Differential equations in automorphic forms

Kim Klinger-Logan

Published 2018-01-02Version 1

Physicists such as Green, Vanhove, et al show that differential equations involving automorphic forms govern the behavior of gravitons. One particular point of interest is solutions to $(\Delta-\lambda)u=E_{\alpha} E_{\beta}$ on an arithmetic quotient of the exceptional group $E_8$. We use spectral theory solve $(\Delta-\lambda)u=E_{\alpha}E_{\beta}$ on the simpler space $SL_2(\mathbb{Z})\backslash SL_2(\mathbb{R})$. The construction of such a solution uses Arthur truncation, the Maass-Selberg formula, and automorphic Sobolev spaces.