Automorphic Spectral Identities from Differential Equations
Abstract: In hindsight, subconvexity results of Diaconu, Garrett, and Goldfeld can be viewed as an application of a spectral identity obtained from an automorphic partial differential equation. In this talk I will discuss examples of spectral identities obtained in this way and the analytic framework necessary for legitimizing their derivations and applications. This approach offers the possibility of operating under somewhat different hypotheses than one would usually take when using trace formula or relative trace formula methods: while trace formulas work best with very smooth data, we use a Poincare series whose data is not smooth nor compactly supported.