CAPÍTULO 2. DISEÑO DEL SISTEMA DE CONEXIÓN/DESCONEXIÓN Y
2.3 Supervisión de los equipos de climatización
2.3.2 Características de la arquitectura cliente/servidor en la presente aplicación
We will first concentrate on the case when our elliptic curves are defined over totally real number fields. As we will see, this will require periods of elliptic curves overR and elliptic logarithms of real points, which can be obtained by Algorithm 4.6.2 or Cohen’s algorithms [Coh93, Algorithm 7.4.7 and 7.4.8]. For the relevant notations, the reader should refer to Chapter 2.
Example 5.1.1. LetE =E1, whereE1is the elliptic curve defined overK =Q(
√
2) given by the Weierstrass equation
E1 : y2 =x3+x+ (1 + 2
√
2).
The discriminant ∆ of E is −3952−1728√2. Moreover, we have h∆i = p8
1p22p3, where p1 =h √ 2i, p2 =h7,3 + √ 2i, p3 =h769,636 + √ 2i,
are prime ideals. Since ordpj(∆) <12 for allj, thenEis given by a globally minimal
model, and so ME = 1.
As explained in Section 3.4, our algorithm, based on Theorem 3.4.1, will start by checking whether a givenµ >0 is a lower bound for the canonical height onEgr(K)
by computingBn(µ) forn = 1, . . . , nmax. If Bn(µ)<1 for somen, thenµis indeed
a lower bound. Otherwise, we proceed to compute Tnmax
n=1 S
(v)
n (−Bn(µ), Bn(µ)) for
every real archimedean place v ∈ Mr
K (here, we do not have to compute any Tn(v),
since K is totally real). If the intersection is empty for some v, then µ is a lower bound. Note that we obtain no conclusion if none of the intersections is empty.
In this example, we define v+, v− to be the real archimedean place of K whose associated real embedding sends √2 7→ ±1.414. . . respectively. By letting µ = 1 and nmax = 5, we have
B1(µ) = 8.117389, B2(µ) = 8.186971×102, B3(µ) = 7.213201×107,
B4(µ) = 5.421641×1012, B5(µ) = 5.685757×1021.
Since none of these is less than 1, we have to compute Tnmax
n=1 S
(v)
n (−Bn(µ), Bn(µ))
for every v ∈ Mr
K. Recall from Section 2.5 that S
(v)
n (ξ1, ξ2) is defined in terms of
ψv(ξ1), ψv(ξ2), where ψv :E0(v)(R)→[1/2,1) is the normalised elliptic logarithm of
the “higher” of the two points on E0(v) with the same x-coordinate. For v = v+,
one can check that the corresponding real embedding E(v) has only one real root at
βv =−1.352786. Using Algorithm 4.6.2, we have (after normalisation)
ψv(B1(µ)) = 0.891227,
which yields1 S(v)
1 (−B1(µ), B1(µ)) = [0.108773,0.891227].
ComputingSn(v)(−Bn(µ), Bn(µ)) for alln = 2, . . . , nmaxin a similar way, we will
eventually see thatTnmax
n=1 S
(v)
n (−Bn(µ), Bn(µ))6=∅. A similar procedure also shows
that another intersection associated tov =v− is non-empty. Hence we fail to show that µ = 1 is a lower bound on Egr(K), in which case we shall repeat the above
computation with a smaller µ (and/or a larger nmax). On the other hand, if µ is
known to be a lower bound, then we can repeat such process with a largerµ to see if it is still a lower bound. This refinement can be done repeatedly as required.
After a number of refinements as shown in Table 5.1, our algorithm finally shows that
ˆ
h(P)> µ= 0.2415
1Onlyψ
Table 5.1: Illustration of the algorithm for Example 5.1.1 Initial Initial Is any Is any intersection Isµa Next Next
µ nmax Bn(µ)<1? empty? lower bound? µ nmax
1.0000 5 No No Fail 0.5000 6 0.5000 6 No No Fail 0.2500 7 0.2500 7 No No Fail 0.1250 8 0.1250 8 Yes Skipped Yes 0.1875 8 0.1875 8 No Yes Yes 0.2187 8 0.2187 8 No Yes Yes 0.2343 8 0.2343 8 No Yes Yes 0.2421 8 0.2421 8 No No Fail 0.2382 9 0.2382 9 No Yes Yes 0.2402 9 0.2402 9 No Yes Yes 0.2412 9 0.2412 9 No Yes Yes 0.2416 9 0.2416 9 No No Fail 0.2414 10 0.2414 10 No Yes Yes 0.2415 10 0.2415 10 No No Fail 0.2415 11 0.2415 11 No Yes Yes
for all non-torsion points P ∈ Egr(K). Nevertheless, the lower bound for Egr(K)
derived from Theorem 2.4.2 is not as good as this one. In this example, we have
αv+ = 1.096562, αv− = 1.001830,
and so αv+αv− = 1.098569. We now choose a prime ideal p whose norm is greater
than √αv+αv−, and set n = ep. To minimise n, we choose p = h
√
2i to obtain
n=ep = 2. Then we have DE(2) = 1.386294, which finally yields the lower bound
µ0 =
DE(n)−log(αv+αv−)
[K :Q]n2 =
1.386294−log(1.098569)
8 = 0.1615.
