Torsion points on elliptic curves (original) (raw)

The paper explores torsion points on elliptic curves, particularly focusing on the Uniform Boundedness Conjecture regarding the orders of torsion subgroups in relation to number fields. It reviews historical results such as those by Manin and Mazur while presenting new findings that affirm the conjecture's strong form for quadratic fields, demonstrating a criterion for the absence of certain rational points. The implications extend to future potential breakthroughs in isogenies and related problems.