A classical question in number theory is: given a positive integer n,
how many ways can we represent n as a sum of k integer squares?
This question has been approached using the theory of modular forms, and in some
cases this approach has yielded some beautiful formulas. In the 1930's, Siegel
generalised this question: given a lattice L with a positive definite quadratic form q,
and given another quadratic form q', on how many sublattices of L does q restrict
to q'? We will explore this question using the theory of Siegel modular forms;
in particular, we will use Siegel's generalised theta series and Siegel-Eisenstein series.
We will discuss how to construct these latter two sorts of objects, and precisely
how they relate to each other.
We will not assume knowledge of modular forms.