[No abstract submitted] / Science, Faculty of / Mathematics, Department of / Graduate

Inaug.-diss.--Rheinischen Friedrich-Wilhelms-Universität, 1921. / Lebenslauf. Bibliographical foot-notes.

Thesis (doctoral)--Bonn, 1984. / Includes bibliographical references (p. 160-162).

Proefschrift--Vrije Universiteit, Amsterdam. / "Literatuur": p. [63].

Cheung Wing-chuen. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2001. / Includes bibliographical references (leaves 80-82). / Abstracts in English and Chinese. / Chapter 1 --- Introduction to CR manifolds --- p.6 / Chapter 1.1 --- Non-equivalence of real analytic hypersurfaces in C2 --- p.6 / Chapter 1.2 --- CR manifold and Levi form --- p.9 / Chapter 1.3 --- The real hyperquadrics --- p.16 / Chapter 2 --- Normal Forms --- p.22 / Chapter 2.1 --- Formal theory of normal forms --- p.22 / Chapter 2.2 --- Geometric theory of normal forms --- p.33 / Chapter 3 --- Connections and Curvatures --- p.45 / Chapter 3.1 --- Solution of the equivalence problem --- p.45 / Chapter 3.2 --- Geometric interpretation of the solution --- p.59 / Chapter 4 --- Chains --- p.65 / Chapter 4.1 --- Identification of the two definitions of chains --- p.65 / Chapter 4.2 --- Chain-preserving maps --- p.73 / Chapter 4.3 --- Some pathological behaviour of chains --- p.76 / Bibliography --- p.80

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

The large sieve method has been used extensively, beginning with Bombieri in 1965, to provide bounds on the number of zeros of Dirichlet L-functions near the line σ = 1. Using the Kuznetsov trace formula and the work of Deshouillers and Iwaniec on Kloosterman sums, it is possible to derive large sieve inequalities for the Fourier coefficients of Maass cusp forms, which may then similarly be used to study the corresponding Hecke-Maass L-functions. Following an approach developed by Gallagher for Dirichlet L-functions, this thesis shows how the large sieve method may be used to prove a zero density estimate, averaged over the Laplace eigenvalues, for Maass cusp forms of weight zero for the congruence subgroup Γ₀(q) for any positive integer q.

Thesis (doctoral)--Universität Zürich, 1905.

Thesis (Ph. D.)--University of Texas at Austin, 2006. / Vita. Includes bibliographical references.

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