Arakelov Invariants of Riemann Surfaces

Robin de Jong

Received: August 25, 2004 Revised: June 16, 2005

Communicated by Thomas Peternell

Abstract. We derive closed formulas for the Arakelov-Green function and the Faltings delta-invariant of a compact Riemann surface.

2000 Mathematics Subject Classification: 14G40, 14H42, 14H55. Keywords and Phrases: Arakelov theory, delta-invariant, Green function, Weierstrass points.

1. Introduction

The main goal of this paper is to give closed formulas for the Arakelov-Green function G and the Faltings delta-invariant δ of a compact Riemann surface. Both G and δ are of fundamental importance in the Arakelov theory of arithmetic surfaces [2] [8] and it is a central problem in this theory to relate these difficult invariants to more accessible ones. For example, in [8] Faltings gives formulas which relate G and δ for elliptic curves directly to theta functions and to the discriminant modular form. Formulas of a similar explicit nature were derived by Bost in [3] for Riemann surfaces of genus 2. As to the case of general genus, less specific but still quite explicit formulas are known due to Bost [3] (for the Arakelov-Green function) and to Bost and Gillet-Soulé [4] [10] (for the delta-invariant). We recall these results in Sections 2 and 4 below. In the present paper we express G and δ in terms of two new invariants S and T. Both S and T are initially defined as the norms of certain isomorphisms between line bundles, but eventually we find that they admit a very explicit description in terms of theta functions. They are intimately related to the divisor W of Weierstrass points. Of these new invariants, the T is certainly the easiest one. We are able to calculate it for hyperelliptic Riemann surfaces [13], where it is essentially a power of the Petersson norm of the discriminant modular form. The invariant S is less easy and involves a certain integral over the Riemann surface. We believe that the approach using S and T is very suitable for obtaining numerical results. An example at the end of this paper, where we compute δ and a special value of G for a certain hyperelliptic Riemann surface of genus 3, is meant to illustrate this belief.

We start our discussion by recalling the definitions of G and δ. From now on until the end of section 4, we fix a compact Riemann surface X. Let g be its genus, which we assume to be positive. The space of holomorphic differentials H^{0}(X, Ω_{X}^{1}) carries a natural hermitian inner product (ω, η) ↦ \frac{i}{2} ∫_{X} ω ∧ \overline{η}. We fix this inner product once and for all. Let {ω_{1}, …, ω_{g}} be an orthonormal basis with respect to this inner product. We have then a fundamental (1,1)-form μ on X given by μ = \frac{i}{2g} ∑_{k=1}^{g} ω_{k} ∧ \overline{ω}_{k}. It is verified immediately that the form μ does not depend on the choice of orthonormal basis, and hence is canonical. Using this form, one defines the Arakelov-Green function G on X × X. This function gives the local intersections “at infinity” of two divisors in Arakelov theory [2].

Theorem 1.1. (Arakelov) There exists a unique function G : X × X → ℝ_{≥0} satisfying the following properties for all P ∈ X:

(i) the function log G(P, Q) is C^{∞} for Q ≠ P; (ii) we can write log G(P, Q) = log |z_{P} (Q)| + f(Q) locally about P, where z_{P} is a local coordinate about P and where f is C^{∞} about P;

(iii) we have ∂_{Q}\overline{∂}_{Q} log G(P, Q)^{2} = 2πiμ(Q) for Q ≠ P; (iv) we have ∫_{X} log G(P, Q)μ(Q) = 0.

Theorem 1.1 is proved in [2]. We call the function G the Arakelov-Green function of X. We note that by an application of Stokes’ theorem one can prove the symmetry relation G(P, Q) = G(Q, P) for any P, Q ∈ X. Importantly, the Arakelov-Green function gives rise to certain canonical metrics on line bundles on X. First, consider line bundles of the form O_{X}(P) with P a point on X. Let s be the canonical generating section of O_{X}(P). We then define a smooth hermitian metric ∥ · ∥_{O_{X}(P)} on O_{X}(P) by putting ∥s∥_{O_{X}(P)}(Q) = G(P, Q) for any Q ∈ X. By property (iii) of the Arakelov-Green function, the curvature form of O_{X}(P) is equal to μ. Second, it is clear that the function G can be used to put a hermitian metric on the line bundle O_{X×X}(Δ_{X}), where Δ_{X} is the diagonal on X × X, by putting ∥s∥(P, Q) = G(P, Q) for the canonical generating section s of O_{X×X}(Δ_{X}). Restricting to the diagonal, we have a canonical adjunction isomorphism O_{X×X}(−Δ_{X})|_{Δ_{X}} ⟶^{∼} Ω_{X}^{1}. We define a hermitian metric ∥ · ∥_{Ar} on Ω_{X}^{1} by insisting that this adjunction isomorphism be an isometry. It is proved in [2] that this gives a smooth hermitian metric on Ω_{X}^{1}, and that its curvature form is a multiple of μ. For the rest of the paper we shall take these metrics on O_{X}(P) and Ω_{X}^{1} (as well as on tensor product combinations of them) for granted and refer to them as Arakelov metrics. Next we introduce the Faltings delta-invariant. Let H_{g} be the generalised Siegel upper half plane of complex symmetric g × g-matrices with positive definite imaginary part. Let τ ∈ H_{g} be a period matrix associated to a symplectic basis of H_{1}(X, ℤ) and consider the analytic jacobian Jac(X) = ℂ^{g}/ℤ^{g} + τℤ^{g} associated to τ. We fix τ for the rest of our discussion. On ℂ^{g} one has a theta function ϑ(z; τ) = ∑_{n∈ℤ^{g}} exp(πi^{t}nτn + 2πi^{t}nz), giving rise to an effective divisor Θ_{0} and a line bundle O(Θ_{0}) on Jac(X). Now consider on the other hand the set Pic_{g−1}(X) of divisor classes of degree g − 1 on X. It comes with a special subset Θ given by the classes of effective divisors. A fundamental theorem of Abel-Jacobi-Riemann says that there is a canonical bijection Pic_{g−1}(X) ⟶^{∼} Jac(X) mapping Θ onto Θ_{0}. As a result, we can equip Pic_{g−1}(X) with the structure of a compact complex manifold, together with a divisor Θ and a line bundle O(Θ). We fix this structure for the rest of the discussion. The function ϑ is not well-defined on Pic_{g−1}(X) or Jac(X). We can remedy this by putting ∥ϑ∥(z; τ) = (det Im τ)^{1/4} exp(−π^{t}y(Im τ)^{−1}y)|ϑ(z; τ)|, with y = Im z. One can check that ∥ϑ∥ descends to a function on Jac(X). By our identification Pic_{g−1}(X) ⟶^{∼} Jac(X) we obtain ∥ϑ∥ as a function on Pic_{g−1}(X). It can be checked that this function is independent of the choice of τ. The delta-invariant is the constant appearing in the following theorem, due to Faltings (cf. [8], p. 402).

