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postgraduate thesis: Trace formulas and their applications on Hecke eigenvalues

TitleTrace formulas and their applications on Hecke eigenvalues
Authors
Advisors
Advisor(s):Lau, YK
Issue Date2012
PublisherThe 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
AbstractThe 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 p-adic 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 Arthur-Selberg 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 L-functions attached to primitive Maass forms in the critical strip. In addition, a large sieve inequality of Elliott-Montgomery-Vaughan 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 L-functions attached to primitive Maass forms.
DegreeDoctor of Philosophy
SubjectTrace formulas.
Eigenvalues.
Dept/ProgramMathematics

 

DC FieldValueLanguage
dc.contributor.advisorLau, YK-
dc.contributor.authorWang, Yingnan-
dc.contributor.author王英男-
dc.date.issued2012-
dc.identifier.citationWang, 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.description.abstractThe 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 p-adic 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 Arthur-Selberg 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 L-functions attached to primitive Maass forms in the critical strip. In addition, a large sieve inequality of Elliott-Montgomery-Vaughan 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 L-functions attached to primitive Maass forms.-
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/B48329526-
dc.subject.lcshTrace formulas.-
dc.subject.lcshEigenvalues.-
dc.titleTrace formulas and their applications on Hecke eigenvalues-
dc.typePG_Thesis-
dc.identifier.hkulb4832952-
dc.description.thesisnameDoctor of Philosophy-
dc.description.thesislevelDoctoral-
dc.description.thesisdisciplineMathematics-
dc.description.naturepublished_or_final_version-
dc.identifier.doi10.5353/th_b4832952-
dc.date.hkucongregation2012-

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