Computations of Automorphic Functions on Fuchsian Groups
This thesis consists of four papers which all deal with computations of automorphic functions on cofinite Fuchsian groups. In the first paper we develop an algorithm for numerical computation of the Eisenstein series. We focus in particular on the computation of the poles of the Eisenstein series. Using our numerical methods we study the spectrum of the Laplace-Beltrami operator as the surface is being deformed. Numerical evidence of the destruction of ?0(5)-cusp forms is presented. In the second paper we use the algorithm described in the first paper. We present numerical investigations of the value distribution and distribution of Fourier coefficients of the Eisenstein series E(z;s) on arithmetic and non-arithmetic Fuchsian groups. Our numerics indicate a Gaussian limit value distribution for a real-valued rotation of E(z;s) as Re s=1/2, Im s ? ? and also, on non-arithmetic groups, a complex Gaussian limit distribution for E(z;s) when Re s > 1/2 near 1/2 and Im s ? ?, at least if we allow Re s ? 1/2 at some rate. Furthermore, on non-arithmetic groups and for fixed s with Re s ? 1/2 near 1/2, our numerics indicate a Gaussian limit distribution for the appropriately normalized Fourier coefficients.In the third paper we develop algorithms for computations of Green's function and its Fourier coefficients, Fn(z;s), on Fuchsian groups with one cusp. Also an analog of a Rankin-Selberg bound for Fn(z;s) is presented.In the fourth paper we use the algorithms described in the third paper. We present some examples of numerical investigations of the value distribution of the Green's function and of its Fourier coefficients on PSL(2,Z). We also discuss the appearance of pseudo cusp forms in a numerical experiment by Hejhal.
Source Type:Doctoral Dissertation
Keywords:Mathematics; 11F72; 11F03; 11F06; 11Y35; 11-04; 11Y16; 33F06; 34B27; 33F05; MATEMATIK
Date of Publication:01/01/2007