Simplifying the model

We continue to consider non-archimedean v and let π =π_v denote a uni-formiser. In the case of elliptic curves it is possible to find explicit formulas forµv depending on the reduction type of the minimal proper regular model of the curve over Spec(Ov) in all cases. This relies on two observations:

1. There are essentially only ten different reduction types and they are well understood and easily distinguishable using Tate’s algorithm.

2. Each elliptic curve has a model such thatε_v andµ_v factor through Φ_v. In contrast to this, there are more than 100 different reduction types for mi-nimal proper regular models of genus 2 curves, classified in [76] and there are curves that have no model satisfying the hypotheses of Theorem 3.30, that is, having rational singularities. Therefore we must look for simplifications.

Because the canonical local height ˆλ_v behaves so nicely under isogenies, in particular under isomorphisms induced by transformations of the under-lying curve, we can simplify the computation of the canonical local height significantly as follows. The idea is to apply transformations until either ε_v(P) becomes trivial or we cannot simplify the model any further. We show that in the latter case we always end up in one of five different si-tuations and we prove simple formulas forµ_v(P) or forµ_v(nP), wheren is small – we always have n≤4 except for one rather exotic reduction type.

If necessary, we first apply a transformation to make sure that the reduc-tion ofCis reduced. This is easy, if we allow field extensions of ramification

index 2, see the proof of Proposition 3.36 below. We may do so because of part (iv) of Proposition 3.23, telling us that the canonical local height is invariant under extensions.

However, we must first discuss how one can define and compute the multiplicity of a pointP lying on the reduction of C.

Definition 3.35. Letlbe a field and letC_F,H be a reduced curve in weighted projective space with respective weights 1, 3, 1 assigned to the variables X, Y, Z that is given by an equation

Y²+H(X, Z)Y =F(X, Z),

whereF, H ∈l[X, Z] are homogeneous of degrees 6 and 3, respectively. The multiplicity δ(P, C_F,H) of P = (X₀ :Y₀ :Z₀)∈C_F,H is defined as follows:

• If P is a singular point of typeA_n, thenδ(P, C_F,H) =n+ 1.

• If P is fixed by the involution (X :Y :Z)7→(X :Y −H(X, Z) :Z), then δ(P, C_F,H) = 1.

• Otherwise δ(P, C_F,H) = 0.

We say thatP is anode of C_F,H if δ(P, C_F,H) = 2 and we callP a cusp of CF,H if δ(P, CF,H) = 3. If S(X, Z) is any binary form, then we also define δ(P, S) to benif we can write

S(X, Z) = (Z₀X−X₀Z)ⁿS^′(X, Z), whereS^′(X₀, Z₀)6= 0.

If the characteristic oflis not equal to 2, then it is easy to compute the multiplicity. Namely, we have

δ(P, C_F,H) =δ(τ(P),4F(X, Z) +H(X, Z)²), where

τ((X :Y :Z)) = (X: 2Y +H(X, Z) :Z).

In particular, ifH = 0, then the multiplicity ofP = (X0:Y0 :Z0) is simply the multiplicity of (Z₀X−X₀Z) in F.

If the residue characteristic is 2, we can use a method due to Liu to compute the multiplicity δ(P, C_F,H). So suppose we are in this situation.

If P = (X₀ :Y₀ :Z₀) ∈ C_F,H, then we see that P must be nonsingular and hence δ(P, C_F,H) ≤ 1 unless H(X₀, Z₀) = 0. So we assume that the latter holds and letG(X, Z) denote the linear form dividingH and satisfy-ing G(X₀, Z₀) = 0. For any binary form S(X, Z) we write δ(P, S) for the multiplicity ofGinS.

If we have 2δ(P, H)≤δ(P, F) or ifδ(P, F) is odd, then we get

δ(P, C_F,H) = min{2δ(P, H), δ(P, F)}. (3.18) Otherwise, let 2n=δ(P, F) and write

F(X, Z) = X

i≥2n

Di(X, Z)G(X, Z)ⁱ.

There is some U(X, Z) ∈ l[X, Z] such that G(X, Z) divides U(X, Z)² − D2n(X, Z). We change our equation using the transformation τ : Y 7→

Y +U(X, Z)G(X, Z)ⁿ. Then we have δ(τ(P), τ(C_F,H)) = δ(P, C_F,H) and δ(τ(P), τ^∗H) =δ(P, H); however, it is clear that

δ(τ(P), τ^∗F)> δ(P, F) (3.19) holds. Hence we can read off the multiplicity δ(P, C_F,H) after applying a finite number of these steps.

