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Descent by 3-isogeny on elliptic curves

Groen, Steven (2019) Descent by 3-isogeny on elliptic curves. Master's Thesis / Essay, Mathematics.

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Abstract

Descent by a rational isogeny has shown to be a useful tool in computing the rank of elliptic curves. After outlining the general theory, we recall the well-known theory of descent by 2-isogeny. A relatively new approach is descent by a 3-isogeny. In this thesis, we formulate an algorithm that computes the Selmer group of any rational 3-isogeny. We apply these techniques to a family of elliptic curves that allow both types of descent. We force elements into the Selmer group of the 3-isogeny, while we show by 2-descent that the elliptic curves have rank zero. This yields a construction of elements of order 3 in a Tate-Shafarevich group.

Item Type: Thesis (Master's Thesis / Essay)
Supervisor:
Supervisor nameSupervisor E mail
Top, J.J.Top@rug.nl
Supervisor (outside RUG):
Supervisor outside RUG nameSupervisor outside RUG E mail
dr. J.S. (Steffen) Müller, SteffenUNSPECIFIED
Degree programme: Mathematics
Thesis type: Master's Thesis / Essay
Language: English
Date Deposited: 15 Jul 2019
Last Modified: 16 Jul 2019 12:36
URI: http://fse.studenttheses.ub.rug.nl/id/eprint/20240

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