Sieving on elliptic curves

Dijk, Thomas (2020) Sieving on elliptic curves. Bachelor's Thesis, Mathematics.

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Abstract

In this project, we extend the results of Appendix A of ``Computing integral points on hyperelliptic curves using quadratic Chabauty''. We do this to come up with an algorithm to compute integral points on elliptic curves with rank 1 and trivial torsion. To do this, the appendix uses quadratic Chabauty combined with the following problem: `Let C be an elliptic curve over Q of rank 1 and trivial torsion. Given a collection of odd primes p1,...pl of good reduction and finite subsets Si of C(Qpi), we want to show that there is no point in C(Q) that belongs to each Si.' To approach this problem, we combine information coming from the group structure of C(Q) with information coming from reducing points modulo pi and information of the logarithm function on C(Qp). To obtain comparable information, we use prime sequences with certain properties. To do this, we first have to introduce the projective plane, and algebraic curves in it. Then work towards Bezout's theorem and the Cayley-Bacharach theorem. Then we introduce elliptic curves and define addition on it for which we give expplicit formulas to compute the coordinates of the addition of two points. We will also state Mordell's theorem and Siegel's theorem. Then, before we start sieving, we look at elliptic curves over finite fields and over the set of p-adic numbers and introduce the reduction modulo p map and the logarithm map, which are important for the sieving.

Item Type:	Thesis (Bachelor's Thesis)
Supervisor name:	Bianchi, F.
Degree programme:	Mathematics
Thesis type:	Bachelor's Thesis
Language:	English
Date Deposited:	01 Sep 2020 13:59
Last Modified:	01 Sep 2020 13:59
URI:	https://fse.studenttheses.ub.rug.nl/id/eprint/23290

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