Differential equations in automorphic forms

TL;DR

This paper explores differential equations involving automorphic forms, linking their solutions on certain spaces to the zeros of the Riemann zeta function, and employs spectral theory and advanced truncation techniques.

Contribution

It establishes conditions for solutions to specific automorphic differential equations on $SL_2(Z)ackslash SL_2(R)$ based on the zeros of the Riemann zeta function and develops methods for their meromorphic continuation.

Findings

Solution existence depends on nontrivial zeros of $\zeta(s)$

Spectral theory is used to solve the equations when solutions exist

Provides proof of meromorphic continuation of solutions

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 $SL_2(\mathbb{Z})\backslash SL_2(\mathbb{R})$ 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 $SL_2(\mathbb{Z})\backslash SL_2(\mathbb{R})$ 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.