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 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. / published_or_final_version / Mathematics / Doctoral / Doctor of Philosophy

Let 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. / published_or_final_version / Mathematics / Master / Master of Philosophy

Thesis (Ph. D.)--University of Missouri-Columbia, 1996. / Typescript. Vita. Includes bibliographical references (leaves 114-118). Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 1996. / Typescript. Vita. Includes bibliographical references (leaves 114-118). Also available on the Internet.

No description available.

We study the problem of the distribution of certain GL(3) Maass forms, namely, we obtain a Weyl’s law type result that characterizes the distribution of their eigenvalues, and an orthogonality relation for the Fourier coefficients of these Maass forms. The approach relies on a Kuznetsov trace formula on GL(3) and on the inversion formula for the Lebedev-Whittaker transform. The family of Maass forms being studied has zero density in the set of all GL(3) Maass forms and contains all self-dual forms. The self-dual forms on GL(3) can also be realised as symmetric square lifts of GL(2) Maass forms by the work of Gelbart-Jacquet. Furthermore, we also establish an explicit inversion formula for the Lebedev-Whittaker transform, in the nonarchimedean case, with a view to applications.

Let F be a local non-Archimedean field of characteristic zero. We prove a Plancherel formula for the symmetric space GL(2,F)\GL(2,E), where E/F is an unramified quadratic extension. Our method relies on intrinsic geometric and combinatorial properties of spherical varieties and constitutes the local counterpart of the global computation of the Flicker-Rallis period as a residue of periods against Eisenstein series. We also give a novel derivation of the Plancherel formula for the strongly tempered variety T\PGL(2) over F (with maximal split torus T) using a canonical smooth asymptotics morphism and a contour shifting method. In this rank one local setting, our proof is similar to Langlands' proof over global fields describing the spectrum of a reductive group in terms of residues of Eisenstein series. Finally, using both L2-decompositions, we develop a local relative trace formula and outline a comparison result in the setting of the unitary rank one Gan-Gross-Prasad conjecture.

In this thesis, we prove that local genericity implies globally genericity for the quasi-split unitary group U3 for a quadratic extension of number fields E/F. We follow [Fli1992] and [GJR2001] closely, using the relative trace formula approach. Our main result is the existence of smooth transfer for the relative trace formulae in [GJR2001], which is circumvented there. The basic idea is to compute the Mellin transform of Shalika germ functions and show that they are equal in the unitary case and the general linear case.

We provide a new relative trace formula approach to the theorem of Waldspurger on toric periods for GL₂, with possible applications to the global Gross-Prasad conjecture for orthogonal groups.

Thesis (Ph. D.)--Oregon State University, 2003. / Typescript (photocopy). Includes bibliographical references (leaves 43-46). Also available on the World Wide Web.