Theorem 1.2. (Faltings) There is a constant δ = δ(X) depending only on X such that the following holds. Let {ω_{1}, …, ω_{g}} be an orthonormal basis of H^{0}(X, Ω_{X}^{1}). Let P_{1}, …, P_{g}, Q be points on X with P_{1}, …, P_{g} pairwise distinct. Then the formula

∥ϑ∥(P_{1} + ⋯ + P_{g} − Q) = exp(−δ(X)/8) · \frac{∥ det ω_{k}(P_{l})∥_{Ar}}{∏_{k<l} G(P_{k}, P_{l})} · ∏_{k=1}^{g} G(P_{k}, Q)

holds.

The definition of the delta-invariant may seem quite complicated, yet it plays an important role in Arakelov intersection theory and in the function theory of the moduli space M_{g} of Riemann surfaces of genus g. In fact, as has become clear from certain asymptotic results [14] [20], the function exp(−δ(X)) can be interpreted as a natural “distance” function on M_{g} measuring the distance to the Deligne-Mumford boundary. As to Arakelov theory, the delta-invariant plays the role of an archimedean contribution in the Noether formula for arithmetic surfaces [8] [18]. The idea that δ(X) gives a distance to the boundary is supported by this formula. The plan of this paper is as follows. In Section 2 we state a proposition and observe that it leads quickly to a formula for G. In Section 3 we prove this proposition. In Section 4 we derive our closed formula for δ. Some applications of our results to Arakelov intersection theory are given in Section 5. We conclude with a numerical example in Section 6.

2. The Arakelov-Green function

As was mentioned in the Introduction, the Weierstrass points of X play an important role in our approach to G and δ. The idea of considering Weierstrass points in the context of Arakelov theory is not new, cf. [6] and [14] for example. We start by recalling how we obtain the divisor of Weierstrass points using a Wronskian differential on X. Let {ψ_{1}, …, ψ_{g}} be an (arbitrary) basis of H^{0}(X, Ω_{X}^{1}). Let P be a point on X and let z be a local coordinate about P. Write ψ_{k} = f_{k}(z)dz for k = 1, …, g. We have a holomorphic function

W_{z}(ψ) = det (\frac{1}{(l − 1)!} \frac{d^{l−1}f_{k}}{dz^{l−1}} )_{1≤k,l≤g}

locally about P from which we build the g(g+1)/2-fold holomorphic differential

ψ̃ = W_{z}(ψ)(dz)^{⊗g(g+1)/2}.

We call ψ̃ the Wronskian differential about P and it is readily checked that ψ̃ is independent of the choice of the local coordinate. In fact, and this is less trivial, the differential ψ̃ extends over X to give a non-zero global section of the line bundle Ω_{X}^{⊗g(g+1)/2}. A change of basis changes the Wronskian differential by a non-zero scalar factor and hence the divisor of a Wronskian differential ψ̃ on X is unique. We denote this divisor by W, the divisor of Weierstrass points. This divisor is effective and we have deg W = g^{3} − g. Writing W = ∑_{P ∈X} w(P) · P we call the integer w(P) the weight at P. This weight can be calculated using gap sequences, but we shall not need this. Now fix for the moment a Q ∈ X. We consider the map ϕ_{Q} : X → Pic_{g−1}(X) given by sending P ↦ [gP − Q]. We put a smooth hermitian metric on O(Θ) by setting ∥s∥ = ∥ϑ∥ where s is the canonical generating section of O(Θ). We shall refer to this metric as the Arakelov metric on O(Θ). It can be verified by a short calculation using Riemann’s bilinear relations that ϕ_{Q}^{∗}O(Θ) is a line bundle on X of degree g^{3} and with curvature form a multiple of μ. In fact we can say more. It is a classical result (cf. for example [9], p. 31) that ϕ_{Q}^{∗}(Θ) = W +g·Q. Hence we obtain the first statement of the next proposition.

Proposition 2.1. We have a canonical isomorphism

σ_{Q} : ϕ_{Q}^{∗}(O(Θ)) ⟶^{∼} O_{X}(W + g · Q)

of line bundles on X. When both sides are equipped with their Arakelov metrics, the isomorphism σ_{Q} has constant norm on X. This norm is independent of the choice of Q.

The proposition will be proven in the next section. Meanwhile, we observe that it leads quite quickly to a closed formula for G.

Definition 2.2. We define S(X) to be the norm of σ_{Q} for any Q ∈ X.

In more concrete terms we have the following formula.

Corollary 2.3. For any P, Q on X we have

G(P, Q)^{g} · ∏_{W ∈W} G(P, W) = S(X) · ∥ϑ∥(gP − Q) ,

where the Weierstrass points are counted with their weights.

It follows from this corollary that the function ∏_{W ∈W} ∥ϑ∥(gP − W) does not vanish if P is not a Weierstrass point. Hence the following formula makes sense.

Theorem 2.4. For any P, Q on X with P not a Weierstrass point we have

G(P, Q)^{g} = S(X)^{1/g^{2}} · \frac{∥ϑ∥(gP − Q)}{∏_{W ∈W} ∥ϑ∥(gP − W)^{1/g^{3}}} .

Here the Weierstrass points are counted with their weights.

Proof. This follows by applying the formula from Corollary 2.3 two times. First, take the (weighted) product over Q running through W. This gives

∏_{W ∈W} G(P, W)^{g^{3}} = S(X)^{g^{3}−g} · ∏_{W ∈W} ∥ϑ∥(gP − W) .

Plug this in again in the formula from Corollary 2.3. This gives

G(P, Q)^{g} · S(X)^{\frac{g^{3}−g}{g^{3}}} · ∏_{W ∈W} ∥ϑ∥(gP − W)^{1/g^{3}} = S(X) · ∥ϑ∥(gP − Q) ,

and a little rewriting gives the result. □

Taking logarithms in Corollary 2.3 and then integrating against μ with respect to the variable P immediately gives the following explicit formula for S(X).

Theorem 2.5. For any fixed Q, the function log ∥ϑ∥(gP − Q) is integrable against μ, and the formula

log S(X) = − ∫_{X} log ∥ϑ∥(gP − Q) · μ(P)

holds.

The invariant S(X) is readily calculated in the case g = 1.

Example 2.6. Suppose that X = ℂ/ℤ + τℤ with Im τ > 0 is an elliptic curve. The form μ is given by μ = \frac{i}{2}(dz ∧ d\overline{z})/Im τ and we have

log S(X) = − ∫_{X} log ∥ϑ∥ · μ .

