TERMODINÁMICA DEL ELEMENTO COBRE EN SOLUCIÓN ACUOSA

Finally, it should be noted that the lower bound we obtain is not model-independent, unlike the one of Hindry and Silverman [HS88, Theorem 0.3]. For example, the values αv in Section 2.2.2 depend on the coefficients of the Weierstrass equation of E. At present, we have not systematically investigated how the bound obtained by our algorithm is affected by a change of model. As mentioned in Chapter 2, however, our formulas can be simplified if E is given by a globally minimal model. Regarding the computational complexity, it can be seen that computing Bn(µ)

is less time-consuming than computing Sn(v), which in turn is less time-consuming

than computing Tn(v). Therefore it is plausible to use Bn(µ) as the first criterion,

Let c be the least common multiple of all Tamagawa indices as in Section 2.1. As pointed out by an anonymous referee of [Tho10], it may be possible to obtain a larger lower bound by making use of the explicit formulas for the local heights at non-archimedean places of bad reduction (see, e.g., [Sil88, Theorem 5.2]), provided thatcis large. This approach, however, is different to ours which uses the subgroup of points of good reduction. In particular, our lower bound on E(K) will be small if cis large. Nevertheless, it might be an interesting area for further study.

In conclusion, we have completed our work on height bound by introducing a method for solving a system of certain inequalities on complex embeddings. Our method involves a number of approximation techniques which eventually yield an approximate region corresponding to each inequality, where finding a solution to the system of these inequalities is equivalent to finding the intersection of all such regions. Together with our results from Chapter 2, we finally obtain an algorithm for computing a lower bound for the canonical height on elliptic curves overnumber fields in general.

Finally, in order to solve a system of inequalities using the methods in Section 2.5 and 3.3, we need to computeperiod lattices of certain real and complex embeddings, as well as elliptic logarithms of certain real and complex points. Nevertheless, algorithms for determining both quantities are currently available only for elliptic curves overR(see Section 1.3 for more discussion). Motivated by this problem, the next chapter will aim to develop a complete method for computing period lattices of elliptic curves over C, and elliptic logarithms of complex points.

Chapter 4

Period Lattices and Complex

Elliptic Logarithms

We will now move on to the second main result of this thesis, where we present a complete method for computing period lattices of elliptic curves over C, and then generalise it to computeelliptic logarithmsof complex points. Based on the complex

arithmetic-geometric mean (AGM) first studied by Gauss, our method will allow one to compute both quantities to a high degree of precision very quickly. For more background on this chapter, see Section 1.3.

The work in this chapter is done in collaboration with Professor John E. Cre- mona at the University of Warwick. Another version of this chapter has been submitted for publication as a joint paper [CT].

4.1

Introduction

In this chapter, we will assume that an elliptic curve E is defined over C, and is given by a Weierstrass equation of the form

E :Y2 _{= 4(X−e}

1)(X−e2)(X−e3),

where all the rootsej ∈Care distinct and P

jej = 0. As mentioned in Section 1.3,

it is well known that there exists an isomorphism (of complex analytic Lie groups) C/Λ→E(C) for some lattice Λ, given by the map

z (mod Λ) 7→ P = (℘Λ(z), ℘0Λ(z))

0 (mod Λ) 7→ O.

(4.1)

Definition. LetE be an elliptic curve defined overC, theperiod lattice of E is the

lattice Λ for which E(C)∼=C/Λ via (4.1).

To be precise, we take Λ to be the lattice of periods of the invariant differential

dX/Y on E. It is a discrete subgroup of C spanned by a Z-basis {w1, w2} with

w2/w1 ∈/ R.

Definition. The inverse map of (4.1) is called theelliptic logarithm. ForP ∈E(C),

we say that a value z such that

P 7→z (mod Λ)

via this inverse is an elliptic logarithm of P (note that z is determined modulo Λ). From this, two natural questions are:

1. Given a Weierstrass equation of E, how can we compute a Z-basis for its period lattice Λ?

2. Given a point P ∈E(C), how can we compute its elliptic logarithmz? For elliptic curves over R, these questions have been answered satisfactorily, since algorithms for computing period lattices of elliptic curves overR and elliptic logarithms of real points are well-known and available in the literature (see, e.g., [Coh93, Algorithm 7.4.7 and 7.4.8] or [Cre97, §3.7]). The theory behind these algorithms, which heavily relies on the AGM of positive real numbers, is explained

succinctly by Bost and Mestre [BM88]. The situation for elliptic curves over C, however, is less satisfactory.

In this chapter, we therefore aim to develop a complete method for computing period lattices and elliptic logarithms for elliptic curves over C. Our approach will closely follow that of [BM88] in the real case, and will also illustrate the connection between the following three classes of objects:

• AGM sequences over C, which were first studied by Gauss and have been explored in depth by Cox [Cox84];

• Chains of lattices inC;

• Chains of 2-isogenies between elliptic curves defined overC.

This connection will allow us to derive an explicit formula for computing the period lattice of E, which yields the first algorithm of this chapter. We then continue further by generalising it to an algorithm for computing elliptic logarithms of points inE(C). Finally, we illustrate the efficiency of both algorithms via some examples. For computational purposes, we have implemented both algorithms inMAGMA

(see Appendix A.1 for the source code); these have been also implemented indepen- dently in Sage (available from version 4.4) by Professor John E. Cremona.