most 2. So there should be a linear relation between the M_{i}.

Let us illustrate this in the case b = −1. We can compute the images of the M_{i} under the regulator pairing (3.10), as explained in Section 9,

α ⟼ (⟨α, γ_{1}⟩, ⟨α, γ_{2}⟩) ∈ ℝ^{2}, (10.2)

where γ_{1} and γ_{2} form a basis of H_{1}(X; ℤ)^{−}. Since we expect the image to be a lattice of rank 2, there should be some integral linear combination of the M_{i} for which the image in ℝ^{2} vanishes. Using LLL to look for a small ℤ-relation between the images in ℝ^{2}, we find that

41M_{1} + 56M_{2} − 44M_{3} ⟼^{?} (0, 0) .

To make the relation more transparent we complete the vector (41, 56, −44) to a unimodular integral matrix and choose the corresponding basis:

(\begin{matrix}M_{1}^{∗} \\ M_{2}^{∗} \\ M_{3}^{∗} \end{matrix}) = (\begin{matrix}0 & 0 & 1 \\ 11 & 15 & 0 \\ 41 & 56 & −44 \end{matrix}) (\begin{matrix}M_{1} \\ M_{2} \\ M_{3} \end{matrix}) .

The numerical values of the regulator pairing (10.2) on the M_{i} and the M_{i}^{∗} are then given by the table

\begin{matrix}α & M_{1} & M_{2} & M_{3} = M_{1}^{∗} & M_{2}^{∗} & M_{3}^{∗} \\ ⟨α, γ_{1}⟩ & −0.519837 & −6.055409 & −8.191279 & −96.549352 & 0.000000 \\ ⟨α, γ_{2}⟩ & 3.243279 & −0.869701 & 1.915254 & 22.630553 & 0.000000 \end{matrix}

This strongly suggests that M_{3}^{∗} is in the kernel of (10.2), in which case M_{3}^{∗} should be zero in Λ_{M} by Beilinson’s conjecture. The regulator R(Λ_{M}) is now obtained as the absolute value of the determinant of the 2 × 2 matrix which consists of the two columns for M_{1}^{∗} and M_{2}^{∗},

R(Λ_{M}) = | \begin{matrix}−8.191279 & −96.549352 \\ 1.915254 & 22.630553 \end{matrix} | = 0.456625 .

Numerically, L^{∗}(C_{−1}, 0) ≈ 0.228312, so L^{∗}(C_{−1}, 0)/R(Λ_{M}) ≈ 0.500000, which we “recognize” as the rational number 1/2. (Here the agreement is to at least 18 digits, which was our working precision in this example).

Recall from Theorem 8.3 that for b = ±1 the element 𝕄 = {−by, −x} also lies in K_{2}(C; ℤ), and that, according to Remark 6.14, 10𝕄 = M_{1} + M_{2} + M_{3}. For b = −1, we can write this as 10𝕄 = −175M_{1}^{∗} + 15M_{2}^{∗} − 4M_{3}^{∗}, so if we assume M_{3}^{∗} = 0, then 2𝕄 = −35M_{1}^{∗} + 3M_{2}^{∗} is in Λ_{M} but 𝕄 is not. In other words, the larger lattice Λ = ⟨M_{1}, M_{2}, M_{3}, 𝕄 ⟩ in K_{2}(C; ℤ) contains Λ_{M} as a sublattice of index 2 and

L^{∗}(C_{−1}, 0)/R(Λ) = 1.000000... =^{?} 1 .

Table 1 summarizes the computations for the curves C_{b} with |b| ≤ 10. In the second column we describe the lattice Λ that we use, and the last column contains the quotient of L^{∗}(C_{b}, 0) and the Beilinson regulator R(Λ). In all cases this quotient appears to be a rational number of relatively small height. For b = ±1 we also include the expected relation between the M_{i}. (We have not included the universal relation 10𝕄 = M_{1} +M_{2} +M_{3} because similar relations would clog up later tables.)

Finally, we remark that the Jacobian of each of the curves appearing in Table 1, except for b = 1, is not isogenous to the product of two elliptic curves, even over \overline{ℚ}. This was verified using the procedure of [6, Chapter 14, Section 4]. In other words, the elements in K-theory and the L-values do not come from elliptic curves, and the verification of Beilinson’s conjecture does not reduce to the case of genus 1. The same holds for all the curves in Table 2 (Example 10.4) and the curves in Tables 6 and 7 (Example 10.7) except the one with v = {2, 5}.

Notation 10.3. In all of the following examples we keep the same notation as above. For a curve

C : y^{2} + f(x)y + x^{d} = 0

as in (8.1) and (8.2), denote by m_{i} the rational factors of the 2-torsion polynomial t(x) = −x^{d} +f(x)^{2}/4. The associated elements of K_{2}^{T} (C)/torsion (see (6.12)) are denoted by M_{i} and we let

Λ′ = \{ \begin{matrix}⟨M_{1}, …, M_{k}⟩, & f(0) ≠ ±1, \\ ⟨M_{1}, …, M_{k}, 𝕄 ⟩, & f(0) = ±1, d = 2g + 1, \\ ⟨M_{1}, …, M_{k}, 2𝕄 ⟩, & f(0) = ±1, d = 2g + 2. \end{matrix}

and take Λ denote the sublattice of integral elements Λ = Λ′ ∩ K_{2}(C; ℤ). In all of the examples below we can determine Λ by applying Theorem 8.3. Then we verify numerically that Λ has rank genus(C) and that the leading coefficient L^{∗}(C, 0) is a rational multiple of the regulator R(Λ) of Λ. To keep the entries for Λ simple, we did not include the universal relation between 𝕄 and the M_{i} of Remark 6.14 in the tables.

Example 10.4: genus 2. We can construct another family of genus 2 curves with enough elements in K_{2}(C; ℤ) in a way similar to Example 7.8, this time using 6-torsion points. The curves are given by

y^{2} + (2x^{3} − 4bx^{2} − x + b)y + x^{6} = 0 , b ∈ ℤ_{>0}.