A calculation (see [15], p. 45 or for a different approach [14], p. 250) yields

log S(X) = − log((Im τ)^{1/4}|η(τ)|) ,

where η(τ) is the usual Dedekind eta-function η(q) = q^{1/24} ∏_{n=1}^{∞}(1 − q^{n}) in q = exp(2πiτ).

In the case that P = W is a Weierstrass point of X, the formula in Theorem 2.4 is still correct, provided that on the right hand side we take a limit for P approaching W. That this limit exists and that it indeed gives G(W, Q)^{g} follows easily from the proof of Theorem 2.4. We finish this section by discussing very shortly several other approaches to G that we know from the literature. First of all, it is quite natural to develop G in terms of the eigenvalues and eigenfunctions of a Laplacian associated to μ on X. This is the approach taken in [8], see especially Section 3 of that paper. Second, it is possible to express G in terms of abelian differentials of the second and third kind, see for example [15], Chapter II. Third, and this is perhaps most close to our approach since it also involves theta functions quite explicitly, there is an integral formula for G derived by Bost, cf. [3], Proposition 1. This interesting result reads as follows: let ν be the curvature form of O(Θ) on Pic_{g−1}(X). Then there is an invariant A(X) of X such that for every P, Q on X the formula

log G(P, Q) = \frac{1}{g!} ∫_{Θ+P −Q} log ∥ϑ∥ · ν^{g−1} + A(X)

holds. It would be interesting to have results that relate A(X) and S(X) to each other in a natural, conceptual way.

3. Proof of Proposition 2.1

Proposition 2.1 follows directly from Lemmas 3.1 and 3.2 below. We will be dealing, among other things, with the line bundle ∧^{g}H^{0}(X, Ω_{X}^{1}) ⊗_{ℂ} O_{X} on X. We equip this line bundle with the constant metric deriving from the hermitian inner product (ω, η) ↦ \frac{i}{2} ∫_{X} ω ∧ \overline{η} on H^{0}(X, Ω_{X}^{1}) that we introduced in Section 1. From now on, this metric will be taken for granted and we shall also refer to it as an Arakelov metric.

Lemma 3.1. There is a canonical isomorphism of line bundles

ρ : Ω_{X}^{⊗g(g+1)/2} ⊗ (∧^{g}H^{0}(X, Ω_{X}^{1}) ⊗_{ℂ} O_{X})^{∨} ⟶^{∼} O_{X}(W)

on X. When both sides are equipped with their Arakelov metrics, the norm of this isomorphism is constant on X.

Proof. The Wronskian differential ψ̃ formed on an arbitrary basis {ψ_{1}, …, ψ_{g}} of H^{0}(X, Ω_{X}^{1}) leads to a morphism of line bundles

∧^{g}H^{0}(X, Ω_{X}^{1}) ⊗_{ℂ} O_{X} ⟶ Ω_{X}^{g(g+1)/2}

by setting

ξ_{1} ∧ … ∧ ξ_{g} ↦ \frac{ξ_{1} ∧ … ∧ ξ_{g}}{ψ_{1} ∧ … ∧ ψ_{g}} · ψ̃ .

This gives a canonical section in Ω_{X}^{⊗g(g+1)/2} ⊗ (∧^{g}H^{0}(X, Ω_{X}^{1}) ⊗_{ℂ} O_{X})^{∨} whose divisor is W and thus we obtain the required isomorphism. The norm is constant on X because both sides have the same curvature form, and the divisors of their canonical sections are equal. □

Lemma 3.2. Let Q be an arbitrary point of X. There is a canonical isomorphism of line bundles

ϕ_{Q}^{∗}(O(Θ)) ⟶^{∼} (Ω_{X}^{⊗g(g+1)/2} ⊗ (∧^{g}H^{0}(X, Ω_{X}^{1}) ⊗_{ℂ} O_{X})^{∨}) ⊗ O_{X}(g · Q)

on X. When both sides are equipped with their Arakelov metrics, the norm of this isomorphism is constant on X and equal to exp(δ(X)/8).

Proof. We are done once we prove that

exp(δ(X)/8) · ∥ϑ∥(gP − Q) = ∥ω̃∥_{Ar}(P) · G(P, Q)^{g}

for arbitrary P, Q on X, where ω̃ is the Wronskian differential formed out of an orthonormal basis {ω_{1}, …, ω_{g}} of H^{0}(X, Ω_{X}^{1}) and where the norm ∥ω̃∥_{Ar} of ω̃ is taken in the line bundle Ω_{X}^{⊗g(g+1)/2} equipped with its Arakelov metric. The required formula follows from the formula in Theorem 1.2, by a computation which is also performed in [14], p. 233 and which runs as follows. Let P be a point on X, and choose a local coordinate z about P. By definition of the Arakelov metric on Ω_{X}^{1} we have that lim_{Q→P} |z(Q) − z(P)|/G(Q, P) = ∥dz∥_{Ar}(P). Letting P_{1}, …, P_{g} approach P in Theorem 1.2 we obtain

lim_{P_{l}→P} \frac{∥ det ω_{k}(P_{l})∥_{Ar}}{∏_{k<l} G(P_{k}, P_{l})} = lim_{P_{l}→P} \{\frac{∥ det ω_{k}(P_{l})∥_{Ar}}{∏_{k<l} |z(P_{k}) − z(P_{l})|} · \frac{∏_{k<l} |z(P_{k}) − z(P_{l})|}{∏_{k<l} G(P_{k}, P_{l})}\}

= \{ lim_{P_{l}→P} \frac{| det ω_{k}(P_{l})|}{∏_{k<l} |z(P_{k}) − z(P_{l})|}\} · ∥dz∥_{Ar}^{g+g(g−1)/2}(P)

= |W_{z}(ω)(P)| · ∥dz∥_{Ar}^{g(g+1)/2}(P)

= ∥ω̃∥_{Ar}(P) .

The required formula is therefore just a limiting case of Theorem 1.2 where all P_{k} approach P. □

4. The Faltings delta-invariant

In the present section we will express the Faltings delta-invariant δ(X) in terms of S(X) and a second invariant T (X). The significance of our formula is that the constant T (X) is in a sense “classical” and easy to calculate numerically. To start our discussion, we observe that it follows from the previous sections that (multiples of) the divisor W of Weierstrass points appear as a divisor of a section of a line bundle in various different situations. We will take advantage of this fact and take combinations until we obtain an isomorphism of line bundles whose norm is easy to measure. First of all, recall (this is Proposition 2.1) that we have for any Q on X a canonical isomorphism

σ_{Q} : ϕ_{Q}^{∗}(O(Θ)) ⟶^{∼} O_{X}(W + g · Q) .

