On low-lying zeros of automorphic L-functions
Abstract (Summary)In this thesis we study assuming the generalized Riemann Hypothesis the distribution of low-lying zeros, zeros at or near the central point s=½, for the family of symmetric power L-functions associated with cusp forms on GL(2) and for those of Hecke L-functions associated with cubic characters. By studying the one-level scaling density statistics for these families of L-functions we provide further evidence to the truth of the density conjecture of Katz and Sarnak which asserts that the statistics of low-lying zeros of a family of L-functions should coincide with the corresponding statistics of the eigenvalues of matrices from a suitable family of classical compact groups determined by the symmetry type of the family. Following the approach of Iwaniec, Luo and Sarnak we determine the corresponding symmetry types for the family of symmetric rth power L-functions where we use the recent works of Kim and Shahidi for r=3 and r=4 while we assume the truth of Langlands Functoriality Conjecture for r > 4. As for the family of Hecke L-functions we identify the symmetry type as unitary and our computations depend on an analog of Polya-Vinogradov inequality due to Heath-Brown and Patterson and the generalized Riemann Hypothesis as well as Patterson's work on the distribution of cubic Gauss sums at prime arguments.
School:The Ohio State University
School Location:USA - Ohio
Source Type:Master's Thesis
Date of Publication:01/01/2005