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Hasse's Theorem on Elliptic Curves

Soeten, M.M.J. (2013) Hasse's Theorem on Elliptic Curves. Master's Thesis / Essay, Mathematics.

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Abstract

In 1924, Artin proposed an estimate for the number of points on an elliptic curve over the finite field with p elements. This was proven by Hasse in 1936. However, in 1948, Weil generalized this theorem to a theorem valid for curves of arbitrary genus over the finite field with q elements. A completely elementary proof of Hasse's theorem for elliptic curves was given by Manin in 1956. In this thesis, the proof of Manin has been studied (for the finite field with q elements). Since this proof only has been done for characteristics greater than 3, also the case of characteristic 2 is included. After this, we succeeded in extending Manin's argument to the more difficult case of characteristic 2. After the theoretical part is done, the proof has been checked in the computer program Magma. Finally, we tried to extend the elementary proof of Manin (valid for curves of genus 1) to curves of genus 2. Here we tried to do some steps of the proof, where again we used Magma to do the calculations.

Item Type: Thesis (Master's Thesis / Essay)
Degree programme: Mathematics
Thesis type: Master's Thesis / Essay
Language: English
Date Deposited: 15 Feb 2018 07:53
Last Modified: 15 Feb 2018 07:53
URI: http://fse.studenttheses.ub.rug.nl/id/eprint/10999

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