Moments of automorphic L-functions

Abstract

This thesis is devoted to investigation of moments of automorphic L-functions, especially on the central values or the edges of the critical strip of automorphic L-functions. There are nine chapters. Chapter 1 is an introduction and provides some background on the analytic theory of automorphic forms. Chapters 2, 3, 4, 5 and 6 are about L-functions associated to the holomorphic cusp forms, while Chapters 7, 8 and 9 are focused on the L-functions associated to the Maass forms.

Chapter 2 is the study of the first moment of the symmetric-square L-functions associated to the holomorphic cusp forms. Asymptotic formulae for the twisted first moment of central values of the symmetric-square L-functions with harmonic weight, in the weight aspect are obtained. The result (in Theorem 2.1.1) extends and improves the known results in the literature. As an application, it is applied to derive an asymptotic formula for the first moment of central values of the symmetric-square L-functions without harmonic weight, under the assumption of the non-negativity of symmetric-square L-functions at the center of critical strip. Analogous new formulae without harmonic weight of the first and second moments of the Hecke L-functions are proved in Chapter 3. Unlike the case in Chapter 2, the results in Chapter 3 are unconditional.

In Chapter 4, complex moments of the symmetric power L-functions of primitive forms at the edge of the critical strip twisted by the central values of the symmetric square L-functions, with or without harmonic weight, are investigated in the weight aspect. The similar problem in the level aspect was treated by Lau, Royer and Wu. The theme of Chapter 5, as well as Chapter 4, is to examine, in the weight aspect, complex moments of the symmetric power L-functions of primitive forms at the edge of the critical strip twisted by the central values of the square L-functions or the square of L-functions, with or without harmonic weight. All the above mentioned results, appeared in Chapters 4 and 5, are in the asymptotic form, that is, given by a formula consisting of a main term and an error term.

Chapter 6 investigates the asymptotic behavior of the main terms of the results in Chapters 4 and 5. In particular, precise expansions of high moments are given.

In Chapters 7, 8 and 9, the previous studies are carried over to Maass forms in the spectral aspect. The first two moments of central values of symmetric square L-functions associated to Maass forms are computed in Chapter 7. The first four moments of central values of L-functions associated to Maass forms are obtained in Chapter 8. Chapter 9 is to research the mixed moments of central values of symmetric square L-functions twisted by the central values of L-functions or the square of L-functions. These investigations for Maass form are not yet done in the literature.