Case (5)

In this case there is a cusp and a node in the reduction ofC. If we explicitly construct the minimal proper regular model, we find that in the notation of Namikawa and Ueno it is of the form [I_m1− K −l], where K is the Kodaira type of the elliptic curve E given by (3.26) and we have l≥0 andm₁ >0.

Note that the Kodaira type I_m1 is the reduction type of the elliptic curve E^′ with equation (3.27).

If char(k_v)6= 2, then we can find, using Hensel’s Lemma, a factorisation F(X, Z) =F₁(X, Z)G(X, Z)F₂(X, Z),

where

F˜₁(X, Z) =Z², F˜₂(X, Z) =X³, G(X, Z˜ ) =X−Z,

and the valuation of the discriminant of G as well as the valuations of the resultants between the different forms vanish. In this case

v(disc(F₁)) =m₁, v(disc(F₂)) =m₂>0,

so that we have v(∆) = m₁+m₂. Note that m₁ is the discriminant of the given equation ofE^′ and m₂ is the discriminant of the given equation of E and this continues to hold for even residue characteristic.

bbbb b b

A

D₁

I₁ I_l−1

D₂ Dm1−2

D_m1−1

C₁ C₂

B

2 2 2

3

Figure 3.6: The special fiber of reduction type [I_m1 −IV^∗−l]

We first suppose, similar to cases (1), (2) and (3) that we have m₁ = min{v(f₆),2v(f₅)}.

In Remarks 3.44 and 3.45 it is explained in some detail how to find a transfor-mation to reduce to this case; there is no additional difficulty in the present case as soon as we have computed m₁.

For now we assume that K is not multiplicative, which always holds whenl= 0. Then we get from Proposition 1.37

Φv ∼=Z/m1Z× G, where

G ∈ {{0},Z/2Z,Z/3Z,Z/2Z⊕Z/2Z,Z/4Z}.

If m₁ = 1, then the special fiber is similar to the special fiber of type [I₀ − K −m], but now A is a genus 0 curve with a node. If m₁ > 1, then the special fiberC_v^min again resembles the special fiber of type [I₀− K −m], but now there is also anm₁-gon, with componentsA=D₀, D₁, . . . , D_m1−1. See Figure 3.6 for the case K=IV^∗.

As before, we denote the other components of multiplicity 1 in the part of the special fiber corresponding to K, if any exist, by C₁, . . . , C_t, where t+ 1 ∈ {1, . . . ,4} is the number of components of K that have simple multiplicity.

We let

χ= (χ₁, χ₂) :J(k_v) −→ Z/m₁Z× G P 7→ (i, j) denote the map induced by these numberings.

Similar to case (4) we have:

Lemma 3.70. The curveC has a model of the form (3.13) whose closureC has rational singularities if and only ifl= 0. If l= 0, then the given model already satisfies this condition.

Proof. See the proof of Lemma 3.59.

Thus we conclude that if l >0, then εv does not factor through Φv in general, for the same reason as in case (4). Therefore we first deal with the case l= 0, which is of course the most common case in practice. This ensures thatε_v andµ_v factor through Φ_v. By the above discussion we know that for any point P ∈J(k_v) there is a multiple nP mapping to [D_i −A], that is χ(nP) = (i,0), where 1≤n≤4 and 0≤i≤m₁−1.

It suffices to find a formula forµ_v(nP), since then we can compute ˆλ_v(P) from ˆλ_v(nP) using Proposition 3.23. We find such a formula upon noticing that points satisfying χ(P) = (i,0) where i6= 0 (the other case is trivial), are characterised by

v(x1)>0, v(x2)≥0, v(x3) = 0, v(x4)>0.

As in (3.24) we define

w(P) := min{v(x1), v(x₄), m₁/2}

and the following lemma is proved similarly to Lemma 3.46. In fact, most of the difficulties in that proof disappear, so the proof is even easier and is omitted.

Lemma 3.71. If χ₂(P) = 0, then we have

ε_v(P) = 2 min{χ1(P), m₁−χ₁(P)}.

The proof of the next lemma is also a simpler version of the proof of Lemma 3.47.

