In this section we try to generalize our construction of canonical local heights from Section 3.4. The discussion is kept brief, because such results are not directly applicable in practice at the moment, since even in the hyperelliptic genus 3 case the results depend on the validity of Conjecture 4.8. Therefore we might as well work in a more general setting. Let C denote a smooth projective curve defined over the completion kv of a number field or one-dimensional function fieldk at a placev∈Mk. LetJ be the Jacobian ofC and let K be its Kummer variety, embedded into projective space using a map
κ:J −→K ֒→P2g−1
such that κ(O) = (0, . . . ,0,1). Let g1, . . . , gN ∈ kv denote the coefficients appearing in the given model of C. Suppose we have homogeneous quartic polynomials
commute and such that
δ(0, . . . ,0,1) = (0, . . . ,0,1).
Example 4.10. These conditions are, of course, satisfied in our constructions in Chapters 2 and 3. They are also satisfied in the situation of a hyperelliptic curve of genus 3 given by a model of the form (4.2) when char(kv) 6= 2, provided Conjecture 4.8 holds; namely, we can pick K′ and δ′ constructed at the end of the previous section asK and δ.
In this situation essentially all our definitions from Chapter 3 carry over.
We defineKummer coordinates on K and the set KA as in Definition 4.4.
Definition 4.11. Let x ∈ KA(kv) be a set of Kummer coordinates on K.
For the next definition recall the definitions of the local normalization constantsNv andnv and of the global normalization constantdkintroduced in Section 1.1. Their purpose is to make the product formula (1.1) work for k.
Definition 4.12. Let x ∈ KA(kv) be a set of Kummer coordinates on K.
Thenaive local height of x is the quantity λv(x) :=−Nv nv
v(x) and thecanonical local height of xis given by
λˆv(x) :=−Nv
nv(v(x) +µv(x)).
It follows that ifkis a number field or function field of dimension 1 and we assume thatC is in fact defined over k, then we have
h(P) = 1
for any choicex of Kummer coordinates for P ∈J(k), where h(P) =h(κ(P))
is the naive height ofP and
h(Pˆ ) = lim
n→∞
1
4nh(2nP)
is thecanonical height of P. If k is a global field, then we can in principle compute the canonical height using the algorithm due to Flynn and Smart as introduced in Section 3.2.2 if we can compute µv(P) for v ∈ Mk∞ and P ∈ J(k). But the method outlined in that section requires a bound on the archimedean local height constant for each v ∈ Mk∞ and we have no such bounds available at the moment, even for our rather special hyper-elliptic genus 3 curves. One possible method for the calculation of height constant bounds – the decomposition of the duplication map into Richelot isogenies as in [42] – is probably difficult to generalize and only leads to rather crude bounds even in genus 2. It seems more promising to use Stoll’s representation-theoretic approach introduced in [92], but we have not at-tempted to do this.
Fortunately we can always compute archimedean canonical local heights using theta functions; we discuss this approach in 4.4.2.
One result from Chapter 3 which generalizes immediately is the follow-ing.
Proposition 4.13. Let α:J →J′ be an isogeny of Jacobians defined over kv and let d = deg(α). Then α induces a map α : K → K′ between the corresponding Kummer varieties. We also get a well-defined induced map α : KA −→ KA′ if we fix a ∈ kv∗ and require α(0,0,0,1) = a(0,0,0,1).
Moreover, we have
ˆλv(α(x)) =dλˆv(x) + log|a|v for any x∈KA(kv).
Proof. See the proof of Proposition 3.24.
In particular we can control how the canonical local height changes when we change the model ofC. For instance, suppose thatC is hyperelliptic and
τ = ([a, b, c, d], e, U)
is a change of model of C as in (3.16). Then τ induces a transformation τ on KA that is a linear map on A2g. Let v(τ) denote the valuation of its determinant; then we get
λˆv(τ(x)) = ˆλv(x)−Nv nvv(τ) in analogy with Corollary 3.25.
