A Class Of Selmer Groups And Mordell-Weil Group Structure Of Elliptic Curves Class Imaginary Quadratic Fields On One

In this paper, we mainly study the weak Mordell-Weil group for elliptic curves defined over fields having class number one, and get some result about the relations between the ranks of its Mordell-Weil groups and Selmer groups mainly by weak Mordell-Weil theorem, descent via two-isogeny method and Hensel lemma. In the foreword we will give the background and significance of this research; in chapter1, some basic symbols and results we need in our proofs are introduced, while in chapter2, the main results of this paper is presented and in the last chapter the proofs are provided.