4.7 Examples
5.1.2 Case II: Number Fields with Complex Embeddings
real number fields. The reader should refer to Chapter 3 for the relevant notations. Let K be a non-totally real number field (i.e., one having non-real complex embeddings), and let E be an elliptic curve defined over K. It can be seen from Theorem 3.4.1 that, in order to determine a positive lower bound for the canonical height onEgr(K), we may have to compute
Tnmax
n=1 T
(v)
n (
p
Bn(µ)) for every complex
archimedean placev ∈Mc K, in addition to Tnmax n=1 S (v) n (−Bn(µ), Bn(µ)) for everyv ∈ Mr K. To obtain T (v)
n (ξ), we first need to construct the approximate corresponding
region S(v)(ξ). Assume that for each v ∈ Mc
E(v) of E is of the form
E(v) :Y2 = 4X3 +AX+B
for some A, B ∈ C. Then it can be seen from Section 3.1.2 that the definition of
S(v)(ξ) requires the quantity
Uξ=|w1|2 µ ξ+ b2 12 ¶ ,
where b2 is an invariant as defined in Chapter 1 for E(v), and w1 ∈ C is one of the
two vectors forming a Z-basis for the period lattice Λ of E(v), in such a way that
Λ =hw1, w2i and τ =w2/w1 satisfies (3.1), i.e.,
|τ| ≥1, |<(τ)| ≤1/2, =(τ)≥√3/2.
One can then use Theorem 4.5.3 (together with some linear transformation if nec- essary) to obtain Λ =hw1, w2i whose τ satisfies (3.1).
Furthermore, one can see from Section 3.2.3 that construction ofS(v)(ξ) requires
a parallelogram C0 containing an elliptic logarithm of a point P ∈ E(v)(C) with
X(P) = 0. Although one can use Algorithm 4.6.2 to compute an elliptic logarithm ofP, it should be noted that this is rarely required in practice, sinceC0 is normally
obtained as one of the parallelograms C satisfying I(C)∩[0, Uξ]6=∅.
We will now illustrate our algorithm for elliptic curves defined over non-totally real number fields with the following examples. For the rest of this chapter, we shall leti=√−1.
Example 5.1.4. LetE =E4, whereE4 is the elliptic curve defined over K =Q(i)
given by the Weierstrass equation
Table 5.4: Illustration of the algorithm for Example 5.1.4 µ nmax Is any Is any
T
Tn(v) Is µa
Bn(µ)<1? empty? lower bound?
0.20 4 No No Fail 0.10 4 No Yes Yes 0.15 4 No Yes Yes 0.18 4 No Yes Yes
The discriminant of E can be factorised into a product of prime ideals as p1p2p83,
where
p1 =h799 + 1124ii, p2 =h7−12ii, p3 =h1 +ii.
Hence the model of E is globally minimal, and soME = 1. Based on a number of
refinements as shown in Table 5.4, our algorithm shows that ˆ
h(P)>0.18
for all non-torsionP ∈Egr(K). Note that in this example we only have to compute
Tn(v)( p
Bn(µ)) but not Sn(v)(−Bn(µ), Bn(µ)), since K has no real embedding. In
addition, we chooseS(v) =S(v,4) for v ∈Mc
K.
It can be checked that the Tamagawa indices cv of E at v = p1,p2,p3 are all
1. Moreover, we have cv = 1 where v is the only complex archimedean place ofK.
Hence c= 1, and by Lemma 2.1.1,
ˆ
h(P)>0.18
for all non-torsion P ∈E(K). One can verify the above results using our MAGMA
code (note that we require elog.m and every file mentioned in Appendix A.3 on- wards) together with the following instructions:
> // Note that all of these files are required > Attach("elog.m"); > Attach("alphas.m"); > Attach("heightbound.m"); > Attach("intersect_real.m"); > Attach("intersect_complex.m"); > Attach("interval_arith.m"); > Attach("interval_wp.m");
> Attach("wp.m");
> // SetVerbose("Bound", 1); // enable this line to see more details > // Define elliptic curve E
> _<x> := PolynomialRing(Integers()); > K<i> := NumberField(x^2+1);
> E := EllipticCurve([91-26*i, -144-323*i]); > // Check if 0.2 is a lower bound on E(K)
> // (to get a lower bound on E_{gr}(K), multiply by a square of lcm of > // all Tamagawa indices)
> IsLowerBound(E, 0.2 : n_max := 4); false
> // Fail to show that 0.2 is a lower bound, so try something smaller > IsLowerBound(E, 0.1 : n_max := 4);
true
> // So 0.15 is a lower bound, try to check a bigger lower bound > IsLowerBound(E, 0.15 : n_max := 4);
true
> IsLowerBound(E, 0.18 : n_max := 4); true
On the other hand, the lower bound on Egr(K) (and also E(K) in this case)
obtained by Theorem 2.4.2 is not as good as this one. In this example, we have
αv = 4.715889.
Choose a prime ideal p with N(p) > αv, say, p = h5,2 +ii, and set n = ep = 5. Then we have DE(5) = 3.218876, which yields the lower bound
µ0 =
3.218876−2 log(4.715889)
2·52 = 2.34×10
−3.
