6.2 Intersection theory
6.2.2 Non-hyperelliptic curves
Because our results from Chapter 5 are not limited to hyperelliptic curves, we would like to give an example of a regulator of (a subgroup of finite index of)
the Mordell-Weil group of a non-hyperelliptic Jacobian. The computation of the non-archimedean local N´eron symbols can be done as in the hyperelliptic case, unless for some v ∈ Mk0 we encounter kv-rational divisors D and E whose supports are defined over local extension fields such that there is no single affine patch containing the entire intersection of the closures of D and E. In this case there are additional complications and we may have to restart our algorithm over an extension field ofk. We refer to Section 5.3.4 for a discussion of this problem.
However, ifk=Qand our divisors are supported in Q-rational points to start with, then this difficulty cannot occur. Many interesting examples are of this form, because in practice one is often lead to regulators of Jacobians of curves that containQ-rational points. For such examples we can use our algorithm to compute the non-archimedean local N´eron symbols.
The archimedean local N´eron symbols are a different matter. See Re-mark 5.24 for a discussion of the current situation of the necessary algo-rithms described in Section 5.3.6. The best we can do is to give a list of explicit computations that yield the desired result once a suitable implemen-tation exists. By virtue of Corollary 5.22 it suffices to list suitable divisors D1,i, D2,i, E1,i, E2,i fori = 1, . . . , N, whereN ≥1 is finite, such that com-puting the local N´eron symbols hD1,i−D2,i, E1,i−E2,ii suffices. Here all D1,i and D2,i should be non-special with disjoint support and all E1,i and E2,i should have degree at most g.
Example 6.8. An example of a curve for which the computation of the re-gulator is interesting and useful is the curveX0dyn(6) considered by Stoll in [97]. It is a quotient of the curveX1dyn(6) which is a smooth projective curve that has an affine patchY1dyn(6) parametrizing 6-cycles, that is pairs (x, c), wherex is periodic of exact order 6 under the iteration
x0=x, xn+1=x2n+c forn≥0. (6.1) It is an interesting problem in arithmetic dynamics to determine whether there are rationalN-cycles for a givenN. The situation forN = 2,3,4,5 is known and it is expected that there are no rationalN-cycles forN >3. See the introduction to [97]. Stoll shows, assuming the existence of an analytic continuation and functional equation of the L-series ofJ and the Birch and Swinnerton-Dyer conjecture for J that there are no rational 6-cycles, that isx, c∈Qsatisfying (6.1). Here J is the Jacobian ofX0dyn(6).
In order to give further evidence for this conditional statement, it would be helpful to verify the second part of the Birch and Swinnerton-Dyer conjec-ture 1.43 for X0dyn(6). Stoll has already computed several terms appearing in that statement and according to [97] it remains to show that
Reg(J/Q)ΩJ#X(J/Q) = 0.03483. . . . (6.2)
Recall that Reg(J/Q) is the regulator, ΩJ is the real period of the N´eron differential andX(J/Q) is the Shafarevich-Tate group ofJ/Q. These terms are defined in [49,§F.4.1].
It can be shown that if X(J/Q) is finite, then in this particular case its order must be a square. The real period ΩJ is probably not too hard to compute; for Jacobian surfaces a method due to Wetherell is reproduced in [44, §3.5]. So in order to check (6.2) up to an integral square, the main problem is the computation of the regulator of a subgroup ofJ(Q) of finite index.
The curve X0dyn(6) is a non-hyperelliptic curve of genus 4 without any special properties. We refer to [97] for the construction of a model C of X0dyn(6) that is given by a curve of bidegree (3,3) in P1×P1 with affine equation
G(u, w) =w2(2 + 1)u3−(5w2+w+ 1)u2−w(w2−2w−7)u+ (w+ 1)(w+ 3).
We will also use the image of this under the Segre embedding intoP3. This yields a modelC′ of X0dyn(6) given by
x10x01+x00x11= 0,
x300−x00x210+x200x01−5x00x10x01+ 2x210x01−x10x201+x210x11
+ 7x10x01x11−x201x11−2x10x211−3x311= 0.