Lemma 3.72. Suppose χ₂(P) = 0. Then we have εv(P) = 2w(P), so in particular

w(P) = min{χ1(P), m₁−χ₁(P)}.

The last two results can be combined into a proof of the following propo-sition, which is the same as the proof of Proposition 3.49.

Proposition 3.73. If χ₂(P) = 0, then we have the formula µ_v(P) = w(P)(m₁−w(P))

m₁ .

As in case (4), the situationl >0 is considerably more complicated. We prove a formula for points satisfyingχ(P) = (i,0), where 0≤i≤m₁−1. If K is not multiplicative, then we can find some n≤4 such that nP satisfies this for allP ∈J(k_v); otherwise,n might be larger. For a point P ∈J(k_v) we define

w₁(P) := min{v(x₁), v(x₄), m₁/2}

and

w₂(P) := min{v(x₃), v(x₄),2l}.

Theorem 3.74. Suppose C has reduction type [I_m1 − K −l], where K is a Kodaira type, m₁, l ≥ 0. Suppose that v(f₀) ≥ 6l, v(f₁) ≥ 4l, v(f₂) ≥2l, v(f₆) 6= 2v(f5) and that applying Tate’s algorithm to E yields no transfor-mations except for(ξ, η) 7→ (π^−2lξ, π^−3lη). Let x = (x₁, x₂, x₃, x₄) be a set of v-integral Kummer coordinates for P ∈J(k_v) satisfying χ₂(P) = 0 such that v(x_j) = 0 for some j. Then we have

µ_v(P) = w₁(P)(m₁−w₁(P))

m₁ +w₂(P).

Proof. See Appendix A.9.

Using the preceding theorem, we can compute µv(nP) for all possible reduction types K, where either n ∈ {1, . . . ,4} or n = m₂ ≥ 1 when K is multiplicative. We might have to apply some transformations, but using the behavior of the canonical local height under changes of the model we can compute µ_v(nP) on the original model. It is not hard to find n from the valuations of the discriminant ∆ and the valuations of the coefficients ofF and H. We could also state an analog of Conjecture 3.66 in case (5) and prove it for residue characteristic 2 as in Proposition 3.67.

It is unfortunate that we haven >4 for [I_m1−Im2−l] andm₁, m₂ >4. At least we can always use a transformation to ensure thatm₂≤m₁. However, this reduction type should hardly ever occur in practice, because if it does, the valuation of ∆(C) must be at least 22.

Remark 3.75. Letk be a one-dimensional function field that is not a global field and let v ∈ M_k. Then the formulas for cases (1) to (5) show that in cases (1), (2) and (3), that is in the semistable cases, the groupU_v of points whichεv vanishes on has finite index inJ(kv), see Remark 3.10. In cases (4) and (5) this may not be true and we have not dared to extend Conjecture 3.66 to Jacobians defined over such fields. However, our methods for the computation of µ_v work perfectly well in such cases, in contrast with the original method due to Flynn and Smart which is not guaranteed to work even theoretically.

Remark 3.76. In case (5) we can only use Corollary 3.32 to bound the height constant β_v in special situations, even if we assume l = 0. For instance, if

m₁ = 1, we can proceed as in Remark 3.60. However, the results of this section are most useful for boundingβ_v when we havec_v(E) = 1, see Remark 3.69. Whenever this is satisfied (so for instance whenK ∈ {I0, II, II^∗}) we can conclude

m₁ even ⇒ β_v ≤ m₁ 4 + 2l, m₁ odd ⇒ β_v ≤ m²₁−1

4m1

+ 2l.

3.7 Archimedean places

Recall from Section 2.4 that in the case of elliptic curves there are essen-tially three methods available for the computation of archimedean canonical local heights, namely Tate’s series, theta (more precisely, Weierstrass σ-) functions and the isogeny trick due to Bost and Mestre. We exhibit analogs of all three of these methods in the case of Jacobian surfaces. Surprisingly, the analog of Tate’s series turns out to be the most efficient.

The setup is a smooth projective curveCof genus 2, given by an equation of the form (3.13) with Jacobian J and Kummer surface K, defined over a number fieldk. We want to compute the canonical local height ˆλ_v(x) for an archimedean place v and a set of Kummer coordinates xon K.