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postgraduate thesis: The basis for space of cusp forms and Petersson trace formula
Title  The basis for space of cusp forms and Petersson trace formula 

Authors  
Advisors  Advisor(s):Lau, YK 
Issue Date  2012 
Publisher  The University of Hong Kong (Pokfulam, Hong Kong) 
Citation  Ng, M. [吳銘豪]. (2012). The basis for space of cusp forms and Petersson trace formula. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR. Retrieved from http://dx.doi.org/10.5353/th_b4717672 
Abstract  Let S2k(N) be the space of cusp forms of weight 2k and level N. AtkinLehner theory shows that S2k(N) can be decomposed into the oldspace and its orthogonal complement newspace. Again, from AtkinLehner theory, it follows that there exists a basis of newspace whose elements are simultaneous eigenforms of all the Hecke operators. Such eigenforms when normalized are called primitive forms.
In 1932, Petersson introduced a harmonic weighted sum of the Fourier coefficients of an orthogonal basis B2k(N) for S2k(N), denoted by _2k;N . Petersson connected _2k;N to Kloosterman sums and Bessel functions, which is now known as the Petersson trace formula. The Petersson trace formula shows that _2k;N is independent of the choice of orthogonal basis. It is known that the oldspace decomposes into the images of newspaces of different levels under the scaling operator Bd where d is a proper divisor of N. It is of interest to derive a Peterssontype trace formula for primitive forms.
In 2001, H. Iwaniec, W. Luo and P. Sarnak obtained an expression of Peterssontype trace formula for primitive forms in terms of _2k;N , when the level N is squarefree. Their method is to construct a special orthogonal basis for S2k(N). Using their approach, D. Rouymi has extended similar results to the case of prime power level in 2011.
In this thesis, the case of arbitrary levels is investigated. Analogously, a special orthogonal basis is constructed and a Peterssontype trace formula for primitive forms in terms of _2k;N is found. The result established in this thesis recovers the results of H. Iwaniec, W. Luo and P. Sarnak, and D. Rouymi respectively for the cases of squarefree and prime power levels. 
Degree  Master of Philosophy 
Subject  Cusp forms (Mathematics) Trace formulas. 
Dept/Program  Mathematics 
Persistent Identifier  http://hdl.handle.net/10722/174338 
HKU Library Item ID  b4717672 
DC Field  Value  Language 

dc.contributor.advisor  Lau, YK   
dc.contributor.author  Ng, Mingho.   
dc.contributor.author  吳銘豪.   
dc.date.issued  2012   
dc.identifier.citation  Ng, M. [吳銘豪]. (2012). The basis for space of cusp forms and Petersson trace formula. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR. Retrieved from http://dx.doi.org/10.5353/th_b4717672   
dc.identifier.uri  http://hdl.handle.net/10722/174338   
dc.description.abstract  Let S2k(N) be the space of cusp forms of weight 2k and level N. AtkinLehner theory shows that S2k(N) can be decomposed into the oldspace and its orthogonal complement newspace. Again, from AtkinLehner theory, it follows that there exists a basis of newspace whose elements are simultaneous eigenforms of all the Hecke operators. Such eigenforms when normalized are called primitive forms. In 1932, Petersson introduced a harmonic weighted sum of the Fourier coefficients of an orthogonal basis B2k(N) for S2k(N), denoted by _2k;N . Petersson connected _2k;N to Kloosterman sums and Bessel functions, which is now known as the Petersson trace formula. The Petersson trace formula shows that _2k;N is independent of the choice of orthogonal basis. It is known that the oldspace decomposes into the images of newspaces of different levels under the scaling operator Bd where d is a proper divisor of N. It is of interest to derive a Peterssontype trace formula for primitive forms. In 2001, H. Iwaniec, W. Luo and P. Sarnak obtained an expression of Peterssontype trace formula for primitive forms in terms of _2k;N , when the level N is squarefree. Their method is to construct a special orthogonal basis for S2k(N). Using their approach, D. Rouymi has extended similar results to the case of prime power level in 2011. In this thesis, the case of arbitrary levels is investigated. Analogously, a special orthogonal basis is constructed and a Peterssontype trace formula for primitive forms in terms of _2k;N is found. The result established in this thesis recovers the results of H. Iwaniec, W. Luo and P. Sarnak, and D. Rouymi respectively for the cases of squarefree and prime power levels.   
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/B47176726   
dc.subject.lcsh  Cusp forms (Mathematics)   
dc.subject.lcsh  Trace formulas.   
dc.title  The basis for space of cusp forms and Petersson trace formula   
dc.type  PG_Thesis   
dc.identifier.hkul  b4717672   
dc.description.thesisname  Master of Philosophy   
dc.description.thesislevel  Master   
dc.description.thesisdiscipline  Mathematics   
dc.description.nature  published_or_final_version   
dc.identifier.doi  10.5353/th_b4717672   
dc.date.hkucongregation  2012   