Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, June 2001. / Includes bibliographical references. Also available on the Internet.

published_or_final_version / Mathematics / Master / Master of Philosophy

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Thesis (Ph. D.)--Ohio State University, 2003. / Title from first page of PDF file. Document formatted into pages; contains vi, 93 p. Includes abstract and vita. Advisor: Stephen Rallis, Dept. of Mathematics. Includes bibliographical references (p. 91-93).

Thesis (M. Phil.)--University of Hong Kong, 2008. / Includes bibliographical references (p. 89-90) Also available in print.

Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität Bonn, 1991. / Includes bibliographical references (p. 72-74).

In this thesis we prove the existence of vector-valued automorphic forms for an arbitrary Fuchsian group and an arbitrary finite dimensional complex representation of this group. For small enough values of the weight as well as for large enough values, we provide explicit formulas for the spaces of these vector-valued automorphic forms (holomorphic and cuspidal). To achieve these results, we realize vector-valued automorphic forms as global sections of a certain family of holomorphic vector bundles on a certain Riemann surface associated to the Fuchsian group. The dimension formulas are then provided by the Riemann-Roch theorem. In the cases of 1 and 2-dimensional representations, we give some applications to the theories of generalized automorphic forms and equivariant functions.

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No description available.

The study of periods of automorphic forms using the theta correspondence and the Weil representation was initiated by Waldspurger and his work relating Fourier coefficients of modular forms of half-integral weight, periods over tori of modular forms of integral weight and special values of L-functions attached to these modular forms. In this thesis, we show that there are general relations among periods of automorphic forms on groups related by the theta correspondence. For example, if G is a symplectic group and H is an orthogonal group over a number field k, these relations are identities equating Fourier coefficients of cuspidal automorphic forms on G (relative to the Siegel parabolic subgroup) and periods of cuspidal automorphic forms on H over orthogonal subgroups. These identities are quite formal and follow from the basic properties of theta functions and the Weil representation; further study is required to show how they compare to the results of Waldspurger. The second part of this thesis shows that, under some restrictions, the identities alluded to above are the result of a comparison of nonstandard relative traces formulas. In this comparison, the relative trace formula for H is standard however the relative trace formula for G is novel in that it involves the trace of an operator built from theta functions. The final part of this thesis explores some preliminary results on local height pairings of special cycles on the p-adic upper half plane following the work of Kudla and Rapoport. These calculations should appear as the local factors of arithmetic orbital integrals in an arithmetic relative trace formula built from arithmetic theta functions as in the work of Kudla, Rapoport and Yang. Further study is required to use this approach to relate Fourier coefficients of modular forms of half-integral weight and arithmetic degrees of cycles on Shimura curves (which are the analogues in the arithmetic situation of the periods of automorphic forms over orthogonal subgroups).