Torsıon Structure Of Elliptic Curves Over Small Number Fields

Although it is well known which groups appear as torsion subgroup of an elliptic curve over a number field K where [K : Q] = 1,2,3, a similar classification is not known for number fields of higher degrees. On the other hand, it is well known which groups can arise as a torsion subgroup for infinitely many Q-isomorphism classes of elliptic curves over a number field K where [K : Q] = 4,5,6. In this thesis, we focus on the torsion subgroups of elliptic curves occurring over a fixed number field K with [K : Q] = 4,5,6. Our approach relies on analyzing the arithmetic structure of the modular curves X1(m,mn), m ≥ 1. First, we investigate the possibility of the growth in torsion subgroups of X1(m,mn) over quartic, quintic and sextic number fields. In the case of growth in torsion, we check the new points and try to answer the following question: ”Do new points give an elliptic curve with the desired torsion?”. Secondly, we check the existence of torsion subgroups over cubic, quartic and quintic number fields with the smallest discriminant and having different Galois groups.