The 2-torsion polynomial has 4 rational factors,

m_{1} = x − b, m_{2} = 2x − 1, m_{3} = 2x + 1, m_{4} = 4bx^{2} + x − b,

so we get 4 elements of K_{2}^{T} (C)/torsion, again denoted by M_{1}, M_{2}, M_{3}, M_{4}. According to Remark 6.15 these satisfy the relation M_{1} +M_{2} +M_{3} −M_{4} = 0 so M_{4} can be dropped. Furthermore, M_{2} and M_{3} are always integral, and M_{1} is for b=1. The numerical results and expected relation between the M_{j} if b = 1 are given in Table 2.

Example 10.5: genus 3. We can look at hyperelliptic genus 3 curves with a rational 7-torsion point and whose 2-torsion polynomial has two given rational root, cf. Example 7.3. A special case of this is the two-parameter family given by

y^{2} + ((4b−3)x^{3} − (4a+5b−1)x^{2} + (5a+b)x − a)y + x^{7} = 0 .

The curves have a rational 7-torsion point (0, 0) and two rational 2-torsion points with x-coordinates 1 and 1/4. The 2-torsion polynomial has 3 rational factors,

m_{1} = x − 1,
m_{2} = 4x − 1,
m_{3} = x^{5} − (4b^{2}−6b+1)x^{4} + (5b^{2}+8ab−2b−6a)x^{3}
−(b^{2}+10ab+4a^{2}−2a)x^{2} + (5a^{2}+2ab)x − a^{2} .

For a = ±1 and b ∈ ℤ we thus get 3 integral symbols and we expect a relation with L^{∗}(C, 0). Moreover, for a=1, b=3 we have a further factorization

m_{3} = m_{3,1}m_{3,2} = (x^{2}−3x+1)(x^{3}−16x^{2}+8x−1).

so we get 4 elements of K_{2}(C; ℤ) and we expect them to be linearly dependent. The numerical results and expected relation for a = 1, b = 3, are summarized in Table 3.

Example 10.6: genus 4, 5. We can also give some sporadic examples of curves of genus g =4 and g =5, given by

y^{2} + f(x)y + x^{10} = 0

with f(x) of degree 5 and

y^{2} + f(x)y + x^{12} = 0

with f(x) of degree 6 respectively. These can be found by a brute search for good polynomials with enough factors in ℤ[x] with constant term ±1. In all cases that we found, we have exactly g + 1 such factors and the lattice Λ = ⟨M_{1}, …, M_{g}, 2𝕄⟩ is of rank g. The examples and the corresponding numerical results are given in Tables 4 and 5.

Example 10.7: arbitrary genus. Finally, we give examples of curves of arbitrary genus g with g integral symbols. Consider a curve of the form

y^{2} + (2x^{g+1} ± ∏_{j=1}^{g}(v_{j}x + 1)) y + x^{2g+2} = 0 , (10.8)

where v_{1} < ⋯ < v_{g} are distinct non-zero integers, and we are assuming that the 2-torsion polynomial t(x) has no multiple roots. Then t(x) has at least g +1 factors in ℤ[x],

v_{1}x + 1, v_{2}x + 1, …, v_{g}x + 1, 4x^{g+1} ± ∏_{j=1}^{g}(v_{j}x + 1) . (10.9)

(This is similar to, but different from the discussion in Remark 7.8, because now, in 4t(x) = −4x^{2g+2} +f(x)^{2} = (f(x)−2x^{g+1})(2x^{g+1} +f(x)), we factor f(x)−2x^{g+1} completely into g linear factors, instead of 2x^{g+1}+f(x), which has degree g+1.) The elements M_{j} associated to all irreducible factors in ℤ[x] are integral by Theorem 8.3, so we get at least g + 1 elements of K_{2}(C; ℤ), but we have to take the universal relation of Remark 6.15 into account.

An equation of the form (10.8) is not unique for a given curve. For even g, we may (and will) assume that the sign ‘±’ in (10.8) is ‘+’, for we can otherwise replace (x, y, {v_{j}}) with (−x, −y, {−v_{g+1−j}}). On the other hand, for odd g the map (x, y) ↦ (−x, y) gives an isomorphism between the two curves with the same sign associated to {v_{j}} and {−v_{g+1−j}}.

Also, if we look at the model at infinity by letting y ↦ y/x^{g+1} and x ↦ 1/x, the equation becomes

y^{2} + (2 ± x ∏_{j=1}^{g}(x + v_{j})) y + 1 = 0 .

Now it is clear that translating x by −v_{j} for some j gives an equation of the same shape. In other words, {v_{j}} and {w_{j}} yield an isomorphic curve whenever {v_{j}}∪{0} is a translate of {w_{j}} ∪ {0}. Thus, after translating by −v_{1} in case v_{1} < 0, we can assume that all v_{j} are positive. Combining this with the above, we see that for odd g, 0 < v_{1} < ⋯ < v_{g} and 0 < v_{g} −v_{g−1} < ⋯ < v_{g} −v_{1} < v_{g} give isomorphic curves, and we have chosen the lexicographically smaller representative in Tables 8 and 9 for genus 3 curves.

We can also look at the cases where the last rational factor in (10.9) is reducible, for instance, when it has a linear factor ax − 1 with a ∈ ℤ. Then 1/a is a root of this polynomial, so

∏_{j=1}^{g}(v_{j}/a + 1) = ∓4a^{−g−1}.

It follows that a(a+v_{1}) ⋯ (a+v_{g}) = ∓4, which leaves only finitely many possibilities for a and the v_{j}. It is easy to see that the only two examples for g > 1 and 0 < v_{1} < v_{2} < … are

v = {2, 5}, a = −1  and  v = {3, 4}, a = −2 .

