北京雁栖湖应用数学研究院

Elliptic curves are among the most fascinating and important objects in modern mathematics, with deep connections to number theory, algebraic geometry, and cryptography. This course aims to provide a comprehensive introduction to the arithmetic properties of elliptic curves, suitable for advanced undergraduate and graduate students in mathematics, as well as anyone interested in exploring this rich and beautiful area of mathematics.

We begin with the fundamental definitions and geometric interpretations of elliptic curves, focusing on their representation through Weierstrass equations and the group law that defines the addition of points on the curve. Following this, we study rational points on elliptic curves, such as Mordell's theorem. We also introduce elliptic curves over finite fields. Additionally, we will discuss several important applications of elliptic curves in both number theory and cryptography.