In this tutorial, we will study the arithmetic theory of quadratic forms in a down-to-earth and self-contained manner. Secretly, this tutorial is meant to be an introduction to some of the techniques of modern number theory. The hope is to show what algebraic number theory looks like outside of the typical undergraduate course.

This tutorial is roughly split into two parts. For the first part, we will look at the classification problem of quadratic forms over number fields, \(p\)-adic fields, and their rings of integers. The big theorems that we will prove (from scratch) are the Hasse-Minkowski theorem (an instance of the very general local-to-global philosophy pervading all of modern number theory), and the full classification of arithmetic lattices. For the second part, we will build up to a statement of the Siegel-Weil mass formula and its applications. For the most part, we will be solely working over \(\mathbb{Q}\) and its completions.

For more detail, as well as for prerequisites, other summer tutorials at Harvard, and requirements for the tutorial, check this out.

Course Notes

I will be writing my own course notes, which are available here.

There were also weekly problem sets. The problem sets are just exercises for those who are interested in becoming more comfortable with the topic. Problem Sets: 1, 2, 3, 4, 5, and 6.