(There is also v = {1, 3, 4}, a = −2 but the resulting curve is singular.) Finally, there is a case for g = 3 where the last factor of (10.9) splits into two quadratic factors over the rationals, namely v = {1, 5, 6}. Thus we have found three examples for which the 2-torsion polynomial t(x) has g +2 rational factors, all with a ‘+’-sign in (10.8),

\begin{matrix}v = \{2, 5\}, & −4t(x) = (x+1)(2x+1)(5x+1)(4x^{2}+6x+1) , \\ v = \{3, 4\}, & −4t(x) = (2x+1)(3x+1)(4x+1)(2x^{2}+5x+1) , \\ v = \{1, 5, 6\}, & −4t(x) = (x+1)(5x+1)(6x+1)(x^{2}+6x+1)(4x^{2}+6x+1) . \end{matrix}

In each of the examples, if we associate to these factors elements M_{1}, …, M_{g+2} of K_{2}(C; ℤ) in the order that the factors are written, we can leave out M_{g+2} and still expect a non-trivial relation between the remaining elements. Such a relation was indeed found numerically in all three cases.

The results are summarized in Tables 6, 7 (genus 2), 8, 9 (genus 3) and 10 (genus 4). The entries in the tables are sorted according to the conductor. Unfortunately, we cannot compute the numerical values of L^{∗}(0) for g ≥ 5 for these curves because the conductors get too large.

Remark 10.10. There are two curves in our tables that are isomorphic over ℚ, with the following isomorphism

(Table 2, b = 1) ≅ (Table 6, v = \{3, 4\}), (x, y) ↦ (−\frac{x}{2x+1}, \frac{y}{(2x+1)^{3}} ).

Except for these two, there are no identical conductors in our tables, so all the other curves are pairwise non-isomorphic.

Remark 10.11. We will not attempt to give any conjectures about the experimental values of L^{∗}(C, 0)/R(Λ) in the various tables. But we note that we often find an integer for L^{∗}(C, 0)/R(Λ). So it might be that L^{∗}(C, 0)/R(K_{2}(C; ℤ)) is always an integer, and that the denominators that occur in L^{∗}(C, 0)/R(Λ) for some of our examples can be explained by the fact that Λ is not the full group K_{2}(C; ℤ) because L^{∗}(C, 0)/R(K_{2}(C; ℤ)) = (K_{2}(C; ℤ) : Λ) · L^{∗}(C, 0)/R(Λ). This would be similar to what happens in the discussion of the case of b = −1 in Example 10.1, where Λ_{M} = ⟨M_{1}, M_{2}, M_{3}⟩ leads to L^{∗}(C, 0)/R(Λ_{M}) =^{?} 1/2, but the larger lattice Λ = ⟨M_{1}, M_{2}, M_{3}, 𝕄 ⟩, with (Λ : Λ_{M}) = 2, gives L^{∗}(C, 0)/R(Λ) =^{?} 1.

Unfortunately, we have not been able to test this as we do not know any systematic ways to construct elements of K_{2}(C; ℤ) other than the ones we discussed. But it would be interesting to do this, especially for the curve defined by

y^{2} + (5x^{3} − 13x^{2} + 7x − 1)y + x^{7} = 0

(which is the example with a = 1 and b = 2 of Table 3), where the denominator is the relatively large prime 19.

b  Conductor  Λ  L^{∗}(0)  L^{∗}(0)/R(Λ)
−10 2^{2}·3·5^{2}·7·11^{2}·1031  ⟨M_{1}, M_{2}⟩  161973.28326750 2^{2}·3·5^{2}·7
−9  3^{3}·23·8461  ⟨M_{1}, M_{2}⟩  2647.35530780278  2^{2}·3^{2}
−8  2^{2}·3·61·6113  ⟨M_{1}, M_{2}⟩  2402.40716016213  3·23/2
−7  3·7^{2}·53·4243  ⟨M_{1}, M_{2}⟩  16142.83542365698  13·19
−6  2^{2}·3^{3}·5·2797  ⟨M_{1}, M_{2}⟩  1090.93388314410  2·3^{2}
−5  3·5^{2}·37·1721  ⟨M_{1}, M_{2}⟩  1602.46040140666  29
−4  2^{2}·3·29·31^{2}  ⟨M_{1}, M_{2}⟩  196.40102935125  2^{2}
−3  3^{3}·7·463  ⟨M_{1}, M_{2}⟩  41.79313813775  1
−2  2^{2}·3·13·173  ⟨M_{1}, M_{2}⟩  16.33803025573  1/2
−1  3·5·37  ⟨M_{1}, M_{2}, M_{3}, 𝕄 ⟩_{41M_{1}+56M_{2}−44M_{3}=0}  0.22831231665  1
1  3·11^{2}  ⟨M_{1}, M_{2}, M_{3}, 𝕄 ⟩_{4M_{1}−26M_{2}+29M_{3}=0}  0.12984963472  1/2
2  2^{2}·3·13·19  ⟨M_{1}, M_{2}⟩  1.90319317513  1/2^{2}·5
3  3^{3}·47  ⟨M_{1}, M_{2}⟩  0.62178975664  1/2·3^{3}
4  2^{2}·3·5·7·223  ⟨M_{1}, M_{2}⟩  42.90813731246  1
5  3·5^{2}·43·569  ⟨M_{1}, M_{2}⟩  802.87262799199  2^{4}
6  2^{2}·3^{3}·17^{2}·67  ⟨M_{1}, M_{2}⟩  1575.43548439889  2^{2}·7
7  3·7^{2}·59·1987  ⟨M_{1}, M_{2}⟩  6154.43484637856  2^{2}·5^{2}
8  2^{2}·3·67·3167  ⟨M_{1}, M_{2}⟩  1788.15229247202  3^{3}
9  3^{3}·5·4733  ⟨M_{1}, M_{2}⟩  281.80083335457  2^{2}
10 2^{2}·3·5^{2}·23·83·293  ⟨M_{1}, M_{2}⟩  85596.822531781  2^{7}·3^{2}
Table 1. Genus 2 curves y^{2} + ((4b − 3)x^{2} − (5b − 1)x + b)y + x^{5} = 0

