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- W1566991888 abstract "This thesis presents some major improvements in the following computations: alower bound for the canonical height, period lattices, and elliptic logarithms.On computing a lower bound for the canonical height, we have successfullygeneralised the existing algorithm of Cremona and Siksek [CS06] to elliptic curvesover totally real number fields, and then to elliptic curves over number fields ingeneral. 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 ellipticcurves over C, whereas this was only possible for elliptic curves over R in thepast. Our method is based on the concept of arithmetic-geometric mean (AGM).In addition, we extend our method further to find elliptic logarithms of complexpoints. This work is done in collaboration with Professor John E. Cremona; anotherversion of this chapter has been submitted for publication [CT].In Chapter 5, we finally illustrate the applications of our main results towardscertain computations which did not work well in the past due to lack of someinformation on elliptic curves. This includes determining a Mordell{Weil basis,finding integral points on elliptic curves over number fields [SS97], and findingelliptic curves with everywhere good reduction [CL07].A number of computer programs have been implemented for the purpose ofillustration and verification. Their source code (written in MAGMA) can be foundin Appendix A." @default.
- W1566991888 created "2016-06-24" @default.
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- W1566991888 date "2011-02-01" @default.
- W1566991888 modified "2023-09-26" @default.
- W1566991888 title "Heights on elliptic curves over number fields, period lattices, and complex elliptic logarithms" @default.
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