Math 844, Arithmetic Geometry - Fall 2017

Nigel Boston

Contact Information

303 Van Vleck Hall

Telephone: 263-4753.

Office Hours: Mondays, 12:00-1:30 (3619 Engineering Hall) and Thursdays, 10:30-12:00 (303 Van Vleck) ; or by appointment.

Sections

Lecture: 9:55-10:45 in B129 Van Vleck.

Useful Web Materials

Elliptic curves resources
Elliptic curve resources
Elliptic curve rank records
Elkies' Computational Number Theory Page
MAGMA

Course Description

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.

Grading

The course grade will be based on 5 homeworks, set every couple of weeks during the semester.