Taking the (weighted) tensor product over the Weierstrass points of X, we obtain a canonical isomorphism

σ_{W} : ⨂_{W ∈W} ϕ_{W}^{∗} (O(Θ)) ⟶^{∼} O_{X}(g^{3} · W) .

Second, recall that by Lemma 3.1 we have a canonical isomorphism

ρ : Ω_{X}^{⊗g(g+1)/2} ⊗ (∧^{g}H^{0}(X, Ω_{X}^{1}) ⊗_{ℂ} O_{X})^{∨} ⟶^{∼} O_{X}(W) .

Thirdly, taking a closer look at Lemma 3.2 we see that the proof in fact implies that we have on X × X a canonical isomorphism

σ : Φ^{∗}(O(Θ)) ⟶^{∼} O_{X×X}(W · X + g · Δ_{X})

where Φ : X × X → Pic_{g−1}(X) is the map sending (P, Q) ↦ [gP − Q] and where again Δ_{X} is the diagonal on X × X. Restricting σ to the diagonal, and using the adjunction isomorphism, we obtain a canonical isomorphism

σ|_{Δ} : Φ^{∗}(O(Θ))|_{Δ_{X}} ⊗ Ω_{X}^{⊗g} ⟶^{∼} O_{X}(W) .

Taking suitable combinations of σ_{W}, ρ and σ|_{Δ} we obtain

Proposition 4.1. There is a canonical isomorphism of (fractional) line bundles

\begin{aligned}τ : & (Φ^{∗}(O(Θ))|_{Δ_{X}} ⊗ Ω_{X}^{⊗g})^{⊗(g+1)} ⟶^{∼} \\ & (⨂_{W} ϕ_{W}^{∗} (O(Θ)))^{⊗(g−1)/g^{3}} ⊗ (Ω_{X}^{⊗g(g+1)/2} ⊗ (∧^{g}H^{0}(X, Ω_{X}^{1}) ⊗_{ℂ} O_{X})^{∨})^{⊗2} \end{aligned}

on X.

Our results thus far imply that τ has a constant norm for the Arakelov metrics on both sides.

Definition 4.2. We define T (X) to be the norm of τ on X.

The constant T (X) admits the following concrete description using a local coordinate.

Proposition 4.3. Let P ∈ X not a Weierstrass point and let z be a local coordinate about P. Define ∥F_{z}∥(P) as

∥F_{z}∥(P) = lim_{Q→P} \frac{∥ϑ∥(gP − Q)}{|z(P) − z(Q)|^{g}} .

This limit exists and is non-zero. Further, let {ω_{1}, …, ω_{g}} be an orthonormal basis of H^{0}(X, Ω_{X}^{1}). Then the formula

T (X) = ∥F_{z}∥(P)^{−(g+1)} · ∏_{W ∈W} ∥ϑ∥(gP − W)^{(g−1)/g^{3}} · |W_{z}(ω)(P)|^{2}

holds, where W_{z}(ω) is the determinant of the Wronskian of {ω_{1}, …, ω_{g}} with respect to z.

In particular, the evaluation of T (X) for a given X only involves the evaluation of certain classical functions at an arbitrary (non-Weierstrass) point of X.

Proof. Let F be the canonical section of Φ^{∗}(O(Θ))|_{Δ_{X}} ⊗ Ω_{X}^{⊗g} coming from the canonical section in Φ^{∗}(O(Θ)) and the canonical generating section of O_{X×X}(Δ_{X}) using the adjunction isomorphism. For its norm we have ∥F ∥ = ∥F_{z}∥ · ∥dz∥_{Ar}^{g} in the local coordinate z. We see from the isomorphism σ|_{Δ} that ∥F ∥(P) does not vanish if P is not a Weierstrass point. Next, the canonical section of ⨂_{W ∈W} ϕ_{W}^{∗} O(Θ) has norm ∏_{W ∈W} ∥ϑ∥(gP − W) at P. Finally, the canonical section of Ω_{X}^{⊗g(g+1)/2} ⊗ (∧^{g}H^{0}(X, Ω_{X}^{1}) ⊗_{ℂ} O_{X})^{∨} has norm ∥ω̃∥_{Ar} = |W_{z}(ω)| · ∥dz∥_{Ar}^{g(g+1)/2}. The proposition follows by taking the appropriate combinations of these norms. □

Considering the norms of the three isomorphisms σ_{W}, ρ and σ|_{Δ} one sees that they are directly expressible in terms of exp(δ) and S(X). Hence the same holds for the norm T (X) of τ. Viewing things a little differently, we obtain a formula for exp(δ) in terms of S(X) and T (X).

Theorem 4.4. The formula

exp(δ(X)/4) = S(X)^{−(g−1)/g^{3}} · T (X)

holds.

Proof. The norm of σ_{W} is equal to S(X)^{g^{3}−g}. The norm of ρ is equal to S(X) exp(−δ(X)/8) as becomes clear by decomposing again the isomorphism from Proposition 2.1, which has norm S(X), into the isomorphisms from Lemmas 3.1 and 3.2. Lastly, the norm of σ|_{Δ} is equal to S(X) since σ has this norm and the restriction to the diagonal using the adjunction isomorphism is an isometry. We obtain the required formula by just combining. □

We want to finish this section with a second formula for T (X), involving only first order derivatives of the theta function. It is based on a function ∥J∥ on Sym^{g}X introduced by Guàrdia in [11]. Let τ ∈ H_{g} be a period matrix associated to a symplectic basis of H_{1}(X, ℤ) and consider again the analytic jacobian Jac(X) = ℂ^{g}/ℤ^{g} + τℤ^{g}. For w_{1}, …, w_{g} ∈ ℂ^{g} we put

J(w_{1}, …, w_{g}) = det (\frac{∂ϑ}{∂z_{k}} (w_{l}))

and

∥J∥(w_{1}, …, w_{g}) = (det Im τ)^{\frac{g+2}{4}} exp(−π ∑_{k=1}^{g}^{t}y_{k}(Im τ)^{−1}y_{k}) · |J(w_{1}, …, w_{g})| .