b  Conductor  Λ  L^{∗}(0)  L^{∗}(0)/R(Λ)
1  2^{4}·3·17  ⟨M_{1}, M_{2}, M_{3}, 2𝕄 ⟩_{10M_{1}−2M_{2}+M_{3}=0} 0.35283625318  1/2^{3}
2  2^{5}·3·5^{2}·13  ⟨M_{2}, M_{3}⟩  22.9767849707  1/2
3 2^{4}·3^{2}·5^{2}·7·29  ⟨M_{2}, M_{3}⟩  347.644931188  2·3
4  2^{5}·3·7·257  ⟨M_{2}, M_{3}⟩  134.428839855  2
5 2^{4}·3·5^{2}·11·401  ⟨M_{2}, M_{3}⟩  2694.74551646  2^{2}·3^{2}
6 2^{5}·3^{2}·11·13·577  ⟨M_{2}, M_{3}⟩  20021.3775652  2·3·41
8 2^{5}·3·5^{2}·17·41  ⟨M_{2}, M_{3}⟩  1106.79592308  2^{2}·3
10 2^{5}·3·5^{2}·7·19·1601  ⟨M_{2}, M_{3}⟩  416121.1462 2^{2}·3·7^{3}
12 2^{5}·3^{2}·5^{2}·23·461  ⟨M_{2}, M_{3}⟩  58663.7341258 2^{2}·3^{3}·5
13 2^{4}·3·5^{2}·13^{2}·541  ⟨M_{2}, M_{3}⟩  46380.28556 2·3^{2}·23
14 2^{5}·3·7^{2}·29·3137  ⟨M_{2}, M_{3}⟩  322837.4973 2·3·467
Table 2. Genus 2 curves y^{2} + (2x^{3} − 4bx^{2} − x + b)y + x^{6} = 0

a, b  Conductor  Λ  L^{∗}(0)  L^{∗}(0)/R(Λ)
−1,−6 3·5·13·62773  ⟨M_{1}, M_{2}, M_{3}, 𝕄 ⟩  157.32845991764  2·3
−1,−5 2·3·5·34543  ⟨M_{1}, M_{2}, M_{3}, 𝕄 ⟩  13.71462235525  1
−1,−4  3^{3}·7^{2}·73  ⟨M_{1}, M_{2}, M_{3}, 𝕄 ⟩  0.99948043865  1/2·3
−1,−2 3·19·1051  ⟨M_{1}, M_{2}, M_{3}, 𝕄 ⟩  0.68841617094  1/2·7
−1, 0 3·5·7·997  ⟨M_{1}, M_{2}, M_{3}, 𝕄 ⟩  1.30028297708  1/2·3
−1, 1 2·3·43·599  ⟨M_{1}, M_{2}, M_{3}, 𝕄 ⟩  1.85769147359  1/2
−1, 2 3^{3}·17·199  ⟨M_{1}, M_{2}, M_{3}, 𝕄 ⟩  0.95009772024 1/2·3·5
−1, 3 2·3·7^{2}·59·599  ⟨M_{1}, M_{2}, M_{3}, 𝕄 ⟩  117.81139857650  2^{2}
1, 0  3·29^{2}·71  ⟨M_{1}, M_{2}, M_{3}, 𝕄 ⟩  1.91931648711  1/3
1, 2  3·13·971  ⟨M_{1}, M_{2}, M_{3}, 𝕄 ⟩  0.41650834991  1/19
1, 3 2·3·5^{2}·229  ⟨M_{1}, M_{2}, M_{3,1}, M_{3,2}, 𝕄 ⟩_{36M_{1}−111M_{2}−41M_{3,1}+71M_{3,2}=0}  0.33653518886  1
1, 4  3^{2}·7877  ⟨M_{1}, M_{2}, M_{3}, 𝕄 ⟩  0.84449758754  2/3·5
1, 5 2·3·11·44071  ⟨M_{1}, M_{2}, M_{3}, 𝕄 ⟩  39.13475965836  2
1, 6 3·5^{2}·19·46273  ⟨M_{1}, M_{2}, M_{3}, 𝕄 ⟩  631.40978672221  2^{2}·5
1, 7 2·3^{2}·479·2011  ⟨M_{1}, M_{2}, M_{3}, 𝕄 ⟩  170.28405530697  2^{2}

Table 3. Genus 3 curves y^{2} + ((4b−3)x^{3} − (4a+5b−1)x^{2} + (5a+b)x − a)y + x^{7} = 0

f(x)  Conductor  L^{∗}(0)  L^{∗}(0)/R(Λ)
2x^{5}+2x^{4}+x^{3}+x^{2}−3x−1  2^{11}·5^{3}·19·29 35.85879769  1/2
2x^{5}+2x^{4}+2x^{3}−3x^{2}−2x+1 2^{11}·3^{2}·17·59 5.336928011  1/2
2x^{5}+3x^{4}−3x^{3}−2x^{2}−x−1 2^{3}·3^{3}·5·19·331 1.865694255  1/2^{2}·3
2x^{5}+3x^{4}−x^{3}−4x^{2}−3x+1 2^{3}·3^{4}·5·7·20759 126.4283012  1
2x^{5}+3x^{4}+x^{3}−3x−1  2^{4}·3^{3}·7·11·4793 41.29358643  1/2
2x^{5}+4x^{4}−3x^{3}−2x+1  2^{4}·5^{3}·7·11·103 3.546483598  1/2^{3}
2x^{5}+4x^{4}−x^{3}−3x^{2}−3x−1  2^{10}·3·7·1051 6.484247251  1/2^{3}
2x^{5}+4x^{4}−x^{3}−3x^{2}−x+1  2^{12}·3^{6}·13  11.50901911  1/2^{2}
2x^{5}+4x^{4}+3x^{3}−5x^{2}−5x−1 2^{12}·3·5·19·79 27.69939565  1/2
2x^{5}+4x^{4}+3x^{3}+3x^{2}−x−1 2^{12}·3·5^{2}·23·43 89.28895569  1
2x^{5}+4x^{4}+5x^{3}+2x^{2}+2x+1 2^{4}·3^{3}·7^{2}·379 2.157167657  1/2^{4}
Table 4. Genus 4 curves y^{2} + f(x)y + x^{10} = 0

