THE THIRD LECTURE: … and Can You Feel Its Form?

Summary

Geometry or Arithmetic?

When we discussed binary quadratic forms in the First Lecture, there was a marked difference between the definite and indefinite cases. This persists in higher dimensions. The values of a positive definite form are best regarded as squared lengths of vectors in a lattice, and we classify such forms by discussing the shape of this lattice geometrically.

In the indefinite case, when the dimension is at least 3, there is a complete classification, due to Eichler, in terms of an arithmetical invariant called the spinor genus, which is defined in terms of a simpler and more important invariant, the genus.

In my Hedrick Lectures, I compressed these two very different topics into one session. In print it has seemed better to separate them. This Third Lecture mainly concerns the geometrical classification of 3-dimensional lattices in terms of the shape of their Voronoi cells. The arithmetical discussion is postponed until the Fourth Lecture, after which we shall describe the Eichler theorem.

The Voronoi cell

We recall from the first lecture that positive definite binary forms can be specified either by three numbers α, β, γ, called the conorms, or three other numbers a, b, c, called the vonorms. We shall now interpret all these numbers geometrically and generalize them to higher dimensions.