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# Communications in Number Theory and Physics

## Volume 12 (2018)

### Number 4

### Differential equations in automorphic forms

Pages: 767 – 827

DOI: https://dx.doi.org/10.4310/CNTP.2018.v12.n4.a4

#### Author

#### Abstract

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 establish that the existence of a solution to $(\Delta-\lambda ) u = E_{\alpha} E_{\beta}$ on the simpler space $SL_2 (\mathbb{Z}) \setminus SL_2 (\mathbb{R})$ for certain values of $\alpha$ and $\beta$ depends on nontrivial zeros of the Riemann zeta function $\zeta(s)$. Further, when such a solution exists, we use spectral theory to solve $(\Delta-\lambda ) u = E_{\alpha} E_{\beta}$ on $SL_2 (\mathbb{Z}) \setminus SL_2 (\mathbb{R})$ and provide proof of the meromorphic continuation of the solution. The construction of such a solution uses Arthur truncation, the Maass–Selberg formula, and automorphic Sobolev spaces.

Received 3 January 2018

Accepted 25 July 2018

Published 14 January 2019