f(x)  Conductor  L^{∗}(0)  L^{∗}(0)/R(Λ)
2x^{6}+2x^{5}−4x^{4}−3x^{3}−2x^{2}+4x−1 2^{8}·3^{4}·5·7·11·19 0.97906446422637 1/2^{2}·3^{3}
2x^{6}+4x^{5}−5x^{4}−3x^{3}+2x^{2}−x−1 2^{7}·3^{2}·5^{2}·107·139 2.86707608488323  1/2^{2}·3
2x^{6}+6x^{5}−5x^{3}−3x^{2}+x+1  2^{16}·3^{2}·5·13^{2}  3.21518014215484  1/2^{2}·3
2x^{6}+2x^{5}+2x^{4}−x^{3}−3x^{2}−3x−1  2^{16}·3·5^{3}·31  4.04748393920751  1/2^{3}·3
2x^{6}+2x^{5}+x^{3}−3x^{2}−x+1  2^{16}·3^{4}·5·181  28.41118880946  1/3
Table 5. Genus 5 curves y^{2} + f(x)y + x^{12} = 0

v  Conductor  Λ  L^{∗}(0)  L^{∗}(0)/R(Λ)
2, 5  2^{3}·3·5^{2}  ⟨M_{1}, M_{2}, M_{3}, 2𝕄 ⟩_{39M_{1}−26M_{2}+M_{3}=0}  0.246481356638  1/2^{2}
2, 4  2^{6}·11  ⟨M_{1}, M_{2}, 2𝕄 ⟩  0.339189596082  1/2^{4}·3
3, 4  2^{4}·3·17  ⟨M_{1}, M_{2}, M_{3}, 2𝕄 ⟩_{9M_{1}−10M_{2}+2M_{3}=0}  0.352836253176  1/2^{3}
1, 8  2·7·59  ⟨M_{1}, M_{2}, 2𝕄 ⟩  0.412953201772  1/2^{3}·7
1, 2  2^{3}·107  ⟨M_{1}, M_{2}, 2𝕄 ⟩  0.372337494391  1/2^{3}
1, 4  2^{3}·3·53  ⟨M_{1}, M_{2}, 2𝕄 ⟩  0.609341856994  1/2^{4}
8, 9  2·3·223  ⟨M_{1}, M_{2}, 2𝕄 ⟩  0.751762610683  1/2^{2}·7
2, 3  2^{3}·3·59  ⟨M_{1}, M_{2}, 2𝕄 ⟩  0.682403286269  1/2^{2}·3
4, 5  2^{3}·5·79  ⟨M_{1}, M_{2}, 2𝕄 ⟩  1.466505046768  1/2^{3}
1, 3  2^{3}·3·139  ⟨M_{1}, M_{2}, 2𝕄 ⟩  1.573797007363  1/2^{2}
1, 5  2^{4}·5·83  ⟨M_{1}, M_{2}, 2𝕄 ⟩  3.296439983269  1/2^{2}
2, 6  2^{6}·3·37  ⟨M_{1}, M_{2}, 2𝕄 ⟩  4.318055838622  1/2^{2}
5, 8 2^{2}·3·5·127  ⟨M_{1}, M_{2}, 2𝕄 ⟩  4.980029782458  1/2·3
1, 9  2^{5}·3·149  ⟨M_{1}, M_{2}, 2𝕄 ⟩  6.585355387353  1/2^{2}
3, 5 2^{3}·3·5·229  ⟨M_{1}, M_{2}, 2𝕄 ⟩  14.988121207002  1
3, 11 2^{2}·3·11·229  ⟨M_{1}, M_{2}, 2𝕄 ⟩  19.618673668593  1/2
4, 8  2^{5}·997  ⟨M_{1}, M_{2}, 2𝕄 ⟩  15.361499842059  1/2
4, 6  2^{6}·3·197  ⟨M_{1}, M_{2}, 2𝕄 ⟩  19.481730347124  1
3, 6  2^{3}·3^{5}·23  ⟨M_{1}, M_{2}, 2𝕄 ⟩  20.448409445891  1
2, 8  2^{6}·3·269  ⟨M_{1}, M_{2}, 2𝕄 ⟩  24.399036028269  1
2, 10  2^{6}·5^{2}·37  ⟨M_{1}, M_{2}, 2𝕄 ⟩  30.231463268350  1
1, 6 2^{3}·3·5·499  ⟨M_{1}, M_{2}, 2𝕄 ⟩  24.839835330300  3/2
6, 9  2^{3}·3^{5}·31  ⟨M_{1}, M_{2}, 2𝕄 ⟩  33.600131091460  1
3, 7 2^{3}·3·7·359  ⟨M_{1}, M_{2}, 2𝕄 ⟩  25.126558038878  1
7, 8  2^{5}·7^{2}·41  ⟨M_{1}, M_{2}, 2𝕄 ⟩  28.545427660372  3/2
5, 6 2^{3}·3·5·733  ⟨M_{1}, M_{2}, 2𝕄 ⟩  43.322633518831  3
3, 12  2^{4}·3^{5}·23  ⟨M_{1}, M_{2}, 2𝕄 ⟩  42.043168670629  1
1, 7 2^{3}·3·7^{2}·101  ⟨M_{1}, M_{2}, 2𝕄 ⟩  59.608745714348  3
2, 7 2^{3}·5·7^{2}·67  ⟨M_{1}, M_{2}, 2𝕄 ⟩  63.124534380486  3
4, 7 2^{4}·3·7^{2}·67  ⟨M_{1}, M_{2}, 2𝕄 ⟩  76.831677676717  3
6, 8  2^{6}·3·829  ⟨M_{1}, M_{2}, 2𝕄 ⟩  79.443510393961  3
1, 10 2^{3}·3·5·1427  ⟨M_{1}, M_{2}, 2𝕄 ⟩  88.599127797882  3
3, 8 2^{5}·3·5^{2}·73  ⟨M_{1}, M_{2}, 2𝕄 ⟩  87.679227492015  3
4, 9 2^{3}·3·5·1907  ⟨M_{1}, M_{2}, 2𝕄 ⟩  105.425224481666  3
6, 7 2^{3}·3·7·1373  ⟨M_{1}, M_{2}, 2𝕄 ⟩  101.061871259886  2·3
2, 9 2^{3}·3·7·11·191  ⟨M_{1}, M_{2}, 2𝕄 ⟩  164.650770360786  2·3
3, 9 2^{3}·3^{5}·199  ⟨M_{1}, M_{2}, 2𝕄 ⟩  197.318826602911  2·3
5, 7 2^{3}·5^{2}·7·353  ⟨M_{1}, M_{2}, 2𝕄 ⟩  208.897958121197  3^{2}
5, 9 2^{4}·3·5·2089  ⟨M_{1}, M_{2}, 2𝕄 ⟩  212.013278144012  2·3

