# Math 844, Arithmetic Geometry - Fall 2017

# Nigel Boston

## Contact Information

303 Van Vleck Hall
Telephone: 263-4753.

E-mail: *boston@math.wisc.edu*

Homepage

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.
## Homeworks