Pólya's shire theorem for automorphic functions (Q1412314)

The author studies the accumulation set of zeros of successive derivatives of automorphic functions in the hyperbolic plane. This study generalizes a classical result by Pólya for meromorphic functions in the complex plane (the so-called \textit{Shire theorem}). A nice ``Shire theorem'' is obtained for functions which are automorphic with respect to a Fuchsian group of the first kind. Under some conditions (e.g. when the group is cocompact), the result agrees exactly with Pólya's.