In order to compute intersection numbers, we need to find a regular model over each Spec(Zp). Stoll has already computed such models. The only primes of singular reduction are p = 2 and p = 8029187; the reduction of C modulo the latter is regular, so only the prime p = 2 remains to be considered. Here Stoll finds a desingularization in the strong sense of the closure of C over Spec(Z2) consisting of two elliptic curves A and B and three rational curvesS, S′ and T. The corresponding intersection matrix is given by:
A B S S′ T
A -4 2 1 1 0
B 2 -2 0 0 0
S 1 0 -2 0 1
S′ 1 0 0 -2 1
T 0 0 1 1 -2
In [97, §3] Stoll lists ten rational pointsP0, . . . , P9 ∈C(Q) (none of which come from a rational 6-cycle) and shows that the divisors supported in them generate a subgroupGofJ(Q) of rank 3. Moreover, he proves that the first part of the Birch and Swinnerton-Dyer conjecture predicts that the rank of J(Q) is exactly 3, which would imply that G has finite index in J(Q).
We have listed coordinates forP0, . . . , P9 onC and onC′ in Table 6.7; here
((U1 :U2),(W1 :W2))∈C (x00:x01:x10:x11)∈C′ cpt
P0 ((0 : 1),(1 : 0)) (0 : 1 : 0 : 0) A
P1 ((0 : 1),(−1 : 1)) (0 :−1 : 0 : 1) B
P2 ((0 : 1),(3 : 1)) (0 : 3 : 0 : 1) B
P3 ((1 : 0),(0 : 1)) (0 : 0 : 1 : 0) A
P4 ((1 : 1),(2 : 1)) (2 : 2 : 1 : 1) T
P5 ((2 : 1),(1 : 1)) (2 : 1 : 2 : 1) B
P6 ((1 : 1),(1 : 0)) (1 : 1 : 0 : 0) A
P7 ((1 : 0),(−1 : 1)) (−1 : 0 : 1 : 0) A P8 ((−1 : 1),(1 : 0)) (−1 : 1 : 0 : 0) A P9 ((−4 : 5),(−1 : 1)) (4 :−5 :−4 : 5) B
Table 6.7: Rational points on models ofX0dyn(6)
(U1 :U2) and (W1 :W2) are the homogenizations of u and w, respectively, and cpt is the component on the regular model of C over Spec(Z2) given above that the respective point maps to. This component can be determined easily by following through the blow-ups necessary for the construction of the regular model.
Lemma 6.9. Let D = (P0) −(P1), let E = (P2) −(P1) and let F = (P4)−(P2). Then the points P, Q and R generate G, where
P = [D], Q= [E] and R= [F].
Moreover, we have
D ∼ (P7) + (P9)−(P6)−(P8) =:D′
E ∼ (P3) + (P5) + (P6)−(P0)−(P7)−(P9) =:E′ 2F ∼ (P3) + 2(P5)−(P0)−2(P6) :=F′
Proof. This follows easily from the six independent linear equivalence rela-tions between the (Pi) and subsequent remarks given in the proof of [97, Lemma 4].
For the next step, we need to compute the intersection multiplicities between different (Pi). It turns out that there are very few non-trivial intersections. Indeed we have (with the obvious abuse of notation)
i2((P6),(P8)) = 1, i2((P1),(P9)) = 1, i5((P7),(P9)) = 1,
and all other intersection multiplicities are trivial. For p 6= 2 we can show this using Lemma 5.17. We also have that P5 reduces to the same singular
point asP1and P9 modulo 2, but blowing up this point separates the image ofP5 from the image of P1 and P9.
Using Lemma 6.9 we can now split up the computation of the terms appearing in the regulator of Gas follows:
ˆh(P) =−hD, Di=−hD, D′i=− X
The correction terms atp= 2 are easily computed using one of the two ap-proaches in Section 5.3.5, because we have the intersection matrix available and we know which components thePi map to. This finishes the computa-tion of the non-archimedean local N´eron symbols.
Recall that for the computation of the archimedean local N´eron sym-bol one of the divisors should be the difference of two non-special divisors with disjoint support. In the present situation this can be arranged easily.
Combining everything, we have:
For the computation of the archimedean local N´eron symbols see Section 5.3.6 and the introduction to the present section. Since the algorithms of Deconinck et al. use Puiseux expansions, we are likely to need a plane curve
inP2 or A2 to construct the Riemann surface of the curve and this was the case in the Maple package. Since the algorithms are supposed to work on singular Riemann surfaces, this is not an essential restriction, since we can use a map to a suitable curve inP2 that is birationally equivalent toC′. This way it will be possible to complete the example once a new implementation of the necessary algorithms exists.