Arithmetic Applications of Hyperbolic Lattice Counting

The classical Gauss circle problem concerns estimating the number of lattice points inside a Euclidean disc of radius $R$.  Equivalently, it asks for the average number of ways to write a positive integer $n<R^2$ as a sum of two squares.  In this talk, we discuss various analogous situations in the hyperbolic space.  Rather than focusing on the underlying geometric results themselves, we treat them as black boxes and concentrate on their arithmetic consequences.  These include results on the distribution of sums of squares, norms of ideals in quadratic fields, and more.