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

TitleThe basis for space of cusp forms and Petersson trace formula
Authors
Advisors
Advisor(s):Lau, YK
Issue Date2012
PublisherThe 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
AbstractLet S2k(N) be the space of cusp forms of weight 2k and level N. Atkin-Lehner theory shows that S2k(N) can be decomposed into the oldspace and its orthogonal complement newspace. Again, from Atkin-Lehner 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 Petersson-type trace formula for primitive forms. In 2001, H. Iwaniec, W. Luo and P. Sarnak obtained an expression of Petersson-type 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 Petersson-type 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.
DegreeMaster of Philosophy
SubjectCusp forms (Mathematics)
Trace formulas.
Dept/ProgramMathematics

 

DC FieldValueLanguage
dc.contributor.advisorLau, YK-
dc.contributor.authorNg, Ming-ho.-
dc.contributor.author吳銘豪.-
dc.date.issued2012-
dc.identifier.citationNg, 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.description.abstractLet S2k(N) be the space of cusp forms of weight 2k and level N. Atkin-Lehner theory shows that S2k(N) can be decomposed into the oldspace and its orthogonal complement newspace. Again, from Atkin-Lehner 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 Petersson-type trace formula for primitive forms. In 2001, H. Iwaniec, W. Luo and P. Sarnak obtained an expression of Petersson-type 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 Petersson-type 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.languageeng-
dc.publisherThe University of Hong Kong (Pokfulam, Hong Kong)-
dc.relation.ispartofHKU Theses Online (HKUTO)-
dc.rightsThe author retains all proprietary rights, (such as patent rights) and the right to use in future works.-
dc.rightsCreative Commons: Attribution 3.0 Hong Kong License-
dc.source.urihttp://hub.hku.hk/bib/B47176726-
dc.subject.lcshCusp forms (Mathematics)-
dc.subject.lcshTrace formulas.-
dc.titleThe basis for space of cusp forms and Petersson trace formula-
dc.typePG_Thesis-
dc.identifier.hkulb4717672-
dc.description.thesisnameMaster of Philosophy-
dc.description.thesislevelMaster-
dc.description.thesisdisciplineMathematics-
dc.description.naturepublished_or_final_version-
dc.identifier.doi10.5353/th_b4717672-
dc.date.hkucongregation2012-

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