Here in the latter formula y_{k} = Im w_{k} for k = 1, …, g. It can be checked that the function ∥J∥(w_{1}, …, w_{g}) depends only on the classes in Jac(X) of the vectors w_{k}. Now let P_{1}, …, P_{g} be a set of g points on X. We take vectors w_{1}, …, w_{g} ∈ ℂ^{g} such that for all k = 1, …, g the class [w_{k}] ∈ Jac(X) corresponds to [∑g_{{l=1 \atop l≠k}} P_{l}] ∈ Pic_{g−1}(X) under the Abel-Jacobi-Riemann correspon-

dence Pic_{g−1}(X) ↔ Jac(X). We then put ∥J∥(P_{1}, …, P_{g}) = ∥J∥(w_{1}, …, w_{g}). One may check that this does not depend on the choice of the period matrix τ. The function ∥J∥ has the following geometrical property: we have ∥J∥(P_{1}, …, P_{g}) = 0 if and only if P_{1}, …, P_{g} are linearly dependent on the image of X under the canonical map X → ℙ(H^{0}(X, Ω_{X}^{1})^{∨}). We refer to [11] for a proof of the following theorem.

Theorem 4.5. Let P_{1}, …, P_{g}, Q be points on X with P_{1}, …, P_{g} pairwise distinct. Then the formula

∥ϑ∥(P_{1} + ⋯ + P_{g} − Q)^{g−1} = exp(δ(X)/8) · ∥J∥(P_{1}, …, P_{g}) · \frac{∏_{k=1}^{g} G(P_{k}, Q)^{g−1}}{∏_{k<l} G(P_{k}, P_{l})}

holds.

A combination of Theorems 2.4, 4.4 and 4.5 yields the following formula for T (X).

Proposition 4.6. Let P_{1}, …, P_{g}, Q be generic points on X. Then the formula

\begin{aligned}T (X) & = (\frac{∥ϑ∥(P_{1} + ⋯ + P_{g} − Q)}{∏_{k=1}^{g} ∥ϑ∥(gP_{k} − Q)^{1/g}})^{2g−2} · \\ & · (\frac{∏_{k≠l} ∥ϑ∥(gP_{k} − P_{l})^{1/g}}{∥J∥(P_{1}, …, P_{g})^{2}} ) · ∏_{W ∈W} ∏_{k=1}^{g} ∥ϑ∥(gP_{k} − W)^{(g−1)/g^{4}} \end{aligned}

holds. Again the Weierstrass points are counted with their weights.

Let us make the invariant T (X) explicit in the case that X is an elliptic curve. Writing X = ℂ/ℤ + τℤ with Im τ > 0 we obtain from either Proposition 4.3 or 4.6 that

T (X) = (Im τ)^{−3/2} exp(πIm τ/2) · |\frac{∂ϑ}{∂z}(\frac{1 + τ}{2} ; τ)|^{−2} .

By Jacobi’s derivative formula (cf. [19], Chapter I, §13) we can rewrite this as

T (X) = (2π)^{−2} · ((Im τ)^{6}|Δ(τ)|)^{−1/4}

where Δ is the discriminant modular form Δ(q) = η(q)^{24} = q ∏_{n=1}^{∞}(1 − q^{n})^{24} in q = exp(2πiτ). Using Theorem 4.4 we obtain

δ(X) = − log((Im τ)^{6}|Δ(τ)|) − 8 log(2π)

which is well-known, see [8], p. 417. In [13] we obtain a generalisation of the above formula for T (X) to the case where X is a hyperelliptic Riemann surface of genus g ≥ 2. The result is expressed in terms of the discriminant modular form φ_{g} on the generalised Siegel upper half plane H_{g} as defined in [17], Section 3. This is a modular form on Γ_{g}(2) = \{γ ∈ Sp(2g, ℤ) : γ ≡ I_{2g} mod 2\} of weight 4r, where r = \binom{2g+1}{g+1}, generalising the usual discriminant modular form Δ in genus 1.

Theorem 4.7. Let X be a hyperelliptic Riemann surface of genus g ≥ 2. Choose an ordering of the Weierstrass points on X and construct a canonical symplectic basis of H_{1}(X, ℤ) starting with this ordering (cf. [19], Chapter IIIa, §5). Let τ ∈ H_{g} be a period matrix of X associated to this canonical basis and put Δ_{g}(τ) = 2^{−(4g+4)n} · φ_{g}(τ) where n = \binom{2g}{g+1}. Then Δ_{g}(τ) is non-zero and the formula

T (X) = (2π)^{−2g} · ((Im τ)^{2r}|Δ_{g}(τ)|)^{− \frac{3g−1}{8ng}}

holds.

It is an intriguing question whether the invariant T (X) admits of a simple description in terms of modular forms for a general Riemann surface X of genus g. We finish this section by remarking that a closed formula of a quite different type can be given for δ using work of Bost [4] and Gillet-Soulé [10]. The point of view leading to this formula is the Riemannian manifold structure on X deriving from μ. Let ds^{2} be the metric on X given in conformal coordinates by ds^{2} = 2h_{z\overline{z}}dzd\overline{z} with h_{z\overline{z}} = ∥dz∥_{Ar}^{−2}. Let det′ Δ_{h} be the zeta regularised determinant of the Laplace operator with respect to this metric, and let vol(X, h) be the volume of X. Then the formula

δ(X) = c(g) − 6 log \frac{det Δ′_{h}}{vol(X, h)}

holds, where c(g) is a constant depending only on g. It would be interesting to know whether the terms occurring in this formula can be naturally related to the constants S(X) and T (X) which are the subject of this paper.

5. Applications to intersection theory

In this section we discuss several applications of our results to Arakelov intersection theory. Let p : X → B be an arithmetic surface over the spectrum B of the ring of integers of a number field K. For us this means that X is a regular scheme and that p is a proper and flat relative curve whose generic fiber is smooth and geometrically connected. We denote this generic fiber by X_{K}. We assume that the reader is familiar with the basic notions and statements in the Arakelov intersection theory on X, as explained in [2] or [8]. We let g be the genus of X_{K}, and assume that it is positive. We fix a Kbasis {ψ_{1}, …, ψ_{g}} of regular differential 1-forms on X_{K}. Looking back at the discussion at the beginning of Section 2, which was purely algebraic, we note that a non-zero Wronskian differential ψ̃ can be formed out of this basis. Its divisor div ψ̃ is an effective K-divisor on X_{K} and we have, completely analogous to Lemma 3.1, a canonical isomorphism

Ω_{X_{K}}^{⊗g(g+1)/2} ⊗_{O_{X_{K}}} (∧^{g}H^{0}(X_{K}, Ω_{X_{K}}^{1} ) ⊗_{K} O_{X_{K}} )^{∨} ⟶^{∼} O_{X_{K}} (div ψ̃)

of invertible sheaves on X_{K}. We denote by W the Zariski closure of div ψ̃ in X. Let ω_{X /B} be the relative dualising sheaf of p : X → B.

Lemma 5.1. The above isomorphism extends to a canonical isomorphism

ρ : ω_{X /B}^{⊗g(g+1)/2} ⊗_{O_{X}} (p^{∗}(det p_{∗}ω_{X /B}))^{∨} ⟶^{∼} O_{X} (V + W)

of invertible sheaves on X, for some effective divisor V whose support is entirely contained in the fibers of p.

