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Proving (in)dependence of rational points on elliptic curves

van Oppen, Yulan (2018) Proving (in)dependence of rational points on elliptic curves. Bachelor's Thesis, Mathematics.


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It is well known that independence of rational points on an elliptic curve may be proved using canonical heights. In this thesis, an alternative method is explained and implemented in the computer algebra system PARI. Given a set of rational points on an elliptic curve points, we can prove their independence by constructing an injective homomorphism from the Mordell-Weil group (modulo doubles) to a binary vector space. If the images of these points by this homomorphism are independent, then the points are. We may also find dependence relations using this homomorphism. This has a great advantage over finding dependence relations using canonical heights, in which case we can only be certain they hold numerically. If a number of rational points on an elliptic curve is shown to be independent, then this number is a lower bound on the rank of the elliptic curve. Finding the rank is a necessary first step for finding the generators of the Mordell-Weil group.

Item Type: Thesis (Bachelor's Thesis)
Supervisor name: Muller, J.S.
Degree programme: Mathematics
Thesis type: Bachelor's Thesis
Language: English
Date Deposited: 17 Jul 2018
Last Modified: 17 Jul 2018 14:16

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