Finally, one can verify that the lower bound obtained by Hindry and Silverman [HS88, Theorem 0.3] is
ˆ
h(P)≥3.0624×10−25
for all non-torsion P ∈E(K). We leave it to the reader to compare the results.
Example 5.1.5. The following elliptic curve is from Cremona’s paper [Cre94, Ex-
ample 2]. Let E =E5, where E5 is the elliptic curve defined over K =Q(i) given
by the Weierstrass equation
One can easily observe that P0 = (0,0) ∈ E(K). Let ∆ be the discriminant of
E. Then we have h∆i = p, where p = h13 + 8ii is prime. Moreover, we have the Tamagawa index cp = 1, and also cv = 1 where v is the only complex archimedean
place of K. Hence c = 1. Using the fact that ˆh(P0) = 0.0230, we set our initial
guess µto be smaller than 0.0230, say,µ= 0.01. Our algorithm shows that
B5(µ) = 0.7772<1.
Thus by Proposition 2.4.1, ˆh(P) > 0.01 for all non-torsion P ∈ Egr(K). Since
c= 1, we also have ˆh(P)>0.01 for all non-torsionP ∈E(K) by Lemma 2.1.1.
Example 5.1.6. Let K = Q(θ) where θ is a root of the polynomial x3−2. Let
E = E6, where E6 is the elliptic curve defined over K given by the Weierstrass
equation
E6 : y2 =x3−(θ2+ 3θ)x+θ2.
Let ∆ be the discriminant ofE. The prime ideal factorisation ofh∆iisp16
1 p2, where
p1 =h2, θi, p2 =h390433,218056 +θi.
It can be verified that the model of E is globally minimal, and so ME = 1. Our
algorithm shows that
ˆ
h(P)>0.25
for all non-torsion P ∈Egr(K), which is obtained after a number of refinements as
shown in Table 5.5. Recall that if TSn(v) = ∅ for some v ∈ MKr, then µ is a lower
bound and so there is no need to computeTTn(v) for any v ∈MKc.
Finally, we note that the Tamagawa indices cv atv =p1,p2 are 2 and 1 respec-
tively. Moreover, since E has only one real embedding, say, E(v1) with three real
Table 5.5: Illustration of the algorithm for Example 5.1.6 µ nmax Is any Is any
T
Sn(v) Is any
T
Tn(v) Is µa
Bn(µ)<1? empty? empty? lower bound?
0.50 3 No No No Fail 0.20 3 No Yes Skipped Yes 0.30 3 No No No Fail 0.25 3 No Yes Skipped Yes
Then againcv2 = 1, and so c= lcm{1,2}= 2. Thus by Lemma 2.1.1, we have
ˆ
h(P)>0.25/22 = 0.0625
for all non-torsionP ∈E(K). Note that we have obtained no additional information from the complex place in this specific example; however, there is no reason to suppose that this would be the case in general.
In the next section, we will explain how to use a lower bound for the canonical height to derive Mordell–Weil bases for elliptic curves defined over number fields. This method will be illustrated when we revisit all the examples recently shown.
5.2
Mordell–Weil Bases
Computing Mordell–Weil bases for elliptic curves over number fields is one of the most difficult computations in the arithmetic of elliptic curves, and so far there is no known procedure which can determine such a basis in general. In this section, we will illustrate an application of a lower bound for the canonical height in assisting such computation. For more background on this section, see Section 1.2.2 or [Cre97, Section 3.5].
Let E be an elliptic curve defined over a number field K. Recall from Sec- tion 1.2.2 that, given some non-torsion points P1, . . . , Pr ∈E(K) whose images in E(K)/Etors(K) generate a subgroup of finite index of E(K)/Etors(K), it is pos-
m-descent for some m ≥ 2) to obtain a full Mordell–Weil basis for E(K). The saturation process consists of the following steps:
1. Determine an upper bound`for the indexn = [E(K)/Etors(K) :hP1, . . . , Pri]
using the geometry of numbers (Theorem 1.2.1), which then requires a positive lower bound for the canonical height onE(K) obtained by Theorem 3.4.1. 2. For each prime p≤`, determine whetherp|n, or equivalently, whether there
existsa1, . . . , ar ∈Z, not all divisible byp, such that r
X
j=1
ajPj =pQ (5.1)
for someQ∈E(K). Without loss of generality, we can assume that|aj| ≤p/2.
3. If there exists a solution to (5.1), letai be the minimal non-zero coefficient (in
absolute value). If ai =±1, then we can simply replace Pi by Q; otherwise,
we find a coefficientaj not divisible byai. Writeaj =aiq+bwith 0< b <|ai|.
Observe that
aiPi+ajPj =ai(Pi+qPj) +bPj.
This then allows us to replace the generator Pi by Pi+qPj, replace aj by b,
and replace i by j. This time, the index of the sublattice generated by the new P1, . . . , Pr inE(K) will be at most `|ai|/p.
4. Repeat the above steps until the index n is not divisible by any primes. The final set{P1, . . . , Pr} will be a Mordell–Weil basis forE(K) modulo torsion.
Nevertheless, the upper bound ` obtained by Theorem 1.2.1 can be very large even though the points P1, . . . , Pr may already form a Mordell–Weil basis, and so
there can be too many primes p to consider. Fortunately, it is possible to quickly eliminate some of p from our consideration before we actually have to solve (5.1).