Torsion Points and Torsion Subgroups of Elliptic Curves Over ?(????) And ?(????????)

Abstract

Within the realm of elliptic curve theory, the count of rational points residing on these curves and the intricate nature of their torsion subgroups hold paramount significance. A comprehensive exploration into the diverse torsion subgroups of elliptic curves across varying number fields not only enriches our comprehension of their inherent properties but also bestows us with tools applicable to intricate mathematical conundrums. This paper embarks on this journey by laying the foundation with Mazur's seminal theorem, which serves as a pivotal classification of these torsion subgroups within the rational number field. Subsequently, our investigation broadens to encompass a discussion of these subgroups across general number fields, including the complex number field. Finally, our exploration culminates with a meticulous examination of the distinct properties characterizing torsion points within quadratic number fields.