Ali, Adil2015-10-132015-10-132015-05https://hdl.handle.net/11299/174859University of Minnesota Ph.D. dissertation. May 2015. Major: Mathematics. Advisors: Paul Garrett, Benjamin Brubaker. 1 computer file (PDF); ii, 81 pages.We apply some ideas of Bombieri and Garrett to construct natural self-adjoint operators on spaces of automorphic forms whose only possible discrete spectrum is λ s = s(s − 1) for s in a subset of on-line zeros of an L-function, appearing as a compact period of cuspidal-data Eisenstein series on GL 4 . These ideas have their origins in re- sults of Hejhal and Colin de Verdi ́ere. In parallel with the GL(2) case, the corresponding pair-correlation and triple-correlation results limit the fraction of on-the-line zeros that can appear in this fashion.enThesis or Dissertation