Arithmetic and Spectral Analysis of Poincare Series

The Arithmetic and Spectral Analysis of Poincare series deals with the spectral properties of Poincare series and their relation to Kloosterman sums. In addition to Poincare series for an arbitrary Fuchsian group of the first kind, the spectral expansion of the Kloosterman-Selberg zeta function is analyzed, along with the adellic theory of Poincare series and Kloosterman sums over a global function field. This volume is divided into two parts and begins with a discussion on Poincare series and Kloosterman sums for Fuchsian groups of the first kind. A conceptual proof of Kuznetsov's formula and its generalization are presented in terms of the spectral analysis of Poincare series in the framework of representation theory. An analysis of the spectral expansion of the Kloosterman-Selberg zeta function is also included. The second part develops the adellic theory of Poincare series and Kloosterman sums over a global function field. The main result here is to show that in this context the analogue of the Linnik conjecture can be derived from the Ramanujan conjecture over function fields. Whittaker models, Kirillov models, and Bessel functions are also considered, along with the Kloosterman-spectral formula, convergence, and continuation. This book will be a valuable resource for students of mathematics.