橢圓曲線在Q上的有限子群與Mazur定理

Abstract

根據Mordell-Weil定理，橢圓曲線群在整數體是有限生成，因此其耦合子群是個有限子群。1977年，Mazur教授給了在有理體上一個很漂亮的結果，他決定了所有的耦合子群之種類。這篇論文就是要探討這漂亮定理的證明以及Mordell-Weil定理。

Let K be a number eld and E=K be an elliptic curve, that is, a smooth projective curve of genus 1 with an distinguished K-rational point chosen. By the Mordell-Weil Theorem, the group of points E(K) is a nitely generated abelian group. Its structure is of the form: E(K) = Etors(K) Zr According to this theorem, we know that Etors(K) is a nite group. In 1977, Mazur [Maz] proved a beautiful theorem for K = Q. It determines all the possible torsion structures of Etors(Q). In this thesis, we try to survey on the proof of this tremendous theorem as well as that of Mordell-Weil Theorem.