4.4.1 Non-archimedean places
Suppose now thatv ∈Mk is non-archimedean. A particularly useful obser-vation in Chapter 3 was Theorem 3.29, stating that the set
Uv :={P ∈J(kv) :εv(P) = 0}
is a subgroup ofJ(kv) if g= 2. Unfortunately, it is not possible to imitate the proof given by Stoll in [94] in the higher genus situation, because we need to have the forms Bij available explicitly. Once these are found, it should not be hard to prove an analog of Theorem 3.29 for hyperelliptic curves of genus 3 when char(k)6= 2.
In any event, the following conditional statement is immediate:
Theorem 4.14. Suppose C is given by an Ov-integral model whose closure C over Spec(Ov) is normal and flat and has rational singularities. Suppose that the set Uv is a subgroup of J(kv). Then εv and µv factor through the component groupΦv of the N´eron model of J.
Proof. This is the same as the proof of Theorem 3.30, since we can de-fine canonical local heights with respect to certain divisors Di, where i ∈ {1, . . . ,2g}as in the previous chapter.
The previous result may also be useful for theoretical investigations. For practical purposes, we first have to find the forms Bij or at least prove Conjecture 4.7, parts (a) and (b). In principle it should also be possible to generalize the simplification procedure introduced in Section 3.4.4 and then find formulas for a small number of models that we can always, using Proposition 4.13, reduce to; yet the difficulties we have encountered handling the much easier genus 2 situation suggest that this should be a very tedious task.
4.4.2 Archimedean places
In order to compute archimedean canonical local height we introduced seve-ral methods in the previous chapters, one of which turns out to geneseve-ralize immediately. Letv∈Mk be archimedean and considerC(C) as a Riemann surface embedded into complex projective space usingv.
Because we can change the model, we suppose that the embedding κ corresponds to L(2Θ), where Θ is the theta divisor corresponding to the Abel-Jacobi map embedding C into J using some fixed base point. Let τv ∈hg such that J(C) is isomorphic to Cg/Λv, where Λv =Zg⊕τvZg and definej by
j:Cg ////Cg/Λv ∼= //J(C).
Let a = (1/2, . . . ,1/2), b = (g/2,(g −1)/2. . . ,1,1/2) ∈ Cg and let θa,b denote the theta function with characteristic [a;b] defined in Section 1.6.
Proposition 4.15. (Pazuki) The functionθa,bhas divisorj∗(Θ). Moreover, the following function is a N´eron function associated with Θ and v:
λˆ′Θ,v(P) =−log|θa,b(z(P))|v+πIm(z(P))T(Im(τv))−1Im(z(P)), where j(z(P)) =P.
Proof. See the proof of Proposition 3.77.
Hence we can use ˆλ′Θ,v to computeλv, because as in Section 3.7.2, there must be a constantdv such that
ˆλv(P) = 2 ˆλ′Θ,v(P) +dv for all P ∈J(C)\supp(Θ), where
λˆv(P) = ˆλv(κ(P)/κ1(P)).
We can find the constantdv using a 3-torsion point as in (3.31) and compute λv(P) for allP ∈J(C)\supp(Θ); other points can be treated similarly. For this we can use the existing implementation of theta functions in Magma mentioned in Section 3.7.2.
Arithmetic intersection theory
133
We have seen in the previous chapter that the computation of canonical heights using the decomposition into canonical local heights becomes quite complicated as we increase the genus. It proved to be rather successful in Chapters 2 and 3, but it runs into problems even in the case of Jacobians of hyperelliptic curves of genus 3 with a rational Weierstrass point. Since the main problems lie in the complexity of the associated Kummer variety and how the group law on the Jacobian is reflected on it, it is not very likely that this situation will improve for other curves of genus at least 3.
In the present chapter we use a different approach to develop a practical algorithm for the computation of canonical heights on Jacobians. However, in contrast to the previous chapters, it does not come with a naive height combining the properties that we can list all points of naive height up to some bound and that the difference between the two heights can be bounded effectively. In the hyperelliptic genus 3 case, it might be possible to com-bine the method that we are about to discuss with the Kummer threefold approach.
