Miao, Xinchen2023-11-282023-11-282023-06https://hdl.handle.net/11299/258629University of Minnesota Ph.D. dissertation. June 2023. Major: Mathematics. Advisor: Dihua Jiang. 1 computer file (PDF); iv, 143 pages.In this thesis, we focus on the local and global theory of automorphic forms and relative trace formulae. In the local aspect, I work on the GL(n) Bessel distributions (functions) over non-archimedean local fields and the local Kloosterman-type orbital integrals. We prove the local integrability of Bessel functions for GL(n) (p-adic case) by using the relations between Bessel functions and local Kloosterman (orbital) integrals proved in several papers of E. M. Baruch [Ba03] [Ba04] [Ba05], the theory of the (relative) Shalika germs established by H. Jacquet and Y. Ye in [JY96] [JY99] and G. Stevens’ approach [Ste87] on estimating certain GL(n) generalized Kloosterman sums. For the global theory, I study the automorphic spectral reciprocity formulae for certain L-functions. We prove a spectral reciprocity formula for the product of GL(n + 1) × GL(n) and GL(n) × GL(n − 1) Rankin-Selberg L-functions (n ≥ 3). This generalizes the work of V. Blomer and R. Khan in [BK17] where the case of n = 2 was established. We extend the method develooped by R. Nunes in [Nun20] from n = 2 to general n ≥ 3, by using the global zeta integrals for the GL(n + 1) × GL(n) Rankin-Selberg L-functions, the spectral theory of square-integrable automorphic forms, the language of automorphic representations and representation theoretic view.enAutomorphic spectral reciprocityBessel distributionsBessel functionsKloosterman sumsRelative trace formulaThesis or Dissertation