Deformations of Maass forms

Abstract

Abstract. We describe numerical calculations which examine the PhillipsSarnak 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 Teichmüller 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.

Computations of Eisenstein series on Fuchsian groups

“…A discussion of the generators of the groups Γ 0 (5), Γ 0 (5), Γ a,r and Γ a,r can be found in [FL05,§3], where a more general case is considered, and in [Ave03] where we have worked out the details for our special case. (Note that the parameters in [FL05] correspond to ours as a = b F L and r = 1/a F L .)…”

“…(Note that the parameters in [FL05] correspond to ours as a = b F L and r = 1/a F L .) Here we simply state the generators of Γ a,r and Γ a,r in Table 1.…”

Computing Maass cusp form on general hyperbolic torus

“…Avelin (2003) and Farmer (2005) continue the study on the deformation of Maass cusp form. The deformed surface is represented by Teichmuller space T (s) and then they used Fricke involution to reduce the number of cusps to one for the congruence subgroup [9,10]. Recently, Chan et.…”

Random polynomials, random matrices andL-functions

“…In addition, eigenvalues of other compact classical matrix groups give a good model of the zeros of various families of L-functions [16,15,24,14,8]. Furthermore, the characteristic polynomials of the matrices provide a good model of the L-functions themselves [19,3,18,4,13].…”

“…In Appendix A we discuss the example of Epstein zeta functions. It is also possible to create Dirichlet series with a functional equation but no Euler product from Maass forms on a nonarithmetic group [7]. Since these functions do not have an Euler product, they are not expected to satisfy the Riemann hypothesis.…”