5.1 Local N´ eron symbols
In this section we discuss the theory of local N´eron symbols whose existence was first proved by N´eron in [78]. We shall present an interpretation that is suitable for explicit computations, following essentially Gross [46] and Hriljac [53]. The content of the latter work is also discussed by Lang in [60]. In order to present these results, we need the definitions and results of Section 1.5, especially the intersection theory on arithmetic surfaces.
Let R be a discrete valuation ring with valuation v, let l be the field of fractions of R and let S = Spec(R). Let C be a smooth projective geometrically connected curve of positive genus g defined over l and let χ:C →S denote aC over S.
Consider divisors on C ∼= Cl. Recall that we denote the group of l-rational divisors on C by Div(C)(l). For each n∈Z the group Divn(C) is defined to be the group of divisors of degree equal tonand we set
Divn(C)(l) := Divn(C)∩Div(C)(l).
We are particularly interested in the casen= 0.
If D ∈Div(C)(l) is prime, then we write DC for the closure of D on C as in Section 1.4. This is a prime horizontal divisor onC and we extend the operationD7→DC to Div(C)(l) by linearity.
For the remainder of this section, we fix a regular modelC′ ofC over S.
In order to define local N´eron symbols we need to deal with fibralQ-divisors.
LetQDivv(C′) denote theQ-vector spaces generated by the irreducible
com-ponents ofC′vand letQCv′ denote theQ-vector space generated by the whole fiber Cv′.
Lemma 5.1. There exists a unique linear map
Φv,C′ : Div0(C)(l)→QDivv(C′)/QCv′,
such that for allD∈Div0(C)(l) the Q-divisor DC′+ Φv,C′(D) is orthogonal to Divv(C′) with respect to iv(·,·).
Proof. Let Cv′ =Pr
i=0niΓiv be the decomposition of Cv′ as a divisor, where Γ0v, . . . ,Γrv are the irreducible components ofCv′. LetMv be the intersection matrix
iv(niΓiv, njΓjv)
0≤i,j≤r of Cv′ and let M :QDivv(C′)−→Qr+1 be the linear map defined by
E 7→ n0iv(E,Γ0v), . . . , nriv(E,Γrv)T
.
Lemma 1.30 implies that the kernel of M is QC′v, hence we get an induced mapMf:QDivv(C′))/QCv′ −→Qr+1 and there is a unique solution of
M(Φf v,C′(D)) =−s(D), wheres(D) = n0iv(DC′,Γ0v), . . . , nriv(DC′,Γrv)T
.
By abuse of notation we denote a representative of Φv,C′(D) also by Φv,C′(D), since in our intended application it does not matter which repre-sentative we choose. Now we have assembled all ingredients necessary to define the central objects of this chapter in the non-archimedean case.
Definition 5.2. The local N´eron symbol on C over lis the pairing hD, Eiv :=iv(DC′+ Φv,C′(D), EC′) logqv,
defined on divisors D, E ∈Div0(C)(l) with disjoint support.
Remark 5.3. The proper regular modelC′that is crucial for the construction of the local N´eron symbol does not show up in this notation. This is justified by part (e) of Proposition 5.7 below. Also note that from the definitions and Lemma 1.30 we immediately get
iv(DC′ + Φv,C′(D), EC′) = iv(DC′ + Φv,C′(D), EC′+ Φv,C′(E))
= iv(DC′, EC′+ Φv,C′(E)).
Next we consider an archimedean local fieldland we want to define local N´eron symbols over l. We can assume l = C (see part (g) of Proposition 5.7 below), so that C(l) is actually a compact Riemann surface. For the construction of local N´eron symbols we need the notion ofGreen’s functions on Riemann surfaces.
