Differential equations in automorphic forms

2018/01/02 by Klinger-Logan, Kim
#FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Number Theory (math.NT) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1801.00838

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 (Δ-λ)u=E_α E_β on an arithmetic quotient of the exceptional group E₈. We establish that the existence of a solution to (Δ-λ)u=E_αE_β on the simpler space SL₂(ℤ)\backslash SL₂(ℝ) for certain values of α and β depends on nontrivial zeros of the Riemann zeta function ζ(s). Further, when such a solution exists, we use spectral theory to solve (Δ-λ)u=E_αE_β on SL₂(ℤ)\backslash SL₂(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.

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