Elliptic Curves over Arithmetic Fields

In this text, we discuss certain results on elliptic curves over arithmetically important fields. In the first section, we show that for a complex elliptic curve, its group of \C-rational points as an abelian group is a torus. In the second section, we prove Hasse’s inequality for an elliptic curve over a finite field. We then explore elliptic curves over the \Q and give a proof of the Mordell-Weil theorem. Finally, we close with a discussion on some extensions of the Mordell-Weil theorem, including the Birch and Swinnerton-Dyer conjecture.