Computing archimedean intersection multiplicities

is [59, Chapter 13]. Suppose that we have embedded C into P^N_C for some N usingv and let C(C) denote the associated Riemann surface. According to Section 5.2 we need to find an almost-Green’s function with respect to a divisor D ∈ Div⁰(C)(C). Notice that we can write any such divisor in the formD =D₁−D₂, where D₁ and D₂ arenon-special, that is they are effective of degree g and their L-spaces have dimension 1. By additivity of Green’s functions it suffices to determine almost-Green’s functions with respect to non-special divisors and any fixed normalized volume form on C(C).

In order to do this it turns out to be useful to work on the analytic JacobianJ. Recall the notation introduced in Section 1.6. We viewJ as an abelian variety over the complex numbers embedded using v. Let τ_v ∈ h_g such thatJ(C) is isomorphic toC^g/Λ_v, where Λ_v =Z^g⊕τ_vZ^g. Let the map j be defined by

j:C^{g ////}C^g/Λ_v ^{∼= //}J(C).

Moreover, we fix an Abel-Jacobi map, that is an embeddingι, defined over C, of the curve into its Jacobian and let Θ ∈ Div(J) denote the theta-divisor with respect toι. LetS : Div(C)−→J denote the summation map associated to ι.

On J(C) we can find the following canonical 2-form: Let η1, . . . , ηg be an orthonormal basis of the differentials of first kind on the Jacobian. Then the canonical 2-form is given by

1

2g(η₁∧η¯₁+. . .+η_g∧η¯_g)

and we define thecanonical volume form dµ on C(C) by pulling this form back using ι, see [59, §13.2]. The details are not important for us as the dependence on dµ disappears because we only want to compute almost-Green’s functions with respect to divisors of degree zero.

For the next theorem, conjectured by Arakelov and proved by Hriljac, recall the definition of N´eron functions from Section 1.3. We use the notation E_P to denote the translation of a divisorE ∈Div(J) by a pointP ∈J.

Theorem 5.20. (Hriljac) Let D ∈ Div^g(C) be non-special, let P = S(D) and D^′ = ([−1]^∗(Θ))P. Let λ_D^′_,v be a N´eron function with respect to D^′ andv. Then λ_D^′◦ιis an almost-Green’s function with respect to Danddµ, where dµ is the canonical volume form on Cv(C)

Proof. See [59, Chapter 13, Theorem 5.2].

Remark 5.21. Additivity of Green’s functions and Theorem 5.20 can be com-bined to give a proof of the existence of Green’s functions for anyD∈Div(C) with respect to dµ, and hence, using [60, Proposition 1.3] with respect to any normalized volume form. See [59, Chapter 13, Theorem 5.1].

The great news is that we already know how to find N´eron functions with respect to Θ in the case of an archimedean place; we show below that this suffices for our purposes. Recall Proposition 4.15, stating that the function

λ^′_Θ,v(P) =−log|θa,b(z(P))|v+πIm(z(P))^T(Im(τ_v))⁻¹Im(z(P)) is a N´eron function associated with Θ and v, where a= (1/2, . . . ,1/2), b = (g/2,(g −1)/2, . . . ,1,1/2) ∈ C^g and θ_a,b denotes the theta function with characteristic [a;b] defined in Section 1.6. Now suppose thatD=D₁−D₂,

where D₁, D₂ ∈Div(C) are non-special divisors with disjoint support, and let E₁ =Pd

i=1(P_i) and E₂ =Pd

i=1(Q_i) be two effective divisors such that supp(E_i)∩supp(D_j) =∅ fori, j∈ {1,2}.

Proof. N´eron functions are invariant under translation of the divisor up to an additive constant, see [59, Chapter 11, Theorem 2.1]. But according to [59, Chapter 5, Theorem 5.8], [−1]^∗(Θ) is just Θ translated byS(K), where Kis a canonical divisor. Hence the desired result follows from Theorem 5.20 and Proposition 4.15.

Remark 5.23. In [50] Holmes gives a more direct proof of Lemma 5.22 using [59, §13.6/7], which relies on the theory of differentials of third kind.

We can use the previous result to compute intersections at archimedean places. In practice we need to be able to do the following:

1) Given D∈Div⁰(C), find non-special D1, D2 such thatD=D1−D2. 2) Compute the period matrix τv.

3) Given P₁∈C(C) andτ_v, determinez∈C^g such that j(z) =ι(P₁).

4) Given τ_v and z∈C^g, compute θ_a,b(z) =θ_a,b(z, τ_v).

The first step is not difficult, because we can, if need be, compute mul-tiples of our divisor and use the bilinearity of the local N´eron symbol. For hyperelliptic curves, steps 2), 3) and 4) are all implemented in Magma by van Wamelen, as already mentioned in Section 3.7.2. In the general case all of the relevant algorithms have been developed (see [29], [7] and [30]) by Deconinck et al. Both approaches are essentially numerical in nature. In contrast to the non-archimedean case the running times of steps 2), 3) and 4) do not crucially depend on the heights of the points in the supports of the respective divisors, since we work with the complex uniformisation. But the amount of work required to find the image of a point P₁ ∈ C on the Riemann surfaceC(C) does depend on this height.

For computational purposes, we want to stress that only D₁ andD₂ are required to be of degree g; E₁ and E₂ can be effective of lower degree. In

many situations the divisor E which we start with is given in such a form, for instance E₁ = (P₁) and E₂ = (Q₁), where P₁, Q₁ ∈ C. Moreover, it is actually desirable to work withE_i of low degree, because this means fewer applications of the Abel-Jacobi map ιand of theta functions are necessary, significantly reducing the running time of the entire algorithm.

Remark 5.24. Deconinck and his collaborators have implemented their algo-rithms inMaple in a package calledalgcurves. Their approach requires the curve to be given as an affine plane curve inA², but because their algorithm can deal with singular Riemann surfaces, this is not an essential restriction.

Since version 11 of Maple, this package has been part of the official Maple distributions. Unfortunately, the Maple developers have since decided to change some of the functions that algcurves uses, in the process destroying some of the package’s functionality. For instance, the implementation of the Abel-Jacobi map is now very unreliable and only occasionally returns the correct value (if indeed it returns anything at all).

This means that the package, which worked perfectly well forMaple 10, is now useless for our purposes. Deconinck [31] is currently working on a long-term project to rewrite all necessary routines in Sage[91]. Once this is completed, steps 1) – 4) should again be possible and we can compute canonical heights on non-hyperelliptic Jacobians in practice. For the mo-ment, however, this is limited to hyperelliptic Jacobians.

Examples and timings

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In this final chapter we investigate how the algorithms developed in this thesis can be applied in practice.

6.1 Jacobian surfaces

We start by discussing practical implications of Chapter 3. Let k be a number field or a one dimensional function field. IfJ is the Jacobian of a genus 2 curveC, then we can use our algorithm introduced in Chapter 3 for the computation of canonical heights onJ and we can also use our results from that chapter to obtain better bounds on the non-archimedean local height constants than those that were previously available.