This method works for the computation of the multiplicity of a point lying on a reduced curve defined by an equation of the form

C_F,H :Y²+H(X, Z)Y =F(X, Z),

in weighted projective space P²_l(1, g + 1,1), where we have g ≥ 1 and H(X, Z), F(X, Z) ∈ l[X, Z] are binary forms of degrees g+ 1 and 2g+ 2, respectively, defined over a field lof characteristic 2.

We assume thatlis algebraically closed and that char(l) = 2, for the mo-ment. As in the case char(l)6= 2, the multiplicity does not change if we act onC_F,H by a transformation of the form (3.16). Recall the list of represen-tatives (i)–(xiii) for each orbit under the action induced by transformations of the form (3.16) given in the proof of Lemma 3.28.

Table 3.2 contains the following information, in parts retrieved from Ap-pendix A.3: For each representative we have listed for a pointx= (x₁:x₂: x₃ : x₄) ∈ K the condition (cond.) that must be satisfied in order for all δi(x) to vanish. Moreover, we have listed under (add.) the condition, if any, that a point x= (x₁ :x₂ : x₃ :x₄) ∈ P³ satisfying (cond.) must satisfy in order to lie onK. Finally we have listed the multiplicities (mults.) that the curve defined by (3.13) has at the points (X :Z) = (1 : 0), (X:Z) = (0 : 1) and (X : Z) = (1 : 1), in case the multiplicities there are greater than 1.

For example, for type (vii) the entry is (4,2), which means that it has mul-tiplicity 4 at (1 : 0 : 0) and mulmul-tiplicity 2 at (0 : 0 : 1).

Now let us return to our original setup of a genus 2 curve defined over k_v, where v ∈ M_k⁰. We want to show that we can always reduce to a

Type cond. add. mults.

(i) x₄= 0

(ii) x₄= 0 (6)

(iii) x₄= 0 x₁= 0 (5) (iv) x₄= 0 x₁= 0 (4) (v) x₄= 0 x₁x₃= 0 (3,2)

(vi) x₁=x₄ = 0 (3)

(vii) x₄= 0 (4,2)

(viii) x₄= 0 (2,2,2)

(ix) x₄= 0 x₁x₃= 0 (2,2)

(x) x₁=x₄ = 0 (2)

(xi) x₁=x₄ = 0 (3)

(xii) x₄= 0 (5)

(xiii) x₄= 0 x₁x₃= 0 (3,3)

Table 3.2: Conditions for the vanishing of δ(x) and multiplicities ofC small number of cases for which we can find simple formulas to compute the canonical local height and then use Corollary 3.25 to find the canonical local height of our original point. The drawback of this approach is that we might have to extend the ground field and this extension may be ramified.

However, the next proposition asserts that at least the primes dividing the ramification index are small and typically the ramification index itself is as well. It is taken from [95] where it was proved for residue characteristic not equal to 2; the proof remains the same in the general case.

Proposition 3.36. (Stoll) There is an extensionk^′/kv of ramification index not divisible by a prime p >5 such that C has a model over k^′ of the form Y²+H(X,1)Y = F(X,1) whose special fiber has no point of multiplicity greater than three and at most one point of multiplicity exactly three. Here multiplicity means multiplicity on the special fiber, introduced in Definition 3.35.

Proof. If the given model ofC is not reduced, we can transform it until we have ˜H = ˜F = 0. Now we simply scale Y by a suitable power of π; we may be required to use a ramified field extension of degree 2 in order to do this.

Hence we may assume that at least one of ˜F and ˜H does not vanish.

There are exactly six ramification points ofC over P¹_kv, namely the Weier-strass points of C. They reduce to six points, counted with multiplicity, on P¹_kv, which we view as the special fiber of a model of P¹ over Spec(Ov).

Whenever several ramification points reduce to the same point on this special fiber, then we can blow up that point. Repeated application of this yields another modelW ofP¹ over Spec(Ov) such that the ramification points of C map to distinct points on the special fiber W_v ofW.

If necessary, we can contract components of W_v in order to get a unique modelW^′ such that the last condition still holds, but we also have the pro-perty that every component of the special fiber of W^′ of self-intersection -1 contains the image of at least 2 of the ramification points. For combina-torial reasons it is always possible to pick one component such that after contracting all other components we get a model W^′′ of P¹ whose special fiber consists of a single component and such that at most three ramification points map to the same point on this special fiber, and this can happen at most once.

However, the generic fiber of W^′′ may not be defined over k_v. Hence we have to analyze the ramification index of the smallest extension k^′/k_v such thatW^′′is defined over the spectrum of the ring of integers ofk^′. This will also provide us with a more explicit description of the process outlined above.

