Multiplicativity of Fourier coefficients of Maass forms for $operatorname{SL}(nhskip -.75pt,mathbb{Z}hskip -.50pt)$ Journal Article

Overview

; The Fourier coefficients of a Maass form; ; phi; ; for; ; operatorname{SL}(n,mathbb{Z}); ; are complex numbers; ; A_{phi}(M); ; , where; ; M=(m_{1},m_{2},dots,m_{n-1}); ; and; ; m_{1},m_{2},dots,m_{n-1}; ; are non-zero integers. It is well known that coefficients of the form; ; A_{phi}(m_{1},1,dots,1); ; are eigenvalues of the Hecke algebra and are multiplicative. We prove that the more general Fourier coefficients; ; A_{phi}(m_{1},dots,m_{n-1}); ; are also eigenvalues of the Hecke algebra and satisfy the multiplicativity relations; ; ; ; A_{phi}(m_{1}m_{1}',m_{2}m_{2}',dots,m_{n-1}m_{n-1}')=A_{phi}(m_{1},m_{2},dots,m_{n-1})cdot A_{phi}(m_{1}',m_{2}',dots,m_{n-1}'); ; ; ; provided the products; ; prod_{i=1}^{n-1}m_{i}; ; and; ; prod_{i=1}^{n-1}m_{i}'; ; are relatively prime to each other.;