Some Series Of Quadratic Twists Of Elliptic Curves

In this paper, we study several series of quadratic twists of elliptic curves. We construct several series of congruent elliptic curves with rank zero and elliptic curves over function fields with rank one.In Chapter 1, we recall the history of BSD conjecture and the congruent number problem, and state our main results.In Chapter 2, we recall basic theory and notations on elliptic curves.In Chapter 3, we give three different series of non-congruent numbers. In Section 1-3, we introduce the definition of congruent numbers and basic notations, state main result and give our strategy. We estimate the image of the Selmer groups of isogenies of degree 2, to ensure the weak Mordell-Weil group is minimal. Then we can obtain congruent elliptic curves with non-trivial Tate-Shafarevich group. In Section 4-8, we calculate the Selmer groups and finish the proof.In Chapter 4, we introduce the modularity of elliptic curves over function fields and modular curves. We give a function field version of Birch lemma. We study the Heegner points and prove it is non-torsion, then we obtain several series of elliptic curves with rank one.We give some perspective at the last chapter.