We parametrize elements in the full Hecke algebra in a way such that the parametrization represents a generic automorphic form. By convolving, we then arrive at pre-trace formulas which are modular in three variables. From here, various identities for higher moments may be derived. We give applications to the sup-norm and fourth-norm of holomorphic Hecke eigenforms as well as Hecke-Maass forms on Γ \ ℍ and furthermore outline future work on higher moments of periods and quantum variance. This is joint work with Ilya Khayutin.
This video is part of the Institute for Advanced Study‘s Number theory seminar.
