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postgraduate thesis: Trace formulas and their applications on Hecke eigenvalues
Title  Trace formulas and their applications on Hecke eigenvalues 

Authors  
Advisors  Advisor(s):Lau, YK 
Issue Date  2012 
Publisher  The University of Hong Kong (Pokfulam, Hong Kong) 
Citation  Wang, Y. [王英男]. (2012). Trace formulas and their applications on Hecke eigenvalues. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR. Retrieved from http://dx.doi.org/10.5353/th_b4832952 
Abstract  The objective of the thesis is to investigate the trace formulas and their applications on Hecke eigenvalues, especially on the distribution of Hecke eigenvalues. This thesis is divided into two parts..
In the first part of the thesis, a review is firstly carried out for the equidistribution of Hecke eigenvalues as primes vary and for the expected size of the error term in this equidistribution problem. Then the Kuznetsov trace formula is applied to prove a result on the size of the error term in the asymptotic distribution formula of Hecke eigenvalues. These eigenvalues become equidistributed with respect to the padic Plancherel measures as Hecke eigenforms vary. Next, this problem is generalized to Satake parameters of GL2 representations with prescribed supercuspidal local representations. Such a generalization is novel to the case of classical automorphic forms. To achieve this result, a trace formula of ArthurSelberg type with a couple of key refinements is used.
In the second part of the thesis, a density theorem is proved which counts the number of exceptional nontrivial zeros of a family of symmetric power Lfunctions attached to primitive Maass forms in the critical strip. In addition, a large sieve inequality of ElliottMontgomeryVaughan type for primitive Maass forms is established. The density theorem and large sieve inequality have many applications. For instance, they are used to prove statistical results on Hecke eigenvalues of primitive Maass forms and the extreme values of the symmetric power Lfunctions attached to primitive Maass forms. 
Degree  Doctor of Philosophy 
Subject  Trace formulas. Eigenvalues. 
Dept/Program  Mathematics 
Persistent Identifier  http://hdl.handle.net/10722/173871 
HKU Library Item ID  b4832952 
DC Field  Value  Language 

dc.contributor.advisor  Lau, YK   
dc.contributor.author  Wang, Yingnan   
dc.contributor.author  王英男   
dc.date.issued  2012   
dc.identifier.citation  Wang, Y. [王英男]. (2012). Trace formulas and their applications on Hecke eigenvalues. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR. Retrieved from http://dx.doi.org/10.5353/th_b4832952   
dc.identifier.uri  http://hdl.handle.net/10722/173871   
dc.description.abstract  The objective of the thesis is to investigate the trace formulas and their applications on Hecke eigenvalues, especially on the distribution of Hecke eigenvalues. This thesis is divided into two parts.. In the first part of the thesis, a review is firstly carried out for the equidistribution of Hecke eigenvalues as primes vary and for the expected size of the error term in this equidistribution problem. Then the Kuznetsov trace formula is applied to prove a result on the size of the error term in the asymptotic distribution formula of Hecke eigenvalues. These eigenvalues become equidistributed with respect to the padic Plancherel measures as Hecke eigenforms vary. Next, this problem is generalized to Satake parameters of GL2 representations with prescribed supercuspidal local representations. Such a generalization is novel to the case of classical automorphic forms. To achieve this result, a trace formula of ArthurSelberg type with a couple of key refinements is used. In the second part of the thesis, a density theorem is proved which counts the number of exceptional nontrivial zeros of a family of symmetric power Lfunctions attached to primitive Maass forms in the critical strip. In addition, a large sieve inequality of ElliottMontgomeryVaughan type for primitive Maass forms is established. The density theorem and large sieve inequality have many applications. For instance, they are used to prove statistical results on Hecke eigenvalues of primitive Maass forms and the extreme values of the symmetric power Lfunctions attached to primitive Maass forms.   
dc.language  eng   
dc.publisher  The University of Hong Kong (Pokfulam, Hong Kong)   
dc.relation.ispartof  HKU Theses Online (HKUTO)   
dc.rights  The author retains all proprietary rights, (such as patent rights) and the right to use in future works.   
dc.rights  This work is licensed under a Creative Commons AttributionNonCommercialNoDerivatives 4.0 International License.   
dc.source.uri  http://hub.hku.hk/bib/B48329526   
dc.subject.lcsh  Trace formulas.   
dc.subject.lcsh  Eigenvalues.   
dc.title  Trace formulas and their applications on Hecke eigenvalues   
dc.type  PG_Thesis   
dc.identifier.hkul  b4832952   
dc.description.thesisname  Doctor of Philosophy   
dc.description.thesislevel  Doctoral   
dc.description.thesisdiscipline  Mathematics   
dc.description.nature  published_or_final_version   
dc.identifier.doi  10.5353/th_b4832952   
dc.date.hkucongregation  2012   