Torsion points on elliptic curves over number fields of small degree

Leiden Repository

Torsion points on elliptic curves over number fields of small degree

Type: Doctoral Thesis
Title: Torsion points on elliptic curves over number fields of small degree
Author: Derickx, M.
Issue Date: 2016-09-21
Keywords: Chabauty, gonality
Curve
Elliptic, torsion
Modular
Number theory
Symmetric power
Abstract: Barry Mazur famously classified the finitely many groups that can occur as a torsion subgroup of an elliptic curve over the rationals. This thesis deals with generalizations of this to higher degree number fields. Merel proved that for all integers d one has that the number of isomorphsim classes of torsion groups of elliptic curves over number fields of degree d is finite. This thesis consists of 4 chapters, the first is introductory and the other tree are research articles. Chapter two deals with the computation of gonalities of modular curves, and the application of these computations to the question which cyclic subgroups can occur as the torsion subgroup of infinitely many non-isomorphic elliptic curves over number fields of degree <7. In the second chapter a general theory for finding rational points on symmetric powers of curves is developed that is similar to symmetric power Chabauty. Application of this theory to symmetric powers of modular curves allows us to determine which primes can divide the order of the torsion subgroup of an elliptic curve over a number field of degree <7. The last chapter studies elliptic curve with a point of order 17 over a number field of degree 4.
Promotor: Supervisor: S.J. Edixhoven Co-Supervisor: L. van Geemen, P. Parent
Faculty: Science
University: Leiden University
Handle: http://hdl.handle.net/1887/43186
 

Files in this item

Description Size View
application/pdf Full Text 2.120Mb View/Open
application/pdf Cover 546.9Kb View/Open
application/pdf Title Pages_Contents 1.299Mb View/Open
application/pdf Chapter 1 472.5Kb View/Open
application/pdf Chapter 2 563.0Kb View/Open Full text at publisher site
application/pdf Chapter 3 705.1Kb View/Open
application/pdf Chapter 4 544.8Kb View/Open
application/pdf Acknowledgements 186.0Kb View/Open
application/pdf Summary in Dutch 204.1Kb View/Open
application/pdf Curriculum Vitae 123.4Kb View/Open
application/pdf Propositions 230.5Kb View/Open

This item appears in the following Collection(s)