3.4 Local heights for general models
3.4.1 Definitions and first properties
t1b′2+k′4 t1b′1+f5k3′ t1b′0+f5k′2 k′1 t0b′2+t1b′3+f3k′3 t0b′1+t1b′2+k4′ t0b′0+t1b′1+f3k1′ k′2 t0b′3+f1k2′ t0b′2+f1k1′ t0b′1+k4′ k′3
W4,1 W4,2 W4,3 k′4
,
where
W4,1 = t0f1b′0+t0f3b′2+t20c+t1f1b′1+f3f1k′1, W4,2 = t0f5b′3+t0t1c+t1f1b′0+f1f5k2′, W4,3 = t0f5b′2+t1f3b′1+t1f5b′3+t21c+f3f5k′3.
It seems curious that our results in this section apparently cannot be com-bined to form a matrix that works in arbitrary characteristic. One pos-sible reason for this is the fact that if char(l) = 2, then an affine point (x, y) invariant under the hyperelliptic involution satisfies H(x,1) = 0 and if char(l)6= 2, then such a point satisfies y= 0. In general, we can only as-sume that 2y+H(x,1) = 0 and this is not a sufficient simplification to make the method used above work. Moreover, if char(l) = 2, then, depending on the number of distinct roots ofH(x,1), we have #J[2]∈ {1,2,4}, whereas otherwise #J[2] = 16. It would be interesting to find out whether there is a matrix WQ representing translation by a point of order 2 in arbitrary characteristic, either by finding such a matrix or by proving that it cannot exist.
3.4 Local heights on Kummer coordinates for ge-neral models
3.4.1 Definitions and first properties
Now we return to our setup of a number field or one-dimensional function fieldk. Letvbe a place ofkand consider a smooth projective genus 2 curve C over kv given as the smooth projective model of an equation
Y2+H(X,1)Y =F(X,1), (3.13) where
F(X, Z) =f0Z6+f1XZ5+f2X2Z4+f3X3Z3+f4X4Z2+f5X5Z+f6X6 and
H(X, Z) =h0Z3+h1XZ2+h2X2Z+h3X3
are binary forms of degrees 6 and 3, respectively, such that the discriminant
∆(C) is nonzero. We can assume without loss of generality that F, H ∈ Ov[X, Z].
We now generalize the definitions of εv and µv (see (3.7) and Definition 3.5, respectively) to include the present case. Let J denote the Jacobian of C and let K be its Kummer surface discussed in Section 3.3. We let δ and B = (Bij)i,j denote the objects defined onK in that section and generalize the notion of Kummer coordinates and of KA introduced in Section 3.1 to the general case in the obvious way. The following definition is similar to Definition 2.7.
Definition 3.19. Let x ∈ KA(kv) be a set of Kummer coordinates on K.
Then we set
εv(x) :=v(δ(x))−4v(x) and
µv(x) = X∞ n=0
1
4n+1εv(δ◦n(x)).
We recall some of the properties of these functions, since they continue to hold in this more general setting. If x and x′ represent the same point in K(kv), then we have εv(x) = εv(x′) and µv(x) = µv(x′), and so we can defineεv andµv on K(kv). If P ∈J(kv), then we define
εv(P) :=εv(x) andµv(P) :=µv(x) for any set of Kummer coordinatesx forκ(P) ∈K(kv).
We will also have occasion to use the following function: Let x, y ∈ KA(kv) and define
εv(x, y) :=v(B(x, y))−2v(x)−2v(y).
If P, Q ∈J(kv), then we have εv(x, y) =εv(x′, y′) for any sets of Kummer coordinates x, x′ forP and y, y′ forQ, respectively. Hence we can set
εv(P, Q) :=εv(x, y) (3.14) for any sets of Kummer coordinatesxandyforP andQ, respectively. This was first defined in [94].
Lemma 3.20. Let x, y, w, z ∈KA(kv) be Kummer coordinates on K satis-fyingw∗z=B(x, y). Then we have
δ(w)∗δ(z) =B(δ(x), δ(y)).
Proof. The proof carries over verbatim from the proof of [94, Lemma 3.2].
Corollary 3.21. Let x, y, w, z ∈KA(kv) be Kummer coordinates onK sa-tisfyingw∗z=B(x, y). Then we have
εv(δ(x), δ(y)) + 2εv(x) + 2εv(y) =εv(w) +εv(z) + 4εv(x, y).
We now refine the notion of the canonical local height. The idea, which is due to Stoll and was first introduced in the unpublished manuscript [95], is to define canonical local heights not for points on the Jacobian or on the Kummer surface as in Definition (3.5), but instead for Kummer coordinates as in Definition 2.8.
Definition 3.22. Let x ∈ KA(kv) be a set of Kummer coordinates on K.
Thenaive local height of x is the quantity λv(x) :=−Nv
nvv(x) and thecanonical local height of xis given by
λˆv(x) :=−Nv nv
(v(x) +µv(x)).
Notice that if k is a number field or function field of dimension 1 and P ∈ J(k) is a point lying on a Jacobian surface J defined over k, then we have
for any choicexof Kummer coordinates forP because of the product formula (1.1). However, our function ˆλv now depends on the choice of Kummer coordinates and not on the choice of a divisor in the class [D1]. As in the case of elliptic curves, the canonical local height ˆλv constructed as above has somewhat nicer properties than the canonical local height defined in Definition (3.5). Compare the following proposition, first stated and proved in [95], to (1.2).
Proof. The validity of (i) can be shown using a straightforward computation:
Property (ii) is also not hard to verify using Lemma 3.20 and Corollary 3.21:
nv
(iii) follows from the fact thatµv(x) is a bounded function, implying λˆv(δ◦n(x)) =−Nv
nvv(δ◦n(x)) +O(1), combined with property (i).
