Nigel Boston, Fall 2017:
This course is an introduction to elliptic curves and arithmetic geometry. Elliptic curves play a central role in modern arithmetic geometry and even in applications to cryptography. On the elliptic curve side, we'll cover elliptic curves over finite fields, over the complex numbers, and over the rationals. We'll prove the Mordell-Weil theorem, discuss torsion, Tate-Shafarevich groups, elliptic curves with CM, and integer points on elliptic curves and Diophantine approximation. We'll discuss associated modular forms and L-functions and the notion of modularity of an elliptic curve. We'll discuss generalizations to other varieties, in particular curves and questions regarding their rational points. Applications and open problems will be mentioned throughout.