Lattices and Codes (a brief introduction)

The goal here to is to introduce irreducible root systems, and through them try to classify certain lattices. As we’ve seen, lattices are just about the best thing there is in number theory (maybe), since it’s easy to pop out a modular form from a lattice under the right circumstances. Luckily, we’ll be working in those very circumstances! To get there, we’re going to have to define what a code is, somewhat unfortunately. I’m going to avoid as much of the code talk as possible, but some of it is pretty useful, such as define the dual lattice and examining un-normalised lattices (i.e. volume not equal to 1). All the things covered in class I will attempt to summarise rather than detail. The source of this talk is Lattices and Codes by Wolfgang Ebeling.