Part (iv) is obvious from the definition of ˆλv.
The canonical local height on Kummer coordinates also behaves well under isogenies. Compare the following to Proposition 2.5.
Proposition 3.24. Let α : J → J′ be an isogeny of Jacobians of di-mension 2 defined over kv and let d = deg(α). Then α induces a map α : K → K′ between the corresponding Kummer surfaces. 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. All assertions except for the last one are trivial. Using part (iii) of Proposition 3.23 it is enough to show
v(δ◦n(α(x))) =dv(δ◦n(x))−4nv(a) +O(1).
However, we have v(α(x))−dv(x) = O(1) by assumption, so it suffices to show
v(δ◦n(α(x))) =v(α(δ◦n(x)))−(4n−1)v(a). (3.15) But since α : J −→ J′ is an isogeny, it is a group homomorphism, so δ◦n(α(x)) and α(δ◦n(x)) represent the same point on K′, hence they are projectively equal. Because they also have the same degree, the factor of proportionality is independent of x. We may therefore check (3.15) for a single x, so we take x = (0,0,0,1) ∈ KA(kv). Because we have δ(x) = x and, by assumption,α(x) =ax′, where x′ = (0,0,0,1), we find
δ◦n(α(x)) =a4nx′ andα(δ◦n(x)) =ax′, thereby proving (3.15) and hence the proposition.
The preceding proposition is particularly useful in order to analyze the behavior of the canonical local height under a change of model, which we also call atransformation, of the curve. Any such transformation τ is given by data ([a, b, c, d], e, U), where
a b c d
∈GL2(kv), e∈kv∗ andU(X, Z)∈ kv[X, Z] is homogeneous of degree 3. If we apply such a transformation τ = ([a, b, c, d], e, U) to an affine point (ξ, η) on the curve, then we get
τ(ξ, η) =
aξ+b
cξ+d,eη+U(ξ,1) (cξ+d)3
. (3.16)
A transformation also acts on the formsF and H by
τ∗F(X, Z) = (ad−bc)−6 e2F′+ (eH′−U′)U′ τ∗H(X, Z) = (ad−bc)−3 eH′−2U′
, where
S′ =S(dX−bZ,−cX+aZ) for any binary formS(X, Z)∈kv[X, Z].
The map which a transformation τ = ([a, b, c, d], e, U) induces on KA
will play a crucial part later on. Therefore we give it here explicitly. Let x= (x1, x2, x3, x4)∈KA and let
U(X, Z) =u0Z3+u1XZ2+u2X2Z+u3X3.
Then τ(x) is equal to the following quadruple, where we have fixed the constant factor to be (ad−bc)−1:
(ad−bc)−1
d2x1+cdx2+c2x3,
2bdx1+ (ad+bc)x2+ 2acx3, b2x1+abx2+a2x3,
(ad−bc)−2(e2x4+ (l1,1+l1,2+l1,3)x1+ (l2,1+l2,2+l2,3)x2 + (l3,1+l3,2+l3,3)x3)
, where fori= 1,2,3 we have
li,1 = e2
(ad−bc)4li,1′ withl′i,1∈Z[f0, . . . , f6, a, b, c, d], li,2 = e
(ad−bc)4li,2′ withl′i,2∈Z[h0, . . . , h3, u0, . . . , u3, a, b, c, d], li,3 = 1
(ad−bc)4li,3′ withl′i,3∈Z[u0, . . . , u3, a, b, c, d].
All of thel′i,j are homogeneous of degree 8 in a, b, c, dand are homogeneous in the other variables. More precisely, the li,1′ are linear in the fj, the l′i,2 are linear in the uj and also linear in the hl, whereas the l′i,3 are quadratic in the uj. So we see that τ acts on kv4 as a linear map whose determinant has valuation v(τ) = 2v(e)−3v(ad−bc). The following corollary was first proved as [95, Proposition 3.2]; in fact we generalized the proof given there in order to prove our Proposition 3.24.
Corollary 3.25. Let τ = ([a, b, c, d], e, U) be a change of model of a genus 2 curve C with associated Kummer surface K. Then we have
ˆλv(τ(x)) = ˆλv(x)−Nv
nvv(τ).
for any x∈KA(kv)
Definition 3.26. LetCbe a genus 2 curve over kv given by a model (3.13) with discriminant ∆(C) and letK be the associated Kummer surface. We call the function
λ˜v :KA(kv) −→ R
x 7→ ˆλv(x)− 1
10log|∆(C)|v
thenormalized canonical local height on KA(kv).
Corollary 3.27. The normalized canonical local height is independent of the given model ofC.
Proof. Letτ be a change of model. Then we have v(∆(τ(C))) =v(∆(C)) + 10v(τ).
Recall that for simplified models (that is models of the form Y2 = F(X,1)) we defined four divisors
Di ={P ∈J :κi(P) = 0} ⊂Div(J)(kv),
see (3.5), where in particular D1 = 2Θ or D1 = Θ+ + Θ−, according to whether C has a unique rational point at infinity or not. We extend this definition to the general case in the obvious way.
LetP ∈J(kv)\supp(Di) and letx= (x1, x2, x3, x4) be a set of Kummer coordinates ofP normalized by
xj = κj(P)
κi(P), j ∈ {1,2,3,4}.
We proved that
λi,v(P) :=λv(x) is a Weil function and
ˆλi,v(P) := ˆλv(x)
is the canonical local height onJ associated withDi, v andgi(P) =δi(x) in the simplified caseH= 0. Both the definitions and the proofs carry over to the general case.