We try to minimise the valuation of the discriminant of the model of C while allowing ramified field extensions with some restriction on the ramifi-cation index. Suppose we have a point, say at (0 : 0 : 1), on the reduction ˜C ofCof multiplicityn≥3. We can find the largestd >0, whered∈Qis not necessarily integral, such that both π^−dF(π^d/nX, Z) and π^−dH(π^d/nX, Z) are integral over the ring of integers of a suitable field extension of ramifi-cation index e, where e is the least common multiple of the denominators ofdand _n^d, written as quotients of coprime integers. Obviouslyecannot be divisible by a prime other than 2, 3 or 5. Let τ denote the transformation corresponding to this; then an application of τ corresponds to moving from one component ofW_v^′ to another.

The effect of the transformation τ on the discriminant ∆(C) is given by v(∆(τ(C))) =v(∆(C))−10d

In case n ≥ 4 the transformation τ thus reduces the valuation of the dis-criminant and hence iterating this process leads to a model whose points all have multiplicity <4 in a finite number of steps. If we have n= 3 and C˜ has another point Qof multiplicity 3, then τ keeps the valuation of the discriminant constant and does not change the multiplicity of Q, but the selected singularity may split up into several points of lower multiplicity. If it does not, we iterate the process; the selected singularity must split up after a finite number of steps.

Remark 3.37. Suppose that n ≥ 4 in the notation of the above proof. If v(2) = 0, then we can assume H = 0 and the effect of τ on the Newton polygon of F indicates that the only way we can have a root of multipli-city n after applyingτ is if the roots are very close v-adically. In general, the number of steps depends on how close the ramification points are

v-adically, since extending the field as above corresponds to “zooming in” on the ramification points.

Ifn= 3, then considering the Newton polygon of ˜F in casev(2) = 0 and H= 0 also shows why the method outlined in the proof of Proposition 3.36 only works if there are two points of multiplicity 3.

Now we distinguish between char(k_v) 6= 2 and char(kv) = 2. If the residue characteristic is not 2, then we can apply a transformation to ensure H = 0. Proposition 3.36 says that over a suitable extension of k_v the reduction of F has no root of multiplicity greater than 3 and at most one root of exact multiplicity 3. This means that we can assume, using Table 3.1, that, possibly after making an unramified field extension, the reduction of F belongs to one of the five cases given below. We leave out the case of square-free reduction, because thenε_v is trivial. Let x = (x₁, x₂, x₃, x₄) denote a set of integral Kummer coordinates for P such that one of the entries is a unit and let ˜x= (˜x₁,x˜₂,x˜₃,x˜₄) be its reduction.

(1) ˜F = X(X −Z)(X−aZ)(X −bZ)Z², a 6= b, a, b 6= 0,1, x˜₁ =

˜ x₄= 0.

(2) ˜F =X²(X−Z)(X−aZ)Z², a6= 0,1, x˜₁x˜₃ = ˜x₄= 0.

(3) ˜F =X²(X−Z)²Z², x˜₄ = 0.

(4) ˜F =X³(X−Z)(X−aZ)Z, a6= 0,1, x˜₃= ˜x₄ = 0.

(5) ˜F =X³(X−Z)Z², x˜₁x˜₃ = ˜x₄ = 0.

In case the residue characteristic is 2, we look at Table 3.2 and find that we can always reduce to one of the following situations, possibly after an unramified extension of the base field.

(1) ˜H = X²Z +XZ², F˜ = aXZ⁵ +bX³Z³, a(a² +a+b+b²) 6=

0, x˜₁= ˜x₄ = 0.

(2) ˜H =X²Z+XZ², F˜ =bX³Z³, b(b+ 1)6= 0, x˜₁x˜₃ = ˜x₄= 0.

(3) ˜H =X²Z+XZ², F˜ =bX³Z³, b(b+ 1) = 0, x˜₄ = 0.

(4) (i) ˜H =X³, F˜ =bX³Z³+aX⁵Z, b6= 0, x˜₃= ˜x₄= 0.

(ii) ˜H =X²Z, F˜ =bX³Z³+aX⁵Z, ab6= 0, x˜₃= ˜x₄ = 0.

(5) ˜H =X²Z, F˜ =bX³Z³, b6= 0, x˜₁x˜₃ = ˜x₄= 0 .

Explicit formulas for all cases will be determined in Section 3.6. Note that for the sake of a consistent normalization, we always move the first node we encounter to∞ and a cusp (unique by construction) to (0 : 0 : 1).

Im Dokument Computing canonical heights on Jacobians (Seite 93-99)