Proof. The idea for the proof is taken from [1], p. 1298, where an analogous result is proven for the function field case. We recall that ψ̃ is given in a local coordinate z by W_{z}(ψ)(dz)^{⊗g(g+1)/2} where

W_{z}(ψ) = det (\frac{1}{(l − 1)!} \frac{d^{l−1}f_{k}}{dz^{l−1}} )_{1≤k,l≤g}

if the ψ_{k} are locally written as ψ_{k} = f_{k}(z)dz for k = 1, …, g. On X_{K} this gives rise to a morphism of invertible sheaves

∧^{g}H^{0}(X_{K}, Ω_{X_{K}}^{1} ) ⊗_{K} O_{X_{K}} ⟶ Ω_{X_{K}}^{⊗g(g+1)/2}

by setting

ξ_{1} ∧ … ∧ ξ_{g} ↦ \frac{ξ_{1} ∧ … ∧ ξ_{g}}{ψ_{1} ∧ … ∧ ψ_{g}} · ψ̃

(cf. the proof of Lemma 3.1). Now note that the construction of ψ̃ is valid for smooth proper curves over any base scheme. As a result, by modifying the basis {ψ_{1}, …, ψ_{g}} if necessary, the above morphism extends canonically at least over the open dense subscheme of X where p is smooth. Automatically it extends then further to give a canonical morphism p^{∗}(det p_{∗}ω_{X /B}) → ω_{X /B}^{⊗g(g+1)/2} on the whole of X. Multiplying by (p^{∗}(det p_{∗}ω_{X /B}))^{∨} we obtain from this a morphism

O_{X} ⟶ ω_{X /B}^{⊗g(g+1)/2} ⊗_{O_{X}} (p^{∗}(det p_{∗}ω_{X /B}))^{∨} .

The image of 1 is a section whose divisor is a divisor V +W where V is effective and has support entirely contained in the fibers of p. This gives the lemma. □

The divisor V is an invariant of the arithmetic surface p : X → B and we shall use it without further mention in the sequel.

Example 5.2. In the case that g = 1, the morphism p^{∗}p_{∗}ω_{X /B} → ω_{X /B} in the above proof is just the natural morphism, as is readily checked. According to [16], Corollary 3.27, if p : X → B is a minimal arithmetic surface, then the natural morphism p^{∗}p_{∗}ω_{X /B} → ω_{X /B} is in fact an isomorphism. Hence we find V = ∅ in this case.

We want to translate the isomorphism ρ of Lemma 5.1 into an equality of Arakelov divisors on X. For this we need a notation for the norm of ρ at the various complex embeddings of K.

Definition 5.3. Let X be a compact Riemann surface of positive genus. We denote by R(X) the norm of the isomorphism ρ from Lemma 3.1.

It follows from our discussion so far that R(X) = S(X) · exp(−δ(X)/8). Now let’s turn back to our arithmetic surface p : X → B. We recall from [2] [8] that both sides of the isomorphism ρ from Lemma 5.1 come equipped with a canonical structure of metrised invertible sheaf, and that to each non-zero rational section of such a sheaf we can associate its Arakelov divisor. For each complex embedding σ of K we denote by X_{σ} the compact Riemann surface (X_{K} ⊗_{K,σ} ℂ)(ℂ) obtained from base changing X_{K} to ℂ along σ. We denote by F_{σ} the corresponding Arakelov fiber. The next proposition follows easily from Lemma 5.1 and from the fact that ρ has constant norm R(X_{σ}) on X_{σ}.

Proposition 5.4. We have an equality

\frac{1}{2}g(g + 1)ω_{X /B} = V + W + ∑_{σ} log R(X_{σ}) · F_{σ} + p^{∗}(det p_{∗}ω_{X /B})

of Arakelov divisors on X. Here the sum runs over the complex embeddings of K.

This proposition can be used to deduce some interesting formulas involving Arakelov intersection numbers.

Definition 5.5. We define a function R on the set of closed fibers of p : X → B as follows. Let s be a closed point of B. If g = 1, we put R(X_{s}) = 0. If g ≥ 2, then we define R(X_{s}) by the equality (2g−2)·log R(X_{s}) = (V_{s}, ω_{X /B})·log #k(s), where (V_{s}, ω_{X /B}) is the usual intersection number of the divisors V and ω_{X /B} above s, and where k(s) is the residue field at s.

As the next proposition implies, the function R can be seen as an analogue of the previously defined R for compact Riemann surfaces. The quantity d̂eg det p_{∗}ω_{X /B} is the usual Arakelov degree of the metrised invertible sheaf det p_{∗}ω_{X /B} on B (i.e., [K : ℚ] times the Faltings height of p : X → B).

Proposition 5.6. Assume that p : X → B is a semi-stable arithmetic surface. Then for the self-intersection of the relative dualising sheaf we have a lower bound

(ω, ω) ≥ \frac{8(g − 1)}{(2g − 1)(g + 1)} ·(∑_{s} log R(X_{s}) + ∑_{σ} log R(X_{σ}) + d̂eg det p_{∗}ω_{X /B}) .

Here the first sum runs over the closed points s ∈ B, and the second sum runs over the complex embeddings of K.

Proof. In the case g = 1, the lower bound is trivially satisfied since we have (ω_{X /B}, ω_{X /B}) = 0 in this case by [8], Theorem 7. So assume that g ≥ 2. We take the equality from Proposition 5.1 and intersect the divisors on both sides with ω_{X /B}. This gives that \frac{1}{2}g(g + 1)(ω_{X /B}, ω_{X /B}) can be written as

(W, ω_{X /B}) + (2g − 2) · (∑_{s} log R(X_{s}) + ∑_{σ} log R(X_{σ}) + d̂eg det p_{∗}ω_{X /B}) .

For the term (W, ω_{X /B}) we have by [8], Theorem 5 the lower bound

(W, ω_{X /B}) ≥ \frac{g^{3} − g}{2g(2g − 2)}(ω_{X /B}, ω_{X /B}) = \frac{1}{4}(g + 1)(ω_{X /B}, ω_{X /B})

since the generic degree of W is g^{3} − g. Using this in the first equality gives the required lower bound. □

We remark that for a semi-stable arithmetic surface p : X → B the numbers log R(X_{s}) are always non-negative. Lower bounds of a similar type for (ω_{X /B}, ω_{X /B}) can be found in [6]. The problem with the above proposition is that the right hand side may be negative, and then the lower bound becomes useless in view of the fundamental inequality (ω_{X /B}, ω_{X /B}) ≥ 0 proved by Faltings [8]. However, for any fixed g ≥ 2 the invariant log R(X) can become arbitrarily large, as the next proposition shows.

