Finite groups of automorphisms on genus one curves without rational points

Roelfszema, Majken (2018) Finite groups of automorphisms on genus one curves without rational points. Master's Thesis / Essay, Mathematics.

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Abstract

Mazur’s theorem concerning rational torsion points on elliptic curves over Q (the rationals) can be reformulated as follows. Given is a genus one curve X over Q. Let σ and τ be involutions in the automorphism group over Q of X, Aut(X), and suppose the group G ⊂ Aut(X) generated by σ and τ is finite. Then G is a dihedral group of order 2n, with n ∈ {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12}. Examples where X = E is an elliptic curve over Q are relatively easy to construct. The aim of this thesis is for each possible n to find curves X where X(Q) = ∅. In order to visualize the situation, the context of Poncelet figures is used. A Poncelet figure containing an n-gon can be made using X exactly when Aut(X) has such a subgroup G of order 2n.

Item Type:	Thesis (Master's Thesis / Essay)
Supervisor name:	Top, J.
Degree programme:	Mathematics
Thesis type:	Master's Thesis / Essay
Language:	English
Date Deposited:	13 Jul 2018
Last Modified:	20 Jul 2018 12:25
URI:	https://fse.studenttheses.ub.rug.nl/id/eprint/17862

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