Table 6. Curves y^{2} + (2x^{3}+(v_{1}x+1)(v_{2}x+1))y + x^{6} = 0 (genus 2)

v  Conductor  Λ  L^{∗}(0)  L^{∗}(0)/R(Λ)
9, 10  2^{3}·3·5·5261  ⟨M_{1}, M_{2}, 2𝕄 ⟩  275.183140339336  2^{2}·3
8, 10  2^{6}·5·2221  ⟨M_{1}, M_{2}, 2𝕄 ⟩  417.499914032374  13
1, 11  2^{3}·5·11·1619  ⟨M_{1}, M_{2}, 2𝕄 ⟩  294.450550333465  3^{2}
4, 12  2^{5}·3·83·103  ⟨M_{1}, M_{2}, 2𝕄 ⟩  412.225445696365  3^{2}
7, 9 2^{3}·3·7^{2}·11·73  ⟨M_{1}, M_{2}, 2𝕄 ⟩  441.365714448209  3·5
2, 11  2^{3}·3·11·6053  ⟨M_{1}, M_{2}, 2𝕄 ⟩  689.050402379536  3·7
5, 12 2^{3}·3·5^{2}·7^{2}·61  ⟨M_{1}, M_{2}, 2𝕄 ⟩  860.242245920864  2·3^{2}
2, 12  2^{6}·3·5·2341  ⟨M_{1}, M_{2}, 2𝕄 ⟩  1267.775571888346  2^{2}·3^{2}
7, 15 2·3·5^{2}·7^{2}·313  ⟨M_{1}, M_{2}, 2𝕄 ⟩  1075.913430726619  2·3^{2}
8, 17  2·3·17·23321  ⟨M_{1}, M_{2}, 2𝕄 ⟩  1201.189781491657  2·3^{2}
5, 10  2^{3}·5^{2}·59·263  ⟨M_{1}, M_{2}, 2𝕄 ⟩  1440.017545464245  2^{2}·3^{2}
4, 10 2^{6}·3·5·17·197  ⟨M_{1}, M_{2}, 2𝕄 ⟩  1562.214360456082  2^{3}·5
6, 10  2^{6}·3·5·3797  ⟨M_{1}, M_{2}, 2𝕄 ⟩  1888.148590577887  2^{4}·3
4, 13 2^{3}·3·11·13^{2}·89  ⟨M_{1}, M_{2}, 2𝕄 ⟩  2048.440414253531  2·3·7
8, 16  2^{6}·109·601  ⟨M_{1}, M_{2}, 2𝕄 ⟩  2029.415672068518  2^{5}
4, 14  2^{6}·5·7^{2}·373  ⟨M_{1}, M_{2}, 2𝕄 ⟩  3298.078842289480  2^{6}
6, 12  2^{6}·3^{5}·431  ⟨M_{1}, M_{2}, 2𝕄 ⟩  3779.698402282094  2·3·13
4, 11 2^{4}·7·11·23·239  ⟨M_{1}, M_{2}, 2𝕄 ⟩  3065.942241509888  2^{3}·3^{2}
3, 10 2^{3}·3·5·7^{2}·1307  ⟨M_{1}, M_{2}, 2𝕄 ⟩  3909.737148728376  2^{2}·3^{3}
7, 10 2^{3}·3·5^{2}·7·43·59  ⟨M_{1}, M_{2}, 2𝕄 ⟩  4653.068964642799  2·3^{2}·7
5, 13 2^{5}·5·13·31·263  ⟨M_{1}, M_{2}, 2𝕄 ⟩  8592.680376054557  2^{3}·3·7
7, 16  2^{5}·3·7^{2}·4493  ⟨M_{1}, M_{2}, 2𝕄 ⟩ 11344.672982180578 2^{2}·3^{2}·5
5, 11 2^{3}·3·5·11·26573 ⟨M_{1}, M_{2}, 2𝕄 ⟩ 16672.060582310110  2·3^{3}·7
6, 14 2^{6}·3·7·41·677  ⟨M_{1}, M_{2}, 2𝕄 ⟩ 20507.523505991035  2^{4}·23
7, 14  2^{3}·7^{2}·117541  ⟨M_{1}, M_{2}, 2𝕄 ⟩ 20242.092778779142 2^{3}·3^{2}·5
3, 13 2^{3}·3·5^{2}·13·6553 ⟨M_{1}, M_{2}, 2𝕄 ⟩ 20094.811949554845 2·3^{2}·5^{2}
6, 16 2^{6}·3·5·73·773  ⟨M_{1}, M_{2}, 2𝕄 ⟩ 26721.552091088677  2^{4}·3^{3}
6, 15 2^{3}·3^{5}·5·59·101  ⟨M_{1}, M_{2}, 2𝕄 ⟩ 28628.778269858471  2·3^{5}
5, 14 2^{3}·3·5·7·95621  ⟨M_{1}, M_{2}, 2𝕄 ⟩ 32866.799798423930 2·3·101
5, 15 2^{3}·3·5^{2}·29·4673 ⟨M_{1}, M_{2}, 2𝕄 ⟩ 41114.005206032957 2^{4}·3^{2}·5
8, 18 2^{6}·3·5^{2}·73·353 ⟨M_{1}, M_{2}, 2𝕄 ⟩ 59468.612113790229 2^{2}·3·71
6, 13 2^{3}·3·7^{2}·13·17·619 ⟨M_{1}, M_{2}, 2𝕄 ⟩ 81849.542063836685 2·3^{3}·29

