Lattice points on determinant surfaces and the spectrum of the automorphic Laplacian

We use classical Fourier analysis along with tools from the spectral theory of Automorphic forms to derive an asymptotic formula with a strong error term for the number of integer solutions $(a,b,c,d)$ inside the expanding box $[-X,X{]}^4$ to the determinant equation $ad-bc=r$, where $r\mathrm{\ne }0$ is a fixed integer. Furthermore, we apply our method to study sums over these solutions where the variables are weighted by periodic arithmetical functions in two of the variables in one case, and by an arbitrary sequence of complex numbers in another.