  The $GL_2$ Poincar\'{e} series giving the subconvexity results of Diaconu and
Garrett is the solution to an automorphic partial differential equation,
constructed by winding-up the solution to the corresponding differential
equation on the free space. Generalizing this approach allows design of higher
rank Poincar\'{e} series with specific number theoretic applications in mind: a
Poincar\'{e} series for producing an explicit formula for the number of lattice
points in an expanding region in a symmetric space, a Poincar\'{e} series
producing moments of $GL_n \times GL_n$ L-functions, and a Poincar\'{e} series
designed for applications involving pseudo-Laplacians.
