This thesis presents some major improvements in the following computations: a lower bound for the canonical height, period lattices, and elliptic logarithms. On computing a lower bound for the canonical height, we have successfully generalised the existing algorithm of Cremona and Siksek [CS06] to elliptic curves over totally real number fields, and then to elliptic curves over number fields in general. Both results, which are also published in [Tho08] and [Tho10] respectively, will be fully explained in Chapter 2 and 3. In Chapter 4, we give a complete method on computing period lattices of elliptic curves over C, whereas this was only possible for elliptic curves over R in the past. Our method is based on the concept of arithmetic-geometric mean (AGM). In addition, we extend our method further to find elliptic logarithms of complex points. This work is done in collaboration with Professor John E. Cremona; another version of this chapter has been submitted for publication [CT]. In Chapter 5, we finally illustrate the applications of our main results towards certain computations which did not work well in the past due to lack of some information on elliptic curves. This includes determining a Mordell{Weil basis, finding integral points on elliptic curves over number fields [SS97], and finding elliptic curves with everywhere good reduction [CL07]. A number of computer programs have been implemented for the purpose of illustration and verification. Their source code (written in MAGMA) can be found in Appendix A.
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