Farrell Brumley: Periods and asymptotic growth of arithmetic eigenfunctions

Given a compact locally symmetric space Y we are interested in the localization properties of sequences of eigenfunctions of the ring of invariant differential operators. When Y is of non-compact type, quantum chaos suggests that such eigenstates should be delocalized. One concrete expression of this is that a generic sequence of L^2 normalized eigenfunctions should have small sup norm. We call these nicely behaved sequences ''tempered'', in analogy with the Ramanujan conjecture from the theory of automorphic forms. We would like to know under what conditions Y admits non-tempered sequences of eigenfunctions, i.e., those whose sup norm grows with a power of the eigenvalue. We provide a fairly complete answer to this question in the arithmetic case, in terms of the recurrence properties of Hecke operators. Our techniques actually pick out the size of certain discrete periods through trace formula methods, and the criterion assuring growth can be read off from the Plancherel measure of an underlying symmetric space G/H. This is joint work with Simon Marshall.