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We prove a new omega result for toric periods of Hecke-Maass forms on compactlocally symmetric spaces associated to forms of PGL(3). This is motivated byconjectures on the maximal growth of L-functions as well as by questions aboutthe size of automorphic periods. We also prove a mean square asymptotic resultfor maximal flat periods on more general locally symmetric spaces ofnon-compact type, which takes as main input bounds for real relative orbitalintegrals.

The Maximal Growth Of Toric Periods and Oscillatory Integrals for Maximal Flat Submanifolds