Proposition 5.7. Let X_{t} be a holomorphic family of compact Riemann surfaces of genus g ≥ 2 over the punctured disk, degenerating as t → 0 to the union of two Riemann surfaces of positive genera g_{1}, g_{2} with two points identified. Suppose that neither of these two points was a Weierstrass point. Then the formula

log R(X_{t}) = −\frac{g_{1}g_{2}}{2g} log |t| + O(1)

holds as t → 0.

For a proof we refer to the author’s thesis [12]. The next application we have in mind is a formula for the self-intersection of a point. In order to derive this formula it is convenient to use the machinery of the determinant of cohomology det Rp_{∗}(·) and the Deligne bracket ⟨·, ·⟩, for which we refer to [7]. We will use that for any section P : B → X and any invertible sheaf L on X we have canonical isomorphisms ⟨O_{X} (P), L⟩ ⟶^{∼} P^{∗}L and ⟨P, ω_{X /B}⟩ ⟶^{∼} ⟨P, P ⟩^{⊗−1}. The latter is sometimes called the adjunction formula. Moreover, we have a Riemann-Roch theorem in the following form: for each invertible sheaf L on X there is a canonical isomorphism

(det Rp_{∗}L)^{⊗2} ⟶^{∼} ⟨L, L ⊗ ω_{X /B}^{−1}⟩ ⊗ (det p_{∗}ω_{X /B})^{⊗2}.

Lemma 5.8. Let P be a section of p, not a Weierstrass point on the generic fiber. Then we have a canonical isomorphism

υ : P^{∗}(O_{X} (V + W))^{⊗2} ⟶^{∼} (det Rp_{∗}O_{X} (gP))^{⊗−2}

of line bundles on B. When restricted to the generic fiber, the left hand side gets identified with O_{SpecK}^{⊗2} and the right hand side gets identified with H^{0}(X_{K}, O_{X_{K}} (gP))^{⊗−2}. The latter has a canonical trivialising section 1 and the isomorphism υ, when restricted to the generic fiber, sends the 1 of O_{SpecK}^{⊗2} to the 1 of H^{0}(X_{K}, O_{X_{K}} (gP))^{⊗−2}.

Proof. The Riemann-Roch theorem applied to the invertible sheaf O_{X} (gP) gives a canonical isomorphism

(det Rp_{∗}O_{X} (gP))^{⊗2} ⟶^{∼} ⟨O_{X} (gP), O_{X} (gP) ⊗ ω_{X /B}^{−1}⟩ ⊗ (det p_{∗}ω_{X /B})^{⊗2} .

By the adjunction formula, the right hand side can be canonically identified with ⟨P, P ⟩^{⊗g(g+1)} ⊗ (det p_{∗}ω_{X /B})^{⊗2}, giving a canonical isomorphism

(det Rp_{∗}O_{X} (gP))^{⊗2} ⟶^{∼} ⟨P, P ⟩^{⊗g(g+1)} ⊗ (det p_{∗}ω_{X /B})^{⊗2} .

On the other hand, pulling back the isomorphism ρ from Lemma 5.1 along P and using once more the adjunction formula we find a canonical isomorphism

⟨P, P ⟩^{⊗−g(g+1)/2} ⟶^{∼} ⟨V + W, P ⟩ ⊗ det p_{∗}ω_{X /B}

and hence

⟨P, P ⟩^{⊗g(g+1)} ⟶^{∼} (P^{∗}O_{X} (V + W))^{⊗−2} ⊗ (det p_{∗}ω_{X /B})^{⊗−2} .

The isomorphism υ follows by combining these isomorphisms. Since P is not a Weierstrass point on the generic fiber, we have that H^{0}(X_{K}, gP) is 1-dimensional and hence is generated by its canonical section 1. The last statement of the lemma follows then by carefully spelling out all the isomorphisms. □

Proposition 5.9. Let P be a section of p, not a Weierstrass point on the generic fiber. Then R^{1}p_{∗}O_{X} (gP) is a torsion module on B and the selfintersection −\frac{1}{2}g(g + 1)(P, P) is given by

− ∑_{σ} log G(P_{σ}, W_{σ}) + log #R^{1}p_{∗}O_{X} (gP) + ∑_{σ} log R(X_{σ}) + d̂eg det p_{∗}ω_{X /B} .

Here σ runs through the complex embeddings of K.

Proof. That R^{1}p_{∗}O_{X} (gP) is a torsion module on B follows since we have H^{1}(X_{K}, gP) = 0 on the generic fiber. As to the formula, we take the equality from Proposition 5.1 and intersect the divisors on both sides with P. By the Arakelov adjunction formula (ω, P) = −(P, P) we obtain

−\frac{1}{2}g(g + 1)(P, P) = (V + W, P) + ∑_{σ} log R(X_{σ}) + d̂eg det p_{∗}ω_{X /B} .

It remains therefore to see that (V + W, P)_{fin} = log #R^{1}p_{∗}O_{X} (gP). For this we invoke Lemma 5.8. It follows from the description of υ on the generic fiber that in fact υ is the natural isomorphism over the open dense subscheme of B where P does not meet V + W. Now for any closed point s of B denote by e_{s} the length at s of R^{1}p_{∗}O_{X} (gP). Then if we let D = ∑_{s} e_{s} · s, the invertible sheaf det R^{1}p_{∗}O_{X} (gP) gets identified with O_{B}(D) and, since det R^{0}p_{∗}O_{X} (gP) is trivialised by the section 1, the determinant of cohomology det Rp_{∗}O_{X} (gP) gets identified with O_{B}(−D). By Lemma 5.8, for any closed point s the length e_{s} coincides with the intersection multiplicity of P and V + W at s and consequently (V + W, P)_{fin} = ∑_{s} e_{s} log #k(s) = log #R^{1}p_{∗}O_{X} (gP). □

6. A numerical example

In this section we use the results of Sections 2 and 4 to calculate the Faltings height and the self-intersection of the relative dualising sheaf of a certain hyperelliptic curve of genus 3 defined over the rationals. We start with two theoretical results, both of which can be proved by methods similar to those used in [5], Section 3. Let K be a number field, and let O_{K} be its ring of integers. For a nonzero element a ∈ O_{K} and a prime ideal ℘ of O_{K} we denote by v_{℘}(a) the exponent of ℘ in the prime ideal decomposition of a · O_{K}. Let f ∈ O_{K}[x] be a monic polynomial of degree 5 with f(0) and f(1) units in O_{K} and put g(x) = x(x − 1) + 4f(x).

