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postgraduate thesis: Moments of automorphic Lfunctions
Title  Moments of automorphic Lfunctions 

Authors  
Issue Date  2016 
Publisher  The University of Hong Kong (Pokfulam, Hong Kong) 
Citation  Ng, M. [吳銘豪]. (2016). Moments of automorphic Lfunctions. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR. Retrieved from http://dx.doi.org/10.5353/th_b5731094 
Abstract  This thesis is devoted to investigation of moments of automorphic Lfunctions, especially on the central values or the edges of the critical strip of automorphic Lfunctions. 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 Lfunctions associated to the holomorphic cusp forms, while Chapters 7, 8 and 9 are focused on the Lfunctions associated to the Maass forms.
Chapter 2 is the study of the first moment of the symmetricsquare Lfunctions associated to the holomorphic cusp forms. Asymptotic formulae for the twisted first moment of central values of the symmetricsquare Lfunctions 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 symmetricsquare Lfunctions without harmonic weight, under the assumption of the nonnegativity of symmetricsquare Lfunctions at the center of critical strip. Analogous new formulae without harmonic weight of the first and second moments of the Hecke Lfunctions 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 Lfunctions of primitive forms at the edge of the critical strip twisted by the central values of the symmetric square Lfunctions, 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 Lfunctions of primitive forms at the edge of the critical strip twisted by the central values of the square Lfunctions or the square of Lfunctions, 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 Lfunctions associated to Maass forms are computed in Chapter 7. The first four moments of central values of Lfunctions associated to Maass forms are obtained in Chapter 8. Chapter 9 is to research the mixed moments of central values of symmetric square Lfunctions twisted by the central values of Lfunctions or the square of Lfunctions. These investigations for Maass form are not yet done in the literature. 
Degree  Doctor of Philosophy 
Subject  Lfunctions Automorphic functions 
Dept/Program  Mathematics 
Persistent Identifier  http://hdl.handle.net/10722/224649 
HKU Library Item ID  b5731094 
DC Field  Value  Language 

dc.contributor.author  Ng, Mingho   
dc.contributor.author  吳銘豪   
dc.date.accessioned  20160411T23:15:18Z   
dc.date.available  20160411T23:15:18Z   
dc.date.issued  2016   
dc.identifier.citation  Ng, M. [吳銘豪]. (2016). Moments of automorphic Lfunctions. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR. Retrieved from http://dx.doi.org/10.5353/th_b5731094   
dc.identifier.uri  http://hdl.handle.net/10722/224649   
dc.description.abstract  This thesis is devoted to investigation of moments of automorphic Lfunctions, especially on the central values or the edges of the critical strip of automorphic Lfunctions. 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 Lfunctions associated to the holomorphic cusp forms, while Chapters 7, 8 and 9 are focused on the Lfunctions associated to the Maass forms. Chapter 2 is the study of the first moment of the symmetricsquare Lfunctions associated to the holomorphic cusp forms. Asymptotic formulae for the twisted first moment of central values of the symmetricsquare Lfunctions 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 symmetricsquare Lfunctions without harmonic weight, under the assumption of the nonnegativity of symmetricsquare Lfunctions at the center of critical strip. Analogous new formulae without harmonic weight of the first and second moments of the Hecke Lfunctions 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 Lfunctions of primitive forms at the edge of the critical strip twisted by the central values of the symmetric square Lfunctions, 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 Lfunctions of primitive forms at the edge of the critical strip twisted by the central values of the square Lfunctions or the square of Lfunctions, 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 Lfunctions associated to Maass forms are computed in Chapter 7. The first four moments of central values of Lfunctions associated to Maass forms are obtained in Chapter 8. Chapter 9 is to research the mixed moments of central values of symmetric square Lfunctions twisted by the central values of Lfunctions or the square of Lfunctions. These investigations for Maass form are not yet done in the literature.   
dc.language  eng   
dc.publisher  The University of Hong Kong (Pokfulam, Hong Kong)   
dc.relation.ispartof  HKU Theses Online (HKUTO)   
dc.rights  Creative Commons: AttributionNonCommerical 3.0 Hong Kong License   
dc.rights  The author retains all proprietary rights, (such as patent rights) and the right to use in future works.   
dc.subject.lcsh  Lfunctions   
dc.subject.lcsh  Automorphic functions   
dc.title  Moments of automorphic Lfunctions   
dc.type  PG_Thesis   
dc.identifier.hkul  b5731094   
dc.description.thesisname  Doctor of Philosophy   
dc.description.thesislevel  Doctoral   
dc.description.thesisdiscipline  Mathematics   
dc.description.nature  published_or_final_version   