Elliptic Curves over Arithmetic Fields
Keigwin, Benjamin, Mathematics, University of Virginia
Rapinchuk, Andrei, Mathematics, University of Virginia
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.
BA (Bachelor of Arts)
Elliptic curves, Mordell-Weil, curve, Hasse