Proposition 6.1. Suppose that the discriminant Δ of g is non-zero, that we have v_{℘}(Δ) = 0 or 1 for each prime ideal ℘ of residue characteristic ≠ 2, and that g mod ℘ has a unique multiple root of multiplicity 2 for prime ideals ℘ with residue characteristic ≠ 2 and with v_{℘}(Δ) = 1. Then the equation

C_{f} : y^{2} = x(x − 1)f(x)

defines a hyperelliptic curve of genus 3 over K. It extends to a semi-stable arithmetic surface p : X → B = Spec(O_{K}). We have that X has bad reduction at ℘ if and only if ℘ has residue characteristic ≠ 2 and v_{℘}(Δ) = 1. For such ℘, the fiber at ℘ is an irreducible curve with a single double point. The differentials dx/y, xdx/y, x^{2}dx/y form a basis of the free O_{B}-module p_{∗}ω_{X /B}. The points W_{0}, W_{1} on C_{f} given by x = 0 and x = 1 extend to disjoint sections of p.

As to the Faltings height and the self-intersection of the relative dualising sheaf of C_{f} we have the following. For a complex embedding σ of K we denote by X_{f,σ} the compact Riemann surface (C_{f} ⊗_{K,σ} ℂ)(ℂ) obtained by base changing C_{f} to ℂ along σ. For each σ, we choose a symplectic basis of H_{1}(X_{f,σ}, ℤ) and form the period matrix Ω_{σ} = (Ω_{1,σ}|Ω_{2σ}) for dx/y, xdx/y and x^{2}dx/y on this basis. We further put τ_{σ} = Ω_{1σ}^{−1}Ω_{2σ}.

Proposition 6.2. The degree of det p_{∗}ω_{X /B} satisfies

d̂eg det p_{∗}ω_{X /B} = −\frac{1}{2} ∑_{σ} log (| det Ω_{1σ}|^{2}(det Im τ_{σ})) .

For the self-intersection of the relative dualising sheaf we have the formula

(ω_{X /B}, ω_{X /B}) = 24 ∑_{σ} log G_{σ}(W_{0}, W_{1}) ,

where G_{σ} denotes the Arakelov-Green function on X_{f,σ}.

We apply these propositions to a concrete example. We choose K = ℚ and f(x) = x^{5} + 6x^{4} + 4x^{3} − 6x^{2} − 5x − 1. It can be checked that g(x) = x(x − 1) + 4f(x) satisfies the conditions of Proposition 6.1. The corresponding hyperelliptic curve C_{f} has bad reduction at the primes p = 37, p = 701 and p = 14717. Let X_{f} be the compact Riemann surface obtained from base changing C_{f} to the complex numbers. We choose an ordering of the Weierstrass points of X and as in [19], Chapter IIIa, §5 this gives us a canonical way to construct a symplectic basis for H_{1}(X_{f}, ℤ). We have computed the periods with respect to this basis of the differentials dx/y, xdx/y and x^{2}dx/y. Using Proposition 6.2 we easily obtain

d̂eg det p_{∗}ω_{X /B} = −1.280295247656532068 …

which is the Faltings height of C_{f}. Next we take a look at the self-intersection of the relative dualising sheaf. According to Proposition 6.2 we need to calculate G(W_{0}, W_{1}). We apply Theorem 2.4 where we carefully take a limit for P approaching W_{0}. Using theory as developed for example in [19], Chapter IIIa it is possible to make the Abel-Jacobi-Riemann correspondence Pic_{2}(X_{f}) ↔ Jac(X_{f}) completely explicit. This makes it easy to carry out the theta function evaluations that are needed to compute G(W_{0}, W_{1}). The calculation of S(X_{f}) is, however, considerably harder. We recall that in our case S(X_{f}) is given by

log S(X_{f}) = − ∫_{X_{f}} log ∥ϑ∥(3P − Q) · μ(P)

where μ is the Arakelov metric and where Q is any point on X_{f}. We want to express μ in terms of the coordinates x, y but then it immediately becomes clear that the integrand will diverge at the Weierstrass point at infinity. However, by taking logarithms in Theorem 2.4 and integrating against μ(Q) we find the alternative formula

log S(X_{f}) = −9 ∫_{X} log ∥ϑ∥(3P − Q) · μ(Q) + \frac{1}{3} ∑_{W ∈W} log ∥ϑ∥(3P − W) ,

valid for any non-Weierstrass point P on X, in which the integrand behaves better. In fact, the integrand now only has a singularity at Q = P. Let Ω = (Ω_{1}|Ω_{2}) be the period matrix of X_{f} referred to above and put τ = Ω_{1}^{−1}Ω_{2}. Let (μ_{kl}) be the matrix given by (μ_{kl}) = (\overline{Ω}_{1}(Im τ)^{t}Ω_{1})^{−1}. An application of Riemann’s bilinear relations yields μ = \frac{i}{6} ∑ μ_{kl} ψ_{k} ∧ \overline{ψ_{l}} with ψ_{1} = dx/y, ψ_{2} = xdx/y and ψ_{3} = x^{2}dx/y. Writing x = u + iv with u, v ∈ ℝ we can rewrite this as the real 2-form

\begin{aligned}μ = & \frac{1}{3} (μ_{11} + 2μ_{12}u + 2μ_{13}(u^{2} − v^{2}) + μ_{22}(u^{2} + v^{2}) \\ & +2μ_{23}u(u^{2} + v^{2}) + μ_{33}(u^{2} + v^{2})^{2}) · \frac{dudv}{|h(u + iv)|} , \end{aligned}

where h(x) = x(x − 1)g(x). Using a computer algebra package, we have evaluated the integral. This is a slow process, because one has to take care of the logarithmic singularity. On the other hand, it is possible to check the answers by trying various choices of P. We found that within reasonable time limits we can only reach an accuracy within ± 0.005. The end result is

log S(X_{f}) = 17.57 …

Using this we find the approximation

G(W_{0}, W_{1}) = 2.33 …

and finally

(ω_{X /B}, ω_{X /B}) = 20.32 …

It is almost no extra effort to compute also the delta-invariant of X_{f}. Using Theorem 4.7 we obtain, first of all,

log T (X_{f}) = −4.44361200473681284 …

With Theorem 4.4 and our value above for log S(X_{f}) we get as a result

δ(X_{f}) = −33.40 …

The reader may check that the Noether formula [18] is verified by our numerical results.

Acknowledgments. The author wishes to thank Gerard van der Geer for his encouragement and helpful remarks.

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Robin de Jong Mathematisch Instituut Universiteit Leiden Postbus 9512 2300 RA Leiden The Netherlands rdejong@math.leidenuniv.nl 330

Documenta Mathematica 10 (2005)