Proposition 5.4. Let X be a compact Riemann surface and let dµ be a positive volume form on X, normalized such that R
Xdµ = 1. For each D∈Div(X) there exists a unique function
gD :X\supp(D)→R,
called theGreen’s function with respect toDanddµ, such that the following properties are satisfied:
(i) The function gD is C∞outside of supp(D) and has a logarithmic sin-gularity along D, that is, if D is represented by a function f on an open subset U of X, then there is some α∈C∞(U) such that
gD(P) =−log|f(P)|+α(P) holds for all P ∈U\supp(D).
(ii)
deg(D)dµ= i π∂∂gD
(iii) Z
X
gDdµ= 0
Proof. See [60] for a proof of existence due to Coleman that uses differentials of third kind. In Section 5.3.6 we use a proof due to Hriljac (see [52]) and reproduced in [59] for the case where dµ is the canonical volume form on X, because it is rather constructive, at least for non-special divisors. For uniqueness, note thatgD is determined uniquely up to an additive constant by (i) and (ii), because the difference of two functions satisfying (i) and (ii) is harmonic everywhere and hence constant. Property (iii) fixes the constant.
Remark 5.5. We call a function satisfying (i) and (ii) an almost-Green’s function with respect to Dand dµ.
Let v be the absolute value on l, normalized as in Section 1.1 and fix a volume form dµv on C(l), normalized as in the theorem above. For two divisors D, E ∈ Div(C)(l) with disjoint support we define the intersection multiplicity ofD and E by
iv(D, E) :=gD(E) :=X
j
mjgD(Qj), whereE =P
jmj(Qj).
Definition 5.6. We call the pairing h·,·iv that associates to all D, E ∈ Div0(C)(l) with disjoint support the numberiv(D, E) thelocal N´eron symbol on C over l.
Notice that in order to compute hD, Eiv for given D, E ∈ Div0(C)(l) with disjoint support, we only need to find an almost-Green’s function with respect to D and that property (ii) reduces to the requirement that gD is harmonic. In particular, this restriction eliminates the dependency of the intersection multiplicity on the choice of dµv.
We list the most important properties of the local N´eron symbol, both archimedean and non-archimedean, in the following proposition. But first we need to introduce further notation. Iff ∈l(C)∗ and E =P
jmj(Qj)∈ Div0(C)(l), then we set
f(E) :=Y
j
f(Qj)mj.
Proposition 5.7. (N´eron, Gross, Hriljac) Let l be a field that is complete with respect to an absolute value v. The local N´eron symbol satisfies the following properties, where D, E ∈Div0(C)(l) have disjoint support.
(a) The symbol is bilinear.
(b) The symbol is symmetric.
(c) If f ∈l(C)∗, then we have hD,div(f)iv =v(f(D)).
(d) Fix D ∈ Div0(l) and P0 ∈ C(l)\ supp(D). Then the map C(l) \ supp(D)−→Rdefined by
P 7→ hD,(P)−(P0)iv
is continuous and locally bounded with respect to the v-adic topology.
(e) If v is non-archimedean, then hD, Eiv is independent of the choice of the proper regular model C′ and of the choice of Φv,C′(D).
(f ) If v is archimedean, then hD, Eiv is independent of the choice of the volume form dµv.
(g) If l′ is an extension of l with valuation v′ extending v, then we have hD, Eiv =hD, Eiv′.
Moreover, the pairing is uniquely determined by properties (a)–(d).
Proof. Existence and uniqueness of a pairing satisfying (a)–(d) was shown by N´eron in [78] whenC(l) is Zariski dense inC. The construction of the pairing using arithmetic intersection theory that is presented in this section and the
proof that the pairing thus constructed coincides with N´eron’s abstractly defined pairing is due to Gross [46] and Hriljac [53].
WhenC(l) is not Zariski dense inC, then N´eron’s proof does not apply.
In this more general situation N´eron shows that any pairing satisfying (a)–
(c) and another condition (d’) similar to (d) must be the local N´eron symbol.
Finally, Hriljac proves that the pairing constructed above satisfies (a)–(c) and (d’). We get (e), (f) and (g) for free because of the uniqueness property.
Remark 5.8. One can define local N´eron symbols for divisors with common support at the loss of some functoriality, see [46,§5].