Deformations of Maass forms

Abstract

We describe numerical calculations which examine the Phillips-Sarnak conjecture concerning the disappearance of cusp forms on a noncompact finite volume Riemann surface $S$ under deformation of the surface. Our calculations indicate that if the Teichmuller space of $S$ is not trivial then each cusp form has a set of deformations under which either the cusp form remains a cusp form, or else it dissolves into a resonance whose constant term is uniformly a factor of $10^{8}$ smaller than a typical Fourier coefficient of the form. We give explicit examples of those deformations in several cases.