In order to obtain a lower bound for the canonical height on E(K), we first note that the Tamagawa indices cv at v = p1,p2,p3 are 4, 2, and 1 respectively.
Moreover, one can easily see that both real embeddings of E have only real root, socv+ =cv− = 1. Hence c= lcm{4,2,1}= 4. By Lemma 2.1.1, we finally have
ˆ
h(P)> µ/c2 = 0.2415/42 = 0.0150
Table 5.2: Illustration of the algorithm for Example 5.1.2 Initial Initial Is any Is any intersection Isµa Next Next
µ nmax Bn(µ)<1? empty? lower bound? µ nmax
1.0000 5 No No Fail 0.5000 6 0.5000 6 No No Fail 0.2500 7 0.2500 7 No No Fail 0.1250 8 0.1250 8 No Yes Yes 0.1875 8 0.1875 8 No No Fail 0.1562 9 0.1562 9 No No Fail 0.1406 10 0.1406 10 No Yes Yes 0.1484 10 0.1484 10 No No Fail 0.1445 11 0.1445 11 No No Fail 0.1425 12 0.1425 12 No No Fail 0.1416 13 0.1416 13 No No Fail 0.1411 14 0.1411 14 No Yes Yes 0.1413 14 0.1413 14 No Yes Yes 0.1414 14 0.1414 14 No Yes Yes 0.1415 14 0.1415 14 No Yes Yes
Example 5.1.2. LetE =E2, whereE2is the elliptic curve defined overK =Q(
√
7) given by the Weierstrass equation
E2 : y2+ (3 + 3
√
7)xy+y=x3+ (26 + 4√7)x2+x.
The discriminant ∆ ofE is −937513−299394√7. Moreover, h∆ican be factorised into a product of prime ideals as p1p2p3, where
p1 =h4219,1083 + √ 7i, p2 =h4657,3544 + √ 7i, p3 =h12799,5358 + √ 7i.
Again, since ordpj(∆)<12 for allj,Eis already given by a globally minimal model,
and thus ME = 1. Our algorithm shows that
ˆ
h(P)>0.1415
for all non-torsion points P ∈ Egr(K). This is obtained after a number of refine-
ments as shown in Table 5.2.
Finally, we note that the Tamagawa indices cv at v = p1,p2,p3 are all 1. In
bothv ∈Mr
K, and soc= 2. Hence by Lemma 2.1.1, we have
ˆ
h(P)>0.1415/22 = 0.0353
for all non-torsion points P ∈E(K).
Example 5.1.3. Let E = E3, where E3 is the elliptic curve defined over K =
Q(√10) given by a Weierstrass equation
E3 : y2 =x3+ 125.
Note thatK has class number 2. Decomposing the discriminant ∆ of E into prime ideals, it can be seen thath∆i=h−243356i=p12
1 p32p33p84, where p1 =h5, √ 10i, p2 =h3,4 + √ 10i, p3 =h3,2 + √ 10i, p4 =h2, √ 10i.
Observe that the model of E is now minimal everywhere except at p1. With the
substitutions
x= (√10)2x0, y= (√10)3y0,
we have a new elliptic curve E0 : y02 = x03 + 1/8. This time, however, the model
of E0 is minimal everywhere except at all prime ideals dividing 2. Thus we let
E(p1) = E0 and E(p) = E for any p 6= p
1 in our computation. Our algorithm then
shows that
ˆ
h(P)>0.2859
for all non-torsion pointsP ∈Egr(K). This is based on a number of refinements as
shown in Table 5.3.
To derive a lower bound onE(K), we first note that the Tamagawa indicescv at v =p1,p2,p3,p4 are 1, 2, 2, and 1 respectively. Moreover, we have cv = 1 for both v ∈ Mr
Table 5.3: Illustration of the algorithm for Example 5.1.3 Initial Initial Is any Is any intersection Isµa Next Next
µ nmax Bn(µ)<1? empty? lower bound? µ nmax
1.0000 5 No No Fail 0.5000 6 0.5000 6 No No Fail 0.2500 7 0.2500 7 Yes Skipped Yes 0.3750 7 0.3750 7 No No Fail 0.3125 8 0.3125 8 No No Fail 0.2812 9 0.2812 9 Yes Skipped Yes 0.2968 9 0.2968 9 No No Fail 0.2890 10 0.2890 10 No No Fail 0.2851 11 0.2851 11 Yes Skipped Yes 0.2871 11 0.2871 11 No No Fail 0.2861 12 0.2861 12 No No Fail 0.2856 13 0.2856 13 Yes Skipped Yes 0.2858 13 0.2858 13 Yes Skipped Yes 0.2860 13 0.2860 13 No No Fail 0.2859 14 0.2859 14 Yes Skipped Yes
and thus by Lemma 2.1.1,
ˆ
h(P)>0.2859/22 = 0.0714
for all non-torsion points P ∈E(K).