Table 7. Curves y^{2} + (2x^{3}+(v_{1}x+1)(v_{2}x+1))y + x^{6} = 0 (genus 2, continued)

v  Conductor  Λ  L^{∗}(0)  L^{∗}(0)/R(Λ)
2, 4, 6  2^{8}·3^{3}·13  ⟨M_{1}, M_{2}, M_{3}, 2𝕄 ⟩  1.237888702787  1/2^{7}
1, 3, 4  2^{8}·3^{2}·7^{2}  ⟨M_{1}, M_{2}, M_{3}, 2𝕄 ⟩  1.319632692018  1/2^{6}
1, 2, 4  2^{6}·3^{3}·67  ⟨M_{1}, M_{2}, M_{3}, 2𝕄 ⟩  1.326260789773  1/2^{5}
1, 2, 3  2^{6}·3^{3}·73  ⟨M_{1}, M_{2}, M_{3}, 2𝕄 ⟩  1.360553792128  1/2^{4}
1, 5, 6  2^{8}·3^{3}·5^{3}  ⟨M_{1}, M_{2}, M_{3}, M_{4}, 2𝕄 ⟩_{M_{1}+M_{2}+11M_{3}−12M_{4}=0}  9.556528296211  1/2
1, 3, 5 2^{6}·3^{3}·5·179  ⟨M_{1}, M_{2}, M_{3}, 2𝕄 ⟩  15.763316182605  1/2^{3}
3, 4, 7 2^{8}·3^{2}·7·113  ⟨M_{1}, M_{2}, M_{3}, 2𝕄 ⟩  21.282782316338  1/2^{3}
1, 2, 5 2^{8}·3^{3}·5·53  ⟨M_{1}, M_{2}, M_{3}, 2𝕄 ⟩  18.574599809025  1/2^{2}
2, 6, 8  2^{14}·3^{2}·17  ⟨M_{1}, M_{2}, M_{3}, 2𝕄 ⟩  41.533490724724  1/2^{3}
2, 3, 5 2^{6}·3^{2}·5^{3}·89  ⟨M_{1}, M_{2}, M_{3}, 2𝕄 ⟩  72.152711581916  1
1, 2, 6 2^{6}·3^{3}·5^{3}·43  ⟨M_{1}, M_{2}, M_{3}, 2𝕄 ⟩  105.836805455409  1
1, 3, 6 2^{6}·3^{2}·5·47·73  ⟨M_{1}, M_{2}, M_{3}, 2𝕄 ⟩  98.432671578138  1/2
2, 3, 6 2^{6}·3^{2}·7^{2}·373  ⟨M_{1}, M_{2}, M_{3}, 2𝕄 ⟩  113.826557165808  1
1, 3, 7 2^{6}·3^{2}·7·2837  ⟨M_{1}, M_{2}, M_{3}, 2𝕄 ⟩  137.788942178887  1
1, 4, 6 2^{6}·3^{2}·5·5669  ⟨M_{1}, M_{2}, M_{3}, 2𝕄 ⟩  198.333924314685  3/2
1, 2, 9 2^{7}·3^{3}·7·883  ⟨M_{1}, M_{2}, M_{3}, 2𝕄 ⟩  206.791570180633  1
3, 4, 8 2^{5}·3^{3}·5·7^{2}·101  ⟨M_{1}, M_{2}, M_{3}, 2𝕄 ⟩  219.415523333470  1
4, 5, 9 2^{8}·3^{3}·5^{3}·29  ⟨M_{1}, M_{2}, M_{3}, 2𝕄 ⟩  278.222780993318  1
2, 4, 10 2^{11}·3^{3}·5·101  ⟨M_{1}, M_{2}, M_{3}, 2𝕄 ⟩  375.379161057633  1
1, 2, 8 2^{6}·3^{3}·7·29·103  ⟨M_{1}, M_{2}, M_{3}, 2𝕄 ⟩  429.330911878658  5/2
1, 4, 9 2^{7}·3^{2}·5·9281  ⟨M_{1}, M_{2}, M_{3}, 2𝕄 ⟩  554.543543287698  2
1, 3, 9 2^{6}·3^{2}·137437  ⟨M_{1}, M_{2}, M_{3}, 2𝕄 ⟩  831.548080326194  2^{2}
1, 5, 8 2^{7}·3^{3}·5·7·659  ⟨M_{1}, M_{2}, M_{3}, 2𝕄 ⟩  798.968120685204  3
1, 4, 7 2^{8}·3^{2}·7·5879  ⟨M_{1}, M_{2}, M_{3}, 2𝕄 ⟩  930.405544167007  5
1, 2, 7 2^{6}·3^{3}·5·7·1999  ⟨M_{1}, M_{2}, M_{3}, 2𝕄 ⟩  1105.646745724194  2^{3}
1, 3, 8 2^{7}·3^{3}·5^{3}·7·41  ⟨M_{1}, M_{2}, M_{3}, 2𝕄 ⟩  1395.300068143365  2^{3}
1, 4, 8 2^{5}·3^{3}·7·21773  ⟨M_{1}, M_{2}, M_{3}, 2𝕄 ⟩  1402.036187242234  2·3
2, 4, 8 2^{11}·3^{3}·4093  ⟨M_{1}, M_{2}, M_{3}, 2𝕄 ⟩  3341.033608754311  2^{2}·3

