8 research outputs found
Torsion of elliptic curves over cyclic cubic fields
We determine all the possible torsion groups of elliptic curves over cyclic
cubic fields, over non-cyclic totally real cubic fields and over complex cubic
fields.Comment: 14 page
Torsion points on elliptic curves over number fields of small degree
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
Torsion points on elliptic curves over number fields of small degree
We determine the set of possible prime orders of -rational points
on elliptic curves over number fields of degree , for and