Table 8. Curves y^{2} + (2x^{4}+(v_{1}x+1)(v_{2}x+1)(v_{3}x+1))y + x^{8} = 0 (genus 3)

v  Conductor  Λ  L^{∗}(0)  L^{∗}(0)/R(Λ)
1, 2, 3  2^{6}·3·5^{3}·11  ⟨M_{1}, M_{2}, M_{3}, 2𝕄 ⟩  2.288179184223  1/2^{3}
1, 2, 4  2^{6}·3·5^{3}·23  ⟨M_{1}, M_{2}, M_{3}, 2𝕄 ⟩  5.368903572985  1/2^{3}
1, 2, 5  2^{7}·3·5·433  ⟨M_{1}, M_{2}, M_{3}, 2𝕄 ⟩  8.855164567689  1/2^{3}
2, 4, 6  2^{12}·3·5^{3}  ⟨M_{1}, M_{2}, M_{3}, 2𝕄 ⟩  19.749659522297  1/2^{3}
3, 4, 8 2^{5}·3·5·7·19·31 ⟨M_{1}, M_{2}, M_{3}, 2𝕄 ⟩  21.559124747863  1/2^{4}
1, 4, 5 2^{10}·3·5·7·23  ⟨M_{1}, M_{2}, M_{3}, 2𝕄 ⟩  22.811436896592  1/2^{2}
1, 3, 5 2^{6}·3·5·3797  ⟨M_{1}, M_{2}, M_{3}, 2𝕄 ⟩  42.967686126549  1/2
2, 3, 7 2^{7}·3·5·7·397  ⟨M_{1}, M_{2}, M_{3}, 2𝕄 ⟩  53.453545184856  1/2^{2}
1, 2, 10 2^{6}·3·5·11·523 ⟨M_{1}, M_{2}, M_{3}, 2𝕄 ⟩  57.955617125753  1/2^{2}
2, 3, 6 2^{6}·3^{2}·5^{3}·7·17 ⟨M_{1}, M_{2}, M_{3}, 2𝕄 ⟩  114.624730232758  3/2^{2}
1, 2, 8 2^{6}·3·5^{3}·7·67  ⟨M_{1}, M_{2}, M_{3}, 2𝕄 ⟩  122.595870053997  3/2^{2}
1, 2, 6 2^{6}·3·5·14731  ⟨M_{1}, M_{2}, M_{3}, 2𝕄 ⟩  125.574116230804  5/2^{2}
1, 5, 6 2^{7}·3·5^{2}·41^{2}  ⟨M_{1}, M_{2}, M_{3}, 2𝕄 ⟩  140.322800930811  1
1, 2, 9 2^{6}·3·5^{3}·7·113 ⟨M_{1}, M_{2}, M_{3}, 2𝕄 ⟩  196.848432921070  1
2, 3, 5 2^{6}·3^{3}·5·13^{3}  ⟨M_{1}, M_{2}, M_{3}, 2𝕄 ⟩  190.825343100486  2
3, 4, 9 2^{7}·3^{2}·5·3847  ⟨M_{1}, M_{2}, M_{3}, 2𝕄 ⟩  227.021533461314  1
3, 4, 7 2^{10}·3^{3}·5^{3}·7  ⟨M_{1}, M_{2}, M_{3}, 2𝕄 ⟩  242.380339993035  1
2, 3, 8 2^{6}·3^{3}·5·29·103 ⟨M_{1}, M_{2}, M_{3}, 2𝕄 ⟩  279.394212255531  1
2, 3, 9 2^{6}·3^{2}·5^{3}·7·61 ⟨M_{1}, M_{2}, M_{3}, 2𝕄 ⟩  357.850908392946  2
2, 4, 8 2^{11}·3·5^{3}·53  ⟨M_{1}, M_{2}, M_{3}, 2𝕄 ⟩  551.668430135999  2
3, 5, 6 2^{6}·3^{2}·5·21737 ⟨M_{1}, M_{2}, M_{3}, 2𝕄 ⟩  756.149101085614  2·3
1, 4, 7 2^{7}·3^{2}·5^{3}·7·109 ⟨M_{1}, M_{2}, M_{3}, 2𝕄 ⟩ 1246.076904950150  2·3
2, 6, 8 2^{15}·3^{3}·5^{3} ⟨M_{1}, M_{2}, M_{3}, 2𝕄 ⟩ 1351.447382022975 2^{2}
1, 4, 6 2^{6}·3^{3}·5·18679 ⟨M_{1}, M_{2}, M_{3}, 2𝕄 ⟩ 1854.823840399464  2^{2}·3
1, 2, 7 2^{6}·3·5·7·47·569 ⟨M_{1}, M_{2}, M_{3}, 2𝕄 ⟩ 1970.441147103680  3·5
1, 3, 9 2^{6}·3^{2}·5^{3}·4273 ⟨M_{1}, M_{2}, M_{3}, 2𝕄 ⟩ 3214.588641563687  2·7

Table 9. Curves y^{2} + (2x^{4}−(v_{1}x+1)(v_{2}x+1)(v_{3}x+1))y + x^{8} = 0 (genus 3)

v  Conductor  Λ  L^{∗}(0)  L^{∗}(0)/R(Λ)
1, 2, 3, 4 2^{9}·3^{2}·17·113 ⟨M_{1}, M_{2}, M_{3}, M_{4}, 2𝕄 ⟩  2.497987723694  1/2^{5}·5
2, 3, 4, 6 2^{8}·3^{2}·80251 ⟨M_{1}, M_{2}, M_{3}, M_{4}, 2𝕄 ⟩ 45.006091920102  1/2^{5}
1, 2, 3, 5 2^{9}·3^{2}·5·10007 ⟨M_{1}, M_{2}, M_{3}, M_{4}, 2𝕄 ⟩ 68.192003860287  1/2^{3}

Table 10. Curves y^{2} + (2x^{5}+(v_{1}x+1) ⋯ (v_{4}x+1))y + x^{10} = 0 (genus 4)

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Department of Mathematical Sciences, University of Durham, Science Laboratories, South Road, Durham DH1 3LE, United Kingdom

Department of Mathematical Sciences, University of Durham, Science Laboratories, South Road, Durham DH1 3LE, United Kingdom

Max-Planck-Institut für Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany and Collège de France, 11 place Marcelin Berthelot, F-75005 Paris, France